Mixed Signal Detection, Estimation, and Modulation ...

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Mixed Signal Detection, Estimation, and Modulation Classification Mixed Signal Detection, Estimation, and Modulation Classification

Yang Qu Wright State University

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MIXED SIGNAL DETECTION, ESTIMATION, ANDMODULATION CLASSIFICATION

A dissertation submitted in partial fulfillmentof the requirements for the degree of

Doctor of Philosophy

by

YANG QUM.S.EG., Wright State University, 2013B.E., Dalian Jiaotong University, 2010

2019Wright State University

Wright State UniversityGRADUATE SCHOOL

November 25, 2019

I HEREBY RECOMMEND THAT THE DISSERTATION PREPARED UNDER MY SU-PERVISION BY Yang Qu ENTITLED Mixed Signal Detection, Estimation, and ModulationClassification BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FORTHE DEGREE OF Doctor of Philosophy.

Zhiqiang Wu, Ph.D.Dissertation Director

Fred D. Garber, Ph.D.Interim Chair, Department of Electrical Engineering

Barry Milligan, Ph.D.Interim Dean, Graduate School

Committee onFinal Examination

Dr. Zhiqiang Wu

Dr. Saiyu Ren

Dr.Vasu Chakravarthy

Dr.Yan Zhuang

Dr. Xiaodong Zhang

ABSTRACT

Qu, Yang. Ph.D., Department of Electrical Engineering, Wright State University, 2019. Mixed

Signal Detection, Estimation, and Modulation Classification.

Signal detection, parameter estimation, and modulation classification are widely ap-

plied to many areas and play a very important role in civilian and military area, such

as bio-science, criminal psychology, communication engineering, radar systems, elec-

tronic warfare and so on. In the civilian field, with the increasing number of wireless

electronic devices and higher transmission data rate demand, the problem of spectrum

congestion becomes more and more highlighted and urgent. In recent years, the wire-

less industry has shown great interest in Cognitive Radio (CR) and Dynamic Spectrum

Access (DSA) networks, whose primary function is to use limited frequency bands to

transmit their signals without any interference to other primary users. Hence, the ac-

curacy of signal detection and parameter estimation is particularly important and can

provide reliable communication performance for cognitive radio users. In the military

field, electronic warfare is a crucially important part of modern war, the outcome of

the war is no longer determined by how many people we have, determined by how

could we securely transmit and receive own signals, how could we successfully detect,

identify, locate and jam enemy’s signals.Thus, in such a non-cooperative environment,

signal detection, parameter estimation, and modulation classification technologies be-

come more and more important and challenging. In the past few decades, several sig-

nal detection methods have been proposed, such as energy-based detection, matched

filter-based detection, and cyclostationary feature-based detection. Energy-based de-

tection is simple to implement, but performs poorly at low SNR. Although the matched

filter-based detection is the optimal detector, it needs to accurately know the prior in-

formation of the detected signal. Hence, matched filter-based detection is impractical

to implement in a real environment, such as a non-cooperative environment. Cyclo-

iii

stationary feature-based signal detection has high computational complexity, but it can

be used for high-precision signal detection, in low SNR environments. In recent years,

many researchers have shown their interest and effort in signal detection, parameter

estimation, and modulation classification technologies. Most of them are working with

single signal detection, parameter estimation, and modulation classification. A few peo-

ple consider time and frequency mixed signals as their target signals. In particular, some

people assume that there is no overlap between co-exited signals in the time domain

and frequency domain. In such a case, we can easily separate those co-existed signals

with a band-pass filter in the frequency domain. Meanwhile, we can easily know the

number of co-existed signals, estimate each signal’s parameters and classify their mod-

ulation types. However, in a spectrum congested environment, such as cognitive radio

and electronic warfare, several signals are often mixed with plenty of overlap in both the

time domain and frequency domain.In some special cases, several signals are entirely

overlapped in both time domain and frequency domain, such as in-band full-duplex

communication signals, they use same carrier frequency to simultaneously transmit and

receive signals. It is more challenging to enumerate and classify those kinds of mixed

signals. Hence, studying mixed-signal detection, parameter estimation, and modula-

tion classification has more practical significance. In this dissertation, we employ sig-

nal energy-based, mainly employ signal cyclostationary features and machine learning

technology-based methods to detect, estimate, and classify mixed-signals, which have

significant overlap in both time domain and frequency domain. In particular, we em-

ploy energy-based detection to preliminary detect the signal is existed or not existed

in the channel and use spectrum analysis to roughly locate the interesting frequency

band. Meanwhile, we employ different order signal cyclostationary features to detect,

estimate, and classify four popular digital communication signals, which includes low-

order modulation type BPSK signal, high-order modulation type QPSK signals, 8-PSK

signals, and 16-QAM signals. According to our previous work, we can use the second-

iv

order cyclostationary feature to detect and classify mixed signals, such as mixed BPSK

signals and mixed QPSK signals. However, since some signals have no second-order

cyclostationary feature, we are unable to precisely estimate and classify them by us-

ing low order cyclostationary features, so we cannot use Spectral Correlation Function

(SCF) to classify mixed QPSK signal and 16-QAM signal. In this dissertation, we con-

sider some more challenging cases, include detecting, estimating, and classifying mixed

higher-order modulation signals, such as 16-QAM and 8-PSK signals, classifying mixed

signals, which have similar cyclostationary features, such as QPSK and 16-QAM mixed

signals, and analyze heavily overlapped mixed signals, such as two signals that have the

same carrier frequency. Moreover, we employ low-order and high order cyclostation-

ary features, i.e., cyclic moment and cyclic cumulants, to detect, estimate and classify

more different combinations of mixed signals, such as BPSK and QPSK mixed-signal,

two QPSK mixed-signals, BPSK, and 16-QAM mixed-signal, etc. In this dissertation, we

also provide a detailed performance analysis to demonstrate our proposed method can

effectively detect mixed signals, estimate mixed signals’ parameters, such as carrier fre-

quency, symbol rate, and power, and classify mixed signals’ modulation types. In addi-

tion, our performance analysis is based on different channels, such as AWGN channels,

flat fading channels and multi-path fading channels.

v

List of SymbolsChapter 1

WWII World War II

EW Electronic Warfare

SCF Spectral Correction Function

STD Signal Detection Theory

AWGN Additive White Gaussian Noise

AR Autoregressive

SAGE Space-alternating Generalized Expectation-maximization

AMC Automatic Modulation Classification

SDR Software Defined Radio

MIMO Multiple-input Multiple-output

OFDM Orthogonal Frequency-division Multiplexing

LB Likelihood-based

FB Feature-based

HOCM Higher-order Cyclic Moment

HOCC Higher-order Cyclic Cumulant

BPSK Binary Phase-shift keying

QPSK Quadrature Phase-shift keying

8PSK 8 Phase-shift keying

16QAM 16 Quadrature Amplitude Modulation

ML Machine Learning

SVM Support Vector Machine

Chapter 2

USAF US Air Force

DARPA US Defense Advanced Research Projects Agency

USRP Universal Software Radio Peripheral

LabVIEW Laboratory Virtual Instrument Engineering Workbench

GRC GNU Radio Companion

AGC Auto Gain Control

vi

NI National Instrument

ADC Analog-to-Digital Converter

DAC Digital-to-Analog Converter

Chapter 3

AI Artificial Intelligence

ML Machine Learning

RL Representation Learning

DL Deep Learning

DM Data Mining

ASR Automatic Speech Recognition

NLP Natural Language Processing

CV Computer Vision

SL Statistical Learning

PR Pattern Recognition

Chapter 4

CCs Cyclic Cumulant

Rαx (τ) Cyclic Autocorrelation Function

Sαx ( f ) Spectral Correlation Function

∆t Measurement interval

α Cyclic Frequency

Cαx ( f ) Spectral Coherent Function

SCF Spectral Correlation Function

SOF Spectral Coherent Function

Fs Sampling Frequency

Fc Carrier Frequency

Fb Symbol Rate

Lx(t ,τ;n,m) n-th order and m-th order conjugate lag product

Rx(t ,τ;n,m) n-th order and m-th order conjugate temporal moment function

Eα general sine-wave extraction operator

Rαx (τ;n,m) nth-order and m-th order conjugate cyclic temporal moment function

vii

Cx(t ,τ;n,m) nth-order and m-th order conjugate temporal cumulant function

Cαx (τ;n,m) nth-order and m-th order conjugate cyclic temporal cumulant function

SOCS Second-order Cyclostationarity

HOS Higher-order Statistics

HOCS Higher-order Cyclostationarity

TMF Temporal Moment Function

CTMF Cyclic Temporal Moment Function

TCF Temporal Cumulant Function

CTCF Cyclic Temporal Cumulant Function

Chapter 5

MSP Mixed Signal Processing

ED Energy Detection

Es Signal Energy

ηE N Energy Detection Threshold

C M Cyclic Moment

NES The number of detected signals by using cyclic moment

ηEC maximum number of entirely overlapped signal

Nt The number of attempts for signal estimation

Chapter 6

s(n) Target Signal

W (n) White Gaussian Noise

x(n) Received data

H0 Hypothesis

Es(x) Energy Detection Statistics

f (x;k)χ2 χ2-distribution

Γ(k/2) Gamma function

P f a False Alarm Rate

η Energy threshold

x(t ) Continue-time cyclostationary process

mx(t ,τ)n n-th order moment function

viii

τ time delay

α cyclic frequency

β nth-order moment cycle frequencies set

mx(α,τ)n n-th order cyclic moment at α

s(t ) Modulated signal

fc Carrier frequency

a(t ) Base-band signal

A Signal Amplitude

s(i ) Transmitted symbol

Ts Symbol duration

p(t ) Signal pulse

t0 Propagation delay

Sa Sinc function

δ Dirac delta function

n(t ) White Gaussian noise

Cs(τ;n,m) n-th order m-th order conjugate cyclic cumulant

ix

Contents

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Overview of Signal Detection, Parameter Estimation, and Modulation Clas-

sification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Signal Parameters Estimation . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Signal Modulation Classification . . . . . . . . . . . . . . . . . . . . . 6

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Problem Statement and Approaches . . . . . . . . . . . . . . . . . . . . . . . 81.5 Dissertation Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Overview of Machine Learning 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Support Vector Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Overview of Cyclostationary Signal Processing 193.1 Second-Order Cyclostationarity . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Higher-Order Cyclostationarity . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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4 Overview of Mixed Signal Processing Diagram 354.0.1 Mixed Signal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.0.2 Mixed Signal Processing Workflow . . . . . . . . . . . . . . . . . . . . 37

5 Cyclostationary Signal Processing theory based Mixed Signal Processing 415.1 Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Signal Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2.1 nth order cyclic moment . . . . . . . . . . . . . . . . . . . . . . . . . 435.2.2 single signal parameter estimation . . . . . . . . . . . . . . . . . . . . 445.2.3 Mixed signal parameter estimation . . . . . . . . . . . . . . . . . . . 51

5.3 Signal Modulation Classification . . . . . . . . . . . . . . . . . . . . . . . . . 575.3.1 Mathematic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3.2 Theoretical Cumulant Value . . . . . . . . . . . . . . . . . . . . . . . 645.3.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Simulations and Performance Analysis 796.1 Energy based Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.1.1 Continues BPSK Modulated signal . . . . . . . . . . . . . . . . . . . . 806.1.2 Pulsed BPSK Modulated signal . . . . . . . . . . . . . . . . . . . . . . 82

6.2 Signal Detection and Parameters Estimation . . . . . . . . . . . . . . . . . . 856.2.1 Single Signal Detection and Parameters Estimation . . . . . . . . . . 856.2.2 Mixed Signal Detection and Parameters Estimation . . . . . . . . . . 87

6.3 Signal Modulation Classification . . . . . . . . . . . . . . . . . . . . . . . . . 956.3.1 Single Signal Classification . . . . . . . . . . . . . . . . . . . . . . . . 956.3.2 Model A: Mixed Signal Classification . . . . . . . . . . . . . . . . . . 1006.3.3 Model B: Mixed Signal Classification . . . . . . . . . . . . . . . . . . 111

7 Conclusion 130

Bibliography 132

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List of Figures

1.1 Signal Detection Theory(SDT) Model . . . . . . . . . . . . . . . . . . . . . . 41.2 Communication System Diagram . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Artificial Intelligence (AL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Machine Learning relevant field . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Machine Learning Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 SVM Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Confusion Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 SCF of BPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 SCF of QPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 SCF of 8-PSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 SCF of 16-QAM signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Cyclic Cumulant of BPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . 283.6 Cyclic Cumulant of BPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . 293.7 Cyclic Cumulant of BPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . 293.8 Cyclic Cumulant of QPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . 303.9 Cyclic Cumulant of QPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . 303.10 Cyclic Cumulant of QPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . 313.11 Cyclic Cumulant of 8PSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . 313.12 Cyclic Cumulant of 8PSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . 323.13 Cyclic Cumulant of 8PSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . 323.14 Cyclic Cumulant of 16QAM signal . . . . . . . . . . . . . . . . . . . . . . . . 333.15 Cyclic Cumulant of 16QAM signal . . . . . . . . . . . . . . . . . . . . . . . . 333.16 Cyclic Cumulant of 16QAM signal . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Mixed Signals Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Mixed Signal Processing flow diagram . . . . . . . . . . . . . . . . . . . . . 374.3 Hierarchical Signal Parameter Estimation Diagram . . . . . . . . . . . . . . 394.4 SVM based Signal Modulation Classification . . . . . . . . . . . . . . . . . . 40

5.1 16-QAM Constellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 16-QAM4 Constellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.3 8-PSK Constellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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5.4 8-PSK8 Constellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.5 Simulation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.6 Pascal’s triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.7 2nd and 4-th order cyclic moment of mixed signal . . . . . . . . . . . . . . 565.8 8-th order cyclic moment of mixed signal . . . . . . . . . . . . . . . . . . . . 575.9 BPSK1:BPSK1+BPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.10 BPSK2:BPSK1+BPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.11 BPSK1+BPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.12 BPSK1+BPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.13 BPSK:BPSK+QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.14 QPSK:BPSK+QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.15 BPSK+QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.16 BPSK+QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.17 BPSK:BPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.18 16QAM:BPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.19 BPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.20 BPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.21 BPSK:BPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.22 8PSK:BPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.23 BPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.24 BPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.25 QPSK1:QPSK1+QPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.26 QPSK2:QPSK1+QPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.27 QPSK1+QPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.28 QPSK1+QPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.29 QPSK:QPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.30 16QAM:QPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.31 QPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.32 QPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.33 QPSK:QPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.34 8PSK:QPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.35 QPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.36 QPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.37 16QAM1:16QAM1+16QAM2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.38 16QAM2:16QAM1+16QAM2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.39 16QAM+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.40 16QAM+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.41 16QAM:16QAM+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.42 8PSK:16QAM+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.43 16QAM+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.44 16QAM+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.45 8PSK1:8PSK1+8PSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.46 8PSK2:8PSK1+8PSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.47 8PSK1+8PSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.48 8PSK1+8PSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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5.49 BPSK1:BPSK1+BPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.50 BPSK2:BPSK1+BPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.51 BPSK1+BPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.52 BPSK1+BPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.53 BPSK:BPSK+QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.54 QPSK:BPSK+QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.55 BPSK+QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.56 BPSK+QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.57 BPSK:BPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.58 16QAM:BPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.59 BPSK:BPSK+QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.60 QPSK:BPSK+QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.61 BPSK:BPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.62 8PSK:BPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.63 BPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.64 BPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.65 QPSK1:QPSK1+QPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.66 QPSK2:QPSK1+QPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.67 QPSK1+QPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.68 QPSK1+QPSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.69 QPSK:QPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.70 16QAM:QPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.71 QPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.72 QPSK+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.73 QPSK:QPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.74 8PSK:QPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.75 QPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.76 QPSK+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.77 16QAM1:16QAM1+16QAM2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.78 16QAM2:16QAM1+16QAM2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.79 16QAM+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.80 16QAM+16QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.81 16QAM:16QAM+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.82 8PSK:16QAM+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.83 16QAM+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.84 16QAM+8PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.85 8PSK1:8PSK1+8PSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.86 8PSK2:8PSK1+8PSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.87 8PSK1+8PSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.88 8PSK1+8PSK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.1 BPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.2 BPSK: Pd versus SNR (dB) with required P f a = 0.01 . . . . . . . . . . . . . . 816.3 BPSK: Pd versus SNR (dB) with required P f a = 0.1 . . . . . . . . . . . . . . . 816.4 BPSK: Pd versus SNR (dB) with required P f a = 0.3 . . . . . . . . . . . . . . . 82

xiv

6.5 Pulsed signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.6 Pulsed/Modulated: Pd versus SNR (dB) with required P f a = 0.01 . . . . . . 836.7 Pulsed/Modulated: Pd versus SNR (dB) with required P f a = 0.1 . . . . . . . 846.8 Pulsed/Modulated: Pd versus SNR (dB) with required P f a = 0.3 . . . . . . . 846.9 QPSK Estimation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.10 Single Signal Fc Detection and Estimation Performance in AWGN Channel 866.11 Pd vs Number of symbols under SNR = 10 dB . . . . . . . . . . . . . . . . . 876.12 Example: Mixed Signal Carrier Frequency (Fc) Estimation . . . . . . . . . . 886.13 Fc Estimation Performance Pd versus SNR in AWGN Channel . . . . . . . . 896.14 Fc Estimation Performance Pd versus SNR in flat fading Channel . . . . . . 906.15 Fc Estimation Performance Pd versus SNR in multi-path fading Channel . 906.16 Mixed Signal Symbol rate (Fb) Estimation . . . . . . . . . . . . . . . . . . . 916.17 Mixed Signal Symbol rate (Fb) Estimation Performance . . . . . . . . . . . 926.18 Mixed BPSK Signal Power Estimation Performance . . . . . . . . . . . . . . 936.19 Mixed QPSK Signal Power Estimation Performance . . . . . . . . . . . . . . 936.20 Mixed 16-QAM Signal Power Estimation Performance . . . . . . . . . . . . 946.21 Mixed 8-PSK Signal Power Estimation Performance . . . . . . . . . . . . . . 946.22 Example: BPSK Training Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.23 Example: QPSK Training Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.24 Classification Performance under SNR = 20 dB . . . . . . . . . . . . . . . . . 966.25 Single signal Pcc vs SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.26 Single signal Classification Pcc . . . . . . . . . . . . . . . . . . . . . . . . . . 996.27 single signal Pcc vs SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.28 The CCs of QPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.29 The CCs of QPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.30 The CCs of QPSK signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.31 The CCs of 16-QAM signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.32 CCs of two signals with 50% spectrum overlap . . . . . . . . . . . . . . . . . 1026.33 Pcc of QPSK and 16-QAM signal . . . . . . . . . . . . . . . . . . . . . . . . . 1036.34 SNR vs Pcc of QPSK and 16QAM mixed signal . . . . . . . . . . . . . . . . . 1036.35 Pcc of mixed QPSK and 16-QAM signal . . . . . . . . . . . . . . . . . . . . . 1056.36 The PDF of normalized C(6,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.37 The PDF of normalized C(8,0) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.38 Pcc of QPSK and 16-QAM at SNR = 20dB (578 symbols) . . . . . . . . . . . . 1086.39 Pcc of QPSK and 16-QAM at SNR = 20dB (925 symbols) . . . . . . . . . . . . 1096.40 Pcc of QPSK and 16-QAM verse SNR . . . . . . . . . . . . . . . . . . . . . . . 1106.41 Example: BPSK Training Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.42 Example: QPSK Training Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.43 BPSK symbol rate vs roll-off factor vs side lobe magnitude . . . . . . . . . . 1136.44 QPSK symbol rate vs roll-off factor vs side lobe magnitude . . . . . . . . . . 1136.45 16QAM symbol rate vs roll-off factor vs side lobe magnitude . . . . . . . . 1146.46 8PSK symbol rate vs roll-off factor vs side lobe magnitude . . . . . . . . . . 1146.47 Mixed Signal Classification Rate . . . . . . . . . . . . . . . . . . . . . . . . . 1166.48 Mixed Signal Classification Rate of 1st signal . . . . . . . . . . . . . . . . . . 1176.49 Feature F1: Power Ratio 1:1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xv

6.50 Feature F1: Power Ratio 3:1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.51 Feature F1: Power Ratio 5:1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.52 Feature F1: Power Ratio 10:1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.53 Feature F2: Power Ratio 10:1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.54 Feature F3: Power Ratio 10:1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.55 Feature F1 values with different power ratio . . . . . . . . . . . . . . . . . . 1206.56 PDF of F1: C (4,2)

C (2,1)2 with Power ratio 1:1 . . . . . . . . . . . . . . . . . . . . . . 121

6.57 PDF of F1: C (4,2)C (2,1)2 with Power ratio 10:1 . . . . . . . . . . . . . . . . . . . . . . 122



6.60 PDF of F3:C (6,3)2


6.61 PDF of F3:C (6,3)2


6.62 PDF of F4:C (8,4)3


6.63 PDF of F4:C (8,4)3




6.66 PDF of F 1 = C (4,2)C (2,1)2 with different power ratio . . . . . . . . . . . . . . . . . . 126

6.67 PDF of F 2 = C (6,3)C (2,1)3 with different power ratios . . . . . . . . . . . . . . . . . 127

6.68 PDF of F 3 = C (6,3)2

C (4,2)3 with different power ratios . . . . . . . . . . . . . . . . . 127

6.69 PDF of F 4 = C (8,4)3

C (6,3)4 with different power ratios . . . . . . . . . . . . . . . . . 128

6.70 PDF of F 5 = C (8,4)C (4,2)2 with different power ratios . . . . . . . . . . . . . . . . . 128

6.71 Classification Rate vs SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xvi

List of Tables

5.1 Theoretical Cumulant value, E is Signal Power . . . . . . . . . . . . . . . . . 64

6.1 Confusion Matrix: Single Signal Classification . . . . . . . . . . . . . . . . . 976.2 Cyclic Cumulants Features (α= (n −2m) fc ) . . . . . . . . . . . . . . . . . . 986.3 classification training table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.4 classification confusion matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.5 Pcc of mixed QPSK signal and 16QAM signal under different conditions . . 1106.6 Cyclic Cumulants Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.7 Mixed co-channel signal cyclic cumulant pattern . . . . . . . . . . . . . . . 1156.8 Classification Confusion Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 1166.9 Cyclic Cumulant Ratio Features . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.10 Cyclic Cumulant Ratio Features . . . . . . . . . . . . . . . . . . . . . . . . . . 121

xvii

Acknowledgment

There are so many people that have earned my gratitude for their contribution to my

PhD study. In particular, I would like to thank a few groups of people, without their sup-

port and help, this dissertation would not be possible.

