1 Blind Modulation Classification: An Idea Whose Time Has Come Octavia A. Dobre Assistant Professor [email protected] Faculty of Engineering and Applied Science Memorial University of Newfoundland Canada
1
Blind Modulation Classification: An Idea Whose Time Has Come
Octavia A. DobreAssistant Professor
Faculty of Engineering and Applied ScienceMemorial University of Newfoundland
Canada
2
Outline
Blind Modulation Classification (MC) :Problem Formulation
Approaches to MCLikelihood-Based (LB) ApproachFeature-Based (FB) Approach
Spatial Receive Diversity for MC
Conclusion
Ongoing and Future Work
3
MC: Problem Formulation
•Preprocessing Tasks: signal bandwidth and carrier frequency estimation, signal and noise power estimation, carrier, timing and waveform recovery, compensation for fading and interferences, etc.
System Block Diagram
Transmitted Signals Channel
Interferenceand Jamming
Receiver Noise
++ SignalPreprocessing
Demodulation
ModulationformatClassification
AlgorithmIntelligent Receiver
Signal Detection & Separation
4
MC: Problem Formulation (cont’d)Requirements for the Modulation Classification Algorithm
Capability to recognize many different modulations in different types
of environments,
High classification performance for low SNR in a short observation
interval,
Rely less on preprocessing,
Robustness to non-ideal conditions, such as carrier frequency offset,
timing errors, residual channel effects,
Real-time functionality and low complexity.
5
Classification Approaches
Approaches to MC
Likelihood-Based (LB) Approach
Requires computation of the likelihood function (LF) of the received signal.Likelihood ratio tests (LRTs) are used for decision-making.
Feature-Based (FB) Approach
Features common to different modulations are used, and the decision is made based on their differences. Such features are selected in an ad-hoc way.
6
Likelihood-Based (LB) Approach
7
LB Approach
iHu
( )( )
( ) ( )( ) ( )
1
1 1 1
2 2 22
11
2 2
| ,|| | ,
H
H H HAA
A H H HH
p H p dHH p H p d
>Ξ= γ
<Ξ∫∫
r u u urr r u u u
( )( )
( )( )
1
11
22 2
11
2 2
max ; ||| max ; |
H
H
HH
GG
G HH
p HHH p H
>Ξ= γ
<Ξu
u
r urr r u
( )( )
( ) ( )( ) ( )
1
1 1 1 11 1
2 2 2 21 2 2
1 2 1 2 21
2 1 2 2 2 2
max ; | ,|| max ; | ,
H
H
HH H H H
HH
H H H H HH
p H p dHH p H p d
>Ξ= γ
<Ξ
∫∫
u
u
r u u u urr r u u u u
the detected signal has the ith modulation format,i=1,…,Nmod .
:iH
is the vector of unknown quantities, i.e., unknown data symbols and parameters (frequency, phase and timing offsets, etc.).
MC is a multiple composite hypothesis-testing problem
Example: Nmod =2LB
Approach
ALRTALRT(Average LRT)(Average LRT)
GLRT GLRT (Generalized LRT)(Generalized LRT)
HLRT HLRT (Hybrid LRT)(Hybrid LRT)
8
The signal samples (taken at the symbol rate) at the output of the receive matched filter are used to compute the LF.
Signal Model
LB Approach (cont’d)
( )( )( ) ( ), 0= − + ≤ ≤∑j iSCLD k Tk
r t e s u t kT w t t KTϕα
We have mostly investigated classification of Single Carrier Linear Digital Modulations (SCLD), in AWGN and Block-Fading Channels,
under the assumption that Waveform recovery, Timing recovery, and Compensation for the Carrier Frequency Offset are performed in the preprocessing step.
Received Baseband Signal
9
LB Approach: ALRTComplexity
( )
2( ) ( ) 2
1
2[ | ] exp Re[ ] | |ik
Ki ij
A AWGN i k ksk
TH E e R sN N
−−
=
Ξ = − ∏r ϕα α
( )1
2( ) ( )
0
2[ | ] | | expi Kk k
i iA CP i K Ks
TH E IN N=
−
Ξ = Ψ −
r α α η
[Abdi, Dobre, Choudhry, Bar-Ness, and Su, 2004]
( )iM KO
( )1[ ]i K
i k ks ==uUnknowns , AWGN
( )1
1 1( ) ( ) ( ) 2
2
| |[ | ] 1 exp 1 .i Kk k
i i iK K K
A Rayleigh i s
T TH EN N N=
− −
−
Ω Ω Ω Ψ Ξ = + +
r η η
( )( )
1,
Kii
K kkR
=Ψ = ∑
0 *
0
( ) ( )mod
( 1)( ) ( ) , 1,..., , 1,..., .
kTi i
k kk T
R r t s t dt k K i N−
= = =∫
( )( ) 2
1| | ,
Kii
K kks
== ∑ηwhere Mi is the number of points in signal constellation,
2[ ]EΩ = α is the average fading power and
( )KiMO( ) †
1[ ]i Ki k ks == α ϕuUnknowns , Rayleigh Fading
( )KiMO
( ) †1[ ]i K
i k ks == ϕuUnknowns , AWGNThe phase ϕ is uniformly distributed over [-π, π).
10
LB Approach: ALRT (cont’d)Remarks on ALRT for MC
With no unknown parameters in AWGN (ideal case), the ALRT-based classifier represents a benchmark, against which performance of other classifiers is compared.
When increasing the number of unknown parameters, the computation of the LF becomes very difficult, even mathematically intractable. Thus, in many cases of practical interest, the ALRT-based classifier becomes impractical.
The ALRT-based classifier requires a priori knowledge of the distribution of the unknown parameters.
In addition, it usually results in structures that may not be applicable to environments other than the ones assumed.
