Transform Domain Acquisition of Spread Spectrum Signals in a Low Signal to Noise
Ratio Environment
A thesis presented to
the faculty of
the Russ College of Engineering and Technology of Ohio University
In partial fulfillment
of the requirements for the degree
Master of Science
Rakesh Kashyap Hassana Ramesh
November 2010
© 2010 Rakesh Kashyap Hassana Ramesh. All Rights Reserved.
2
This thesis titled
Transform Domain Acquisition of Spread Spectrum Signals in a Low Signal to Noise
Ratio Envirionment
by
RAKESH KASHYAP HASSANA RAMESH
has been approved for
the School of Electrical Engineering and Computer Science
and the Russ College of Engineering and Technology by
Jeffrey C. Dill
Professor of Electrical Engineering and Computer Science
Dennis Irwin
Dean, Russ College of Engineering and Technology
3
ABSTRACT
HASSANA RAMESH, RAKESH KASHYAP, M.S., November 2010,
Electrical Engineering
Transform Domain Acquisition of Spread Spectrum Signals in a Low Signal to Noise
Ratio Environment
Director of Thesis: Jeffrey C. Dill
Signal acquisition in direct sequence spread spectrum (DSSS) communication
systems determines the efficiency of the receiver. Problems like Doppler shift and timing
uncertainty further reduces the performance of the acquisition process. This research
presents a performance based comparative study of transform domain and time domain
signal acquisition algorithms on asynchronous DSSS signals in an additive white
Gaussian noise (AWGN) channel with Doppler uncertainty. The lowest possible signal to
noise ratio (SNR) that can be detected perfectly for a 128 chip preamble is analyzed.
Approved: _____________________________________________________________
Jeffrey C. Dill
Professor of Electrical Engineering and Computer Science
4
ACKNOWLEDGMENTS
I would like to thank Dr. Jeffrey Dill for giving me an opportunity to work on this
thesis topic. His encouragement, support and indefinite patience have helped me to learn
and understand new concepts in this field. I have always relished sharing new ideas with
him.
I would like to thank Dr. David Matolak for his support and encouragement. I
would also like to thank Dr. Skidmore and Dr. Kruse for being a part of my committee
and dedicating their valuable time.
I would like to thank Dr. Katherine Milton and Nathaniel Berger from The
Aesthetic Technologies lab for funding my studies. It has been my pleasure to work with
them for the past three years.
I would like to thank all my friends for their support and advice.
Finally, I would like to thank my parents and my sister for their support and
encouragement even while living far apart. It is this strength that keeps me focused and
working towards my goals.
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TABLE OF CONTENTS
Page
Abstract ............................................................................................................................... 3
Acknowledgments............................................................................................................... 4
List of Figures ..................................................................................................................... 8
Chapter 1: INTRODUCTION........................................................................................... 10
1.1 Spread Spectrum: History and Present ............................................................. 10
1.2 Signal Acquisition: History and Present ........................................................... 11
1.3 Thesis Outline ................................................................................................... 12
Chapter 2: BACKGROUND............................................................................................. 14
2.1 Direct Sequence Spread Spectrum (DSSS) ............................................................ 14
2.2 Signal Acquisition: Definition ................................................................................ 15
2.3 Methods of Acquiring Signals in Different Domains ............................................. 16
2.4 Operating in Transform Domain ............................................................................. 17
2.5 Software and Tools ................................................................................................. 18
2.6 Definition ................................................................................................................ 18
Chapter 3: DESIGN .......................................................................................................... 19
3.1 Wireless Communication System ........................................................................... 19
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3.2 The Transmitter ....................................................................................................... 20
3.2.1 The Preamble ................................................................................................... 20
3.2.2 Timing Offset ................................................................................................... 21
3.2.3Over Sampling .................................................................................................. 22
3.3 The Channel ............................................................................................................ 23
3.3.1 Modeling AWGN Noise .................................................................................. 23
3.3.2 Modeling Doppler Effects ................................................................................ 25
3.4 The Receiver ........................................................................................................... 28
3.4.1 Cross Correlation ............................................................................................. 28
3.4.2 Threshold ......................................................................................................... 33
3.4.3. Quantization .................................................................................................... 34
3.4.4. Doppler Estimation ......................................................................................... 35
3.4.5 Signal Detection and Timing Estimation ......................................................... 36