First, I am indebted to my dissertation advisor, Dr.Zhiqiang Wu. Since my first day in

PhD program, he has provided me with endless support. On the academic level, Dr. Wu

encouraged me and guided me. On a personal level, he inspired me by his hardworking

and passionate attitude.

Second, besides my advisor, I would like to thank the rest of my dissertation committee

members, Dr.Vasu Chakravarthy, Dr.Saiyu Ren, Dr.Yan Zhuang and Dr.Xiaodong Zhang,

for their great support and invaluable advice.

Third, I would like to thank Dr.Zhiping Zhang and Dr.Chad Spooner, who are also very

important persons in my PhD study. Every time I encountered difficulties, they patiently

helped me and overcame difficulties with me.

Last but not least, I am also grateful to my family and friends. This dissertation would

not have been possible without their love and endless support.

xviii

Introduction

1.1 Motivation

Signal detection, which originated in the study of radar during World War II(WWII), is

used to detect enemy targets (aircraft, ships, vehicles) on the battle field. In the 1950s,

due to the development of modern mathematics, a relatively systematic and compre-

hensive signal detection theory was established, which is widely used in many areas,

such as military, geology, physics, electronics, communications, cosmology and so on.

One good example is University of Michigan psychologists W.P. Tanner and J.A. Schweitz,

who first applied signal detection theory to human perception processing of the psy-

chological study in 1954. They took the development of psychophysical law to a new

stage. Initially, signal detection theory was specifically used for dealing with separating

signal from noise background, the main purpose being solving the randomness prob-

lem in signal transmission[1]. Nowadays, with the rapid development of science and

technology, signal detection is deeply developed and widely applied to many different

fields, such as biology, criminal psychology, wireless communication, radar systems, etc.

Moreover, signal detection is extended for more purposes, such as signal parameter es-

timation and signal recognition/classification. In recent years, signal detection has be-

come more and more important in civil and military wireless communications. In the

civilian area, due to the increasing number of wireless electronic devices and the re-

quirements for higher data rates, the problem of spectrum congestion has become more

1

and more highlighted and urgent. In recent years, the wireless industry has shown great

interest in Cognitive Radio (CR) and Dynamic Spectrum Access(DSA) networks. The pri-

mary function of cognitive radio and dynamic spectrum access networks is to discover

available spectrum resources and guarantee that the signal transmission of secondary

users does not interfere with the primary users’ signal transmission or other CR users’

signal transmission. Hence, cognitive radios place higher demands on secondary users

for signal detection and estimation accuracy. In other words, the accuracy of signal de-

tection and estimation affects the performance of multi-user transmissions in the same

channel.[2][3]. In the military area, people have higher demands for the performance of

signal detection. It is well known that we need to hide and safely transmit and receive

our signals, and also detect and identify the enemy’s signals. However, under such a

non-cooperative environment, signal detection, estimation, and classification become

more challenging, such as when in order to successfully get the enemy’s information,

some signal classification receiver requires more precise prior knowledge about the tar-

get signal. Therefore, signal detection becomes particularly important[4]. In 1950, Y.

W. Lee, T. P. Cheatham and J. B. Wiesner published a paper [5]. After their paper, many

people started to study signal detection, signal parameters estimation, and signal mod-

ulation classification. Today, many researchers only study single signal detection and

identification, and do not consider co-existed signals detection and identification.[6][7].

In particular, there is no spectral overlap between co-existing signals in frequency do-

main. Therefore, it is very easy to detect how many signals existed. Meanwhile, it is

easier to employ a band-pass filter to filter out non-target signal, and then use parame-

ter estimation and modulation classification technology to locate and identify the target

signals. However, in some spectrum congested environments, such as cognitive radio

and dynamic spectrum access networks, or in non-cooperative environments such as

Electronic Warfare (EW), many signals are mixed with plenty of overlap in both time do-

main and frequency domain[8]. It is hard to know through general spectrum analysis

2

how many signals are mixed together. Moreover, some signals may possess the same

features or no features due to special signal processing methods. It is hard to detect

and classify those signals, such as QPSK signals and 16-QAM signals, which have simi-

lar cyclostationary features. Besides, in some extreme cases, several signals have been

heavily overlapped in both time domain and frequency domain, such as in-band full-

duplex communication signals. It is more challenging to enumerate and classify those

signals. Hence, it is highly desirable to find an effective way to detect mixed signals, esti-

mate mixed signals’ parameters and classify mixed signals’ modulation type in complex

and realistic environments.

1.2 Overview of Signal Detection, Parameter Estimation,

and Modulation Classification

1.2.1 Signal Detection

Signal Detection Theory(SDT) is based on probability theory and mathematical statis-

tics as its theoretical basis. According to the theories of parameter estimation, statistical

distribution theory and statistic judgment of stochastic phenomena in probability the-

ory and mathematical statistics, signal detection can accurately identify and judge signal

and noise[9]. Signal detection theory assumes that noise is always present in the system

and cannot be eliminated. Hence, a hypothesized model for signal detection is defined

as[10][11],

H0 : x(n) = w(n)

H1 : x(n) = s(n)+w(n)

3

Where H0 represents the hypothesis corresponding to "no signal transmitted", H1 rep-

resents "signal transmitted", s(n) is transmitted signal, w(n) is an Additive White Gaus-

sian Noise (AWGN) with zero mean and variance σ2n .

Figure 1.1: Signal Detection Theory(SDT) Model

Signal distribution and noise distribution must have a certain overlap. The subjects

decide whether a signal was present based on a criterion (C). If the stimulus intensity is

greater than C, the subject responds "yes", that is, a signal was present; otherwise, the

subject responds "No", a signal was absent. In the SDT, the subjects can have four kinds

of results to determine the signal was presence or absence[12],

(1) . Hit. When the signal s(n) appears, the subject responds "yes".

(2) . False Alarm. When only noise w(t ) appears, the subject responds "yes".

(3) . Miss. When the signal s(n) appears, but the subject responds "No".

(4) . Correct Reject. When only noise w(t ) appears, the subject responds "No".

In the last few decades, several signal detection methods have been proposed, such

as energy-based detection proposed in [13–15], Matched Filter based detection pro-

posed in [16][17] and Cyclostationary Feature-based Detection proposed in [18–21]. Energy-

based detection is non-coherent and non-optimal detection. It compares the signal

4

energy with a threshold, which is created over the noise floor, to judge if the signal is

present or absent and does not need any prior knowledge of the target signal. Although

it is less expensive and more simple to implement, energy-based detection has poor

performance at low Signal-to-Noise Ratio(SNR) levels and could lead to false alarms.

Matched filter-based detection is a coherent and optimal detection method, which re-

quires comprehensive knowledge of the target signal. Hence, match filter-based detec-

tion could be impractical to implement and cost more. In particular, if no accurate prior

information about the target signal is available, match filter-based detection is much

weaker. Cyclostationary feature-based detection employs the cyclostationary features of

signal to identify if the target is present or absent. It could be used for very low SNR de-

tection. However, the disadvantage is that Cyclostationary feature-based detection has

high computational complexity and require high cyclic frequency resolution because it

has to deal with all frequencies to generate a spectral correlation function.

1.2.2 Signal Parameters Estimation

Signal estimation is a very important link in signal detection and classification. The de-

tected signal contains unknown parameters, such as frequency, phase, power, time de-

lay, etc. If these parameters can be accurately estimated, signal classification becomes

possible. Generally speaking, signal estimation techniques can be classified into two

main categories: the parametric method and the non-parametric method[22][23]. The

parametric method uses a Fourier algorithm based on spectral estimation without prior

assumed model, and it has low computational complexity. The drawbacks of the para-

metric method are limited frequency resolution and tendency to suffer spectral leakage

effects. On the contrary, the non-parametric method is a high resolution estimation with

prior assumed signal structure. The non-parametric methods include autoregressive

(AR) process model, Prony algorithm and space-alternating generalized expectation-

maximization (SAGE) algorithm[24–26].

5

1.2.3 Signal Modulation Classification

Figure 1.2: Communication System Diagram

Automatic Modulation Classification (AMC) is the important bridge between signal

preprocessing and demodulation. The signal preprocessing includes some functions,

such as signal detection, signal parameters estimation and so on. Based on different

algorithm is implemented in classifier, preprocessor needs to provide different levels

of accuracy for signal detection and signal parameters estimation. In the civilian area,

blindly automatic modulation classification plays a key role in many different types of

communications, such as commercial systems and Software Defined Radio(SDR). In the

military area, a friendly signal should be safely transmitted and received, and difficult

to identify. However, the enemy’s signal should be as easy as possible to detect, iden-

tify and demodulate. Hence, automatic modulation classification becomes an impor-

tant and difficult problem in many non-cooperative environments[27]. In recent years,

since some new technologies have been developed in the wireless communication area,

people have become more interested in the multiple-carrier signals, like OFDM signals,

and multiple-input multiple-output (MIMO) systems. It undoubtedly brings more chal-

lenges for the people who design the intelligent receivers. Nowadays, there are two gen-

eral algorithms for modulation classification, likelihood-based (LB) method[28–30], and

feature-based (FB) method[31–33]. The LB method employs a likelihood function of a

received signal-based likelihood ratio to compare with a threshold value to make a deci-

6

sion for the classification. The FB method makes a decision depending upon observed

feature values. The LB method is optimal in the Bayesian sense, but it suffers from high

computational complexity. On the other hand, although the FB method is not optimal,

it has low computational complexity.

1.3 Literature Review

From 1975 to 1995, William A. Gardner and Chad M. Spooner published many papers

about cyclostationary signal processing, such as [34] and [35]. They comprehensively

and profoundly indicated that signal cyclostationarity and statistics play an important

role in signal detection, parameter estimation, and modulation classification. After 1996,

more and more people became dedicated to studying cyclostationarity and statistics-

based signal detection, estimation and classification. Beijing University applied higher-

order statistics to detect non-Gaussian stochastic signals in their paper [36]. In 1997

and 1998, Pierre Marchand and Jean-Louis Lacoume proposed a modulation classifica-

tion method based on cyclic cumulant of different orders in [37]. They could classify

single QPSK, 16-QAM, and 64-QAM signals. However, they needed to use at least 4096

symbols and keep SNR above 10dB for the probability of correct modulation classifi-

cation to reach 90%. In 2000, the paper [38] was published during the 10th European

Signal Processing Conference. The authors employed 4th order cumulant to estimate

wideband signal parameters. In 2004, [39] was proposed by Xinzheng Lv, who used high

order cumulant to classify 2ASK, 4ASK, 4PSK, 2FSK, and 4FSK signals. However, he also

considered a single signal classification problem. [40] was published in 2008. The au-

thors built a spectral correlation function and SVM based classifier. Since some signals

have no second-order cyclostationary features, the authors only considered some lower

order modulation signals. In recent years, the paper [41] employs cumulant theory to

classify single signals’ modulation type and multiple signals’ modulation types. How-

7

ever, the authors assumed that they knew the number of existing signals.

1.4 Problem Statement and Approaches

In this dissertation, our main objective is to solve detection, parameter estimation and

modulation classification of mixed signals, which overlap in the time domain and fre-

quency domain. In our previous research, we implemented second-order cyclostation-

ary processing to detect and estimate the mixed signal. However, since some signals

have the same or no second-order cyclostationary features, we have to limit our mod-

ulation type pool to BPSK and QPSK signals only. For example, the QPSK signal, 8-PSK

signal, and 16-QAM signal have no second-order cyclostationary feature in the cyclic

frequency domain. Hence, the low-order cyclostationary features cannot be used to dis-

tinguish mixed signals if it contains QPSK signal, 8-PSK or 16-QAM signal. In some ex-

treme situations, if two signals are entirely overlapping in the frequency domain, such

as one BPSK signal mixed with one QPSK signal, with the same carrier frequency, the

second-order cyclostationary processing would not work as well. Hence, in the past,

we mainly focused on detecting, estimating, and classifying BPSK and QPSK mixed sig-

nals with different carrier frequency and symbol rate. Now, in this dissertation, we will

mainly use high-order cyclostationary processing to blindly detect, estimate, and clas-

sify mixed signals, which are heavily overlapped in both time domain and frequency

domain. Our modulation pool includes four popular modulation types: BPSK modula-

tion, QPSK modulation, 16-QAM modulation, and 8-PSK modulation. Specifically, we

employed 2nd, 4th, and 8th-order Cyclic Moment (HOCM) to enumerate the signals, es-

timate the signals’ parameters, such as carrier frequency and symbol rate, and classify

some of their modulation types. Since QPSK and 16-QAM have similar 4-th order cyclic

moment features, we will use different order cyclic cumulants to further classify their

modulation types. For other modulation types, we can identify the BPSK signals by us-

8

ing 2-nd order cyclic moment and identify the 8-PSK signals by using 8th order cyclic

moment. In particular, since the QPSK signal and 16QAM signal have very similar cyclic

cumulant features and they have a large overlap in the time domain and frequency do-

main, we cannot use the traditional threshold method to distinguish them. In this case,

Machine Learning (ML) and Support Vector Machine (SVM) is used to assistant us to

classify QPSK signal and 16QAM signals. We also employ the low-cost energy-based de-

tection method to predict whether there is any signal in the channel or not. Meanwhile,

we employed spectrum analysis to roughly locate the mixed signal and reduce the none

target signals’ interference. In this dissertation, we analyzed the mixed-signal detection,

parameter estimation, and classification performance in different channels, such as the

Additive White Gaussian Noise (AWGN) channel, flat fading channel, and multi-path

fading channel. In particular, we analyzed the signal detection, estimation, and mod-

ulation classification performance under different conditions, such as different Signal-

to-Noise (SNR), different symbols, or different power ratios, etc. For the mixed-signal

classification, a confusion matrix is also provided to show the detailed performance of

our classifier.

1.5 Dissertation Contributions

This dissertation is first to use both low-order and high-order communication signals’cyclostationarity

to solve heavily overlapped mixed-signal problems, including solving mixed signals de-

tection, parameter estimation, and modulation classification problems. Meanwhile, it

combines signal cyclostationarity and machine learning techniques to classify different

combinations of mixed signal’s modulation types. Some very challenging mixed-signal

cases are also analyzed in this dissertation. In addition, the performance analysis of

mixed-signal detection, parameter estimation, and modulation classification is com-

prehensively provided.

9

1.6 Dissertation Organization

In this dissertation, there is a total of seven chapters. Chapter 1 comprehensively in-

troduces the motivation of this dissertation and introduces the background and devel-

opment of signal detection, parameter estimation, and modulation classification tech-

niques. In addition, this chapter describes the problem statement, approach, and con-

tributions. Chapter 2 introduces the machine learning technique and in particular, in-

cludes one of the most popular machine learning techniques, i.e., Support Vector Ma-

chine (SVM). Chapter 3 presents lower-order and higher-order cyclostationary theory-

based signal processing, includes signal detection, signal estimation, and signal mod-

ulation classification. In particular, this chapter includes detailed second-order cyclo-

stationary and high-order cyclic cumulants theoretical knowledge and simulations of

signals. In Chapter 4, we introduce our two signal-processing models, mixed-signal

model A and special mixed-signal case model B. Model A is multiple signals that are

heavily overlapped in both time domain and frequency domain with different parame-

ters, such as carrier frequency, power, .etc. Model B is multiple signals that possess the

same carrier frequency. In addition, this chapter also introduces the detailed workflow

of mixed-signal detection, parameter estimation, and modulation classification. Chap-

ter 5 mainly describes the theoretical derivation of the cyclic moment and cyclic cu-

mulant of a single signal and mixed-signal. This chapter also includes some simulations

about signal detection, parameter estimation, and modulation classification. In Chapter

6, we analyze the performance of our proposed signal detection, parameter estimation,

and modulation classification methods under different channels, such as AWGN chan-

nel, flat fading channel, and multi-path fading channel. Chapter 7 is the conclusion.

10

Overview of Machine Learning

In this chapter, we will introduce basic knowledge of machine learning, includes its

background, definition, algorithms, and development. Meanwhile, we will describe one

of the most popular machine learning algorithms, which is the Support Vector Machine

(SVM). In this dissertation, we will employ SVM to help us solve the signal detection

problem and modulation classification problem.

11

2.1 Introduction

Figure 2.1: Artificial Intelligence (AL)

2.1.1 History

Machine Learning (ML) originated in Artificial Intelligence (AI), a new technology that

studies and develops the theories, methods, techniques, and applications for simulat-

ing, extending and extending human intelligence. Machine Learning is the core of Arti-

ficial Intelligence. In 1959, the term "Machine Learning" was defined by Arthur Samuel,

who proposed machine learning as a way to give a machine the ability to learn so that it

could perform functions that cannot be directly programmed[42].

2.1.2 Definition

Machine Learning (ML) is an interdisciplinary subject involving multiple theories such

as statistics and probability theory. It studies how computers could acquire new knowl-

edge like humans do and how to use large amounts knowledge obtained by computers

to improve problem-solving performance[43]. In practice, machine learning is a way of

using the data to train the model and then use the model to predict future outputs.

12

2.1.3 Development

Machine learning is a relatively young branch of artificial intelligence research, and its

development process can be divided into four stages as follows[44],

1. The first stage was from the mid-1950s to the mid-1960s, and it was an active pe-

riod.