The ALRT-based classifier provides an optimum solution, in the sense that it minimizes the probability of misclassification.
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LB Approach: GLRT and HLRT
( ) †1[ ]i K
i k ks == ϕuUnknowns , AWGN
( )* ( )1 2 21
[ | ] = Re[ ] 2 | | .K i ijG CP i k k kk
H s r e T s− ϕ −− =
Ξ − α∑ (i)ks
r max maxϕ
( )* ( )1 2 1 21
[ | ] exp 2 Re[ ] | | .K i ijH CP i k k kk
H N s r e TN s− − ϕ −− =
Ξ = α − α ∏ (i)ks
r Emaxϕ
GLRT [Panagiotou, Anastasopoulos, and Polydoros, 2000]
HLRT [Panagiotou, Anastasopoulos, and Polydoros, 2000]
12
LB Approach: GLRT and HLRT (cont’d)
Remarks on GLRT and HLRT for MC
Averaging over data symbols in HLRT removes the nested constellations problem of GLRT.
GLRT and HLRT do not depend on the distribution chosen for the unknown parameters (usually, with HLRT, average is performed over unknown symbols only).
GLRT displays some implementation advantages over ALRT and HLRT, as it avoids the calculation of exponential functions and does not require the knowledge of noise power to compute the LF.
However, with GLRT maximization over data symbols can lead to equal LFsfor nested signal constellations, e.g., 16-QAM and 64-QAM, which in turn leads to incorrect classification [Panagiotou, Anastasopoulos, and Polydoros, 2000] .
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LB Approach: HLRT (cont’d)
12 ( )21[ | ]
|| || (1 )
Ki
KM
H Rayleigh i m i Ki m
KHeM− =
Ξ = −
∑rrπ ρ
Complexity ( )KiMO
( ) ( ) ( )| | /(|| || || ||)i H i im m m= ⋅r s r sρ
Unknowns , Block-Fading Channel( ) †1[ ]i K
i k kN s == α ϕu
HLRT [Dobre and Hameed, 2006]
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LB Approach: HLRT (cont’d)
Further Remarks on HLRT for MC
Low-complexity estimators can be used instead,leading to the so-called Quasi-HLRT classifiers.
With several unknown parameters, HLRT is not a good solution either, as it suffers of high computational complexity .
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LB Approach: QHLRT( ) †
1[ ]i Ki k kN s == α ϕu
( )
( )ˆ ( )( ) 2
,1 ( ) ( )1 1 ˆ[ | ] exp | | ˆ ˆ
i
ik
K ii jQH Rayleigh i k k mk s i iH E r e s
N Nϕ
− = Ξ = − −α
∏rπ
( )1 42 2 1 2( )( ) 242 21
( )
ˆ ˆ2ˆ E[| | ]2
iiki
M Ms
b
− −α = − 1 22 2
( ) 42 2121 ( )
ˆ ˆ2ˆ ˆ2
ii
M MN M
b
−= − −
( ) 1- 1
ˆ arg( )iK MiiM PSK kkM r−
=ϕ = ∑
With Method-of-Moments (MoM) estimates of the unknown parameters, the LF is
Unknowns , Block-Fading Channel
QHLRT [Dobre and Hameed, 2006]
( ) 1 4- 1
ˆ 4 arg( )KikM QAM k r−
=ϕ = ∑
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LB Approach: QHLRT (cont’d)HLRT Versus QHLRT in Rayleigh Fading, K=10 (BPSK/QPSK)
0 5 10 150.65
0.7
0.75
0.8
0.85
0.9
0.95
1
SNR (dB)
Ave
rage
P cc
ALRT (Perfect estimates)HLRT (ML estimates)QHLRT (MoM estimates)
5.25 dB 8.5 dB 9 dB
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LB Approach: QHLRT
Remarks on QHLRT for MC
The QHLRT-based classification algorithm is less complex, applicable to any distribution of the unknown parameters, yet providing a good classification performance.
Methods for joint parameter estimation, less complex, yet accurate, need to be devised.
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Carrier frequency offset, ∆fc
BPSK, QPSK, 8-PSK, 16-PSK
LB Approach: Sensitivity Analysis
( ) ( ) ( )( )1
Kj iSCLD k Tk
r t e s u t kT w tϕ=
= α − +∑2π cj ∆f te
( )1[ ]i K
i k ks ==uUnknowns , AWGN
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
∆fcT
Ave
rage
Pcc
K=50K=100K=300
SNR=12dB
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LB Approach: Sensitivity Analysis (cont’d)
Synchronization errors, ε
SNR=12dB
( )1[ ]i K
i k ks ==uUnknowns , AWGN
( ) ( ) ( )( )1
Kj iSCLD k Tk
r t e s u t kT w tϕ=
= α − − +∑ εT
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
ε
BPSK, QPSK, 8PSK, 16PSK
K = 50K = 100K = 300
Ave
rage
Pcc
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Remarks on LB for MC
The LB-based classifier is sensitive to model mismatches, such as carrier frequency and timing errors. Here we have presented the individual effect of model mismatches on the classification performance. Apparently, the performance will degrade further under the cumulative effect of model mismatches.
LB Approach: Sensitivity Analysis (cont’d)
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LB Approach: Conclusion
Remarks on the LB Approach to MC
The ALRT-, GLRT- and HLRT- based algorithms suffer of highcomputational complexity.
The QHLRT-based classification algorithm is less complex, applicableto any distribution of the unknown parameters, yet providing a good classification performance.
The LB-based classifier is sensitive to model mismatches, such ascarrier frequency and timing errors.
The LB-based classifier is not suitable to classify different modulationtypes, such as digital against analog modulations.
Methods for joint parameter estimation, less complex, yet accurate,need to be devised.
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Feature-Based (FB) Approach
23
FB Approach
The features show unique characteristics for every specific modulation.