Chapter 4: SIMULATION RESULTS.............................................................................. 39
4.1 Channel Simulations ............................................................................................... 39
4.2 Receiver Simulations .............................................................................................. 40
4.2.1 Transform Domain Simulations ....................................................................... 41
4.2.2 Time Domain Simulations ............................................................................... 45
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Chapter 5: CONCLUSION ............................................................................................... 48
REFERENCES ................................................................................................................. 50
Appendix A: DOPPLER ANALYSIS .............................................................................. 53
Appendix B: THRESHOLD ESTIMATION .................................................................... 55
Appendix C: QUANTIZATION ....................................................................................... 56
Appendix D: CHANNELS ............................................................................................... 57
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LIST OF FIGURES
Page
Figure 3.1: Basic wireless communication system representation ……………….. ...19
Figure 3.2: Block diagram of the DSSS transmitter ……………….. ……………….20
Figure 3.3: Autocorrelation for 128 length preamble for 10,000 trials …...…………21
Figure 3.4: Modeling preamble offset ……………………………………………….22
Figure 3.5: Preamble and its sampling ………………………………………….…...23
Figure 3.6: Modeling AWGN noise …………………………………………………25
Figure 3.7: Modeling Doppler ……………………………………………………….27
Figure 3.8: Block diagram of acquisition at the receiver…………………………….28
Figure 3.9: Time domain linear correlation ………………………………………….30
Figure 3.10: Extraction of linear correlated bits from circular correlation…………..31
Figure 3.11: Transform domain linear correlation…………………………………. .33
Figure 3.12: De-Multiplexing to 16 channels and Rx window………………………35
Figure 3.13: Doppler frequency estimation…………………………………………..36
Figure 3.14: Signal acquisition ad timing estimation………………………………...37
Figure 4.1: Example of channel offset……………………………………………….39
Figure 4.2: Transmitter output ………………………….…………………………...39
Figure 4.3: Plot of channel at SNR = 2 dB………………………………. …………40
Figure 4.4: Received waveform before quantization…………………….………… 41
Figure 4.5: Received waveform after quantization…………….…………………….41
Figure 4.6: Map of Doppler correlation, SNR = 5 dB……………………………….42
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Figure 4.7: Map of timing estimation and acquisition……………………………….43
Figure 4.8: Probability of detection and false alarm. ∆f = 0.625 Hz……………….. 43
Figure 4.9: Timing error (Transform Domain Acquisition) ……………………….44
Figure 4.10: Doppler correlation map, Time domain, SNR = 2 dB,
Peak at channel 14…………………………………………………...45
Figure 4.11: Correlation map of signal acquisition and timing detection…………...46
Figure 4.12: Probability of Detection and Probability of false alarm,
at 0.625 Hz Doppler. ……………………………………………….47
Figure 4.13: Timing Error (Time Domain Acquisition)…………………………….47
Figure A.1: ∆f = 0.1 cycles/preamble..........................................................................53
Figure A.2: ∆f = 0.6 cycles/preamble..........................................................................54
Figure A.3: ∆f = 0.9 cycles/preamble .........................................................................54
Figure B.1: Correlation plot at -10 dB CNR ..............................................................55
Figure C.1: Receiver inputs at 5 dB CNR with no quantization,
8 bits and 4 bit quantization......................................................................................56
Figure D.1: Assignment of pre-determined preambles for
the 16 channels in steps of 0.125 Hz........................................................................57
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CHAPTER 1: INTRODUCTION
This chapter introduces the reader to Signal Acquisition in direct sequence spread
spectrum (DSSS) communication systems. Also, the concept of acquiring signals in a
transformed domain is discussed, followed by the thesis outline.
1.1 Spread Spectrum: History and Present
A spread spectrum communication system uses excess bandwidth as compared to a
normal communication system. Here, the transmitted signal is spread over the entire
bandwidth of operation using a spreading code [1]. “Sometimes the transmitted bandwidth
is as much as 105
times the information bandwidth” [2]. Fundamentally there are two types
of spread spectrum communication systems, direct sequence and frequency hopping.
Although the concept of spread spectrum was demonstrated in the early 1900s,
commercial use of the spread spectrum technology only began during the 1980s in the US
[2].Today Wi-Fi and code division multiple access (CDMA) are two of the many
technologies based on the DSSS technique, and these have become the backbone of
wireless internet and cell phone networks, respectively. The latest WPANs like ZigBee
(IEEE 802.15.4) have gained popularity in developing smart home-appliances and
distributed wireless networks in recent years [3]. Cordless phones working at 900 MHz,
2.4 GHz and 5.8 GHz are examples of frequency hopping spread spectrum (FHSS)
technologies [4].
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1.2 Signal Acquisition: History and Present
Signal acquisition has been a topic of discussion since the first DSSS receivers were
designed during the 1920s [17]. Published research papers on acquisition of pseudo-noise
signals dates back to 1965 [18]. Ever since, several researchers have proposed, designed
and prototyped acquisition algorithms to acquire signals at the lowest possible signal to
noise ratio (SNR). In 1965, Cooley and Tukey developed an efficient technique to
compute discrete Fourier Transform (DFT) which lead to the development of many fast
and efficient algorithms called the Fast Fourier Transforms (FFT) [7] [8]. This opened
up opportunities to efficiently convert time-domain discrete data into frequency domain
data.
