Blind Modulation Classification: An Idea Whose Time Has Come...Feature-Based (FB) Approach ... BPSK,...

1

Blind Modulation Classification: An Idea Whose Time Has Come

Octavia A. DobreAssistant Professor

[email protected]

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)

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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]

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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


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


( ) ( ) ( )( )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.

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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


( )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


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 ∑ ∑


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


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


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


|cr^(

K)( β

' )1,

0|

First-order CC of Discrete-time Signals (cont’d)

37

SNR=20dB1 sec observation interval


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


|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


|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


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


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


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


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−×


42



• 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)


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


44


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


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.


46


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


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


49

SN R (dB)

Probability of correct classification

Classification Performance


50


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.


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


,[ 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.


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


-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


=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;


-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


58


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


[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


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)


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.


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.


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


-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.


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

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