Decision-making is based on the difference of the features for diversemodulations.
Feature Extraction
Decision -
Making
Feature-Based Classifier
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FB Approach (cont’d)
FB Method
InstantaneousAmplitude, Phase, andFrequency
Signal Statistics:Moments, Cumulants,Cyclic Cumulants (CC)
WaveletTransform
Information in the Zero-Crossing
Sequence
Examples of features:
- Euclidian distance between estimated and prescribed values of the features, - Correlation between estimated and theoretical features,
- The probability density function of a feature estimator.
Examples of decision criteria:
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We have explored signal cyclostationarity for MC.
FB Approach: Signal Cyclostationarity
Why Signal Cyclostationarity?
- Cyclic statistics (CS) provide supplementary
information through the cycle frequency domain.
- CS-based features can be used to classify a large
number of modulation types.
- CS-based features can be robust to model mismatches.
26
Signal Cyclostationarity: Definitions
Exploitation of Signal Cyclostationarity for MC
FB Approach: Signal Cyclostationarity
Signal Cyclostationarity
for Modulation Classification (MC)
27
Signal Cyclostationarity: Definitions
FB Approach:Signal Cyclostationarity (cont’d)
28
FB Approach: Signal Cyclostationarity - Definitions
Cyclostationary Signals• Definition: A stochastic process r(t) is said to be cyclostationary of
order n (for a given conjugation configuration, i.e., q conjugate) if its cumulants up to order n (assuming they exist) are (almost)-periodic functions of time.
The nth-order/q-conjugate moments are also (almost)-periodic functions of time.
• Time-Variant and Cyclic Statistics [Spooner and Gardner, 1994]
Time-varying nth-order/q-conjugate cumulant
The nth-order/q-conjugate cyclic cumulant(CC) the cycle frequency (CF) β(Used in Our Work)
The (nth-order) cycle frequencieswhere is the delay-vector.
,
2, ,( ; ) (β; )
n q
j tr n q r n qc t c e πβ
κ
= ∑τ τ
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, ,2
1( ; ) ( ; )limT
j tr n q r n q
T T
c c t e dtT
− πβ
→∞ −
β = ∫τ τ
, , | ( ; ) 0κ = β β ≠τn q r n qc†1 1[ ... ]nτ τ −=τ
29
FB Approach: Signal Cyclostationarity - Definitions
Interrelationships between time-, cyclic-, and frequency-domain
FT
FS
( ) ,;τr n q
c t
( ) ,;fr n q
P t
( ) ,;β fr n q
P
( ) ,r n qc β;τ
FS
FT
↔τ f
↔τ ft ↔ β
t ↔ β
The nth-order/q conjugate time-varying cumulant (q conjugations)
The nth-order/q conjugate cyclic cumulant (CC) at cycle frequency (CF) β
The nth-order/q conjugate time-varying cumulant polyspectrum
The nth-order/q conjugate cyclic cumulant polyspectrum (CP) at CF β
Cyclostationary Signals (cont’d)
( ) ,;fr n q
P t
( ) ,;β fr n q
P
( ) ,;τr n q
c t
( ) ,;β τr n q
c
30
FB Approach: Signal Cyclostationarity (cont’d)
Exploitation of
Signal Cyclostationarity
for Modulation Classification (MC)
31
FB Approach: Signal Cyclostationarity for MC (cont’d)
Exploitation offirst-order
cyclostationarity
Exploitation of second-order
cyclostationarity
OFDMExploitation of higher-order
cyclostationarity
Exploitation of higher-order
cyclostationarity
AM (detection ofa single CF)
M-ary FSK(detection of
M CFs)
GOAL:DEVELOP A GENERAL CLASSIFIER BASED ON SIGNAL CYCLOSTATIONARITY
CP-SCLD SCLD
Modulation Type
OFDM, CP-SCLD, SCLD
(no first-order CFs)
Modulation Type
SCLD: Single Carrier Linear Digital Modulation
CP-SCLD: Cyclically Prefixed SCLD
OFDM: Orthogonal Frequency Division Multiplexing
FSK: Frequency Shift Keying
AM: Amplitude Modulation
32
Signal Models
FB Approach: Signal Cyclostationarity for MC (cont’d)
( )2 ( )2( ) ( ) ( )i
kc dj f s t kT Tj tjFSK k
fr t e e e g t kT T w tπ − −εϕ π∆α ε= − − +∑
is complex zero-mean additive Gaussian noisetime delay
w(t)t0
is the symbol period
T
denotes the modulation format iis the signal powerα
is the pulse shape µA modulating indexis the zero-mean modulating signal convolved with Rx filter impulse responseis the symbol transmitted within the kth period, with values drawn from an alphabet AMFSK=±1, ±3,…, ±(M-1)
g(t)m(t)
is the carrier frequency offset∆fcis the timing offsetε
is the frequency deviationfdis the phase
ϕ
( )iks
02( ) (1 ( )) ( )cj t
A Afj
Mr t e e m t t w tπ∆ϕα= + µ − +
( )iks
ϕ
33
Signal Models (cont’d)
SCLD2 (∆ ) ε( ) ( ) ( )
∞π
=−∞
ϕα= − − +∑cj t ik
k
fjr t e e s g t k w tT T
FB Approach: Signal Cyclostationarity for MC (cont’d)
is the symbol transmitted within the lth symbol period of block b, with modulation i
is the symbol index within a blocklis the block indexb
is the number of symbols in the cyclic prefix
Lis the number of information data symbols in a block
N
( ) ( ) ( ) ( ) ( ). 1 , ,1 ,
cyclic prefix information data symbols
[ ]i i i i ib b N L b N b b Ns s s s− +=s L L
144424443 1442443
12 ( )
CP-SCLD0
∆,( ) ( ( ) ) ( )
∞ + −π
=− =
ϕ
∞
= − + − −α ε +∑ ∑cN L
j tj ib l
b
f
lr t e e s g t b N L T lT w tΤ
( ),i
b ls
34
Signal Models (cont’d)
OFDM
12 2 ∆ ( )∆
,0
ε( )( ) ( ( )ε )c VV
j t j kv f t kT Tj iu k
k v
fr t e e s e g t k w tT T∞ −
π π − −
=−∞
ϕ
=
= − − +α ∑ ∑
FB Approach: Signal Cyclostationarity for MC (cont’d)
frequency separation between two adjacent subcarriers
∆fVnumber of subcarriers
V
is the symbol transmitted over the kth symbol interval and vth subcarrier,with i denoting the modulation type on each subcarrier.