In the past decade scientists have extensively worked on FFT based techniques for
acquiring signals. In 1993, T. A. Brown et.al, presented a paper at the MILCOM
conference discussing the advantages of transform domain signal acquisition for the
DSSS communication system [13]. Multiple DSP techniques were covered and
improvement of processing time was highlighted. In 1999, P. G. Temple et.al presented a
performance based evaluation for transform domain communication system (TDCS) for
CDMA signals. SNR as low as 0 dB with multiple channels were discussed using a
MATLAB® simulated model. In 2000, M. L. Roberts et.al, presented a performance
based study for transform domain acquisition. Asynchronous conditions were assumed
during modeling and a 7 bit Barker code was used as the preamble [6]. Apart from
communication systems, several researchers have also worked on signal acquisition in the
transform domain for GPS signals [2, 15, 16 and 18].
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In this research we will be presenting an algorithm for signal acquisition in the
transform domain for a DSSS communication system which works in a very low SNR
environment. Performance of the model, based on Doppler correction, signal detection
and timing estimation, will be assessed.
1.3 Thesis Outline
This thesis is organized as follows:
Technical details and definitions of DSSS and signal acquisition are discussed in
Chapter 2. This chapter also provides background information which includes
relevant mathematical equations to support the design.
Chapter 3 discusses the design in detail; each section of the communication
system developed is investigated. Two designs are presented, one in the transform
domain using linear FFT correlation and the other in time domain. Both designs
use linear correlation and parallel search. Concepts leading to the acquisition are
initially discussed and each unit in the design is discussed in the order of the
signal propagation. The explanations in both the domains are simultaneously
discussed. Concepts are illustrated with diagrams for clarity.
Chapter 4 presents the simulated results for the design presented in Chapter 3. An
example of transmitted preamble is considered and the design is examined
presenting the simulated results. Both transform domain and time domain models
are presented and discussed together. Performance graphs including probability of
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detection, probability of false alarm and timing error with respect to signal to
noise ratio (SNR) are presented and discussed.
Chapter 5 provides a conclusive discussion and possible future work.
Appendix A provides the experimental justification for the analytical computation
of the Doppler model.
Appendix B describes the selection of the pre-determined threshold for the
receiver.
Appendix C presents a simulated plot of various quantization intervals considered
for the receiver.
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CHAPTER 2: BACKGROUND
This chapter discusses the technical aspects of the topics introduced in Chapter 1.
Basic principles of direct sequence spread spectrum (DSSS) and signal acquisition with
respect to this research are covered. The concept of transform domain processing is
explained with equations. Finally, details of software and functions used to model this
design are discussed.
2.1 Direct Sequence Spread Spectrum (DSSS)
DSSS is a communication technique designed to transmit signals over the entire
bandwidth of operation. A spreading code with a code chip rate higher than the data
signal bandwidth is multiplied by the data, and then at the receiver, a synchronized
replica of the code is used to de-spread the signal. De-spreading is achieved by
multiplying the stored copy of the spreading code with the received signal. Multiple
harmonics of the signal of interest will occur as a result of this operation. A filter with the
bandwidth equal to the spreading code is used to remove unnecessary components to
retrieve the data [4].
Shannon’s capacity theorem provides a clear picture of the relation between the
bandwidth and the signal during a DSSS transmission.
1 … … … … … 1.1
C: Channel capacity in bits/sec
W: Bandwidth in Hz
S: Signal power
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N: Noise power
So, to increase the bandwidth, the SNR has to be increased hence transmitting more
information.
By re-arranging equation 1.1 and considering natural logarithms we get,
0.6913
1 … … … … … 1.2
At, SNRs < -7dB , Equation 1.2 can be approximated as,
0.6913 0.6913
… … … … … 1.3
Note:
-7dB ~ 0.2 ……………1.4
1 0.2 0.1823 ~ 0.2 …………… 1.5
When the capacity of the channel is fixed and the channel is modeled as additive
white Gaussian noise (AWGN) [2]. So, if the channel capacity and the signal strength is
fixed, The only way to counter the noise will be by increasing the bandwidth. Observing
equation 1.3 it can be inferred that, by increasing the length of the spreading code, we can
detect signals with lower SNR.
2.2 Signal Acquisition: Definition
In practice, the received signal and the transmitted signals are out of synchronization,
so the receiver has to first align the reference (de-spreading code) and the transmitted
16
code in order to faithfully reproduce the transmitted data. The receiver searches the
region of time – frequency uncertainty to match the stored preamble with the incoming
code. This process is termed Acquisition [5].