2. The second stage was in the mid-1960s to mid-1970s, known as the cool period.

3. The third stage is from the mid-1970s to the mid-80s, known as the revival period.

4. The latest phase of machine learning began in 1986.

Figure 2.2: Machine Learning relevant field

Nowadays, machine learning is used in many areas, such as Data Mining(DM), Au-

tomatic Speech Recognition (ASR), Natural Language Processing (NLP), Computer Vi-

sion (CV), Statistical Learning (SL), and Pattern Recognition (PR). In particular, Data

13

Mining combines machine learning and databases. Most data mining algorithms are

machine learning algorithms that are optimized for the database[45]. Automatic Speech

Recognition combines machine learning and speech processing, the most popular speech

recognition product is Apple’s Siri[46]. Natural Language Processing is make up of ma-

chine learning and text processing. The objective of natural language processing is to let

the machine understand human language[47]. Computer vision is made up of machine

learning and image processing. The image processing technique is used to process the

image into suitable input for the machine learning model. Machine learning is respon-

sible for identifying the relevant patterns from the image[48]. Statistical learning is a

discipline, which has significant overlaps with machine learning because most of the

methods in machine learning come from statistics[49]; Pattern Recognition is similar to

machine learning. The main difference between them is that pattern recognition is used

in industry, and machine learning is mainly used in computer science[50].

14

2.1.4 Algorithm

Figure 2.3: Machine Learning Algorithm

Machine learning has two techniques: supervised learning and unsupervised learning.

Supervised learning can learn or build a model from training materials and speculate

on new examples. Training data is composed of input data and expected output data.

The output of a function can be a continuous value (regression analysis), or a classifi-

cation label (classification analysis). Unsupervised learning is a technique for machine

learning that aims to categorize raw data to understand the internal structure of that

data. Unlike supervised learning, unsupervised learning does not know whether their

classification results are correct, that is, they are not subject to supervision to tell them

15

what kind of learning is correct. It is characterized only by the input data, and it will

automatically use this input data to find its potential category rules. When the study is

completed and tested, it can also be applied to the new case. A typical example of un-

supervised learning is clustering. The purpose of clustering is to bring together similar

things, and we do not care which class is. Therefore, a clustering algorithm usually only

needs to know how to calculate the similarity of data to begin working[51][52].

There are a lot of different algorithms for supervised and unsupervised learning,

such as Support Vector Machine, Discriminant Analysis, and Neural Networks. However,

there is no best, universal algorithm. Find the right algorithm is just a matter of trying

[52].

2.2 Support Vector Machine

In the field of machine learning, Support Vector Machine is a supervised learning model

usually used for pattern recognition and classification. In 1963, Vladimir N. Vapnik and

Alexey Ya. Chervonenkis invented the SVM algorithm[53]. SVM works for linearly sep-

arable situations. In the case of linear indivisibility, SVM transforms the linearly in-

divisible sample into a high-dimensional feature space by using the nonlinear map-

ping algorithm to make it linearly separable, which makes it possible to linearly analyze

the nonlinear characteristics of the samples by using the linear algorithm in the high-

dimensional feature space.

16

0 100 200 300 400 500 600 700 800 900

Value

0

0.002

0.004

0.006

0.008

0.01

0.012

PD

F

Data0: Mu = 500, Var = 50

Data1: Mu = 700, Var = 40

Data2: Mu = 300, Var = 60

TraingSamples = 9000; Test Samples = 1000

Data 0 Data 1 Data 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Accura

cy r

ate

Figure 2.4: SVM Example

Figure 2.4 shows an example of the SVM algorithm to classifying three sets of Gaus-

sian random data with different mean values and variance values. Data0 is the Gaussian

random data vector with a mean value of 500 and a variance value of 50. Data1 is a Gaus-

sian random data vector with a mean value of 700 and variance value of 40. Data2 is a

Gaussian random data vector with a mean value of 300 and variance value of 60. For the

SVM algorithm, the length of the training data is 1000 samples. There are 9000 sets of

training data for Data0, Data1, and Data2, respectively. 1000 sets of data are used to be

testing data. From the 2.4(b), we see that the classification accuracy of Data0 is 97.4%

and, the classification accuracy of Data1 and Data2 is 100%. We also get the confusion

matrix as following,

Figure 2.5: Confusion Matrix

17

2.3 Conclusion

In this dissertation, we employed the SVM algorithm of machine learning to classify sig-

nal modulation types, which have pretty similar features. It is possible but not viable to

classify signals if they provide a single training signal or mixed training signal to SVM

directly. Hence, in order to effectively classify mixed-signals, we used different higher-

order cyclic cumulants as training features for classification. This method not only re-

duces the length of time spent processing data but also makes the distinction between

each modulated mixed signal are more obvious.

18

Overview of Cyclostationary Signal

Processing

This chapter includes the introduction of low order cyclostationary signal processing

to higher-order cyclostationary signal processing. Section 3.1 briefly introduces the

second-order cyclostationarity theory. In section 3.2, we describe the higher-order cy-

clostationarity of signal, including its background, mathematical definition, and devel-

opment.

19

3.1 Second-Order Cyclostationarity

3.1.1 Introduction

In signal processing, the statistics of the signal play an important role. It is well known

that the most commonly used statistics are the mean (first-order statistics), the corre-

lation function and the power spectral density function (second-order statistics), and

the third and higher-order statistics. If the statistics of the signal vary periodically or

poly-periodically, the signal is called a cyclostationary signal[54]. Cyclostationarity is

an important characteristic of the cyclostationary signal. It gives cyclostationary sig-

nals a common property, that is, the spectral correlation feature, which can be used

for distinguishing non-cyclostationary signals. Moreover, cyclostationarity can be used

to estimate the single observation record and obtain time-varying statistics. Besides,

cyclostationarity can also suppress any stationary colored noise. High-order cyclosta-

tionarity can even suppress non-stationary Gaussian colored noise. Therefore, cyclosta-

tionarity is widely used in varied signal processing tasks, such as signal detection, signal

parameters estimation, and signal modulation classification[34, 35]. To compare it with

other signal processing technologies such as energy-based detection technology or car-

rier recovery technology, second-order cyclostationarity performs much better under

low Signal-to-Noise Ratio (SNR) channels and has high performance for blind param-

eter estimation. However, second-order cyclostationarity suffers from high computa-

tional complexity. Meanwhile, some higher-order modulation signals, such as QPSK,

8-PSK, and 16QAM, have the same second-order cyclostationarity. Hence, the modula-

tion classification performance of second-order cyclostationarity is constrained, which

led to it not being widely used in actual signal modulation classification.

Second-order cyclostationarity has been widely developed in the last ten years. In

2008, this paper[55] applied cyclic autocorrelation function into performance analysis

of the ranging system. Later, spectral correlation was used for signal detection and clas-

20

sification in Cognitive Radio[56, 57]. As people pay more attention to the multi-carrier

signals, second-order cyclostationarity is also used to analyze multi-carrier signals, such

as OFDM signal[58].

3.1.2 Definition

Assume x(t ) is a waveform, its cyclic autocorrelation function is given by

Rαx (τ) = lim

∆t−→∞1

∆t

∫ ∆t/2

−∆t/2x(t +τ/2)x(t −τ/2)e−i 2παt d t (3.1)

Where ∆t is measurement interval, τ is time delay, and α is cyclic frequency. When

α = 0, the Equation 3.1 become general autocorrelation function Rx(τ). If all α 6= 0 and

Rαx (τ) = 0, then the waveform x(t ) is purely stationary. If x(t ) is periodic with period T

and Rαx (τ) 6= 0 for α= i nteg er /T , then we say x(t ) is purely cyclostationary with period

T[34].

According to cyclic Wiener relation[59], we could get the spectral correlation func-

tion.

Sαx ( f ) =∫ ∞

−∞Rα

x (τ)e−i 2π f t dτ (3.2)

The spectral correlation function is the Fourier transformation of the cyclic auto-

correlation function. When α= 0, the Equation 3.2 becomes the power spectral density

function.

The normalized version of the spectral correlation function is called the spectral

autocoherence function, which is given by

Cαx ( f ),

Sαx ( f )

[Sx( f +α/2)Sx( f −α/2)]1/2(3.3)

21

Where |Cαx ( f )| É 1. When |Cα

x ( f )| = 0, x(t ) is completely incoherent at f and α. When

|Cαx ( f )| = 0, x(t ) is completely coherent at f and α.

3.1.3 Examples

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

α (Hz) ×104

0

1

2

3

4

5

6

SC

F

×106 SCF of BPSK (α Domain)

Figure 3.1: SCF of BPSK signal

Fig.3.1 shows the SCF of BPSK signal with rectangular pulse shape. The sampling

frequency Fs is 18000 Hz, the carrier frequency Fc is 3000 Hz and symbol rate Fb is 600

Hz. It is clear that BPSK has second-order cyclostationary feature atα= 2Fc±kFb , where

k = 0,1,2,3...

-8000 -6000 -4000 -2000 0 2000 4000 6000 8000

α (Hz)

0

1

2

3

4

5

6

SC

F

×106 SCF of QPSK (α Domain)

Figure 3.2: SCF of QPSK signal

22

-8000 -6000 -4000 -2000 0 2000 4000 6000 8000

α(Hz)

0

1

2

3

4

5

6

SC

F

×106 SCF of 8-PSK (α Domain)

Figure 3.3: SCF of 8-PSK signal

-8000 -6000 -4000 -2000 0 2000 4000 6000 8000

α(Hz)

0

1

2

3

4

5

6

SC

F

×106 SCF of 16-QAM (α Domain)

Figure 3.4: SCF of 16-QAM signal

23

From Fig.3.2 to Fig.3.4, we can see that higher-order modulated signals, such as

QPSK, 8-PSK, and 16-QAM signals do not exhibit second-order cyclostationary features

if the observation interval is infinity. Hence, it is able to use SCF to distinguish BPSK

among higher-order modulated signals. However, we cannot distinguish QPSK, 8-PSK,

and 16-QAM by using second-order cyclostationarity. Therefore, the higher-order cyclo-

stationarity is desirable.

3.2 Higher-Order Cyclostationarity

3.2.1 Introduction

Higher-Order Cyclostationarity (HOCS) was created from cumulant theory, which itself

was developed in the probabilistic theory of stochastic processes and generated from

mathematical statistics[60]. The term cumulant was first proposed by John Wishart in

his paper[61], which was published in 1928. 20 years later, the cumulant theory was for

the first time applied in engineering. Stratonovich, Kuznetsov, and Tikhonov employed

cumulant theory for stochastic processes in their published paper[62]. In the late 1960s,

cumulant theory was integrated into the detection techniques[63–65]. From the early

1970s, high order statistics (HOS) had been proposed because of no phase information

if the signal is stationary or does not show second-order cyclostationarity (SOCS)[66].

Since then, higher-order cyclostationarity (HOCS) has been widely used for some cyclo-

stationary signals whose statistic is varying periodically with time. William A. Gardner

and his student Chad M. Spooner contributed a lot of research about signal cyclosta-

tionarity, such as [67–70]. In their paper[60], they indicate that HOCS deal with fea-

tures of time-domain and frequency domain of the intensity of sine wave elements of

the periodically time-varying higher-order probability function of cyclostationary sig-

nals. HOCS touches on many of the momentous developments in communications that

24

occurred in the first 100 years from the twentieth century: synchronization[71], signal

reconstruction[72], and extracting radar signal’s range and doppler information[73].

3.2.2 Definition

Assume x(t) is a complex value sine wave, we could get its n-th order lag product from

Lx(t ,τ;n,m) =n∏

j=1x(∗) j (t +τ j ) (3.4)

Where (∗) represents the j-th item’s conjugation value, and m is the number of conjugate

items. Since higher-order (greater than 2) products contain the product of the lower-

order sine wave, we have to remove the lower-order product items from the result. In

order to characterize all the possible combinations of low-order sine waves, we have

to set partitions to do the following calculations of higher-order Cyclostationarity. For

example, when n = 4, we can get 4 partitions instead of 15 partitions because of all odd-

order moments are equal to zero[60]. The partitions are as follows,

{1,2,3,4}, p = 1

{{1,2}, {3,4}}, p = 2

{{1,3}, {2,4}}, p = 2

{{1,4}, {2,3}}, p = 2

Based on the partitions, we could easily subtract all of the products of pure lower-

order sine waves from the n-th order product.

25

Based on Equation(3.4), we can get the nth-order temporal moment function(TMF)[60],

Rx(t ,τ;n,m) = Eα[Lx(t ,τ;n,m)] = Eα[n∏

j=1x(∗) j (t +τ j )] (3.5)

Where Eα[·] is the general sine-wave extraction operator,

Eα[x(t )] =∑β

⟨x(t )e−i 2πβt ⟩e i 2πβt (3.6)

< x(t ) >= limz→∞

1

z

∫ z/2

−z/2x(t )d t (3.7)

β is frequency for ⟨x(t )e−i 2πβt ⟩ 6= 0, < · > is time-averaging operator, which is expressed

by Equation.(3.7). Then, we could get the nth-order cyclic temporal moment func-

tion(CTMF), which is Fourier coefficient of the temporal moment function[60],

Rαx (τ;n,m) = ⟨Rx(t ,τ;n,m)e−i 2παt ⟩ (3.8)

The relationship between nth-order temporal moment function and nth-order tem-

poral cumulant function(TCF) is given by[60],

Cx(t ,τ;n,m) =∑P

(−1)p−1(p −1)!p∏

j=1Rx(t ,τ j ;n j ,m j ) (3.9)

Where P is the number of partition set. Lower case p is the number of sub-set in each

partition set. Now, we can get the nth-order cyclic temporal cumulant function (CTCF)

or cyclic cumulant via Fourier coefficient of the TCF[60],

26

Cαx (τ;n,m) =∑

P[(−1)p−1(p −1)!

∑β†1=α

p∏j=1

Rβ jx (τ j ;n j ,m j )] (3.10)

The vector of cycle frequencies β = [β1β2β3...βp ] is the vector of cyclic temporal mo-

ment cycle frequencies, and their summation must be equal to cyclic cumulant cycle

frequency α.

3.2.3 Examples

(i) 2nd order cumulant

C (2,0) =Cum(x, x) = M(2,0)

C (2,1) =Cum(x, x∗) = M(2,1)

(ii) 4th order cumulant

C (4,0) =Cum(x, x, x, x) = M(4,0)−3M(2,0)2

C (4,1) =Cum(x, x, x, x∗) = M(4,1)−3M(2,1)M(2,0)

C (4,2) =Cum(x, x, x∗, x∗) = M(4,2)−M(2,0)2 −2M(2,1)2

(iii) 6th order cumulant

C (6,0) =Cum(x, x, x, x, x, x) = M(6,0)−15M(4,0)M(2,0)+30M(2,0)3

C (6,1) =Cum(x, x, x, x, x, x∗) = M(6,1)−5M(4,0)M(2,1)−10M(2,0)M(4,1)+30M(2,1)M(2,0)2

C (6,3) =Cum(x, x, x, x∗, x∗, x∗, ) = M(6,3)−6M(4,1)M(2,0)−9M(2,1)M(4,2)+18M(2,1)M(2,0)2

+12M(2,1)3

(iv) 8th order cumulant

27

C (8,0) =Cum(x, x, x, x, x, x, x, x) = M(8,0)−35M(4,0)2 −28M(6,0)M(2,0)

+420M(4,0)M(2,0)2 −630M(2,0)4

3.2.4 Simulations

2 PSK

Fc = 3000; Fb = 400; Power = 1; Roll-off factor = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (8,0) (8,1) (8,2) (8,3) (8,4) (8,5) (8,6) (8,7) (8,8)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

ha

rmo

nic

nu

mb

er

20

40

60

80

100

120

140

160

180

200

Figure 3.5: Cyclic Cumulant of BPSK signal

28



Fig.3.5, 3.6, and 3.7 show the nth order and mth conjugate cyclic cumulant magnitude

of 1 BPSK signal with raised cosine filter. Sampling frequency Fs = 18000Hz, carrier fre-

quency Fc = 3000Hz, symbol rate Fb = 400Hz, power is 1 and roll-off factor = 0.5.

29

4 PSK


(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (8,0) (8,1) (8,2) (8,3) (8,4) (8,5) (8,6) (8,7) (8,8)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

ha

rmo

nic

nu

mb

er

0

5

10

15

20

25

Figure 3.8: Cyclic Cumulant of QPSK signal


30


Fig.3.8, 3.9 and 3.10 shows the nth order and mth conjugate cyclic cumulant magni-

tude of 1 QPSK signal with raised cosine filter. Sampling frequency Fs = 18000Hz, carrier

frequency Fc = 3000Hz, symbol rate Fb = 400Hz, power is 1 and roll-off factor = 0.5.

8 PSK


(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (8,0) (8,1) (8,2) (8,3) (8,4) (8,5) (8,6) (8,7) (8,8)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

ha

rmo

nic

nu

mb

er

0

5

10

15

20

Figure 3.11: Cyclic Cumulant of 8PSK signal

31



Fig.3.11, 3.12 and 3.13 shows the nth order and mth conjugate cyclic cumulant mag-

nitude of 1 8PSK signal with raised cosine filter. Sampling frequency Fs = 18000Hz, car-

rier frequency Fc = 3000Hz, symbol rate Fb = 400Hz, power is 1 and roll-off factor = 0.5.

32

16 QAM


(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (8,0) (8,1) (8,2) (8,3) (8,4) (8,5) (8,6) (8,7) (8,8)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

ha

rmo

nic

nu

mb

er

0

1

2

3

4

5

6

7

8

9

10

Figure 3.14: Cyclic Cumulant of 16QAM signal


33


Fig.3.14, 3.15 and 3.16 shows the nth order and mth conjugate cyclic cumulant mag-

nitude of 1 16QAM signal with raised cosine filter. Sampling frequency Fs = 18000Hz,

carrier frequency Fc = 3000Hz, symbol rate Fb = 400Hz, power is 1 and roll-off factor =

0.5.

34

Overview of Mixed Signal Processing

Diagram

In this chapter, we will introduce the detailed Mixed-Signal Processing workflow and our

proposed mixed detection, parameter estimation, and modulation classification meth-

ods. In addition, we will also propose a mechanism to solve the problem if the system

failed to detect a signal, estimate signal parameters, or classify signal modulation types.

35

4.0.1 Mixed Signal Models

Figure 4.1: Mixed Signals Models

In order to more clearly analyze the mixed-signal estimation and identification prob-

lem, we propose two different mixed-signal models, model A and model B, which are

shown in Figure 4.1. Model A is a more general case, with multiple signals mixed in both

time domain and frequency domain. They have different carrier frequencies and may

have the same or different symbol rates. They may also have different power, different

carrier phase offset, or different time delays. Moreover, these signals cannot be identi-

fied and enumerated via general spectrum analysis. For this model, our target signal is

two mixed signals, but the approach can also be extended to solve more mixed signals.

Model B is the most challenging case, with multiple signals possess the same carrier fre-

quency and different symbol rates. They may also have different power, different carrier

phase offset, or different time delay. If we add more and more statistically independent

signals together, we will get a signal that is more and more like a Gaussian signal, so it

is impractical to try to classify infinite overlapped signals. Hence, for analysis purposes,

we limit the maximum number of time and frequency domain overlapped mixed signals

to two. But note that, although we limit the maximum number of mixed signals, our

proposed approach is still valid for detecting and estimating more than two overlapped

36

signals with different symbol frequency. However, the estimation and classification per-

formance will become worse with as the number of mixed signals increases.

4.0.2 Mixed Signal Processing Workflow

Figure 4.2: Mixed Signal Processing flow diagram

Figure 4.2 shows a general working flow diagram of the Mixed Signal Processing

(MSP) system, which includes signal detection processing, signal estimation processing,

and signal modulation classification processing. Some system response mechanisms

37

are also shown in the figure. The detailed procedure is as follows,

[1] Energy-based Signal Detection. It is well known that Energy-based Detection(ED)

is widely used for transmitter detection, such as in cognitive radio and electronic

warfare. Compared with other signal detection methods, such as Matched Filter

Detection and Cyclostationary Feature Detection, Energy detection is less optimal

but is simple to implement. It has a high computational speed, low computa-

tional complexity, and can be implemented in both time domain and frequency

domain[74]. Hence, Energy-based Detection is the best first step for mixed-signal

processing. In particular, we judge the existence or existence of the signal based

on the signal energy value Es and noise-based threshold value ηE N . If Es is greater

than the threshold ηE N , the system will start the next step and perform spectrum

sensing. Otherwise, it will stop to process the received signal and display the mes-

sage: No Signal.