symbol period; OFDM: T = Tu+Tcp, Tu=1/∆fV and Tcp is the cyclic prefix.T
( ),i
v ks
The signals are oversampled at the receive-side.
35
First-order Cyclic Cumulant (CC) of Discrete-time Signals
FB Approach: Signal Cyclostationarity for MC (cont’d)
1 21,0( ) − πγ ε− ϕαβ = s
FSK
j f Tjrc e eM
1 1 11,0
1
,..., ( 1 [ 1/ 2,1/ 2) | , ,
if
)
.
− − −
−
κ = β∈ − β = + γ γ =∆ = ± ± −
=
FSKsc s
d
f pT f
f l
p l
T
f l M
The first-order CC of the other signals, i.e., SCLD, CP-SCLD, and OFDM, equals zero. In other words, there are no first-order cycle frequencies.
Discriminating Signal Feature: Number of Detected First-Order Cycle Frequencies [Dobre, Rajan, and Inkol, 2007]
1,0( ) = ϕαβAM
jrc e
11,0 [ 1/ 2,1/ 2) | AM
scf f −∆κ = β∈ − β =
[Dobre, Rajan, and Inkol, 2007]
M-FSK: M first-order cycle frequencies.
AM: A single first-order cycle frequencies.
36
SNR=20dB 1 sec observation interval
FB Approach: Signal Cyclostationarity for MC (cont’d)
2-FSK
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Candidate cycle frequency, β'
|cr^(
K)( β
' )1,
0|
8-FSK1,01| ( ) | , 1, 2
FSKr Mc M−α α == =β1,0
1| ( ) | , 1, 8FSKr Mc M−α α == =β
1 1 1s sc f T ff − − −β = ±∆ 1 1 1, 1, 3, 5, 7s sc f pT ff p− − −β = + = ± ± ± ±∆
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
Candidate cycle frequency, β'
|cr^(
K)( β
' )1,
0|
First-order CC of Discrete-time Signals (cont’d)
37
SNR=20dB1 sec observation interval
FB Approach: Signal Cyclostationarity for MC (cont’d)
2-PSK
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Candidate cycle frequency, β'
|cr^(
K) ( β
' )1,
0|
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Candidate cycle frequency, β'
|cr^(
K)( β
' )1,
0|
AM 1,0
1
| ( ) | = , =1AMr
c sf
c
f −
β α
β ∆=
α2 1,0| ( ') | =0
any 'PSKrc
−β
β
First-order CC of Discrete-time Signals (cont’d)
38
Second-order/ One-conjugate CC of Discrete-time Signals
SCLD
21 22 2
2,1 2,,2, 11( ; ) ( ) ( ) ( ; )π
τ− − πβ ρ ∗ − πβρ
∆εβ τ = ρ + τ + β τα ∑
cj Tj j m
r wm
f
scc e e m m cg eg
FB Approach: Signal Cyclostationarity for MC (cont’d)
SCLD 12,1κ [ 1/ 2,1/ 2) | , is an integer and is the oversampling factorl l−= β∈ − β = ρ ρ
CP-SCLD
21 2 * 2
2,1
1
21 2 *
2,2,1
2,2
2,1 2
2,1
( ) ( ) ( ; ) ,
for delays around 0 and , integer
[( ) ] ( ) ( ) ( ; )( ; )
π ∞− τ− − πβ ρ − πβρ
=−∞
−
π− τ
− − πβ ρ − πβρ
∆ε
∆ε
ρ + τ + β τ
τ = β = ρ
+ ρ − ρ − ρ − ρ +
α
α τ + β τβ τ =
∑c
c
f
s
f
s
j Tj j m
wm
j Tj j m
wr
e e m m e c
k k
N L e e m l m l
c g g
c g g N e cc
2
1
,2,1
,10
1
2 11 2 * 2
2,10
1
,
for delays around and [( ) ] , integer
[( ) ] ( ) ( ) ( ; ) ,
for delays around and [( ) ] ,
− ∞
= =−∞
−
π − ∞− τ− − πβ ρ − πβρ
= =−
−
ε
∞
∆
τ = ρ β = + ρ
+ ρ − ρ − ρ + ρ + τ + β τ
τ = −ρ β = + ρ
α
∑ ∑
∑ ∑cf
L
l m
Lj Tj j m
wl
sm
N b N L b
N L e e m lc g g m l N e c
N b N L b integer
[Dobre, Zhang, Rajan, and Inkol, 2008]
39
Second-order/ One-conjugate CC of Discrete-time Signals(cont’d)
The additional factor yields significant peaks in the second order/one conjugate CC at delays equals to
( )Ξ τV
, int e .eg rτ = ρv V v
FB Approach: Signal Cyclostationarity for MC (cont’d)
OFDM
21 22
,2,2
,1 112 2,( ; ) ( ( ) () ) ( ; )∆
επ
τ− − πβ ∗ − πβρ Ξβ τ = + τ + β τα τ ∑
c ufj Tj D j mV
Vr wm
sc D e e m g m e cc g
OFDM 12,1 [ 1/ 2,1/ 2) | , κ is an integer and is the oversampling factor lD l D−− β =β∈=
2 ( 11
0
) sin( / )( )sin( / )
π− τρ
=
π− τ
ρ= =πτ ρ
Ξ τπτ ρ∑
j VV
V j vV
Vv
eeV
40
FB Approach: Signal Cyclostationarity for MC (cont’d)
The magnitude of second-order / one-conjugate CC versus cycle frequency and delay (in the absence of noise) for a) SCLD and b) CP-SCLD signals.
