On the Classification of Linearly Modulated Signals in ...

On the Classification of Linearly Modulated Signals in Fading Channels O. A. Dobre1, J. P. Zarzoso1, Y. Bar-Ness1 and W. Su2 1Center

for Communications and Signal Processing Research Department of Electrical and Computer Engineering New Jersey Institute of Technology, Newark, New Jersey 2HQ

US Army CECOM Forth Monmouth, New Jersey 03/19/04

CISS, Princeton University, NJ

Outline  Automatic modulation classification (AMC): problem formulation  Decision-theoretic (DT) approach to AMC  Hybrid likelihood ratio test (HLRT) for classification of linear modulations in fading channels  Simulation results  Conclusion 03/19/04

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Automatic modulation classification (AMC): problem formulation System model Signal model Performance measure

03/19/04

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AMC: problem formulation  System model Input symbols Modulator

Interference and jamming Noise

Channel

Output symbols

Preprocessor

+

Demodulator

+

Classification algorithm

Modulation format

 Preprocessing tasks may be: Estimation of signal power, symbol period, etc. Carrier, timing and waveform recovery Compensation for fading and interferences

 Classification approaches: Decision-theoretic (DT) Pattern-recognition (PR) 03/19/04

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AMC: problem formulation (cont’d)  Baseband signal model r  t   s (t ; ui ,0 )  n  t 

s (t ; ui ,0 )  a0e

j 2f 0t

e

j 0 0 ( t ) 

N

(i ) s  k p0 t  (k  1)T0   0T0  k 1

ui ,0 : [a0 f 0 0 0 (t ) T0 0 p0 (t ) {sk(i ) }]† vector of unknown parameters. 0

is the timing offset

p0(t)

is the pulse shape

a0

is the amplitude

f0

is the carrier frequency offset

0

is the phase offset

sk(i )

0(t)

is the phase jitter

i

denotes the modulation format of the incoming signal

T0

is the symbol period

N

is the number of observed symbols

n t 

is aggregate noise: receiver noise, cochannel interference and jamming

is the symbol transmitted within the kth period

Rectangular M-QAM

sk( M QAM )  sk( ,MI QAM )  jsk( ,MQ QAM ) ,

M-PSK

sk( M  PSK )  e jk ,

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sk( ,MI QAM ) , sk( ,MQ QAM ) {1, 3,..., ( M 1/ 2  1)}

 k  2 m M , m  0,...M  1 CISS, Princeton University, NJ

AMC: problem formulation (cont’d)  Performance measure i  1...Nmod

is the number of candidate modulations (modulation library) . denote the candidate modulations.

Pc( i| j )

is the probability to declare the modulation i when modulation j is present.

N mod

Confusion matrix

i

Example: BPSK versus QPSK

BPSK ( BPSK | BPSK ) c

BPSK

P

QPSK

Pc( BPSK |QPSK )

QPSK Pc(QPSK | BPSK )

Pc(QPSK |QPSK )

Average probability of correct classification

Pcc(1,..., Nmod ) 

1 N mod



N mod i 1

Pc(i|i )

We seek algorithms which provide high classification performance for low SNR in a short observation interval. 03/19/04

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Decision-theory (DT) approach to AMC

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DT approach to AMC Within the decision theoretic (DT) framework AMC is a multiple composite hypothesis-testing problem, and the maximum likelihood principle is used to solve it.

H i : the incoming signal has the ith modulation format

i=1,…,Nmod .

The conditional likelihood function NT  2     1 * (r (t ) | ui , H i )  exp  Re   r (t ) s (t; ui )dt      N0 0  N0

NT

 0

 s (t; ui ) dt   2

ui : [a f  (t ) T  p(t ) {sk(i ) }]† 03/19/04

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DT approach to AMC (cont’d) The likelihood function can be obtained by: Averaging over unknown parameters (ALRT-Average Likelihood Ratio Test). Using ML estimates (MLE) of the unknown parameters (GLRT-Generalized Likelihood Ratio Test). A combination of the afore-mentioned methods (HLRT-Hybrid Likelihood Ratio Test). Log-likelihood

ratio

test

for

a

(e.g., BPSK vs. QPSK):

two

hypothesis

problem

Hi

 (r (t ))   (r (t )) (i )

( j)







 is a threshold.