[2] Spectrum Sensing. The objective of this step is to sense the spectrum of the target

signals and select the pre-processed frequency range. It easily senses the spec-

trum of the target signals via spectral analysis. Hence, we can filter out non-target

signals in the frequency domain to reduce non-target signal interference and in-

crease system performance.

[3] Parameters Estimation. In this step, we employ n-th order Cyclic Moment (CM)

to estimate the signal’s parameters, include carrier frequency, symbol rate, and

signal pattern types, all of which determines the specific set of cycle frequencies

that the signal will exhibit in Cyclic Cumulants (CCs). The cyclic moment is a more

straightforward way to estimate signal parameters. It can locate target signals’ car-

rier frequency and symbol rate. To compare CM to CCs, since the CCs are the sum

of products of lower-order moments plus the nth-order moment itself, they gener-

ally have a higher variance than the CM. In this dissertation, we employed a cyclic

38

moment based hierarchical estimation method to estimate signal parameters. De-

tails in Figure 4.3.

Figure 4.3: Hierarchical Signal Parameter Estimation Diagram

We employed 2nd order cyclic moment to estimate the BPSK signal’s parameters,

employed 4th-order cyclic moment to estimate the QPSK and 16-QAM signal’s pa-

rameters and employed 8th-order cyclic moment to estimate the 8-PSK signal’s

parameters. We can now consider the two cases, which are shown in Figure 4.1. In

the mixed-signal model A, we can estimate mixed-signals’ carrier frequencies. If

there are more than two mixed signals with different carrier frequencies and sym-

bol rates, we are able to estimate their parameters. In model B, we can get one

carrier frequency and multiple symbol rate.

[4] Signal Number Acquisition. Based on step [3], we can get the number of signals

mixed together by counting the number of different carrier frequencies or symbol

rates. For example, in model A, we could enumerate the mixed signals by counting

the number of different carrier frequencies. In model B, we could enumerate the

mixed signals by counting the number of different symbol rates.

[5] Modulation Classification. In this part, we mainly employ n-th order cyclic mo-

ment, n-th order cyclic cumulants, and machine learning technology to classify

the modulation types of mixed signals. In step [3], for the model (A), we could

39

classify some of the modulation types by using different order cyclic moment, i.e.,

we can employ 2nd order cyclic moment to classify BPSK signals and non-BPSK

signals, we can employ 4th order cyclic moment to classify QPSK/16-QAM and

8-PSK signals. We can also use 8th order cyclic moment to classify 8-PSK signal.

Here the hardest part is to classify QPSK and 16-QAM signals because they have

similar cyclostationary features. In particular, if they have different power, their

cyclostationary features will be mixed up. Hence, in order to solve this difficult

problem, we use cyclic cumulant non-correlation items to estimate each signal’s

power and normalize that power, and then feed each signal’s cyclic cumulant fea-

tures into a pre-trained Support Vector Machine (SVM) model for classification.

Compared with traditional threshold value based classification, SVM can provide

more classification accuracy.

Figure 4.4: SVM based Signal Modulation Classification

Fig.4.4 shows the workflow of SVM based signal modulation classification. We use

CCs based and non-CCs based training data to train a SVM classification model.

Then, we employ this SVM model to classify signal modulation types. We do this

with the training data which is under SN R = 20 dB noise environment. Mean-

while, we also consider other data, such as the CCs of noise and the uncorrected

CCs data in the training data in order to make the results more reliable.

[6] Display Results The number of signals, the signals’ parameters, and the modula-

tion types will be shown in the output.

40

Cyclostationary Signal Processing

theory based Mixed Signal Processing

In this chapter, we will process the mixed-signals by using energy-based signal detec-

tion and cyclostationary signal processing. In particular, we will employ n-th order

cyclic moment functions to preliminary enumerate and estimate mixed-signals, and

then employ higher-order cyclic cumulant functions to enumerate and classify mod-

ulation types of mixed-signals.

41

5.1 Signal Detection

If we assume that the target signal is s(n), white Gaussian Noise is w(n) ∼ N (0,1). x(n) is

the received signal, the energy-based detection can be considered a binary hypothesis

test problem:

H0 : x(n) = w(n)

H1 : x(n) = s(n)+w(n)

Where H0 represents a signal being absent, H1 represents a signal being present, and

n = 0,1..., N −1. Hence, the signal energy detection statistics is computed by,

Es(x) =N−1∑n=0

|x(n)|2 (5.1)

Where Es(x) follows a Chi-square or χ2-distribution with N degrees of freedom. The

probability density function (PDF) of χ2-distribution is shown in Eq.(5.2) as[75]:

f (x;k)χ2 = 1

2

k

2 Γ(k

2)

x(

k

2−1)

e−x

2 (5.2)

where Γ(k/2) denotes the Gamma function, which has closed-form values for integer k.

x ≥ 0. Since a false alarm rate (P f a) is only related to noise, the best threshold value (η)

for a certain noise environment can be determined by a required P f a .

P f a = Pr [Es(x) > η|H0] (5.3)

Hence, if required P f a is known, we can easily get the best threshold value (η) from

42

Eq.(5.4) or a large amount of Gaussian random variables.

P f a =∫ ∞

η

1

2

k

2 Γ(k

2)

x(

k

2−1)

e−x

2 d x (5.4)

5.2 Signal Estimation

5.2.1 nth order cyclic moment

In chapter 4, we introduced the cyclic temporal moment theory-based signal parameter

estimation, such as in Equation (3.2.2), which is the Fourier coefficient of the tempo-

ral moment function. Assume x(t ) is a continue-time cyclostationary process, the n-th

order temporal moment function of x(t ) is the expected of lag product of x(t )[76],

mx(t ,τ)n = E {x(t +τ1)x(t +τ2)...x(t +τn)}

=∑β

mx(α,τ)ne j 2παt(5.5)

E {x(t )} = 1

N

N−1∑k=0

x(t +kT ) (5.6)

Where E {·} is the expected value operator, which is expressed as Equation.(5.6), and τi

is the time delay, i = 0,1,2...n. β is the nth-order temporal moment cycle frequency set,

α ∈ β. The nth order cyclic moment function is given by,

mx(α,τ)n =F {mx(t ,τ)n}

= limT→∞

1

T

∫ T /2

−T /2mx(t ,τ)ne− j 2παt d t

(5.7)

Where F is Fourier Transform operator, and T is time interval.

43

5.2.2 single signal parameter estimation

Assume s(t) is a modulated signal, whose pass band form is given by,

s(t ) = a(t )e j (2π fc t+θ) (5.8)

Where fc is carrier frequency, θ is carrier phase. And a(t ) is base band signal, which is

expressed as,

a(t ) = AN∑

i=1s[i ]p(t − i Ts − t0) (5.9)

Where A is the amplitude of the signal, s[i ] = si + j sq is the i-th transmitted symbol in

complex-value form, Ts is the symbol duration and p(t ) is the pulse, t0 is propagation

delay.

BPSK signal parameters estimation

Assume s(t ) is a BPSK modulated signal with rectangular pulse shape, amplitude A = 1,

carrier phase θ = 0, propagation delay t0 = 0, and the symbol s[i ] is -1 or 1. Here, we

employ reduced-dimension temporal moment and cyclic moment to estimate signal

parameters, where τ= 0. The second-order temporal moment of BPSK signal is

ms(t ,0)2 = E {s(t )s(t )}

= E {a(t )2e j 2π(2 fc )t }

= E {(N∑

i=1s[i ]2p(t − i Ts)2)e j 2π(2 fc )t }

= (N∑

i=1E {s[i ]2}p(t − i Ts)2)e j 2π(2 fc )t

(5.10)

Since

44

E {s[i ]2} ={ 1, for b[i ] = 1 (5.11)

1, for b[i ] =−1

Where b[i ] is information bits. So,

ms(t ,0)2 =N∑

i=1p(t − i Ts)2e j 2π(2 fc )t (5.12)

The second-order cyclic moment function of BPSK is given by,

ms(α,0)2 =F {mx(t ,0)2}

=F {N∑

i=1p(t − i Ts)2}⊗F {e j 2π(2 fc )t }

=F {N∑

i=1p(t − i Ts)2}⊗δ( f −2 fc )

(5.13)

Where δ is Dirac delta function. We know that p(t − i Ts)2 is always equal to 1. We can

get the F {p(t − i Ts)2} as following,

F {p(t − i Ts)2} =∫ Ts /2

−Ts /2e− j 2π f t d t

= TsSa(π f Ts)

(5.14)

Where Sa(t ) = si nt/t , which is Sinc function. Hence, we can get the absolute value of

second-order cyclic moment of BPSK signal is,

|ms(α,0)2| =∣∣TsSa(π f Ts)⊗δ( f −2 fc )

∣∣ (5.15)

So, we can find multiple peaks at α = 2 fc ±k fb . Where α is cyclic frequency, fb is

45

symbol rate, k = 0,1,2,3....

QPSK signal parameters estimation

Assume s(t ) is a QPSK modulated signal with rectangular pulse shape, amplitude A = 1,

carrier phase θ = 0, propagation delay t0 = 0, the symbol s[i ] is e j π4 , e j 3π4 , e j 5π

4 or e j 7π4 .

Hence, the 4th-order temporal moment function of QPSK is,

ms(t ,0)4 = E {s(t )s(t )s(t )s(t )}

= E {a(t )4e j 2π(4 fc )t }

= E {(N∑

i=1s[i ]4p(t − i Ts)4)e j 2π(4 fc )t }

= (N∑


(5.16)

Since

E {s[i ]4} =

−1, for s[i ] = e j π4 (5.17)

−1, for s[i ] = e j 3π4

−1, for s[i ] = e j 5π4

−1, for s[i ] = e j 7π4

So,

ms(t ,0)4 =−N∑

i=1p(t − i Ts)4e j 2π(4 fc )t (5.18)

46

the fourth-order cyclic moment of QPSK signal is given by,

ms(α,0)4 =F {ms(t ,0)4}

=F {−N∑

i=1p(t − i Ts)4e j 2π(4 fc )t }

=−F {N∑

i=1p(t − i Ts)4}⊗δ( f −4 fc )

(5.19)

According to Equation.(5.14), the absolute value of 4th-order cyclic moment of QPSK

signal is

|ms(α,0)4| =∣∣TsSa(π f Ts)⊗δ( f −4 fc )

∣∣ (5.20)


symbol rate, k = 0,1,2,3....

16-QAM signal parameters estimation

Since 16-QAM and QPSK are in the same modulation category, we can also employ 4th-

order cyclic moment to estimate the parameters of 16-QAM signals. Meanwhile, accord-

ing to the cyclic moment of BPSK and QPSK, we can easily know that the complex-valued

symbol s[i ] plays a very important role in signal estimation. In other words, the symbol

s[i ] or signal constellation decides the cyclic moment’s form.

47

-1 -0.5 0 0.5 1

I

-1

-0.5

0

0.5

1

Q

16-QAM Constellation

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Figure 5.1: 16-QAM Constellation

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5

I

-1

-0.5

0

0.5

1

Q

16-QAM4 Constellation

0, 2, 8, 10

3, 4, 9, 14

5, 7, 13, 15

1, 6, 11, 12

Figure 5.2: 16-QAM4 Constellation

Figure.5.1 shows the constellation symbols of the 16-QAM signal. Figure.5.2 shows

the 4-th power of 16-QAM symbol values, which are not symmetrical. So, it will have a

non-zero component in its 4-th order temporal moment and 4-th order cyclic moment.

s[i ]4 =

−3.24, for j = 0,2,8,10 (5.21)

−0.04, for j = 5,7,13,15

0.28+0.96 j , for j = 3,4,9,14

0.28−0.96 j , for j = 1,6,11,12

Where j is the symbol index in the constellation map, and j = 0,1,2, ...,15. Since all 16

different symbols have the same probability of occurring, the expected value is

E {s[i ]4} = [(−3.24)+ (−0.04)+ (0.28+0.96 j )+ (0.28−0.96 j )]/4 =−0.68 (5.22)

Hence, we can get the absolute value of 4th-order cyclic moment of 16-QAM signal

by using Eq.(5.14) and the above equations,

48

|ms(α,0)4| = 0.68∣∣TsSa(π f Ts)δ( f −4 fc )

∣∣ (5.23)

The peaks are at α= 4 fc ±k fb , k = 0,1,2,3....

8-PSK signal parameters estimation

For the 8-PSK modulated signal, we employed 8th-order cyclic moment to estimate its

carrier frequency and symbol rate. The 8-th order temporal moment of 8-PSK is,

ms(t ,0)8 = E {s(t )8}

= E {a(t )8e j 2π(8 fc )t }

= (N∑


(5.24)

-1 -0.5 0 0.5 1

I

-1

-0.5

0

0.5

1

Q

8-PSK Constellation

0

1

2

3

4

5

6

7

Figure 5.3: 8-PSK Constellation

-1 -0.5 0 0.5 1

I

-1

-0.5

0

0.5

1

Q

8-PSK8 Constellation

0, 1, 2, ..., 7

Figure 5.4: 8-PSK8 Constellation

In Figure.5.3 and Figure.5.4, we can see that, the 8-th order of 8-PSK Constellation

symbols s[ j ]8 = −1, where j = 0,1,2,...,8. Hence, we can get the absolute value of 8th-

49

order cyclic moment of 8-PSK signal as

|ms(α,0)8| =∣∣TsSa(π f Ts)δ( f −8 fc )

∣∣ (5.25)


symbol rate, k = 0,1,2,3....

Simulations

3700 3800 3900 4000 4100 4200 4300

Cyclic Frequency (α)

0

0.2

0.4

0.6

0.8

1

1.2

Nor

mal

ized

Mag

nitu

de

2nd-order cyclic moment of BPSK

X: 4000Y: 1

X: 4100Y: 0.04532

7700 7800 7900 8000 8100 8200 8300


0

0.2

0.4

0.6

0.8

1

1.2

Nor

mal

ized

Mag

nitu

de

4th-order cyclic moment of QPSK

X: 8000Y: 1

X: 8100Y: 0.2794

7700 7800 7900 8000 8100 8200 8300


0

0.2

0.4

0.6

0.8

1

1.2

Nor

mal

ized

Mag

nitu

de

4th-order cyclic moment of 16-QAM

X: 8000Y: 1

X: 8100Y: 0.2664

1.57 1.58 1.59 1.6 1.61 1.62 1.63

Cyclic Frequency (α) ×104

0

0.2

0.4

0.6

0.8

1

1.2

Nor

mal

ized

Mag

nitu

de

8-th order cyclic moment of 8-PSK

X: 1.6e+04Y: 1

X: 1.61e+04Y: 0.4627

Figure 5.5: Simulation result

Figure.5.5 shows the cyclic moment of BPSK, QPSK, 16-QAM, and 8-PSK signals, respec-

tively. In particular, all of the signals’ carrier frequency is 2000 Hz and the symbol rate is

100 Hz. We can easily find out the signal’s carrier frequency and symbol rate at nFc and

nFc±kFb , respectively. Where n is the order number of cyclic moment, k = 1,2,3, ... Here,

50

since we only consider the above 4 different signal modulation types above and rectan-

gular pulse shape as the examples, we have to mention that the constellation symbols

s[i ] influence the features of temporal moment and cyclic moment. Meanwhile, signals

with different pulse shapes may cause different amplitudes of temporal moment and

cyclic moment. However, the features of temporal moment and cyclic moment will be

the same.

5.2.3 Mixed signal parameter estimation

Second-order cyclic moment

Now, assume that we have a mixed signal y(t ), which is made up of two different modu-

lated signals s(t). Hence, we could get the math expression of y(t ) as,

y(t ) = s1(t )+ s2(t )

= a1(t )e j (2π fc1t+θ1) +a2(t )e j (2π fc2t+θ2)(5.26)

The second-order moment of y(t ) is given by

my (t ,0)2 = E [(x1(t )+x2(t ))2]

= E [a21(t )e j 2π(2 fc1)t +a2

2(t )e j 2π(2 fc2)t +2a1(t )a2(t )e j 2π(2 fc1+2 fc2)t ](5.27)

Then, the absolute value of second-order cyclic moment of y(t ) is given by,

∣∣my (α,0)2∣∣=F {my (t ,0)2}

=∣∣∣F {E [a2

1(t )e j 2π(2 fc1)t +a22(t )e j 2π(2 fc2)t +2a1(t )a2(t )e j 2π(2 fc1+2 fc2)t ]}

∣∣∣ (5.28)

Since,

E [2a1(t )a2(t )e j 2π(2 fc1+2 fc2)t ] = 0 (5.29)

51

Then,

∣∣my (α,0)2∣∣= ∣∣∣F {E [a2

1(t )e j 2π(2 fc1)t +a22(t )e j 2π(2 fc2)t }

∣∣∣= ∣∣F {a2

1(t )}⊗δ( f −2 fc1)+F {a22(t )}⊗δ( f −2 fc2)

∣∣ (5.30)

According to the Equation(5.30), we can find the carrier frequency of s1(t ) and s2(t )

at α = 2 fc1 and α = 2 fc2, respectively. Meanwhile, we can find the symbol rate of s1(t )

and s2(t ) at α= 2 fc1 ± fb1 and α= 2 fc2 ± fb2, respectively.

4th-order cyclic moment

The 4-th order moment of y(t ) is given by

my (t ,0)4 = E [(s1(t )+ s2(t ))4]

= E [a41(t )e j 2π(4 fc1)t +a4

2(t )e j 2π(4 fc2)t +6a21(t )a2

2(t )e j 2π(2 fc1+2 fc2)t

+4a31(t )a2(t )e j 2π(3 fc1+ fc2)t +4a1(t )a3

2(t )e j 2π( fc1+3 fc2)t ]

(5.31)

Hence, we can get the value of the 4th-order cyclic moment of y(t ) as

my (α,0)4 =F {my (t ,0)4}

=F {E [a41(t )e j 2π(4 fc1)t +a4

2(t )e j 2π(4 fc2)t +6a21(t )a2

2(t )e j 2π(2 fc1+2 fc2)t


2(t )e j 2π( fc1+3 fc2)t ]}

=F {E [a41(t )]}δ( f −4 fc1)+F {E [a4

2(t )]}δ( f −4 fc2)

+6F {E [a21(t )a2

2(t )]}δ( f −2( fc1 + fc2))

+4F {E [a31(t )a2(t )]}δ( f − (3 fc1 + fc2))+4F {E [a1(t )a3

2(t )]}δ( f − ( fc1 +3 fc2))

(5.32)

52

Since

E [a31(t )a2(t )] = 0

E [a1(t )a32(t )] = 0

Hence,

my (α,0)4 =F {E [a41(t )]}δ( f −4 fc1)+F {E [a4

2(t )]}δ( f −4 fc2)

+6F {E [a21(t )a2

2(t )]}δ( f −2( fc1 + fc2))

QPSK /16Q AM ⇒F {E [a41(t )]}δ( f −4 fc1)+F {E [a4

2(t )]}δ( f −4 fc2)

(5.33)

From the above derivation, we know that if two signals have different carrier frequencies,

the high order cyclic moment has some cross-terms. With the increasing number of

signals and orders, there will be more cross-terms. For the 4-th order cyclic moment,

if the mixed-signal y(t ) is make up of two QPSK or 16-QAM, the estimation result is

reliable. However, if the mixed-signal y(t ) is made up of two BPSK signals, the cross

term will be kept. Therefore, we have to filter out the 2( fc1 + fc2) component from the

mixed-signal before we do 4-th order cyclic moment estimation, and it is easy to find the

fc1 and fc2 by using second-order cyclic moment. For the 8-PSK signal, the 4-th order

cyclic moment will be 0 with an infinity length signal.