Even after achieving acquisition the receiver gradually looses alignment in time due
to various parameters, the most important of which are listed below:
Noise in the channel
Doppler shift due to relative motion between the transmitter and the receiver
Tolerance parameters of the receiver hardware
Multipath from the surrounding environment
Drift or difference in individual clocks maintained by the transmitter and the
receiver
The receiver has to incorporate a continuous tracking process to maintain the required
alignment. Together, the concept of detecting the signal using acquisition and
maintaining the alignment through Tracking is called Synchronization [4], [5].
2.3 Methods of Acquiring Signals in Different Domains
Acquisition can be performed by different techniques. Matched filters and active
correlators are a natural choice to acquire DSSS signals. Different versions of both have
been discussed and simulated by Rappaport and Grieco [5]. Serial search, parallel search
and their hybrids are most common and accurate in terms of performance. Since the
primary focus of this thesis is to evaluate performance and not speed, parallel search has
been used to model both the time domain and the transform domain designs.
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Furthermore, if the digital signal is converted to a different domain (e.g. wavelet,
Laplace or Fourier) for processing and is not processed as it is (in time domain), then the
process is said to be in the Transformed Domain and the system is called Transform
Domain Communication System (TDCS) [6]. Since the computations of the acquisition
are being modeled and examined in the frequency domain, the terms “Transform
Domain” and “Frequency Domain” are used interchangeably in this thesis.
2.4 Operating in Transform Domain
The Discrete Fourier Transform (DFT) is a method to compute frequency samples
from discrete time signals. The formula to calculate the DFT of the signal x(n) is:
… … … … … 2.1
0,1 … 1
Hence, a sequence of 0 , 1 , 2 , … , 1 in time domain gets transformed
to 0 , 2 , 3 , … , 1 in frequency domain.
Similarly, the inverse discrete Fourier transform (IDFT) can be calculated as shown:
1
… … … … … 2.2
0,1 … 1
Equation 2.1 computes an N point DFT for a digital sequence x (n).
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2.5 Software and Tools
This thesis has been modeled in MATLAB®. The Communication ToolboxTM and
Signal Processing ToolboxTM have been used extensively. The function fft ( ) has been
used to compute the DFT during simulation. MATLAB® uses FFT algorithms from
http://www.fftw.org, which provides open source algorithms created by M. Frigo, and S.
G. Johnson for increased speed. As this is a performance-oriented study, methods of
computing FFT are not in the scope of this thesis [9] [10]. MATLAB®, Communication
ToolboxTM and Signal Processing ToolboxTM are trademarks of the MathWorks, Inc. [12].
2.6 Definition
Before we start with the design aspects, let us define acquisition with respect to
this design.
"
12
"
In this thesis, we present a frequency domain approach to acquire DSSS signals
and analyze its detection, time synchronization and Doppler correction capabilities. The
final results gives us an algorithm for detecting signals at its lowest possible signal to
noise ratio (SNR) with Doppler shifts, noise, detection threshold, and length of preamble
as parameters.
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CHAPTER 3: DESIGN
This chapter discusses the design, algorithm and the working principles of the
acquisition model to be presented. The communication system is discussed in the order of
the signal propagation. A basic block diagram is presented and each section of the
diagram is described. Furthermore, since acquisition-models of both time and transform
domains have been developed together, concepts related to these models are discussed in
parallel as we proceed.
3.1 Wireless Communication System
Any wireless communication system can be fundamentally described by a transmitter,
channel and a receiver. Figure 3.1 represents a DSSS receiver. Each section will now be
examined in the order of signal propagation.
Figure 3.1.Basic wireless communication system representation
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3.2 The Transmitter
The concept of spreading the data was explained in section 1.2 but the reception
of data occurs only after reliable acquisition. Hence, only the preamble is transmitted
during the acquisition phase. Throughout this research, the data is switched off or can be
considered to be transmitting only +1. Figure 3.2 below illustrates the DSSS transmitter.
Figure 3.2.Block diagram of the DSSS transmitter
3.2.1 The Preamble
Acquisition at the receiver is basically measuring the strength of the correlation of
the stored preamble with the incoming signal. It is high when both the received and the
stored preambles are perfectly aligned. Hence, to obtain a better correlation, an
assessment of the Autocorrelation function is necessary. Figure 3.3 shows the
autocorrelation for preamble of a length of 128 chips. A zero mean random sequence of
anti-podal chips of length 128 are generated. An auto correlation is performed on the
sequence and the mean square is calculated. The sequence with the lowest mean square
for a 10,000 trials is selected as the preamble for the simulation.