Hj

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HLRT for classification of linear modulations in fading channels

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HLRT  Signal model Slowly-varying flat fading:

r  t    0 2S0 e

j0

N

(i ) s  k uT0 t  (k  1)T0   n t  k 1

0   0 e j0 is fading-induced multiplicative noise

0 , 0

S0

uT0 (t ) n(t )

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are the channel amplitude and phase, respectively

is the signal power is the rectangular pulse shape is additive white Gaussian noise (AWGN) with a two-sided power spectral density N0

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HLRT (cont’d)  Log-likelihood function N

 (r (t ))  max max ln  E s ( i ) (i ) H



ui  [  {sk(i ) }]†



k=1

k

  2 S0 S0T0 2 (i ) 2     ( i )*  j  exp  2 Re r s e  2  | s |       k k k N N  0   0   

vector of unknown parameters.

   + E s( i ) {.}

denotes averaging with respect to input data signal symbol.

k

rk : 

kT0

 k 1T0

r (t )uT0 (t  (k  1)T0 )dt output of the matched filter at time t=kT0.

 Decision rule Hi

 (Hi ) (r (t ))   (Hj ) (r (t ))





H

Hj

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Simulation results HLRT, N=100 symbols BPSK vs. QPSK

BPSK vs. QPSK 1

1

ALRT, AWGN

0.95

0.95

P

(BPSK,QPSK)

0.85 0.8

HLRT, Rayleigh fading

0.65 0.6 0.55

HLRT, Fading HLRT, Fading HLRT, Fading ALRT, AWGN ALRT, AWGN ALRT, AWGN ALRT, Sensitivity

cc

N=50, N=100, N=300, N=50, N=100, N=300, N=100,

0.7

0.9

0.85

0.8

P

0.75

cc

(BPSK,QPSK)

0.9

0.75

0.7 Ricean fading, K=10dB Rayleigh fading

ALRT, sensitivity to Rayleigh fading

0.5 -5

0

E{ 2}SNR (dB) E /N(dB) s

0

5

10

0.65 -5

0

2

(dB)(dB) E{ SNR }E /N s

5

10

0

The experimental results are obtained by performing a number of 300 Monte Carlo trials.

SNR : E{2 }S0T0 / N0 03/19/04

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Simulation results (cont’d) N=100 symbols i=BPSK, QPSK, 8-PSK, 16-PSK

i=16-QAM and 64-QAM

1

1

0.9

0.9

0.8

0.8

0.7

P (i |i ) c

0.6

c

P (i|i)

0.7

0.5

0.6 0.5

0.4 0.4

0.3

BPSK QPSK 8-PSK 16-PSK

0.2 0.1 -5

0

2

10

5

(dB) } E /N E{ SNR (dB) s

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0

0.3

15

0.2 -5

16-QAM 64-QAM 0

5

2

10

15

20

E{ SNR } E /N (dB) (dB) s 0

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Conclusion  ALRT ALRT results in an optimal classifier when the true probability density function of the unknown parameters coincides with the hypothesized one.

When the number of unknown parameters increases, only approximations of the optimal solution are possible, which leads to suboptimal classifiers [Polydoros et al., 90,94,95,96]. Thus, the difficulty of performing a multidimensional integration for a large number of unknown parameters and the need for prior knowledge can render the ALRT impractical. ALRT classifier may not be applicable to environments different from those for which the classifier is designed. 03/19/04

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Conclusion (cont’d)  HLRT Although the HLRT is suboptimal, compared with ALRT, it could be implemented with less complexity, achieving a reasonable performance. In addition, it has the advantage of applicability in different environments, e.g., Rayleigh and Ricean fading.

Moreover, the actual values of the unknown parameters are of interest for data demodulation.

03/19/04

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