8th-order cyclic moment

According to Pascal’s triangle[77], which is shown here,

53

Figure 5.6: Pascal’s triangle

The 8-th order moment of y(t ) is

my (t ,0)8 = E [(s1(t )+ s2(t ))8]

= E [a81(t )e j 2π(8 fc1)t +a8

2(t )e j 2π(8 fc2)t

+28a61(t )a2

2(t )e j 2π(6 fc1+2 fc2)t +28a21(t )a6

2(t )e j 2π(2 fc1+6 fc2)t


2(t )e j 2π( fc1+7 fc2)t

+56a51(t )a3

2(t )e j 2π(5 fc1+3 fc2)t +56a31(t )a5

2(t )e j 2π(3 fc1+5 fc2)t

+70a41(t )a4

2(t )e j 2π4( fc1+ fc2)t

= E [a81(t )e j 2π(8 fc1)t +a8

2(t )e j 2π(8 fc2)t

+28a61(t )a2

2(t )e j 2π(6 fc1+2 fc2)t +28a21(t )a6

2(t )e j 2π(2 fc1+6 fc2)t

+70a41(t )a4

2(t )e j 2π4( fc1+ fc2)t ]

(5.34)

The 8-th order cyclic moment of y(t ) is given by

54

my (α,0)8 =F {my (t ,0)8}

=F {E [a81(t )]}δ( f −8 fc1)+F {E [a8

2(t )]}δ( f −8 fc2)

+28F {E [a61(t )a2

2(t )]}δ( f − (6 fc1 +2 fc2))+28F {E [a21(t )a6

2(t )]}δ( f − (2 fc1 +6 fc2))

+70F {E [a41(t )a4

2(t )]}δ( f −4( fc1 + fc2))

8PSK ⇒F {E [a81(t )]}δ( f −8 fc1)+F {E [a8

2(t )]}δ( f −8 fc2)

(5.35)

Based on Section 5.2.2, we know that any (n < 8)-order cyclic moment cannot esti-

mate a 8-PSK signal. Hence, if y(t ) is made up of two 8-PSK signals, all cross-terms will

be 0, and we can easily find the peaks at 8 fc1±k fb1 and 8 fc1±k fb2, where k = 0,1,2,3, ...

Simulations

Here, we provide a mixed signal model y(t ) = s1(t ) + s2(t ) + n(t ). Where s1(t + τ1) =a1(t )e j (2π fc1t+θ1), s2(t +τ2) = a2(t )e j (2π fc2t+θ2) and n(t ) is white Gaussian noise. In the

simulation, we set fc1 = 2000 Hz, fc2 = 2050 Hz, fb1 = 100 Hz, fb2 = 100 Hz, SN R = 20 dB

and all signals are generated by a raised-cosine pulse filter with roll-off factor 0.5.

55

3200 3400 3600 3800 4000 4200 4400 4600 4800

α(Hz)

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic m

omen

t

2-nd orde cyclic moment: BPSK+BPSK

X: 4100Y: 1

X: 4000Y: 0.9995

X: 4200Y: 0.06687

X: 3900Y: 0.06726

7200 7400 7600 7800 8000 8200 8400 8600 8800

α(Hz)

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic m

omen

t

4-th orde cyclic moment: 16QAM+BPSK

X: 8200Y: 1

X: 8000Y: 0.3841

X: 8300Y: 0.03838

X: 7900Y: 0.1199

7200 7400 7600 7800 8000 8200 8400 8600 8800

α(Hz)

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic m

omen

t

4-th orde cyclic moment: QPSK+BPSK

X: 8200Y: 1

X: 8000Y: 0.5851

X: 8300Y: 0.0372

X: 7900Y: 0.1763

7200 7400 7600 7800 8000 8200 8400 8600 8800

α(Hz)

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic m

omen

t

4-th orde cyclic moment: QPSK+QPSK

X: 8000Y: 0.9917

X: 8200Y: 1

X: 8300Y: 0.2985

X: 7900Y: 0.2988

7200 7400 7600 7800 8000 8200 8400 8600 8800

α(Hz)

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic m

omen

t

4-th orde cyclic moment: QPSK+16QAM

X: 8000Y: 1

X: 8200Y: 0.6789

X: 8300Y: 0.2039

X: 7900Y: 0.3056

7200 7400 7600 7800 8000 8200 8400 8600 8800

α(Hz)

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic m

omen

t

4-th orde cyclic moment: 16QAM+BPSK

X: 8200Y: 1

X: 8000Y: 0.3841

X: 8300Y: 0.03838

X: 7900Y: 0.1199

Figure 5.7: 2nd and 4-th order cyclic moment of mixed signal

56

1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68

α(Hz) ×104

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic m

omen

t

8-th orde cyclic moment: BPSK+8PSK

X: 1.6e+04Y: 1

X: 1.59e+04Y: 0.2945

X: 1.64e+04Y: 0.1882

1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68

α(Hz) ×104

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic m

omen

t

8-th orde cyclic moment: QPSK+8PSK

X: 1.6e+04Y: 1

X: 1.64e+04Y: 0.5528

X: 1.65e+04Y: 0.2935

X: 1.58e+04Y: 0.1616

1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68

α(Hz) ×104

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic m

omen

t

8-th orde cyclic moment: 16QAM+8PSK

X: 1.6e+04Y: 1

X: 1.64e+04Y: 0.3589

X: 1.59e+04Y: 0.3556

X: 1.65e+04Y: 0.2024

1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68

α(Hz) ×104

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic m

omen

t

8-th orde cyclic moment: 8PSK+8PSK

X: 1.6e+04Y: 1 X: 1.64e+04

Y: 0.9496

X: 1.65e+04Y: 0.5638

X: 1.59e+04Y: 0.535

Figure 5.8: 8-th order cyclic moment of mixed signal

5.3 Signal Modulation Classification

5.3.1 Mathematic Model

Single Modulated Signal

Assume s(t ) is a communication signal, which is expressed by Equation (5.8) and (5.9).

Here, we just take Cαs (τ;4,0) as an example, where time delay τ= 0. According to Equa-

tion.[3.5-3.10], we could get the following results. The 2nd order and 4th order temporal

moment function of s(t ) is defined as,

57

Rx(t ,0;2,0) = Eβ[Lx(t ,0;2,0)]

= Eβ[s(t )s(t )]

= Eβ[a(t )2e j 2π(2 fc )t ]

=∑β

⟨a(t )2e j 2π(2 fc )t e− j 2πβt ⟩t e j 2πβt

(5.36)

Rx(t ,0;4,0) = Eα[Lx(t ,0;4,0)]

= Eα[s(t )s(t )s(t )s(t )]

= Eα[a(t )4e j 2π(4 fc )t ]

=∑α

⟨a(t )4e j 2π(4 fc )t e− j 2παt ⟩t e j 2παt

(5.37)

Where β and α is cyclic frequency of 2-nd order and 4-th order temporal moment func-

tion, respectively. β = 2 fc ± k fb and ⟨a(t )2e j 2π(2 fc )t e− j 2πβt ⟩t 6= 0. α = 4 fc ± k fb and

⟨a(t )4e j 2π(4 fc )t e− j 2παt ⟩t 6= 0. k = 0,1,2, ... . ⟨·⟩t represents time average operator. Here,

since process s(t) is cycloergodic, we can also employ expected value operator E [·] in-

stead of general sine-wave extraction operator Eα[·].As Equation (3.2.2) and cycloergodic property, the 2nd order and 4th order cyclic

temporal moment function of s(t ) is inferred as,

58

Rβs (0;2,0) = ⟨Rs(t ,0;2,0)e−i 2πβt ⟩t

= ⟨E [a(t )2e j 2π(2 fc )t ]e− j 2πβt ⟩t

c ycloer g odi c ⇒⟨a(t )2e− j 2π(β−2 fc )t ⟩t

= ⟨[∑γ

Rγa (0;2,0)e j 2πγt +e(t )]e− j 2π(β−2 fc )t ⟩t

=∑γ

Rγa (0;2,0)⟨e− j 2π(β−2 fc−γ)t ⟩t

(5.38)

Rαs (0;4,0) = ⟨Rs(t ,0;4,0)e−i 2παt ⟩t

= ⟨E [a(t )4e j 2π(4 fc )t ]e− j 2παt ⟩t

c ycloer g odi c ⇒⟨a(t )4e− j 2π(α−4 fc )t ⟩t

= ⟨[∑γ

Rγa (0;4,0)e j 2πγt +e(t )]e− j 2π(α−4 fc )t ⟩t

=∑γ

Rγa (0;4,0)⟨e− j 2π(α−4 fc−γ)t ⟩t

(5.39)

Where e(t ) is aperiodic residual part and ⟨e(t )e− j 2πγt ⟩ = 0. γ is n-th order impure cycle

frequency of s(t ), γ = ±K fb , K = 0,1,2,3... Hence, when β = 2 fc +γ, Equation (5.38) is

non-zero. Whenα= 4 fc +γ, Equation (5.39) is non-zero. Rγa (0;2,0) = ⟨a2(t )e− j 2πγt ⟩t and

Rγa (0;4,0) = ⟨a4(t )e− j 2πγt ⟩t .

Now, according to the Equation (3.10), the 4th order cyclic temporal cumulant func-

tion of s(t ) is given by

59

Cαx (0;4,0) =∑

P[(−1)p−1(p −1)!

∑β†1=α

p∏j=1

Rβ jx (0;n j ,m j )]

= Rαx (0;4,0)−3

∑β†1=α

Rβ1x (0;2,0)Rβ2

x (0;2,0)

=∑γ

Rγa (0;4,0)⟨e− j 2π(α−4 fc−γ)t ⟩t

−3∑

β†1=α(∑γ1

Rγ1a (0;2,0)⟨e− j 2π(β1−2 fc−γ1)t ⟩t

∑γ2

Rγ2a (0;2,0)⟨e− j 2π(β2−2 fc−γ2)t ⟩t )

(5.40)

Where β1 +β2 = α, i.e., γ1 +γ2 = γ. For a BPSK modulated signal, when α = 4 fc , β1 =2 fc ±K fb and β2 = 2 fc ∓K fb . For a higher order modulated signal, such as QPSK, 16-

QAM, or 8-PSK signals, no β1 +β2 = α, their 4th order cyclic cumulant is equal to 4th

order cyclic moment. The time average over the symbol sequence raised to the fourth is

non-zero.

Composite Modulated Signal

The mixed signal model is expressed as,

x(t ) = s1(t )+ s2(t )

= a1(t )e j (2π( fc1)t+θ1) +a2(t )e j (2π( fc2)t+θ2)(5.41)

The 2nd order and 4th order temporal moment function of x(t ) is given by

60

Rx(t ,0;2,0) = Eα[Lx(t ,0;2,0)]

= Eα[x(t )x(t )]

= Eα[a21(t )e j 2π(2 fc1)t +a2

2(t )e j 2π(2 fc2)t +2a1(t )a2(t )e j 2π( fc1+ fc2)t ]

= E [a21(t )e j 2π(2 fc1)t +a2

2(t )e j 2π(2 fc2)t +2a1(t )a2(t )e j 2π( fc1+ fc2)t ]

= E [a21(t )e j 2π(2 fc1)t +a2

2(t )e j 2π(2 fc2)t ]

(5.42)

Rx(t ,0;4,0) = E [(s1(t )+ s2(t ))4]

= E [a41(t )e j 2π(4 fc1)t +a4

2(t )e j 2π(4 fc2)t +6a21(t )a2

2(t )e j 2π(2 fc1+2 fc2)t


2(t )e j 2π( fc1+3 fc2)t ]

= E [a41(t )e j 2π(4 fc1)t +a4

2(t )e j 2π(4 fc2)t +6a21(t )a2

2(t )e j 2π(2 fc1+2 fc2)t ]

(5.43)

Then, we can get the 2nd order and 4th order cyclic moment of x(t ),

Rβx (0;2,0) = ⟨Rx(t ,0;2,0)e−i 2πβt ⟩t

= ⟨E [a21(t )e j 2π(2 fc1)t +a2

2(t )e j 2π(2 fc2)t ]e− j 2πβt ⟩t

= ⟨E [a21(t )e j 2π(2 fc1)t ]e− j 2πβt ⟩t +⟨E [a2

2(t )e j 2π(2 fc2)t ]e− j 2πβt ⟩t

c ycloer g odi c ⇒⟨a21(t )e j 2π(2 fc1−β)t ⟩t +⟨a2

2(t )e j 2π(2 fc2−β)t ⟩t

= ⟨[∑γ1

Rγ1a1(0;2,0)e j 2πγ1t +e(t )]e− j 2π(β−2 fc1)t ⟩t

+⟨[∑γ2

Rγ2a2(0;2,0)e j 2πγ2t +e(t )]e− j 2π(β−2 fc2)t ⟩t

=∑γ1

Rγ1a1(0;2,0)⟨e− j 2π(β−2 fc1−γ1)t ⟩t +

∑γ2

Rγ2a2(0;2,0)⟨e− j 2π(β−2 fc2−γ2)t ⟩t

(5.44)

61

From Equation (5.44), we know that when β = 2 fc1 +γ1, the first term is non-zero,

and when β= 2 fc2 +γ2, the second term is non-zero.

Rαx (0;4,0) = ⟨Rx(t ,0;4,0)e−i 2παt ⟩t

= ⟨E [a41(t )e j 2π(4 fc1)t +a4

2(t )e j 2π(4 fc2)t +6a21(t )a2

2(t )e j 2π(2 fc1+2 fc2)t


2(t )e j 2π( fc1+3 fc2)t ]e− j 2παt ⟩t

= ⟨E [a41(t )e j 2π(4 fc1)t +a4

2(t )e j 2π(4 fc2)t +6a21(t )a2

2(t )e j 2π(2 fc1+2 fc2)t ]e− j 2παt ⟩t

= ⟨E [a41(t )e j 2π(4 fc1)t ]e− j 2παt ⟩t +⟨E [a4

2(t )e j 2π(4 fc2)t ]e− j 2παt ⟩t

+⟨E [6a21(t )a2

2(t )e j 2π(2 fc1+2 fc2)t ]e− j 2παt ⟩t

c ycloer g odi c ⇒⟨a41(t )e− j 2π(α−4 fc1)t ⟩t +⟨a4

2(t )e− j 2π(α−4 fc2)t ⟩t +6⟨a21(t )a2

2(t )e− j 2π(α−2 fc1−2 fc2)t ⟩t

= ⟨[∑γ1

Rγ1a1(0;4,0)e j 2πγ1t +e(t )]e− j 2π(α−4 fc1)t ⟩t

+⟨[∑γ2

Rγ2a2(0;4,0)e j 2πγ2t +e(t )]e− j 2π(α−4 fc2)t ⟩t

+6⟨[∑γ1

Rγ1a1(0;2,0)e j 2πγ1t +e(t )][

∑γ2

Rγ2a2(0;2,0)e j 2πγ2t +e(t )]e− j 2π(α−2 fc1−2 fc2)t ⟩t

(⟨e(t )e− j 2πγt ⟩ = 0) ⇒∑γ1

Rγ1a1(0;4,0)⟨e− j 2π(α−4 fc1−γ1)t ⟩t +

∑γ2

Rγ2a2(0;4,0)⟨e− j 2π(α−4 fc2−γ2)t ⟩t

+6⟨[∑γ1

Rγ1a1(0;2,0)e j 2πγ1t ][

∑γ2

Rγ2a2(0;2,0)e j 2πγ2t ]e− j 2π(α−2 fc1−2 fc2)t ⟩t

=∑γ1

Rγ1a1(0;4,0)⟨e− j 2π(α−4 fc1−γ1)t ⟩t +

∑γ2

Rγ2a2(0;4,0)⟨e− j 2π(α−4 fc2−γ2)t ⟩t

+6[∑γ1

Rγ1a1(0;2,0)⟨e− j 2π(α/2− fc1− fc2−γ1)t ⟩t ][

∑γ2

Rγ2a2(0;2,0)⟨e− j 2π(α/2− fc1− fc2−γ2)t ]⟩t

(5.45)

From Equation.(5.45), we could know that when α= 4 fc1 +γ1, the first term is non-

zero, when α= 4 fc2 +γ2, the second term is non-zero, and when α= 2 fc1 +2 fc2 +2γ1 &

α= 2 fc1 +2 fc2 +2γ2, γ1 = γ2, the third term is non-zero.

62

Hence, we can get the 4th order cyclic cumulant of x(t ),

Cαx (0;4,0) =∑

P[(−1)p−1(p −1)!

∑β†1=α

p∏j=1

Rβ jx (0;n j ,m j )]

= Rαx (0;4,0)−3

∑β†1=α

Rβ1x (0;2,0)Rβ2

x (0;2,0)

=Cαs1(0;4,0)+Cα

s2(0;4,0)

(5.46)

63

5.3.2 Theoretical Cumulant Value

Cumulant\Modulation BPSK QPSK 16QAM 8PSK Noise

C(2,0) E 0 0 0 0

C(2,1) E E E E E

C(4,0) −2E 2 E 2 −0.68E 2 0 0

C(4,1) −2E 2 0 0 0 0

C(4,2) −2E 2 −E 2 −0.68E 2 −E 2 0

C(6,0) 16E 3 0 0 0 0

C(6,1) 16E 3 −4E 3 2.08E 3 0 0

C(6,2) 16E 3 0 0 0 0

C(6,3) 16E 3 4E 3 2.08E 3 4E 3 0

C(8,0) −272E 4 −34E 4 −13.98E 4 E 4 0

C(8,1) −272E 4 0 0 0 0

C(8,2) −272E 4 34E 4 −13.98E 4 0 0

C(8,3) −272E 4 0 0 0 0

C(8,4) −272E 4 −34E 4 −13.98E 4 −33E 4 0

Table 5.1: Theoretical Cumulant value, E is Signal Power

5.3.3 Simulation

In this section, we show some examples of (n,m)th order cyclic cumulant of two mixed

signals, which are 50% overlap and 100% overlap, respectively. Since the higher order

cyclic cumulant magnitude will include more variance, the maximum cyclic cumulant

order is set to 6. For special case B, which is described in Chapter 4, since all mixed

64

signals have the same carrier frequency and different symbol rates, all cyclic frequency

α = (n − 2m) fc become correlated terms. Since the symbol rate is different, the terms

α = (n − 2m) fc ± k fb , where, k = 1, 2, 3 ... and m 6= n/2 can be used to estimate each

mixed signal. However, those terms are relative to symbol rate and raised cosine filter

roll-off factors. We will have a more detailed discussion about this in the next chapter.

For the model A, we could use α= (n −2m) fc , where m 6= n/2, cyclic cumulant features

to classify each mixed signals. All following simulation results are generated by using

1456 symbols for each mixed signal.