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The magnitude of any autocorrelation is equal to the number of points in the
sequence [2]; hence, weaker signals can be acquired by increasing the length of the
preamble.
Figure 3.3. Autocorrelation for 128 length preamble for 10,000 trials
3.2.2 Timing Offset
To account for the timing offset the preamble is positioned at a random location
between a sequence of zeros. The receiver has to detect the preamble and accurately
determine the offset shown in figure 3.5. Since the data collection window discussed in
section 3.4.1 514 samples long, the length of the channel is considered 128*16*7 to
provide sufficient number of shifts for the window during the acquisition process.
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Figure 3.4. Modeling preamble offset
3.2.3Over Sampling
Since the aim of this research is to detect and synchronize the time of the
incoming preamble, the preamble is over-sampled 16 times before transmission as
described in Figure 3.5. Over-sampling allows for accurate measurement of “slip” in each
chip during acquisition. In this case 8 samples = ½ chip.
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Figure 3.5. Preamble and its sampling
3.3 The Channel
The receiver is expected to detect the signal, and estimate the timing offset and
the drift in frequency due to the Doppler shift. The timing offset is modeled in the
transmitter design and the channel is modeled for AWGN and Doppler.
3.3.1 Modeling AWGN Noise
Next, to account for the thermal and the environmental noise, additive zero-mean
white Gaussian noise (AWGN) is added to the entire sequence. When designing optimum
0 200 400 600 800 1000 1200 1400 1600 1800 2000-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Plot of Preamble (128 chips) sampled at 16 samples/chip
Time
Am
plitu
de
Samples
24
receivers, AWGN is consistently used as the first approximation in modeling wireless
communication systems [4].
Since the SNR at the receiver is of interest, the channel is modeled to obtain the required
statistics at the receiver. Equations 3.1 to 3.5, shows the computation of the noise
variance used to model the AWGN. Figure 3.6 shows the addition of noise to the signal.
,
10 …………… 3.1
, 2
2
2 … … … … … 3.2
, 1,
……………3.3
The chip energy is normalized to 1 and only the noise energy is varied. Each chip
sample in the channel can be represented as the sum of the Gaussian variable and a DC
signal as it is shown in equation 3.4 [4].
… … … … … 3.4
:
:
: [4]
25
Hence, to model an AWGN channel, a zero mean ramdom valued sequence equal
to the length of the channel is multiplied with the noise variance . Two independent
sequences are computed to account for the real and imaginary parts of the noise. Figure
3.6 describes the modeling of AWGN channel.
Figure 3.6. Modeling AWGN noise
3.3.2 Modeling Doppler Effects
The channel was then modeled for Doppler effects during propagation. Any shift
in the frequency greater than ± ½ a cycle over the length of the preamble cannot be
26
detected by the receiver correlator. Also, the Doppler frequency shift is assumed to be
constant throughout the channel.
The following set of equations explain the computation of sample and chip timing.
128
∆ 1 ⁄
16 ⁄
∆ 128 ⁄
∆ 16 128 2048 ⁄
1
16 128 488.28
1
1287.8125
1
Furthermore the modeling of Doppler was determined my simulation. To do this,
a range of Doppler frequencies were tested on a given length of preamble to determine
the point at which the signal cannot be recovered (see Appendix B). Then, different
Doppler frequencies were chosen near the limits of operation. Equations 3.6, 3.7 and 3.8
show the mathematical expressions for Doppler addition. Figure 3.7 below shows the
modeling of Doppler shift to an AWGN channel and the resulting phasor rotation in
successive samples.
1 … … … … … 3.6
i.e. 1
27
0:∆
:
1 , 1
… 3.8
Figure 3.7. Modeling Doppler
Now, the channel is asynchronous with AWGN noise and has a Doppler unknown
to the receiver.
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3.4 The Receiver
The initial task of the receiver is to estimate and correct the Doppler shift i.e., the
receiver has to search for the exact frequency before acquiring the signal. The linear
correlation technique was used for both Doppler detection and acquisition. Both the
transform domain and the time domain models are presented. In this section, some
important concepts leading to acquisition are discussed followed by the actual design.
Figure 3.8 shows a block diagram of the receiver.
Figure 3.8 Block diagram of acquisition at the receiver
3.4.1 Cross Correlation
The difference between transform domain acquisition and time domain
acquisition is fundamentally the mechanism of correlation. Hence, we will analyze
correlation techniques for both domains.
The preamble is transmitted only once and the receiver is expected to acquire it.
The receiver has to employ a linear correlation in order to achieve an acquisition under
this circumstance. Processing 1 iteration of circular convolution of a 128 point sequence
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in frequency domain is equivalent to processing 128 iterations to achieve circular
correlation in time domain. Since we are interested in linear correlation, 128 iterations of
linear correlation in time domain can be obtained in frequency domain by 1 iteration of
zero-padded circular convolution. The zero padding helps to recover data samples which
would otherwise be aliased due to the circular convolution process [13].