a) b)
β τ
SCLD
2,1
|(
;)
|rc
βτ
β τC
P-SC
LD2,
1|
(;
)|
rcβ
τ
Second-order/ One-conjugate CC of Discrete-time Signals (cont’d)
41
FB Approach: Signal Cyclostationarity for MC (cont’d)
The magnitude of second-order/ one-conjugate CC versus cycle frequency and delay (in the absence of noise) for OFDM signals.
c)β τ
OFD
M2,
1|
(;
)|
rcβ
τ
310−×
Second-order/ One-conjugate CC of Discrete-time Signals (cont’d)
42
Second-order/ One-conjugate CC of Discrete-time Signals (cont’d)
FB Approach: Signal Cyclostationarity for MC (cont’d)
• The second-order/ one-conjugate CC of the SCLD signals is non-zero only for delays around zero. This differs from the case of CP-SCLD and OFDM signals, in which non-zero values are also obtained for delays around and , respectively, due to the existence of the cyclic prefix (CP).
• For the SCLD signals, peaks in the CC magnitude appear at zero and CFs for delays around zero. The CC and CFs of the CP-SCLD signals are the same as for SCLD signals for delays around zero. For the OFDM signals, the CC magnitude at non-zero CFs and zero delay is close to zero.
• For the OFDM signals, the peaks in the CC magnitude which appear at delays around , are at CFs integer multiples of . For the CP-SCLD signals, such peaks appear at delays around and CFs integer multiples of .
N± ρ V±ρ
1−±ρ
N±ρ1[( ) ]N L −+ ρ
1D−V±ρ
43
τ
CP-
SCLD
2,1
ˆ |(0
;)
|rc
τ
b)
FB Approach: Signal Cyclostationarity for MC (cont’d)
τ
OFD
M2,
1ˆ |
(0;
)|
rcτ
c)
τ
SCLD
2,1
ˆ |(0
;)
|rc
τ
a)
The magnitude of estimated second-order / one-conjugate CC versus positive delays, at zero CF and 0 dB SNR, for a) SCLD, b) CP-SCLD, and c) OFDM signals.
SCLD against CP-SCLD and OFDM
Second-order/ One-conjugate CC of Discrete-time Signals (cont’d)
44
FB Approach: Signal Cyclostationarity for MC (cont’d)
β
CP-
SCL
D2,
1ˆ |
(;0
)|
rcβ
a)β
OFD
M2,
1ˆ |
(;0
)|
rcβ
b)
The magnitude of estimated second-order (one-conjugate) CC over the cycle frequency domain, at zero delay and 0 dB SNR, for a) CP-SCLD and b) OFDM signals.
CP-SCLD against OFDM
Second-order/ One-conjugate CC of Discrete-time Signals (cont’d)
45
CC-Based Classification Algorithm (Example: SCLD against CP-SCLD and OFDM )
Step 1:
Step 2: A cyclostationarity test [Dandawate and Giannakis, 1995] is used to check whether or not zero is indeed a CF for the delay selected in Step 1.If is found to be a CF, then we decide that the signalbelongs to the class OFDM and CP-SCLD; otherwise, we declare it as SCLD.
0=β
Over the considered delay range, we select that delay value for which the CC magnitude reaches a maximum.
The magnitude of the second-order/ one-conjugate CC of the baseband received signal is estimated at zero CF and over a range of positive delay values.
FB Approach: Signal Cyclostationarity for MC (cont’d)
46
FB Approach: Signal Cyclostationarity for MC (cont’d)
Test to Verify the Presence of a Cycle Frequency [Dandawate and Giannakis 95]
0 :H the tested candidate CF ' 0β = is not a CF.
the tested candidate CF1 :H ' 0β = is a CF.
Steps (test applied for zero CF):Step 1:
( ) ( )2,1 2,2 1,1 ,ˆ ˆ: [Re ( '; ) Imˆ ( '; ) ]K K
r rr c c= β τ β τcEstimation of the CC at candidate CF, β’, from K samples,
and calculation of( )2,1ˆ ( '; )K
rc β τ1 †
,2,1 ,2,1 ,2,1 ,1,1 ,22ˆ ˆˆ ˆ , where is the estimate ofr r r rK −= c cT Σ Σ
2,0 2,1 2,0 2,1,2,1
2,0 2,1 2,0 2,1
Re ( ) / 2 Im ( ) / 2, with
Im ( ) / 2 Re ( ) / 2r
Q Q Q Q
Q Q Q Q
+ − = + −
Σ
( ) ( )2,0 2,1 2,1ˆ ˆ: lim Cum[ ( '; ) , ( '; ) ]K K
r rKQ c c
→∞= β τ β τ
( ) *2,1 , 2,1ˆ ˆ: lim Cum[ ( '; ) , ( '; ) ]K
r n q rKQ c c
→∞= β τ β τ
Step 2: Calculate
47
FB Approach: Signal Cyclostationarity for MC (cont’d)
Test to Verify the Presence of a Cycle Frequency (cont’d)Step 3:
2,1 ≥ ΓT
The statistic is compared again a threshold for decision-making.
If
⇓One decides that the tested candidate CF is indeed a CF for the selected delay.
Otherwise it is not declared a CF.