Model A: Mixed Signals

2 PSK + 2 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

5

10

15

20

25

30

35

Figure 5.9: BPSK1:BPSK1+BPSK2

2 PSK + 2 PSK

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

10

20

30

40

50

60

70

80

90

100


0

20

40

60

80

100

120

Cyclic

Cum

ula

nt M

agnitude

2 PSK + 2 PSK = (n-2m)fc

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

BPSK 1

BPSK 2

Figure 5.11: BPSK1+BPSK2

0

0.5

1

1.5

2

2.5

harm

onic

num

ber

2 PSK + 2 PSK = (n-2m)fc + fb

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

BPSK 1

BPSK 2


65

2 PSK + 4 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

5

10

15

20

25

30

35

Figure 5.13: BPSK:BPSK+QPSK

2 PSK + 4 PSK

Fc2 = 3000; Fb

2 = 400; Power

2 = 1; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

5

10

15

20

25

Figure 5.14: QPSK:BPSK+QPSK

0

5

10

15

20

25

30

35

40

Cyclic

Cum

ula

nt M

agnitude

2 PSK + 4 PSK alpha = (n-2m)

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

BPSK

QPSK

Figure 5.15: BPSK+QPSK

0

0.2

0.4

0.6

0.8

1

1.2

1.4harm

onic

num

ber


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

BPSK

QPSK


2 PSK + 16 QAM

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

2

4

6

8

10

12

14

16

18

20

Figure 5.17: BPSK:BPSK+16QAM

2 PSK + 16 QAM

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

2

4

6

8

10

12

14

Figure 5.18: 16QAM:BPSK+16QAM

66

0

5

10

15

20

25

Cyclic

Cum

ula

nt M

agnitude

2 PSK + 16 QAM alpha = (n-2m)

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

BPSK

16 QAM

Figure 5.19: BPSK+16QAM

0

0.2

0.4

0.6

0.8

1

1.2

harm

onic

num

ber

2 PSK + 16 QAM alpha = (n-2m)+fb

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

BPSK

16 QAM

Figure 5.20: BPSK+16QAM

2 PSK + 8 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

5

10

15

20

25

30

Figure 5.21: BPSK:BPSK+8PSK

2 PSK + 8 PSK

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

5

10

15

20

Figure 5.22: 8PSK:BPSK+8PSK

0

5

10

15

20

25

30

35

Cyclic

Cum

ula

nt M

agnitude


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

BPSK

8PSK

Figure 5.23: BPSK+8PSK

0

0.2

0.4

0.6

0.8

1

1.2

harm

onic

num

ber

2 PSK + 8 PSK alpha = (n-2m)+fb

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

BPSK

8PSK


67

4 PSK + 4 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

5

10

15

20

25

Figure 5.25: QPSK1:QPSK1+QPSK2

4 PSK + 4 PSK

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

5

10

15

20

25


0

5

10

15

20

25

30

Cyclic

Cum

ula

nt M

agnitude


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

QPSK 1

QPSK 2

Figure 5.27: QPSK1+QPSK2

0

0.5

1

1.5harm

onic

num

ber

4 PSK + 4 PSK = (n-2m)fc+fb

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

QPSK 1

QPSK 2


4 PSK + 16 QAM

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

2

4

6

8

10

12

14

Figure 5.29: QPSK:QPSK+16QAM

4 PSK + 16 QAM

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

5

10

15

Figure 5.30: 16QAM:QPSK+16QAM

68

0

2

4

6

8

10

12

14

16

Cyclic

Cum

ula

nt M

agnitude

4 PSK + 16 QAM = (n-2m)fc

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

QPSK

16 QAM

Figure 5.31: QPSK+16QAM

0

0.2

0.4

0.6

0.8

1

harm

onic

num

ber

4 PSK + 16 QAM = (n-2m)fc+fb

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

QPSK

16QAM


4 PSK + 8 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

5

10

15

20

Figure 5.33: QPSK:QPSK+8PSK

4 PSK + 8 PSK

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

5

10

15

20

Figure 5.34: 8PSK:QPSK+8PSK

0

5

10

15

20

25

Cyclic

Cum

ula

nt M

agnitude


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

QPSK

8 PSK

Figure 5.35: QPSK+8PSK

0

0.2

0.4

0.6

0.8

1

1.2

harm

onic

num

ber


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

QPSK

8 PSK


69

16 QAM + 16 QAM

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

2

4

6

8

10

12

Figure 5.37: 16QAM1:16QAM1+16QAM2

16 QAM + 16 QAM

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

2

4

6

8

10

12


0

2

4

6

8

10

12

14

Cyclic

Cum

ula

nt M

agnitude

16 QAM + 16 QAM = (n-2m)fc

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

16 QAM 1

16 QAM 2

Figure 5.39: 16QAM+16QAM

0

0.2

0.4

0.6

0.8

1

harm

onic

num

ber

16 QAM + 16 QAM = (n-2m)fc+fb

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

16 QAM 1

16 QAM 2


16 QAM + 8 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

5

10

15

20

25

Figure 5.41: 16QAM:16QAM+8PSK

16 QAM + 8 PSK

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

5

10

15

20

25

Figure 5.42: 8PSK:16QAM+8PSK

70

0

5

10

15

20

25

30

Cyclic

Cum

ula

nt M

agnitude

16 QAM + 8 PSK = (n-2m)fc

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

16 QAM

8 PSK

Figure 5.43: 16QAM+8PSK

0

0.2

0.4

0.6

0.8

1

harm

onic

num

ber

16 QAM + 8 PSK = (n-2m)fc+fb

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

16 QAM

8 PSK


8 PSK + 8 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

5

10

15

20

25

Figure 5.45: 8PSK1:8PSK1+8PSK2

8 PSK + 8 PSK

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

5

10

15

20

25


0

5

10

15

20

25

30

Cyclic

Cum

ula

nt M

agnitude


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

8 PSK 1

8 PSK 2

Figure 5.47: 8PSK1+8PSK2

0

0.2

0.4

0.6

0.8

1

1.2

harm

onic

num

ber


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3300; Fb

2 = 400; Power

2 = 2; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

8 PSK 1

8 PSK 2


71

Model B: Mixed Signals

2 PSK + 2 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

2

4

6

8

10

12

14

16

18


2 PSK + 2 PSK

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

2

4

6

8

10

12

14

16

18


0

5

10

15

20

Cyclic

Cum

ula

nt M

agnitude

2 PSK + 2PSK, = (n-2m)fc

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

BPSK 1

BPSK 2


0

0.1

0.2

0.3

0.4

0.5

0.6

harm

onic

num

ber

2 PSK + 2PSK, = (n-2m)fc+fb

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

BPSK 1

BPSK 2


2 PSK + 4 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

2

4

6

8

10

12

14


2 PSK + 4 PSK

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

2

4

6

8

10

12

14


72

0

2

4

6

8

10

12

14

16

Cyclic

Cum

ula

nt M

agnitude

2 PSK + 4 PSK =(n-2m)fc

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

BPSK

QPSK


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

harm

onic

num

ber

2 PSK + 4 PSK =(n-2m)fc + fb

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

BPSK

QPSK


2 PSK + 16 QAM

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

1

2

3

4

5

6

7

8

9

10

11

Figure 5.57: BPSK:BPSK+16QAM

2 PSK + 16 QAM

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

1

2

3

4

5

6

7

8

9

10

Figure 5.58: 16QAM:BPSK+16QAM

0

2

4

6

8

10

12

Cyclic

Cum

ula

nt M

agnitude


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

BPSK

16QAM


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

harm

onic

num

ber


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

BPSK

16QAM


73

2 PSK + 8 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

2

4

6

8

10

12

14

16

Figure 5.61: BPSK:BPSK+8PSK

2 PSK + 8 PSK

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

2

4

6

8

10

12

14

Figure 5.62: 8PSK:BPSK+8PSK

0

2

4

6

8

10

12

14

16

18

Cyclic

Cum

ula

nt M

agnitude


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

BPSK

8PSK


0

0.1

0.2

0.3

0.4

0.5

0.6harm

onic

num

ber


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

BPSK

8PSK


4 PSK + 4 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

1

2

3

4

5

6


4 PSK + 4 PSK

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

1

2

3

4

5

6


74

0

1

2

3

4

5

6

7

Cyclic

Cum

ula

nt M

agnitude


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

QPSK 1

QPSK 2


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

harm

onic

num

ber


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

QPSK 1

QPSK 2


4 PSK + 16QAM

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 5.69: QPSK:QPSK+16QAM

4 PSK + 16 QAM

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 5.70: 16QAM:QPSK+16QAM

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Cyclic

Cum

ula

nt M

agnitude


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

QPSK

16QAM


0

0.05

0.1

0.15

0.2

0.25

0.3

harm

onic

num

ber


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

QPSK

16QAM


75

4 PSK + 8 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 5.73: QPSK:QPSK+8PSK

4 PSK + 8 PSK

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 5.74: 8PSK:QPSK+8PSK

0

1

2

3

4

5

6

Cyclic

Cum

ula

nt M

agnitude


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

QPSK

8PSK


0.05

0.1

0.15

0.2

0.25

0.3

0.35harm

onic

num

ber


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)


16 QAM + 16 QAM

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

3

3.5


16 QAM + 16 QAM

Fc2 = 3000; Fb

2 = 400; Power

2 = 1; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

3

3.5


76

0

0.5

1

1.5

2

2.5

3

3.5

4

Cyclic

Cum

ula

nt M

agnitude

16 QAM + 16 QAM = (n-2m)fc

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

16 QAM 1

16 QAM 2


0

0.05

0.1

0.15

0.2

0.25

harm

onic

num

ber

16 QAM + 16 QAM = (n-2m)fc+fb

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

16 QAM 1

16 QAM 2


16 QAM + 8 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

3

3.5

Figure 5.81: 16QAM:16QAM+8PSK

16 QAM + 8 PSK

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

3

3.5

Figure 5.82: 8PSK:16QAM+8PSK

0

0.5

1

1.5

2

2.5

3

3.5

4

Cyclic

Cum

ula

nt M

agnitude

16 QAM + 8 PSK = (n-2m)fc

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

16 QAM

8 PSK


0

0.05

0.1

0.15

0.2

0.25

harm

onic

num

ber

16 QAM + 8 PSK = (n-2m)fc+fb

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)

16 QAM

8 PSK


77

8 PSK + 8 PSK

Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

3

3.5

4


8 PSK + 8 PSK

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

3

3.5

4


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Cyclic

Cum

ula

nt M

agnitude


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)


0

0.05

0.1

0.15

0.2

0.25harm

onic

num

ber


Fc1 = 3000; Fb

1 = 400; Power

1 = 1; Roll-off

1 = 0.5;

Fc2 = 3000; Fb

2 = 500; Power

2 = 1; Roll-off

2 = 0.5;

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4)

(n,m)


78

Simulations and Performance Analysis

In this chapter, we will analyze the performance of energy-based signal detection, and

focus on analyzing the performance of mixed-signal detection, mixed-signal parame-

ters estimation, and mixed-signal modulation classification. In particular, we will ana-

lyze the performance of signal carrier frequency estimation, symbol rate estimation and

power estimation. Meanwhile, the performance analysis will be for different channels,

such as AWGN channel, flat fading channel, and multi-path fading channel.

79

6.1 Energy based Signal Detection

In this section, a continuous/pulsed BPSK modulated signal is the target signal. We em-

ployed energy-based detection to analyze the detection performance.

6.1.1 Continues BPSK Modulated signal

The signal model is expressed with the following equation,

s(t ) = AN∑

i=1b[i ]p(t − i Ts)cos(2π fc t ) (6.1)

Where A is the amplitude, b is the data symbol, i is the index for the data symbol, N is

the total number of symbols, Ts is the symbol duration, and fc is the carrier frequency.

0 0.2 0.4 0.6 0.8 1

time(second)

-1.5

-1

-0.5

0

0.5

1

1.5

Am

plitu

de

BPSK signal

Figure 6.1: BPSK signal

Fig.6.1 shows a continue BPSK modulated signal. Apparently, energy based detec-

tion rate of this signal will approach 100% under the high SNR.

80

-40 -30 -20 -10 0 10

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pro

babi

lity

Required Pfa = 0.01

Pd, K = 10

Pd, K = 100

Pd, K = 1000

Pfa

Figure 6.2: BPSK: Pd versus SNR (dB) with required P f a = 0.01

-40 -30 -20 -10 0 10

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pro

babi

lity

Required Pfa = 0.1

Pd, K = 10

Pd, K = 100

Pd, K = 1000

Pfa


81

-40 -30 -20 -10 0 10

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pro

babi

lity

Required Pfa = 0.3

Pd, K=10

Pd, K=100

Pd, K=1000

Pfa


In Fig.6.2, Fig.6.3 and Fig.6.4, we analyzed the detection rate versus SNR with dif-

ferent K degrees of freedom and false alarm rate (P f a), such as 0.01, 0.1 and 0.3. From

the figures, we know that if we increase the detection time window K or SNR, we can get

better detection performance. When SNR is low, and then the detector can only detect

noise. Hence, the detection rate Pd is the same as required P f a when the SNR is too low.

Since our target signal is occupying continuous time, the detection rate can approach

100% under the high SNR.

6.1.2 Pulsed BPSK Modulated signal

In many RF applications, the signal is not continuously transmitting. The signal is peri-

odic with short pulses.

82

time(second)0 0.01 0.02 0.03 0.04 0.05 0.06

Am

plitu

de

-1.5

-1

-0.5

0

0.5

1

1.5Pulsed Signal

τ

T

Figure 6.5: Pulsed signal

Fig.6.5 shows the pulsed signal in time domain. Where τ is pulse duration, T is the

pulse repetition period. It is clear that the detection performance lies with observed win-

dow size and signal-to-noise ratio (SNR). If the observed window size is not big enough,

the detection rate Pd could not reach 100% even though the SNR is high enough.

-40 -30 -20 -10 0 10

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pro

babi

lity

Pulsed/BPSK: Required Pfa = 0.01

Pd, K = 10

Pd, K = 100

Pd, K = 1000

Pfa

Figure 6.6: Pulsed/Modulated: Pd versus SNR (dB) with required P f a = 0.01

83

-40 -30 -20 -10 0 10

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pro

babi

lity


Pd, K = 10

Pd, K = 100

Pd, K = 1000

Pfa


-40 -30 -20 -10 0 10

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pro

babi

lity


pd, K = 10

pd, K = 100

pd, K = 1000

Pfa


Fig.6.6, Fig.6.7 and Fig.6.8 show the signal detection rate with different false alarm

rates. The detection rate cannot reach 100% because the signal does not occupy continuous-

time and the obverse window is not big enough.

84

6.2 Signal Detection and Parameters Estimation

In the following section, we will provide a performance analysis of signal detection and

parameter estimation. We use Pd to represents signal detection and parameter estima-

tion performance. Pd represents the correct rate for two different parameters, carrier

frequency and symbol rate.

6.2.1 Single Signal Detection and Parameters Estimation

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

α (Hz) ×104

0

0.2

0.4

0.6

0.8

1

Nor

mal

ized

4th

ord

er C

yclic

mom

ent

X: 1.16e+04Y: 0.3185

Figure 6.9: QPSK Estimation Example

Figure.6.9 shows the example of QPSK signal carrier frequency and symbol rate detec-

tion and estimation. The x-axis is cyclic frequencyα and the y-axis is the normalized 4th

order cyclic moment value. The QPSK signal carrier frequency is 3000 Hz, and the sym-

bol rate is 400 Hz. Hence, its 4-th order cyclic moment feature will be shown at 4 fc ±k fb ,

where k = 0,1,2,3.... For the peaks detection, we employed MATLAB’s "findpeaks" func-

tion to find local maximum values. When compared with traditional fixed value thresh-

old peak detection, the "findpeaks" function provides more accuracy. It confirms the

peak by detecting the neighboring values. In the figure 6.9, we could find three peaks,

85

which represents carrier frequency and symbol rate.

-20 -15 -10 -5 0 5 10 15 20

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pd

BPSK

QPSK

16QAM

8PSK

Figure 6.10: Single Signal Fc Detection and Estimation Performance in AWGN Channel

In Figure.6.10, we use 115 symbols and 100 Monte Carlo experiments to generate the

carrier frequency detection and estimation rate under different SNR. The Pd is the de-

tection and estimation rate of carrier frequency. We employed 2nd order cyclic moment

to estimate the BPSK signal’s carrier frequency, 4th order cyclic moment to estimate the

QPSK/16QAM signal’s carrier frequency and 8th order cyclic moment to estimate the

8PSK signal’s carrier frequency. From the figure, we can easily see that low order modu-

lation types have better detection performance. If same number of symbols is used, with

the cyclic moment order increasing, the detection and estimation performance become

worse. The 2nd order cyclic moment has the best estimation performance. We could

get 100% detection and estimation rate at SN R =−15dB . The 4th order cyclic moment

still has very good performance. The QPSK and 16-QAM signal’s carrier frequency esti-

mation rate is similar. The QPSK and 16QAM carrier frequency detection rate can reach

up to 100% at SN R =−4dB and SN R = 4dB , respectively. For the 8-PSK signal, the car-

rier frequency estimation rate will be 100% at SN R = 10dB . The cyclic moment is more

straight-forward for estimating signal parameters. However, it also include lower-order

86

cyclic moment components and noise. Hence, the estimation performance becomes

worse as the order number increase.

0 20 40 60 80 100 120

Number of Symbols

0

0.2

0.4

0.6

0.8

1

Pd

BPSK

QPSK

16QAM

8PSK

Figure 6.11: Pd vs Number of symbols under SNR = 10 dB

Figure 6.11 shows the single signal carrier frequency detection and estimation rate

verse the number of symbols. The SNR is 10 dB. The sampling frequency is equal 17000

Hz, the symbol rate is 60 Hz. In the figure, we can see that BPSK can easily achieve a 100%

carrier frequency estimation rate with 4 symbols and QPSK just needs 18 symbols to get

a 100% carrier frequency estimation rate. Meanwhile, 16QAM and 8PSK need about 100

symbols to get a 100% carrier frequency estimation rate.

6.2.2 Mixed Signal Detection and Parameters Estimation

In this section, we will introduce the performance of mixed-signal detection and pa-

rameter estimation. In particular, we consider a 50% spectrum overlapping two mixed

signals as our analyzing signal. Moreover, we analyze the mixed-signal detection and pa-

rameter estimation performance in AWGN channel, flat fading channel and multi-path

fading channel.

87

Carrier frequency detection and estimation

3500 4000 4500 5000 5500 6000

Cyclic frequency α (Hz)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

4-th

ord

er c

yclic

mom

ent

BPSK+QPSK 50% overlap

BPSK

QPSK

Figure 6.12: Example: Mixed Signal Carrier Frequency (Fc) Estimation

Figure 6.12 shows an example of BPSK and QPSK mixed-signal parameter estimation. If

both carrier frequencies are correctly estimated, we believe that the parameter estima-

tion is successful. For example, if there are two signals in the channel, we just need to

find the highest 2 peaks, which are the carrier frequency of each mixed signal. In Figure

6.12, the carrier frequency of the BPSK signal is 932 Hz, and the carrier frequency of the

QPSK signal is 1000 Hz. The symbol rate of the BPSK signal and the QPSK signal is 100

Hz. The sampling frequency is 17000 Hz. SNR = 10 dB. From the figure, we can easily

find the carrier frequency of BPSK and QPSK signals based on the top two peaks with a

red color circle. The peaks located at 4 f c.

88

-20 -15 -10 -5 0 5 10 15 20

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pd

BPSK: BPSK+BPSK

BPSK: BPSK+QPSK

BPSK: BPSK+16QAM

BPSK: BPSK+8PSK

QPSK: QPSK+BPSK

QPSK: QPSK+QPSK

QPSK: QPSK+16QAM

QPSK: QPSK+8PSK

16QAM: 16QAM+BPSK

16QAM: 16QAM+QPSK

16QAM: 16QAM+16QAM

16QAM: 16QAM+8PSK

8PSK: 8PSK+BPSK

8PSK: 8PSK+QPSK

8PSK: 8PSK+16QAM

8PSK: 8PSK+8PSK

Figure 6.13: Fc Estimation Performance Pd versus SNR in AWGN Channel

Figure 6.13 shows the carrier frequency estimation performance of 50% spectrum

overlapped signals under different SNR values. The result is generated by 100 Monte

Carlo experiments. We can clearly see that the lower-order modulation type has better

parameter estimation performance than the high-order modulation type. If the mixed

signals contain higher-order modulation types, such as 8-PSK signals, we need more

symbols/samples to get a more accurate estimation rate. For the mixed BPSK signals,

we can achieve 100% estimation rate using 115 symbols when SNR is greater than -10

dB. We need 346 symbols to achieve 100% estimation rate of a QPSK signal’ carrier fre-

quency when SNR is greater than 0 dB . For the 16QAM signal, if we have 346 symbols, we

can get above 90% carrier frequency estimation rate at SNR=5dB. For the higher-order

modulation type 8-PSK signal, we need at least 9250 symbols to get more than 80% esti-

mation rate when the SNR is greater than 12 dB. We also find in Figure 6.13 that regard-

less of which signal is mixed with which, the estimation rates are similar. For example,

two mixed BPSK signals have a similar estimation rate to mixed BPSK and QPSK signals.

Hence, in the following simulation, we just show four different signals’ performance, i.e.,

mixed BPSK signal, mixed QPSK signal, mixed 16QAM signal, and mixed 8-PSK signal.

89

-20 -15 -10 -5 0 5 10 15 20

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pd

BPSK

QPSK

16QAM

8PSK

Figure 6.14: Fc Estimation Performance Pd versus SNR in flat fading Channel

Figure 6.14 shows the mixed signal carrier frequency estimation performance in flat

fading channel. Sampling frequency Fs = 17000 Hz, Fc1 = 1000 Hz, Fc2 = 1045 Hz, sym-

bol rate Fb = 60H z and raised cosine filter roll-off factor is 0.5. The result is generated

by using 100 Monte Carlo experiments. For mixed BPSK signals, we used 115 symbols.