Few authors, such as Jing Pang [15] and Thomas Brown [13], have discussed and
prototyped their frequency domain acquisition in this method. But, most researchers
perform only a circular correlation due to the periodic nature of the spreading sequence
under consideration [1, 2, 6, 10 and 16].
Linear correlation is implemented easier in the time domain. However, further
computations are necessary while operating in the transform domain. Let us first examine
the time domain correlation.
“The cross-correlation sequence for two wide-sense stationary random process, x (n) and y (n) is:
where * denotes the complex conjugate and the expectation is over the ensemble of
realizations that constitute the random processes” [12]. Figure 3.9 shows an illustration
of the linear correlation in the time domain. The data collection window is 128 samples.
Each time the received samples are correlated with the stored preamble and the
correlation peak is measured against the pre-determined threshold. Then, the window is
shifted 1 sample and the process is repeated until the peak is strong enough to declare a
detection.
30
Magnitude of real and imaginary parts are computed and then
compared with the threshold. and where is the arbitrary
channel phase when the Doppler shift frequency .
Figure 3.9. Time domain linear correlation
“The circular correlation over all time shifts between two sequences can be
implemented by the circular convolution of one sequence with a time reversed version of
the other sequence” [13]. In the transformed domain, circular correlation is achieved by
multiplying the FFT of the received window with the complex conjugated (time reversed)
FFT of the stored preamble and then, by taking IFFT of the product. If the preamble is
transmitted periodically, then the circular correlation is enough to obtain acquisition. In
our case, since the preamble is transmitted only once, linear correlation has to be used.
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The vector overlap save method is a simple and effective method to get linear correlated
samples from circular correlated sequences. This method is based on overlap save
convolution for FIR filters [14]. The idea is to consider a window larger enough to
process a lengthier point FFT and excise linear correlated bits. But first, we need to
calculate an optimum window that is larger than 128 points and is long enough not to
alias the required linear correlated bits. Computations and figure 3.10 illustrates the
algorithm.
L = 128 (Length of the preamble)
N = Smallest integer power of 2 > Length of 2 * Preamble in chips (2*L)
N = 4*L
N = 512 (Length of the window)
Figure 3.10. Extraction of linear correlated bits from circular correlation
32
Hence, if we have to perform a linear correlation on a 128 bit sequence in the
transform domain, the minimum length of the receiver window has to be 512 samples.
The received signal is segmented into overlapping windows of 512 samples and
transformed into frequency bins by FFT. Then, the following steps are undertaken to
obtain a 128 point linear correlation:
Step 1. The stored preamble is zero padded to 512 samples and transformed to
frequency samples by FFT.
Step 2. A complex conjugate is taken to account for the time reversal.
Step 3. Take FFT of length -512 windowed samples
Step 4. Both the sequences in frequency domain are point wise multiplied.
Step 5. The product is transformed back to time domain using IFFT
Step 6. Magnitude is computed. This accounts for the in-phase and
the quadrature-phase detection.
Step 7. The first 128 samples are extracted: this is the required 128 point linear
sequence of correlated samples.
Step 8. The window is shifted by 128 samples and the process is repeated
Figure 3.11 below illustrates the steps.
33
Figure 3.11. Transform domain linear correlation
The samples collected from the receiver window are separated into in-phase and
quadrature-phase channels and correlated. A magnitude of both the channels is computed
for each sample.
3.4.2 Threshold
After the correlation process, the correlated peaks are compared against a pre-
determined threshold. If the magnitude is greater than the threshold, it will be considered
a correct detection. Experimental justification for the selection of threshold is provided in
34
Appendix C. Any value above this threshold is recorded on a map as show in Figure 3.13
and 3.14.
3.4.3. Quantization
The term quantization is used to describe mapping a continuous set of values to a
relatively finite set of discrete vales. In a practical receiver an analog-to-digital converter
(ADC) performs the mapping [4]. In our case since the samples are already discrete, we
examined 2 different quantization intervals (see Appendix D) and choose to proceed with
4bit quantization. The quantization interval is set using the dynamic range which depends
on the SNR. We assume that is known for simulation purposes. All the receive
signal samples. The data collected at the receiver window are quantized and saturated
using the dynamic range to 4 bits before de- multiplexing.