Threshold Setting: 2,1 0Pr | FP H= ≥ ΓT
2,1T has an asymptotic 2χ
distribution with two degrees of freedom under 0.H
48
Simulation Setup• SCLD and CP-SCLD modulations: BPSK, QPSK, 8-PSK, 16-QAM and 64-QAM.
• The transmit filter is root-raised cosine, with 0.35 roll-off factor. The signal bandwidth is 40 kHz.
• We simulate unit variance constellations.• For OFDM signal generation we use raised cosine windowing function with 0.025 roll-off factor. The number of subcarriers is set to 128, the bandwidth to 800 kHz, the useful time period to 160 µs and the cycle prefix to 40 µs.
• The observation interval available at the receiver is 0.1 and 0.05 seconds, respectively.
• At the receive-side, a low-pass filter is used to eliminate the out of band noise.• The oversampling factor per symbol per subcarrier is set to 4. The sampling frequency for SCLD and CP-SCLD is set to 160 kHz, whereas to 3.2 MHz for OFDM.
• The phase θ is uniformly distributed in the interval [- π, π). The carrier frequency offset is set to , and the timing offset to 1 0.1c sf f −∆ = =0.75.ε
FB Approach: Signal Cyclostationarity for MC (cont’d)
49
SN R (dB)
Probability of correct classification
Classification Performance
FB Approach: Signal Cyclostationarity for MC (cont’d)
50
FB Approach: Signal Cyclostationarity for MC (cont’d)
Remarks
We are currently investigating the effect of the number of processed symbols on the classification performance, i.e., what is the minimum number of symbols required to achieve a certain performance at a given SNR.
In addition, we are extending the algorithm(s) to time-dispersive channels, and looking into the complexity of the algorithm(s). Results for classifying OFDM against SCLD in time dispersive channel has been already reported [Dobre, Punchihewa, Rajan, and Inkol, 2008].
51
We have investigated higher-order cyclic cumulants (CCs) of the baseband signal at the output of the receive filter as features to classify SCLD (CP-SCLD) signals in AWGN and block fadingchannels.
FB Approach: Signal Cyclostationarity for MC (cont’d)
First- and second-order cyclostationarity cannot be used to classify SCLD (CP-SCLD) signals (see results presented in slide #40).
Higher-order CCs have particular properties, which make them attractivefor MC, e.g.,- tolerance to stationary noise- CC-based features are robust to phase and timing offsets.
Why Higher-Order Cyclic Cumulants?
52
( )2 ( )( ) ( )∆= − − +∑cj tj iSCLD kk
fr t e e s g t kT T w tπϕα εSignal Model
FB Approach: Signal Cyclostationarity for MC (cont’d)
,[ 1/ 2;1/ 2) | ( 2 ) / , / , integer, ( ) 0 γ ∈ − γ = β + − ρ β = ν ρ ν γ ≠∆ τSCLDc r n qn q T cf ;
The Cycle Frequencies are given by
†
1 0[ ,..., ]
nn ==τ
ττ τwhere ν is an integer and is the delay-vector.
(
1
)1
,
2 ( )1 2 ( 2 ),
(*) 2
,
1
( )
( ) ,
−=
π ρ − τ− − πβ ρ −
−
∆
β=
ε
π
ϕα ∑γ = ρ
× + τ∑ ∏
τn
uci
uu
SCLD
u
fns
j Tj j n qr n q
n j mum u
n qc e e e
m e
c
g
;
/( ) ( ) | = ρ= t mTr m r tThe nth-order Cyclic Cumulant (q conjugations, q=0,…,n) of the discrete-time signal (no aliasing condition)
[Dobre, Bar-Ness, and Su, 2003]
53
We need to choose the:
-- order norder n-- number of conjugations qnumber of conjugations q-- delaydelay--vector vector ττ-- cycle frequency cycle frequency ββ
to achieve the best discrimination capability for a specific pool of modulations.
FB Approach: Signal Cyclostationarity for MC (cont’d)
The nth-order/ q-conjugate CC Magnitude
( )(*)1 2
, 1, ,| ( ) | | | ( )p
SCL iD
n j mr n q pp
nn q ms
c mg ec − − πβ=
γ = ρ τα +∑ ∏τ;
is robust to a time-invariant phase and timing errors.
54
FB Approach: Signal Cyclostationarity for MC (cont’d)
-111.8464
-111.8464
-111.8464
-111.8464
-111.8464
8.32
8.32
8.32
8.32
-1.36
-1.36
-1.36
1
1
4ASK4ASK
-92.018
-92.018
-92.018
-92.018
-92.018
7.1889
7.1889
7.1889
7.1889
-1.2381
-1.2381
-1.2381
1
1
8ASK8ASK
-33
0
0
0
1
4
0
0
0
-1
0
0
1
0
8PSK8PSK
-34
0
34
0
-34
4
0
-4
0
-1
0
1
1
0
QPSKQPSK
-272
-272
-272
-272
-272
16
16
16
16
-2
-2
-2
1
1
BPSKBPSK
-33
0
0
0
0
4
0
0
0
-1
0
0
1
0
16PSK16PSK
-11.5022-13.7862-13.9808cc8484
000cc8383
-11.5022-3.8446-13.9808cc8282
000cc8181
-11.5022-1.9926-13.9808cc8080
1.79722.11002.08cc6363
000cc6262
1.79720.57002.08cc6161
000cc6060
-0.619-0.6900-0.6800cc4242
000cc4141
-0.619-0.1900-0.6800cc4040
111cc2121
000cc2020
64QAM64QAM32QAM32QAM16QAM16QAM
Cumulants of the Normalized Noise-Free Signal Constellations
55
FB Approach: Signal Cyclostationarity for MC (cont’d)
=τ 0
(*)1 2, 1
( , , ) ( )pn j mn q pm p
F r g m e− − πβ=
β = ρ + τ∑ ∏τ
CC magnitude dependency on the delay-vector and roll-off factor
• maximum reached at zero delay vector.• maximum increases with the roll-off factor.• maximum decreases as increasing . β
56
Selected Features
γ=1/ρ+(n-2q)∆fcT /ρ(discrimination distance decreases with an increase in CF)
Cycle frequencyCycle frequency
τ=08 (the features reach maximum)DelayDelay--vectorvector
- q=0,…,8* q=n/2
Number of Number of conjugationsconjugations
- n=8 (M-ASK, M-PSK (maximum order M=8), M-QAM
* n=4,6,8 (Rectangular QAM)
Order Order
( )(*)1 2
, 1, ,| ( ) | | | ( )p
SCL iD
n j mr n q pp
nn q ms
c mg ec − − πβ=
γ = ρ τα +∑ ∏τ;
FB Approach: Signal Cyclostationarity for MC (cont’d)
-Robust to carrier phase, ϕ, and timing errors, ε.