For mixed QPSK signals, we used 346 symbols. For mixed 16-QAM signals, we used 578

symbols. For mixed 8-PSK signals, we used 9250 symbols.

-20 -15 -10 -5 0 5 10 15 20

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pd

BPSK

QPSK

16QAM

8PSK

Figure 6.15: Fc Estimation Performance Pd versus SNR in multi-path fading Channel

90

Figure 6.15 shows the mixed-signal carrier frequency estimation performance in a

multi-path fading channel. We still use same signal configuration as Figure 6.14. We

used the MATLAB provided "rayleighchan" function to simulated two-path fading chan-

nels. From the figure, we can see that the 8-PSK signal carrier frequency estimation rate

is pretty low because the 8-PSK signal is a higher-order modulation type which needs

8-th order cyclic moment to extract its cyclostationary feature. However, we know that

higher order cyclic moment/cumulant requires more symbols to make the features sta-

ble. In the AWGN channel, we know that if we want to estimate a 8-PSK signal, we need

about 9000 symbols. However, in the multi-path fading channel, the channel is chang-

ing through time, so we cannot get enough symbols for the 8-PSK signal to get perfect

carrier frequency estimation. Hence, 8-PSK is difficult to get high estimation rate for in

a multi-path fading channel.

Symbol rate detection and estimation

-200 -150 -100 -50 0 50 100 150 200

frequency(Hz)

0

0.5

1

1.5

2

C(2

,1)

cyclic

mom

ent

107

X -69.96

Y 6.71e+05

X -50

Y 6.134e+05

Figure 6.16: Mixed Signal Symbol rate (Fb) Estimation

For the signal symbol rate, we can use the 2nd order 1st conjugate cyclic moment term,

M(2,1), to estimate the signal’s symbol rate in the frequency domain, such as in Figure

91

6.16. In the figure, there are two signals mixed together, with symbol rates of 70 and 50

respectively. After we compute the Fourier transform of M(2,1), we can easily detect and

estimate the signals’ symbol rate.

-20 -15 -10 -5 0 5 10 15 20

SNR(dB)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pd

Symbol Rate (Fb) Detection

BPSK

QPSK

16QAM

8PSK

Figure 6.17: Mixed Signal Symbol rate (Fb) Estimation Performance

Figure. 6.17 shows the detection and estimation performance of mixed-signal sym-

bol rate. This result is generated by 925 symbols and 100 Monte Carlo experiments.

92

Mixed Signal Power estimation

Figure 6.18: Mixed BPSK Signal Power Estimation Performance

Figure 6.19: Mixed QPSK Signal Power Estimation Performance

93

Figure 6.20: Mixed 16-QAM Signal Power Estimation Performance

Figure 6.21: Mixed 8-PSK Signal Power Estimation Performance

Figures 6.18, 6.19, 6.20 and 6.21 show the mixed-signal power estimation performance

of BPSK, QPSK, 16-QAM, and 8-PSK respectively. In the figures, DEA represents the dif-

ference between estimated power and actual power. In particular, actual signal power is

2. We used 115 symbols and 2nd order cyclic moment/cumulant to estimate the BPSK

signal’s power. We used 345 symbols and 4th order cyclic moment to estimate the QPSK

signal’s power, which is equal top

M(4,0). We used 578 symbols and 4th order cyclic

94

moment to estimate the 16-QAM signal’s power, which is equal top

M(4,0)/0.68. If we

want to estimate mixed 8-PSK signal power, we need to use 5780 symbols and 8-th order

cyclic moment to estimate the 8-PSK signal’s power, which is equal to M(8,0)1/4. From

the figures, we can clearly see that the estimated power is close to the actual power val-

ues when the signal is BPSK or QPSK signal. For the 16-QAM and 8-PSK signals, the

power estimation performance is better when SNR is higher. In addition, we know that

if we have more symbols, the power estimation is more accurate.

6.3 Signal Modulation Classification

In this section, we use cyclic cumulant features and SVM based classifiers to do mixed

signal classification. We will focus on analyzing two different mixed-signal models, which

were proposed in Chapter 4. Detailed single signal classification performance is also

shown in this chapter.

6.3.1 Single Signal Classification

For the single signal classification, we can simply classify individual signals with cyclic

cumulants of different orders.

1 (M = 2)PSKfs = 16000 Hz, f

c = 3000 Hz, f

b = 400 Hz

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

1

2

3

4

5

6

7

8

9

10

Figure 6.22: Example: BPSK Training Data

1 (M = 4)PSKfs = 16000 Hz, f

c = 3000 Hz, f

b = 400 Hz

(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

Figure 6.23: Example: QPSK Training Data

95

Figure.6.22 and Figure.6.23 shows the examples of SVM classifier training data, which

use both low order and high order Cyclic Cumulant (CCs) features including 2nd order,

4th order, and 6th order CCs features. Each training data set is only 15 samples long,

which makes SVM training model becomes not heavy. Here, we do not choose 8th order

cyclic cumulant for training data because higher-order cyclic cumulants have a larger

variance in CCs values. For the higher-order modulation type 8-PSK, its 2nd, 4th, and

6th order cyclic cumulant features are already different from the other modulation types.

BPSK QPSK 8PSK 16QAM 0

0.2

0.4

0.6

0.8

1

Pcc

Figure 6.24: Classification Performance under SNR = 20 dB

In figure 6.24, there are 4 totally different classes of classification targets, including

BPSK, QPSK, 8PSK, and 16QAM signals. We used 100 sets of training samples to train the

SVM model and used 100 test samples to calculate the classification correction rate(Pcc).

The training data and test data for each set was generated using 115 symbols under SNR

= 20 dB. From the figure, we can easily classify single BPSK, QPSK, 8PSK, and 16QAM

signals with only a small number of symbols. In particular, the cyclic cumulant features

of QPSK and 16QAM are very similar, but they are also easily distinguished.

96

Predict\Actual BPSK QPSK 8PSK 16QAM

BPSK 100 0 0 0

QPSK 0 100 0 0

8PSK 0 0 100 0

16QAM 0 0 0 100

Table 6.1: Confusion Matrix: Single Signal Classification

Table 6.1 is the single signal classification confusion matrix, which is corresponds

to figure 6.24. From the table, we know the detailed classification performance for each

class. In the following sections, we will use this evaluation method to analyze mixed-

signal classification performance.

-20 -15 -10 -5 0 5 10 15 20

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pcc

BPSK

QPSK

8PSK

16QAM

Figure 6.25: Single signal Pcc vs SNR

Figure 6.25 shows the classification rate Pcc vs different SNR. We can clearly see that

all modulation types have a very good classification rate when SNR is larger than 0 dB. In

particular, the BPSK modulation type is very easily distinguished with other modulation

types at low SNR.

97

C(2,0) C(2,1) C(2,2) C(4,0) C(4,1) C(4,2) C(4,3) C(4,4)

pX X

p pX X X

C(6,0) C(6,1) C(6,2) C(6,3) C(6,4) C(6,5) C(6,6) C(8,0)

p p pX X X X —–

Table 6.2: Cyclic Cumulants Features (α= (n −2m) fc )

According to Chapter 6.2.2, we know that we can detect, estimate, and classify BPSK

signals, QPSK/16QAM signals and 8PSK signals with 2nd-order, 4th-order, and 8th-order

cyclic moments, respectively. However, since QPSK and 16QAM have very similar 4th or-

der cyclic features, we cannot classify them using only 4th order cyclic moment/cumu-

lants. The classification of QPSK and 16QAM signals are as a result more of a concern to

us. In the following sections, we will focus on classifying mixed QPSK signals and mixed

16QAM signals.

Since our target signals are mixed, we cannot use all the different order cyclic cumu-

lants for classification like in single signal classification. Some cyclic cumulant terms

will turn into corrected terms, such as C (n,n/2), which is shown in table 6.2 in red.

Those red terms include all mixed signals CCs features and cannot be used to estimate

an individual signal’s CCs features. Hence, we will not use all C (n,n/2) terms to do clas-

sification. In the table 6.2, we also tick off repeated terms, i.e., yellow color terms, whose

magnitude is equal to the corresponding blue terms. For the higher-order C (8,0), we

will keep C (8,0) as reserve term now. If we can get good classification performance with

CCs features below 6th order, then we don’t need the 8th order CCs feature, which will

reduce the computational complexity. We will have a detailed discussion about this in

the following sections.

98

BPSK QPSK 16QAM 8PSK 0

0.2

0.4

0.6

0.8

1

Pcc

Figure 6.26: Single signal Classification Pcc

Figure 6.25 shows the single signal classification rate at SNR=20 dB. We can see that

the classification rate is above 98% with only 6 CCs features. Hence, we can believe that

6 CCs feature can also provide better classification performance. Meanwhile, we just

need less amount of CCs features to train the SVM model.

-20 -15 -10 -5 0 5 10 15 20

SNR(dB)

0

0.2

0.4

0.6

0.8

1

Pcc

BPSK

QPSK

8PSK

16QAM

Figure 6.27: single signal Pcc vs SNR

In figure 6.27, we still use 115 symbols to generate different test data sets at different

SNR. Compared with all order CCs feature trained SVM classifiers, non-correlated CCs

99

features based classifiers can easily classify BPSK and 8PSK signals when SNR is smaller

than 0 dB. The reason for this is that all of 8PSK’s non-correlated lower-order CCs values

are 0, so it only sensitives to C (8,0). Meanwhile, it is well known that white noise only

has value at C (2,1). Hence, 8PSK can still be identified at very low SNR.

6.3.2 Model A: Mixed Signal Classification

For the single signal classification, we already know that we can easily classify BPSK,

QPSK, 16QAM, and 8PSK signals with few symbols. However, if those signals are mixed,

that is a different story. With the overlap of mixed signals, the estimation accuracy of the

cyclic cumulants is affected, and there will be a lot of interference affecting the classifi-

cation results. In model A, we assume that the two mixed signals have different carrier

frequencies, and that they overlap heavily in both time domain and frequency domain,

i.e., 50% spectrum overlap. Since we can use 2nd order cyclic moment to classify the

BPSK signal and 8th order cyclic moment to classify the 8PSK signal, we will focus on

classifying mixed QPSK signals and 16QAM signals in this section. There are 7 totally dif-

ferent combinations of mixed QPSK signals and mixed 16QAM signals, i.e., QPSK+BPSK,

QPSK+QPSK, QPSK+16QAM, QPSK+8PSK, 16QAM+BPSK, 16QAM+16QAM, 16QAM+8PSK.

We need to find out which signals are QPSK signal or 16QAM signals.

Phase I: Identical power mixed-signal classification

In the initial stages, we just simply assume two mixed QPSK or 16QAM signals have iden-

tical power. In this case, we can easily know the characteristics of cyclic cumulant based

mixed signal classification.

100

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic C

umul

ant M

agni

tude

CCs of QPSK. QPSK1 + QPSK2

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

No overlap25% overlap50% overlap75% overlap

Figure 6.28: The CCs of QPSK signal

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic C

umul

ant M

agni

tude

CCs of QPSK. QPSK+16QAM

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)



Fig. 6.28 shows the cyclic cumulant of the QPSK signal when two QPSK signals are

mixed with different spectrum overlap. Fig.6.29 shows the cyclic cumulant of the QPSK

signal when one QPSK signal and one 16-QAM signal is mixed with different amounts of

spectrum overlap. From the figures, we could know that the CC features will not change

too much in different amounts of spectrum overlap. The 6th order cyclic cumulant val-

ues change slightly because the higher-order cyclic cumulant takes off more low-order

cyclic moment, causing more variance. According to the figures, it is a good to employ

QPSK and 16-QAM based non-correlated CCs features to classify QPSK and 16-QAM sig-

nals because they are not sensitive to the amount of spectrum overlap.

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic C

umul

ant M

agni

tude

CCs of 16QAM. QPSK+16QAM

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)



0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic C

umul

ant M

agni

tude

CCs of 16QAM. 16QAM+16QAM

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)


Figure 6.31: The CCs of 16-QAM signal

101

Fig. 6.30 and Fig.6.31 show the cyclic cumulant values of 16-QAM when QPSK and

16-QAM signal mixed or two 16-QAM signals mixed. We also find that there is no signif-

icant difference between the different amount of spectrum overlap.

0

0.2

0.4

0.6

0.8

1

1.2

Cyc

lic C

umul

ant M

agni

tude

CCs 50% overlap

(2,0) (2,1) (2,2) (4,0) (4,1) (4,2) (4,3) (4,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(n,m)

CCs QPSK (QPSK+QPSK)CCs QPSK (QPSK+16QAM)CCs 16QAM (16QAM+16QAM)CCs 16QAM (QPSK+16QAM)

Figure 6.32: CCs of two signals with 50% spectrum overlap

In figure 6.32, we see the CCs values when QPSK signals and 16-QAM signals are

mixed with different modulation types. From the figure, we know that the difference in

CCs features between QPSK and 16-QAM is very small. It is impractical to use a thresh-

old value to distinguish between them, so, we need a more intelligent classifier to clas-

sify them. Since Support Vector Machine (SVM) is good for small size features training

and classification, we chose SVM as our classifier to classify QPSK signals and 16QAM

signals. Figure 6.32 also verifies that we have to use non-correlated CCs terms,i.e., blue

color terms in table 6.2, to do the mixed-signal classification.

102

QPSK 16QAM0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pcc

Figure 6.33: Pcc of QPSK and 16-QAM signal

In figure 6.33, we employ all 2nd order, 4th order, and 6th order cyclic cumulant

magnitudes for our training data. We use 500 sets of training data to train the SVM clas-

sification model and 100 sets of testing data to get classification rate Pcc . Each data set

is generated with 578 symbols. From the figure, we know that the classification rate of

QPSK is around 90% and the 16QAM classification rate is about 91%.

0 2 4 6 8 10 12 14 16 18 20

SNR(dB)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pcc

Figure 6.34: SNR vs Pcc of QPSK and 16QAM mixed signal

Figure.6.34 shows the average classification rate of mixed QPSK and 16QAM signals

103

at different SNR. When SNR is greater than 5 dB, the classification rate is greater than

80%.

Phase II: Non-power Estimation based Classification

In real-world communication, we know that mixed signals may come from different

channels with different power. Hence, this poses a significant challenge to the CCs

feature-based classification approach because signal power directly affects cyclic cu-

mulant feature magnitude. Therefore, knowing how to test the power changed test data

in a fixed training model is critical for classification. In order to deal with mixed signal

power problems, we proposed a method to avoid directly estimating individual signal

power.

First, we normalize mixed-signal total power to 1 using a cyclic cumulant C(2,1)

feature, which represents the total power of mixed-signal. Then the power of every single

signal is between 0 and 1. We can use Equation (6.3.2) to represent normalized power

for each mixed signal.

E1 = E1/(C (2,1))

E2 = E2/(C (2,1)) (6.2)

Where, E1 and E2 is the original power of two mixed signals, respectively. The normal-

ized power E1 and E2 ranges from 0 to 1. In this case, we designed the SVM training

model with 8 different classes. There are 4 different classes for the QPSK signal and 4

different classes for the 16 QAM signal. The classification training table is as following,

104

Ratio to total power(%)

QPSK 20 40 60 80

16QAM 20 40 60 80

Table 6.3: classification training table

Table 6.3 actually represents eight different hypothesize about the power of the mixed-

signals, such as assuming that the power of QPSK signal is 20 percent of the total mixed-

signal power.

QPSK(0.2) QPSK(0.4) QPSK(0.6) QPSK(0.8) 16QAM(0.2)16QAM(0.4)16QAM(0.6)16QAM(0.8)

x(100%)

0

0.2

0.4

0.6

0.8

1

Pcc

Figure 6.35: Pcc of mixed QPSK and 16-QAM signal

In figure 6.35, we used 1000 sets training data and 100 sets of test data to generate

the classification result. Both training data and test data are generated with 512 symbols

under SNR = 20 dB. From the figure we can see that some of the power ratios of mixed

QPSK signal and 16QAM signal have very good classification rates, such as QPSK signal

with above 60% power of total mixed-signal power. However, there are some cases that

have a low classification rate, such as when the QPSK signal’s power is 20% of the total

mixed signal’s power.

105

QPSK 16QAM

Predict/Actual (20%) (40%) (60%) (80%) (20%) (40%) (60%) (80%)

QPSK(20%) 39 1 0 0 17 10 0 0

QPSK(40%) 0 78 0 0 0 14 18 0

QPSK(60%) 0 0 85 0 0 0 6 28

QPSK(80%) 0 0 0 100 0 0 0 0

16QAM(20%) 61 0 0 0 83 2 0 0

16QAM(40%) 0 16 0 0 0 74 1 0

16QAM(60%) 0 5 1 0 0 0 75 0

16QAM(80%) 0 0 14 0 0 0 0 72

Table 6.4: classification confusion matrix

Table 6.4 shows a more detailed classification performance for mixed QPSK signals

and mixed 16QAM signals at different power ratios. We can clearly see that the overall

classification rate of mixed QPSK signals and 16QAM signals is 75.5% and 76% respec-

tively. For the QPSK signal, if its power is higher, the classification rate is higher. If QPSK

takes 20% of the total power, the QPSK signal is more like a 16QAM signal with 20% of to-

tal power. If 16QAM has 80% of the total power, the 16QAM signal is more like the QPSK

signal that has 60% of total power. Overall, we are able to use this method to classify

the QPSK signal and 16QAM signal. However, the classification performance is not very

ideal.

Phase III:Power Estimation based Classification

Based on chapter 6.2.2 we know that we can use C (2,0) to estimate BPSK signal’s power,

use C (4,0) to estimate QPSK and 16QAM signal’s power, and use C (8,0) to estimate 8PSK

106

signal’S power. After we estimated each mixed signal’s power, we just need to normalize

each mixed signal’s power to 1 before we send CC features to the SVM classifier model.

In this case, to compare with the Phase II method, we only need to train two class based

classifier models, one for power normalized QPSK signals, and another for power nor-

malized 16QAM signals. Note that since our main work is classifying the QPSK signals

and 16QAM signals, we need to estimate C(4,0) for power normalizing purposes. After

we normalize each signal power with C(4,0), the new normalized signal’s C(4,0) becomes

1. According to table 5.1, we can know that only non-correlated terms C(6,1) and C(8,0)

are not equal to 0 for QPSK signals and 16QAM signals. So, in order to increase the clas-

sification rate, we also employed 8th order cyclic cumulant to classify QPSK signals and

16QAM signals. For power estimation based QPSK and 16QAM signal classification, the

detailed procedures are as following,

1. Computing the magnitude of cyclic cumulant C(4,0) of target signal.

2. Normalizing the target mixed signal with C (4,0)14 . There are two hypotheses,

H0 : Sqpsk = R/C (4,0)14

H1 : S16qam = R/C (4,0)14 (6.3)

3. Computing the cyclic cumulant values of Sqpsk or S16qam .

4. Classifying Sqpsk /S16qam via two classes based training model.

5. Check classification results. For example, if input signal is Sqpsk and the classi-

fication result is a QPSK signal, then we believe that the signal is a QPSK signal.

If input signal is S16qam , the classification result is a 16QAM signal, and then we

know that the input signal is a 16QAM signal.

107

2.8 3 3.2 3.4 3.6 3.8 4 4.2

Cyclic Cumulant Feature C(6,1)

0

1

2

3

4

5

6

7

Pro

babili

ty D

ensity F

unction

QPSK

16QAM

Figure 6.36: The PDF of normalized C(6,1)

24 25 26 27 28 29 30 31 32 33 34

Cyclic Cumulant Feature C(8,0)

0

0.5

1

1.5

2

2.5

3

3.5

4

Pro

babili

ty D

ensity F

unction

QPSK

16QAM

Figure 6.37: The PDF of normalized C(8,0)

Figure 6.36 and Figure 6.37 shows the pdf of two normalized CCs features of one

QPSK signal and one 16QAM signal. From the figures, we can see that QPSK and 16QAM

have many different CCs features.