The following steps are used to carry out the quantization:
Step 1. Calculate the dynamic range by using the equation
3
Step 2. Calculate the quantization interval using
2
Step 3. Round off each value in the received window to the nearest multiple of Q
(Saturation)
Step 4. Using the interval , map all the value in the window as shown
,
35
3.4.4. Doppler Estimation
The first signal processing unit in the receiver is a Doppler estimator. This resolves
frequency uncertainty and also confirms coarse acquisition. Detection is defined for only
± ½ chip of the true correlation (see Section 2.6). Hence, the Doppler search is conducted
only 1 chip over the possible acquisition.
The figure 3.13 below illustrates the design for Doppler estimation. The received
signal is de-multiplexed to 16 channels as shown in Figure 3.12 and processed through
cross-correlators, depending on the domain of operation as discussed in Section 3.4.1.
The window is 128 samples long for the time domain and 512 samples for the transform
domain Doppler estimation.
Figure 3.12. De-Multiplexing to 16 channels and Rx window
The windowed data is correlated with Doppler shifted copies of the stored
preamble. A high peak correlation is observed when there is a close match in the received
and stored Doppler shifted preambles. This concept is sometimes called Ambiguity
36
Function [2]. Once the
best estimate is found, the system adjusts the Rx local oscillator frequency to resolve the
Doppler Effect by choosing the stored preamble with the Doppler shift of the closest
Doppler to perform a fine acquisition and timing estimation.
Figure 3.13. Doppler frequency estimation
3.4.5 Signal Detection and Timing Estimation
Once the Doppler shift has been estimated, the receiver proceeds to acquire the
signal. Doppler estimation also gives us the coarse acquisition, i.e. the location of the
37
signal within the search-range of a chip. This signal processing unit will find the exact
location of a signal and approximate the offset previously modeled in the channel.
The stored copy of the preamble with the nearest estimated Doppler is duplicated
16 times and parallel processed with the data collected from the Rx window for signal
detection. Data collection for the Rx window and the correlation, work according to the
domain in which they are processed as discussed in section 3.4.1. The correlated values
are weighed against the threshold and a map of correlated peaks is developed. The
acquisition unit then searches for a particular pattern as show in Figure 3.13 to confirm
the detection. Furthermore, the timing offset is determined by identifying the location of
shift in the peak values on the map as shown below.
Figure 3.14. Signal acquisition and timing estimation
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The overall process of de-multiplexing the received digital signal to 16 Doppler
offset channels, preparing the stored preambles, correlating and checking for the strength
of the correlation-peak against the threshold values, plotting the maps and searching them
for Doppler and timing acquisition constitutes the process of signal acquisition. The
domain of operation determines the receiver window and the correlation method.
Simulations and results are discussed in the next chapter.
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CHAPTER 4: SIMULATION RESULTS
This chapter presents the results obtained using simulations. Using examples
helps comprehend the results better.
4.1 Channel Simulations
Let us consider the following setup shown in Figure 4.1.
Figure 4.1. Example of channel offset
Preamble: 128 Chips sampled at 16 samples/ chip
Channel: 128 × 16 × 7 = 14336 samples
Offset: 500 samples
Figure 4.2.Transmitter output
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Now, let us trace each step in the design described in chapter 2 and observe the
developments. The second step was to introduce AWGN and Doppler shift to the
channel. Figure 4.3 shows the simulated plot of simulated waveform at Ec/No = 2 dB and
Doppler f = 0.625 Hz.
Figure 4.3. Plot of received waveform at SNR = 2 dB
4.2 Receiver Simulations
The receiver simulations can be discussed in two parts, the time domain and the
transform domain. Results for both Doppler estimation and timing detections will be
presented. Before acquisition, the received signal is quantized. Four-bit quantization is
modeled and, the simulated received waveforms before and after quantization are shown
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below in figures 4.5 and 4.6. Furthermore, acquisition results show that 4 bits, i.e. 24
quantization levels, are sufficient to successfully acquire signals with required accuracy.
Figure 4.4. Received waveform before quantization
Figure 4.5. Received waveform after quantization
4.2.1 Transform Domain Simulations
Two important results are worth mentioning here, the correlation map of the
Doppler search unit and the correlation map of the timing detection. In Figure 4.7, peak
correlation occurs at “Channel” 14, which corresponds to 0.625 Hz (see Appendix D), the
initial Doppler introduced into the channel.
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Figure 4.6. Map of Doppler correlation, SNR = 5 dB
Figure 4.8 is the map to correlation peaks obtained by frequency domain correlation
using the Doppler corrected preamble.
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Figure 4.7. Map of timing estimation and acquisition
Since a fairly high SNR (5 dB) is considered, the probability of detection is 100%.
To analyze the design and its performance, the model was simulated for 1000 trials over a
range of SNRs (-35 dB to 5 dB). Figure 4.9 shows the probability of detection and false
alarm with respect to SNR. Also, this graph gives us a measure of Transform Domain
Signal Acquisition. Figure 4.10 on the other hand, shows a plot of timing error with
respect to SNR. This graph plots the number of errors made by the algorithm over the
range of SNRs. It is evident that, at the region of 100% detection which is about -10dB,
the errors are almost zero.