*Also robust to carrier frequency offset ∆fc and phase noise [Dobre, Bar-Ness, and Su, 2004].
57
A feature vector is formed for each i =1,…, Nmod and saved in a look-up table.
( )iϒ
Estimates of the signal amplitude and pulse shape are needed to compute these features. We assume perfect (error free) estimates.
An estimate of the feature vector is computed for the receivedsignal from Kρ samples (K symbol length observation interval) taken at the output of the receive filter.
ϒ
mod
( )
1,...,arg min ( , )i
i Ni i d
== ϒ ϒChoose as the received modulation if
where d(.,.) is the Euclidian distance.
Decision Criterion
FB Approach: Signal Cyclostationarity for MC (cont’d)
58
FB Approach: Signal Cyclostationarity for MC (cont’d)
The CC-based features have the capability of classifying a largenumber of modulations. In addition, these have the advantage ofrelying less on preprocessing, being robust to carrier phase and timingerrors (see, for example, classification of M-ASK, M-QAM, and M-PSKsignals).
First-order cyclostationarity can be used to classify AM, FSK, and SCLD,CP-SCLD, and OFDM (as a signal class), second-order cyclostionarity to distinguish between SCLD, CP-SCLD, and OFDM signal classes, andhigher-order cyclostationarity to identify signals within SCLD and CP-SCLD signal classes.
The CC-based features are also robust to carrier frequency offset andphase noise when classifying rectangular QAM constellations [Dobre, Bar-Ness, and Su, 2004].
When higher-order statistics are employed, a large observationinterval is required to obtain accurate feature estimates.
Remarks on CC-based Features for MC
59
Spatial Receive Diversity for Modulation Classification
60
Spatial Receive Diversity for MC
( ) ( ; ) ( ), 0 , 1,2,...,ir t s t w t t KT L= + ≤ ≤ =ul l l l
Let us consider an L branch antenna array.
Signal and noise models
• Independent AWGN processes from branch to branch.• Independent fading among L branches.
11
je ϕα
LjLe
ϕα
.
.
.
GOAL: improve the performance of classifiers inGOAL: improve the performance of classifiers infading channels with multiple receive fading channels with multiple receive antennas.antennas.
• Channel amplitudes and phases are constant over K symbols (block fading model).
61
LB and Spatial Receive Diversity
( )( ) ( )2 2,1 11
2[ | ] exp Re | | .ik
K L Li ij
A Array i k ksk
TH E e R sN N
−−
= ==
Ξ = − ∑ ∑∏r ll l l
l l
ϕα α
( )1[ ]i K
i k ks ==uUnknowns , AWGN
[Abdi, Dobre, Choudhry, Bar-Ness, and Su, 2004]
( )1 1 1[ ]iL L K
i k ks= = == α ϕu l l l lUnknowns , Rayleigh Fading
The Maximal Ratio Combiner is used to combine the replicas of the received signal.
( )( ) ( )2 2,1 11
2[ | ] exp Re | | ,ik
K L Li ij
QH Array i k ksk
TH E e R sN N
−−
= ==
Ξ = − ∑ ∑∏r l)
l l ll l
) )ϕα α
When there are unknown parameters, e.g., , integration over the unknown parameters becomes more difficult for the array classifier.
1 1 and L L= =l l l lα ϕ
62
QHLRT and Spatial Receive Diversity (cont’d)
Effect of Number of Antennas
1 2 3 4 5 0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
AWGN
Rayleigh fading
16QAM, 32QAM, 64QAM
Ave
rage
Pcc
AWGN (perfect estimate, ALRT)Rayleigh fading (perfect estimate, ALRT)Rayleigh fading (MOM estimate, QHLRT)
Number of antennas, L
SNR=12dB
500 symbols
Average P
cc
63
QHLRT and Spatial Receive Diversity (cont’d)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
Correlation coefficient between two branches
Pcc
16QAM, 32QAM, 64QAM, N=500 symbols, 1000 trials for each modulation, L=2
Rayleigh fading (perfect estimate, SNR = 15 dB, ALRT)Rayleigh fading (perfect estimate, SNR = 10 dB, ALRT)Rayleigh fading (MOM estimate, SNR = 15 dB, QHLRT)Rayleigh fading (MOM estimate, SNR = 10 dB, QHLRT)
10dB
5dB
The correlation coefficient between two branches
*12 1 2[ ]E z z=ς
11 1 ,jz e= ϕα 2
2 2jz e ϕα=
are two zero-meancomplex Gaussian variables.
where
Effect of the Correlation among the Antennas
Average P
cc
64
-infinity 5 10 150.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
Rice factor (dB)
Pcc
16QAM, 32QAM, 64QAM, N=500 symbols, 1000 trials for each modulation, SNR=10 dB
Rice fading (perfect estimate, L=1, ALRT)Rice fading (perfect estimate, L=2, ALRT)Rice fading (perfect estimate, L=4, ALRT)Rice fading (MOM estimate, L=1, QHLRT)Rice fading (MOM estimate, L=2, QHLRT)Rice fading (MOM estimate, L=4, QHLRT)
QHLRT and Spatial Receive Diversity (cont’d)
L=1
L=2L=4
Effect of the Rice Factor
SNR=10dB
500 symbols
Average P
cc
L=1
L=2L=4
65
With an antenna array at the receiver, the complexity of the LBclassifier increases further.