QPSK(1:2) QPSK(1:1) QPSK(2:1) 16QAM(1:2) 16QAM(1:1) 16QAM(2:1)0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pcc

Figure 6.38: Pcc of QPSK and 16-QAM at SNR = 20dB (578 symbols)

108

QPSK(1:2) QPSK(1:1) QPSK(2:1) 16QAM(1:2) 16QAM(1:1) 16QAM(2:1)0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pcc

Figure 6.39: Pcc of QPSK and 16-QAM at SNR = 20dB (925 symbols)

Figure 6.38 shows the classification rate of mixed QPSK and 16QAM signals with

different power ratios. In this figure, we use 512 symbols. We see that the classifica-

tion performance is still not ideal, and the classification rate is similar to the Phase II

method. However, the classification rate is more stable and not sensitive to the power

ratio. Moreover, compared with the Phase II method, the Phase III method is only based

on two classes. In figure 6.39, we used 2000 sets training data with each training data set

made up of 925 symbols. Figure 6.39 shows the classification rate of mixed QPSK signals

and 16QAM signals with different power ratios. For example, "QPSK(1:2)" means the tar-

get signal is QPSK signal, and the power ratio between the target signal and the reference

signal is 1:2. The reference signal modulation types is random. From the figure, we can

see that the classification rate is not very sensitive to the mixed-signal power ratio. Un-

der different power ratios, the mixed QPSK signal and 16QAM signal classification rate

is nearly 90%.

109

0 2 4 6 8 10 12 14 16 18 20

SNR(dB)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pcc

Figure 6.40: Pcc of QPSK and 16-QAM verse SNR

Figure 6.40 shows the overall classification performance of the QPSK signal and the

16QAM signal under different SNR. We used 100 sets testing data for QPSK signals and

16QAM signals respectively. Each testing data set is generated with 925 symbols. The

mixed signals’ maximum power ratio is 3dB difference.

Channel\SNR SNR=10dB SNR=15dB SNR=20dB

AWGN 0.85 0.89 0.89

Flat Fading 0.82 0.84 0.85

Table 6.5: Pcc of mixed QPSK signal and 16QAM signal under different conditions

Table 6.5 shows the mixed QPSK signal and 16QAM signal classification performance

comparison on AWGN channel and flat fading channel. We can see that this method can

work well in both AWGN channel and the flat fading channel. However, since it does not

work well in a multi-path fading channel, we not provide classification results in a multi-

path fading channel. There are a couple of reasons for this. First, the QPSK signals and

16QAM signals belong to the high-order modulation signals, and higher-order modu-

lation signals’ higher-order cumulant values include more variance. Second, the cyclic

110

cumulant features of single QPSK signals and 16QAM signals are very similar. Under our

model, there is a 50% spectrum overlap, which leads to mixed signals THAT are pretty

close to each other. In this case, we need more symbols to obtain relatively stable cu-

mulative values for classification. This has been reflected in figure 6.38 and figure 6.39.

Therefore, for multi-path fading channels with faster channel variation, all methods that

rely on cumulative features to classify mixed signals are challenging to use.

6.3.3 Model B: Mixed Signal Classification

In model B, our target signal is two mixed signals with the same carrier frequency and

different symbol rates. With our symbol rate detection we can know how many signals

are overlapped. Hence, in this section, we assume that we already know the number

of mixed signals and their symbol rates. Also, since we have discussed signal carrier

frequency estimation in Section 6.2.2, we assume that we already know the carrier fre-

quency of mixed signals.

Challenge I

2 PSK


(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

2

4

6

8

10

12

Figure 6.41: Example: BPSK Training Data

4 PSK


(2,0)(2,1)(2,2)(4,0)(4,1)(4,2)(4,3)(4,4)(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

(n,m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

harm

onic

num

ber

0

0.5

1

1.5

2

2.5

3

Figure 6.42: Example: QPSK Training Data

111

We have to admit that model B is a very special and challenged case because of mixed

signals possessing the same carrier frequency. This causes all CC items at the cycle fre-

quency α = (n −2m) fc become correlated terms. Hence, the mixed signal’s individual

CC features are moving to the cycle frequency α= (n −2m) fc ±k fb , where k = 0,1,2,3...,

such as the red square area in Figure 6.41 and Figure 6.42.

α= (n −2m) fc ± fb α= (n −2m) fc

C(2,0) C(2,1) C(2,2) C(4,0) C(4,1) C(4,2) C(4,3) C(4,4) C(2,0) C(2,1) C(2,2) C(4,0) C(4,1) C(4,2) C(4,3) C(4,4)

pX X

p pX X X X X X X X X X X

C(6,0) C(6,1) C(6,2) C(6,3) C(6,4) C(6,5) C(6,6) C(8,0) C(6,0) C(6,1) C(6,2) C(6,3) C(6,4) C(6,5) C(6,6) C(8,0)

p p pX X X X

pX X X X X X X X

Table 6.6: Cyclic Cumulants Features

Table 6.6 shows the available CC features, which can be used to represent each

mixed signals with the same carrier frequency. The challenging thing is all CC features at

α= (n−2m) fc±k fb relative to symbol rate, signal power, signal filter type, e.g., RRC filter,

Hamming filter, Hanning filter, etc. Theoretically speaking, if we know the signal power

and have enough signal length, we can find and estimate the relationship between cyclic

cumulant sidelobe magnitude, signal filter coefficient, and symbol rate. Note that we as-

sume all mixed signals use the most popular Root-raised-cosine filter (RRC).

112

Figure 6.43: BPSK symbol rate vs roll-off factor vs side lobe magnitude

Figure 6.44: QPSK symbol rate vs roll-off factor vs side lobe magnitude

113

Figure 6.45: 16QAM symbol rate vs roll-off factor vs side lobe magnitude

Figure 6.46: 8PSK symbol rate vs roll-off factor vs side lobe magnitude

From figure 6.43 to Figure 6.46, we can see the relationship between signal symbol

rate, raised-cosine roll-off factor, and cyclic cumulant magnitude when cyclic frequency

is at α = 4 fc + fb . We can also see that the magnitude of the cyclic cumulants at the

side lobe does not change too much under different symbol rates and different roll-off

factors, but for the cyclic cumulant that is higher-order, the variance is bigger. Mean-

while, it is hard to find a relatively stable and uniform pattern to explain the relationship

114

between the symbol rate, roll-off factor, and single cyclic cumulant magnitude value.

Challenge II

Pattern BPSK Pattern QPSK Pattern 8PSK Pattern 16QAM

1 BPSK+BPSK 1 QPSK+BPSK 1 8PSK+BPSK 1 16QAM+BPSK

1 BPSK+QPSK 2 QPSK+QPSK 2 8PSK+QPSK 2 16QAM+QPSK

1 BPSK+8PSK 2 QPSK+8PSK 3 8PSK+8PSK 2 16QAM+8PSK

1 BPSK+16QAM 2 QPSK+16QAM 2 8PSK+16QAM 2 16QAM+16QAM

Table 6.7: Mixed co-channel signal cyclic cumulant pattern

Table 6.7 shows the cyclic cumulant estimation pattern for mixed co-channel signals.

"1" is a BPSK-like pattern; "2" is QPSK/QAM-like pattern; "3" is an 8PSK-like pattern.

Since mixed signals have the same carrier frequency, some of them need to share the

same pattern. In such a case, the cyclic cumulant estimation will have reduced accu-

racy. For example, if two signals are mixed, signal 1 is the BPSK signal, and signal 2 is

QPSK/16QAM signal, these two signals have different carrier frequencies, we can use

2nd-order and 4th-order cyclic moment to classify them respectively. Meanwhile, we

know the signal 1 pattern is BPSK-like pattern and signal 2 pattern is QPSK/16QAM-like

pattern. However, if signal 1 and signal 2 have the same carrier frequency, we can use

2nd order cyclic moment to detect their carrier frequency, and know that there is a BPSK

signal. However, we do not know which modulation type the BPSK signal is mixed with.

In such a case, we have to use a BPSK-like pattern to estimate their cyclic cumulant.

Hence, for mixed QPSK/16QAM/8PSK signals, some zero terms cannot be directly set

as zero, such as C (4,1) and C(6,0). This will decrease the classification rate of the BPSK

signals and 8PSK signals.

115

BPSK QPSK 16QAM 8PSK 0

0.2

0.4

0.6

0.8

1

Pcc

Figure 6.47: Mixed Signal Classification Rate

Predict\Actual BPSK QPSK 8PSK 16QAM

BPSK 100 0 0 0

QPSK 0 50 0 57

8PSK 0 5 100 1

16QAM 0 45 0 42

Table 6.8: Classification Confusion Matrix

Figure 6.47 and table 6.8 show the classification performance of mixed co-channel

signals. From this result, we assume that we know each mixed signal’s cyclic cumulant

patterns. We used 7 different features, i.e., C(2,0),C(4,0),C(4,1),C(6,0),C(6,1),C(6,2) and

C(8,0), to train the SVM classifier model. We can clearly see that BPSK and 8PSK have as

perfect classification rate. The classification performance of QPSK and 16QAM is poor.

Note that, we only use CC features when α= (n −2m) fc + fb .

116

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pcc

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

BPSK+16QAM

QPSK+16QAM

BPSK+BPSK

16QAM+BPSK

BPSK+QPSK16QAM+QPSK

BPSK+8PSK16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM

Figure 6.48: Mixed Signal Classification Rate of 1st signal

In figure 6.48, we simulate a more realistic situation. All mixed signal’s cyclic cu-

mulant patterns in the following table 6.7. We can see that only two mixed 8PSK signals

have a very high classification rate (96%). The classification performance of BPSK be-

comes very low because some other mixed types share the BPSK-like pattern. Note that,

in this result, we only use 400 sets of training data and 100 sets of test data. Meanwhile,

we still use the same 7 different cyclic features as in Figure 6.47.

Classification

For Challenge I, we know that if mixed signals have the same carrier frequency, we can-

not easily employ available non-corrected side-lobe terms to classify the mixed signals’

modulation types. We need to consider the signal power, symbol rate, and signal filter

coefficient. So, could we find a way to ignore the difficulties caused by these factors? We

did some analysis in the following section. For Challenge II, in some mixed signals cases,

since we cannot know their cyclic cumulant patterns in advance, the only way that we

can improve is by increasing the number of symbols, which make cumulant more pre-

117

cise.

Phase I

In order to avoid the influence of signal power, signal symbol rate and signal filter coeffi-

cient on the cyclic cumulants on the side lobes, we use the ratio value of different order

cyclic cumulants instead of the single cyclic cumulant value to look for classification

features. Based on table 6.6, we choose some ratio values as following,

α= (n −2m) fc + fb

F1 F2 F3

C (4,1)C (4,0)

C (2,0)2

C (4,0)C (4,0)2

C (8,0)

Table 6.9: Cyclic Cumulant Ratio Features

Note that, fb is the first signal’s symbol rate. For example, for BPSK+QPSK, fb is BPSK’s

symbol rate.

0 50 100 150 200 250 300 350 400 450 500

c(4,1)/c(4,0)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

c(4

,1)/

c(4

,0)

Power Ratio 1:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM

Figure 6.49: Feature F1: Power Ratio 1:1

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

c(4

,1)/

c(4

,0)

Power Ratio 3:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM


118

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

c(4

,1)/

c(4

,0)

Power Ratio 5:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM


0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

c(4

,1)/

c(4

,0)

Power Ratio 10:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM


From figure 6.49 to figure 6.52, we can see that when the signal power ratio is larger

than 5:1, we can use feature F1 to classify BPSK signal, but the classification rate is low.

0 50 100 150 200 250 3000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

c(2

,0)2

/c(4

,0)

Power Ratio 10:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM


0 50 100 150 200 250 3000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

c(4

,0)2

/c(8

,0)

Power Ratio 10:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM


Figure 6.53 and Figure 6.54 shows the feature F2 and F3 values for different mixed

types at power ratio 10:1. We can see that the F2 and F3 features do not work.

119

1:1 3:1 5:1 10:1 20:1 25:1

Power Ratio

0

0.2

0.4

0.6

0.8

1

1.2

1.4

c(4

,1)/

c(4

,0)

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM

Figure 6.55: Feature F1 values with different power ratio

Figure 6.55 shows the Feature F1 values with different power ratios. We can see that

the mixed BPSK signal’s F1 values are nearly equal to 1. However, the F1 value of other

mixed signals like 8PSK+BPSK, its F1 value is also equal to 1. In this case, we know that

the side-lobe cyclic cumulant value cannot represent the individual mixed signals.

Phase II

Since the variance of the side-lobe value of the cyclic cumulant value is too large, we

did not get any good classification features in the first stage, so we tried to return to the

main lobe value of the cyclic cumulant values. Meanwhile, since all the cyclic cumulant

values of the main lobe are correlated terms, we do not expect to be able to classify the

modulation type of all mixed signals. We do expect to classify the modulation type of

the main component in the mixed signal. In other words, we expect to classify the signal

which has the highest power of the mixed signals.

120

α= (n −2m) fc

F1 F2 F3 F4 F5

C (4,2)C (2,1)2

C (6,3)C (2,1)3

C (6,3)2

C (4,2)3C (8,4)3

C (6,3)4C (8,4)

C (4,2)2

Table 6.10: Cyclic Cumulant Ratio Features

Table 6.10 shows some cyclic cumulant ratio values, based on theoretical cumulant

values, which could possibly be used to classify signal power cancellation.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

C(4,2)/C(2,1)2

0

5

10

15

20

25

Pro

ba

bili

ty

Power Ratio: 1:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM

Figure 6.56: PDF of F1: C (4,2)C (2,1)2 with Power ratio 1:1

121

0.2 0.4 0.6 0.8 1 1.2 1.4

C(4,2)/C(2,1)2

0

5

10

15

20

25

30

35

40

45

50

Pro

ba

bili

ty

Power Ratio 10:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM


0 0.5 1 1.5 2 2.5 3 3.5 4

c(6,3)/c(2,1)3

0

1

2

3

4

5

6

Pro

babili

ty

Power Ratio: 1:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM


122

0 1 2 3 4 5 6 7 8 9 10

c(6,3)/c(2,1)3

0

1

2

3

4

5

6

7

8

9

10

Pro

ba

bili

ty

Power Ratio: 10:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM


0 2 4 6 8 10 12 14 16 18 20

c(6,3)2/c(4,2)

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Pro

ba

bili

ty

Power Ratio 1:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM

Figure 6.60: PDF of F3:C (6,3)2

C (4,2)3 with Power ratio 1:1

123

5 10 15 20 25 30 35

c(6,3)2/c(4,2)

3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Pro

babili

ty

Power Ratio 10:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM



-50 0 50 100 150 200 250 300

c(8,4)3/c(6,3)

4

0

0.01

0.02

0.03

0.04

0.05

0.06

Pro

ba

bili

ty

Power Ratio 1:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM



124

120 140 160 180 200 220 240 260 280 300 320

c(8,4)3/c(6,3)

4

0

0.1

0.2

0.3

0.4

0.5

0.6

Pro

babili

ty

Power Ratio 10:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM



-10 0 10 20 30 40 50

c(8,4)/c(4,2)2

0

0.05

0.1

0.15

0.2

0.25

0.3

Pro

ba

bili

ty

Power Ratio 1:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM


125

20 25 30 35 40 45 50 55 60 65 70

c(8,4)/c(4,2)2

0

0.5

1

1.5

2

2.5

Pro

ba

bili

ty

Power Ratio 10:1

BPSK+BPSK

BPSK+QPSK

BPSK+8PSK

BPSK+16QAM

QPSK+BPSK

QPSK+QPSK

QPSK+8PSK

QPSK+16QAM

16QAM+BPSK

16QAM+QPSK

16QAM+8PSK

16QAM+16QAM

8PSK+BPSK

8PSK+QPSK

8PSK+8PSK

8PSK+16QAM


From figure 6.56 to figure 6.65, we see the distribution of different features in differ-

ent mixed cases and with different power ratios. We know that if two mixed signals have

identical power, we are unable to classify their modulation types. If two mixed signals

have different power ratios, we can use feature F1, F2, F3, and F5 to classify BPSK sig-

nal, 16QAM signal, and QPSK/8PSK signal, and we can use features F4 to classify QPSK

signal and 8PSK signal.

1:1 2:1 3:1 5:1 10:1 20:1 25:1

Power Ratio

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

F1:c

(4,2

)/c(2

,1)2

BPSK

QPSK

16QAM

8PSK

Figure 6.66: PDF of F 1 = C (4,2)C (2,1)2 with different power ratio

126

1:1 2:1 3:1 5:1 10:1 20:1 25:1

Power Ratio

0

1

2

3

4

5

6

7

8

9

10

F2:c

(6,3

)/c(2

,1)3

BPSK

QPSK

16QAM

8PSK

Figure 6.67: PDF of F 2 = C (6,3)C (2,1)3 with different power ratios

1:1 2:1 3:1 5:1 10:1 20:1 25:1

Power Ratio

5

10

15

20

25

30

35

F3:c

(6,3

)2/c

(4,2

)3

BPSK

QPSK

16QAM

8PSK

Figure 6.68: PDF of F 3 = C (6,3)2

C (4,2)3 with different power ratios

127

1:1 2:1 3:1 5:1 10:1 20:1 25:1

Power Ratio

100

150

200

250

300

350

F4:c

(8,4

)3/c

(6,3

)4

BPSK

QPSK

16QAM

8PSK

Figure 6.69: PDF of F 4 = C (8,4)3

C (6,3)4 with different power ratios

1:1 2:1 3:1 5:1 10:1 20:1 25:1

Power Ratio

10

20

30

40

50

60

70

F5:c

(8,4

)/c(4

,2)2

BPSK

QPSK

16QAM

8PSK

Figure 6.70: PDF of F 5 = C (8,4)C (4,2)2 with different power ratios

From figure 6.66 to figure 6.70, we can see the trend of the average features of dif-

ferent mixed signals according to the power ratio. We find that for the feature F1 and F2

when the power ratio is greater than 20:1, the feature tends to be stable. For the features

F3, F4, and F5, when the power ratio is greater than 5:1, the features tends to be stable.

Hence, since F1 and F2 only work when the ratio of the two mixed-signal powers is very

128

large, and both F1 and F2 rely on low-order C(2,1), which includes the total signal power

and noise power, we will not consider F1 and F2 for our classification features.

0 2 4 6 8 10 12 14 16 18 20

SNR(dB)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pcc

Power Ratio 5:1

BPSK

QPSK

16QAM

8PSK

Figure 6.71: Classification Rate vs SNR

In the figure 6.71, we see the classification performance when the mixed-signal

power ratio is 5:1. The classification rate of mixed QPSK signals and 8PSK signals is

similar. Meanwhile, we know that BPSK signals and 16QAM signals are more easily be

classified.

129

Conclusion

The purpose of this dissertation is to solve the problem of mixed-signal detection, pa-

rameter estimation, and modulation classification. We analyzed two different mixed sig-

nals models, one is a more general case, mixed-signal have different parameters, such as

carrier frequency, symbol rate, and power. Meanwhile, those mixed signals are heavily

overlapped in both time domain and frequency domain. Another mixed-signal model

is multiple mixed signals with different symbol rates but the same carrier frequency. In

this model, the most challenging thing is all mixed signals are sharing the same cyclic

frequency. We cannot easily classify those signals with general cyclostationary based

classification method. In this dissertation, we mainly employed cyclostationary pro-

cessing to detect, estimate and classify the modulation types of mixed signals. In par-

ticular, we used 2nd-, 4th-, and 8th order cyclic moments to detect and estimate mixed

signals’ parameters, such as carrier frequency, symbol rate, and power. Meanwhile, we

used 4th-, 6th-, and 8th -order cyclic cumulant to classify the mixed signals’ modulation

types, which has similar cyclostionary features, such as mixed QPSK signals and mixed

16QAM signals. Moreover, machine learning is also used through the Support Vector

Machine to classify the mixed signals’ modulation types. For mixed signals detection

and parameter estimation, 8-PSK is the most challenging, especially when it is mixed

with other signals. For mixed signals modulation classification, it is easy to classify most

mixed modulation types. QPSK and 16-QAM mixed signals classification is more dif-

ficulty. All in all, our approaches can solve most of the mixed-signal cases in different

130

channels, such as AWGN channels, flat fading channels, multi-path fading channel. Our

future work will be to improve the signal detection, estimation, and modulation classi-

fication performance while using as few symbols as possible.

131

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