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Figure 4.8. Probability of detection and false alarm. ∆f = 0.625 Hz
Figure 4.9. Timing error (Transform Domain Acquisition)
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4.2.2 Time Domain Simulations
Similar to the transform domain, the time domain simulations are shown below. First, the
Doppler correlation map is shown in figure 4.11 for CNR = 5dB, f = 0.625 Hz and τ
500 samples.
Figure 4.10. Doppler correlation map, Time domain, CNR = 5 dB, Peak at channel 14
Next, the map of correlated peaks obtained by time domain correlation using the
Doppler corrected preamble is shown in Figure 4.12.
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Figure 4.11. Correlation map of signal acquisition and timing detection
Similar to transform domain, a probability of detection and false alarm versus
CNR graph and a timing error graph are shown in Figures 4.13 and 4.14. Again, the
timing error is almost zero at about -10 dB in Figure 4.14 corresponding to the 100%
detection region in Figure 4.13.
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Figure 4.12. Probability of detection and probability of false alarm, at 0.625 Hz Doppler.
Figure 4.13. Timing Error (Time Domain Acquisition)
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CHAPTER 5: CONCLUSION
This chapter discusses the results obtained in chapter 4 and concludes the thesis
with a note of future work that can be conducted.
It can be clearly observed from Figures 4.9 and 4.13 that the preamble of length
128 chips can be detected at SNR= -10 dB with 100% accuracy. A Doppler of ± 1Hz can
be successfully acquired and any random timing offset that might occur (e.g. drift in
individual clock) can be accurately determined using this design.
Also, from Figures 4.8 and 4.12 it is evident that the transform domain technique
attains 100% accurate detection at -11dB which is an advantage of 2 dB over the time
domain (which attains 100% detection at -9 dB.)
This design is a framework to develop further DSSS-based signal acquisition models.
It can be observed that almost all the parameters in this design are generic. A few
parameters of interest are:
Length of the preamble
Modulation technique
Sampling rate
Timing offset
Doppler shift
Number of parallel processing channels
By varying one or all of these generic values in the model, a completely different set
of transform domain and time domain performance curves can be analyzed.
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Since this design has been developed from first principles, no part of the design
assumes or uses data from any other research for the simulations. Hence, further
complications can be easily added and studied.
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APPENDIX A: DOPPLER ANALYSIS
In this section, an experimental analysis is presented to corroborate the analytical
description of Doppler modeling. The maximum amount of Doppler that can be
recovered for a preamble of length 128 chips is investigated. It is obvious form section
3.3.2 that ∆f = +1 cycle /preamble is the maximum Doppler frequency that can be
introduced to the channel model. Figures C.1, C.2 and C.3 show the detection curves for
various Doppler frequencies. It is clear from C.3 that at values near to 1 cycle/preamble,
the receiver will stop detecting the signal justifying the Doppler selection.
Figure A.1. ∆f = 0.1 cycles/preamble
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Figure A.2. ∆f = 0.6 cycles/preamble
Figure A.3. ∆f = 0.9 cycles/preamble
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APPENDIX B: THRESHOLD ESTIMATION
This section provides the approach and justification to the computation of the pre-
determined threshold.
The threshold is has to be set to an optimum CNR above the noise floor to
minimize the false alarm and maximize the probability of detection. Reducing the
threshold below this optimum level would improve detection at the cost of increased false
alarms. The threshold is normally set at 0.01 Pfa [6]. To choose this, the signal is turned
off, i.e. the Preamble is not transmitted and the Probability of false alarm is simulated for
various thresholds. The value at which Pfa = 0.01 at -10 dB CNR (which is the optimum
CNR for the Pd) is selected as the pre-determined threshold. After series of simulations
the threshold value of 120 was achieved. Form Figure C.1 it can be observed that, at -10
dB CNR, the threshold is about 10 dB above the noise floor.
Figure B.1. Correlation plot at -10 dB CNR
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APPENDIX C: QUANTIZATION
This section presents the simulated results with different quantization intervals. Two
quantization intervals 8 bits and 4 bits were simulated for comparison. Figure C.1 shows
the receiver input with no quantization, 8 bit quantization and 4 bit quantization.
Figure C.1. Receiver inputs at 5 dB CNR with no quantization, 8 bits and 4 bit
quantization.
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APPENDIX D: CHANNELS
Diagrammatic justification for channel 14 corresponding to 0.625 Hz is provided in
Figure D.1
Figure D.1.Assignment of pre-determined preambles for the 16 channels in steps of
0.125 Hz