By adding only a second antenna, a large performance improvement is achieved in Rayleigh fading.
Further improvement is obtained by using more antennas.
The multi-antenna QHLRT classifier is reasonably robust to correlations among branches, as well as the Rice factor.
QHLRT and Spatial Receive Diversity (cont’d)
Remarks on Multi-Antenna ALRT and QHLRT for MC
66
Multi-Antenna CC-based Classifier
' ( ) ( )SCr m r m= l1
' arg maxL≤ ≤
= ηll
l
ηlwhere is the received SNR on each branch.
1' ' 1
( ),
2 ( )2 ( 2 )( ) 1, ' , ,
(*) 21
( )
( )
nc u ud u
iSC
u
j f Tj j n qi nr n q s n q
n j mum u
c c e e e
p m e
−=
π∆ ρ − τ− πβ ε ρ − ϕ−
− πβ=
∑γ = α ρ
× + τ∑ ∏
τ l l
l;
Signal at the output of the Selection Combiner (SC)
The nth-order CC of the signal at the output of the SC
Feature Vector : the same as for the single antenna case.
mod
( )
1,...,arg min ( , )i
SC SCi N
i i d=
= ϒ ϒChoose as the received modulation ifDecision Criterion
ϒ
CC-based Classifier and Spatial Receive Diversity
[Dobre, Abdi, Bar-Ness, and Su, 2005]
67
Single- and Multi-Antenna CC-based Classifiers in Rayleigh Fading
By using two and four two antenna elements with a CC-SC classifier, we get a 4dB and 7dB SNR improvement to attain Pcc=0.9, respectively.
0 5 10 150.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR(dB)
Ave
rage
P
cc
CC-SC, L=4CC-SC, L=2CC-SA
118 15
4dB
7dB
L=1
L=2
L=4
4000 symbols
CC-based Classifier and Spatial Receive Diversity (cont’d)
• 4ASK, 8ASK, BPSK, QPSK, 8PSK, 16PSK, 16QAM, 32QAM, 64QAM
68
Correlation coefficient between two branches
*12 1 2[ ]E z z=ς
11 1 ,jz e= ϕα 2
2 2jz e ϕα=
are two zero-meancomplex Gaussian variables.
where
As expected, the Pcc decreases when the correlation coefficient increases. The performance degradation seems to be less at high SNRs. As can be noticed from the flat portion of these curves, the array classifiers appear to be reasonably robust to some possible correlations that may exist between the branches.
CC-based Classifier and Spatial Receive Diversity (cont’d)
0 0.2 0.4 0.6 0.8 10.75
0.8
0.85
0.9
0.95
1
ς12
Ave
rage
P
cc
CC-SC, L=2, SNR=15dBCC-SC, L=2, SNR=10dB
SNR=15dB
SNR=10dB
4000 symbols, L=2
Effect of the Correlation among the Antennas
69
It is noteworthy to mention the significant performance enhancement by adding only one extra antenna, i.e., L=2 comparing to L=1 particularly at low values of the Rice factor (For K=0 and ∞, Rice fading reduces to Rayleigh fading and no fading, respectively.)
L=1
L=2
L=44000 symbols
Effect of the Rice Factor
CC-based Classifier and Spatial Receive Diversity (cont’d)
-infinity 5 10 150.75
0.8
0.85
0.9
0.95
1
Rice factor (dB)
Ave
rage
Pcc
CC-SC, L=4CC-SC, L=2CC-SA, L=1
70
By adding only a second antenna, a large performance improvement is achieved in fading channels.
Further improvement is obtained by using more antennas.
The multi-antenna CC-SC classifier is reasonably robust to some level of correlation that may exist between branches and to the Rice factor.
CC-based Classifier and Spatial Receive Diversity (cont’d)
Remarks on Multi-Antenna CC-based Classifier
71
ConclusionLB Approach to MCQHLRT seems to be the approach to follow, as it is relatively simple to implement, yet providing a good classification performance.
However, it suffers of sensitivity to model mismatches, such as timing errors.
It cannot be applied to a large pool of modulation types, e.g., analogagainst digital modulations.
FB Approach to MCCC-based MC algorithms are robust to carrier phase and timing errors, and applicable to a large pool of modulations. Higher-order CCs used to discriminate SCLD signals require a large observation interval for accurate estimations.
Spatial Diversity for MCWith multiple receive antennas and proper combining at the receive side, performance of the LB and FB classifiers improves.
The price is the increase in complexity.
72
Ongoing and Future Work
Cyclostationarity-based Approach to MCInvestigation of new criteria of decision, which lead to maximumprobability of correct classification.
Extension to more complex environments (an extension of the algorithm to identify OFDM against SCLD in time-dispersive channels has been already developed).
Study of the algorithm complexity.
Extension of the cyclostationarity-based MC algorithm to identify other modulation types, such as CPM.
Study of the minimum number of symbols required to achieve a certain probability of correct classification at a given SNR.
Ongoing Work
73
Ongoing and Future Work (cont’d)
FB Approach to MC
Investigation of other signal features, such as wavelet transform.
Handle new classification problems raised by the emerging wireless technologies, such as classification of signals received from single and multiple transmit antennas, identification of space-time modulation format, etc.
Future Work