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MAXIMUM-LIKELIHOOD MODULATION CLASSIFICATION FOR PSK/QAM

J.A. Sills

Signal Exploitation and Geolocation DivisionSouthwest Research Institute

San Antonio, Texas 78238-5166

ABSTRACT

This paper addresses automatic modulation classification for PSK

and QAM signals under coherent and noncoherent conditions. Inparticular, the paper extends previous results by treating the classi-fication of higher-state QAM signals. A maximum-likelihood al-

gorithm is presented for coherent classification of PSK and QAMsignals. We evaluate the algorithms performance for various PSKand QAM modulation types including 64-state QAM and then

compare it with a psuedo maximum-likelihood noncoherent clas-sification technique in terms of error rate, false alarm rate, and

computational complexity. The application of these results to thedesign and performance of an automatic signal recognizer is dis-

cussed throughout the paper.

1. INTRODUCTION

Automatic modulation recognition is a rapidly evolving area of

signal exploitation with applications in DF confirmation, monitor-ing, spectrum management, interference identification, and elec-

tronic surveillance. Generally stated, a signal recognizer is used toidentify the modulation type (along with various parameters suchas baud rate) of a detected signalfor the purpose of signalexploita-

tion. For example, a signal recognizer could be used to extractsignal information useful for choosing a suitable counter measure,such as jamming.

In recent years interest in modulation recognition algorithms

has increased with the emergence of new communication tech-

nologies. In particular, there is growing interest in algorithms thattreat quadrature amplitude modulated (QAM) signals, which areused in the HF, VHF, and UHF bands for a wide variety of appli-

cations including FAX, modem, and digital cellular.Many techniques for modulation identification have been pub-

lished in the literature. Early work in modulation identification isfound in a report by Weaver, Cole, and Krumland [1] in whichfrequency-domain parameters were used to distinguish between

six candidate modulation types. One of the well-known early pa-pers treating digital modulation types was by Liedtke [2] in which

he presents results based on a statistical analysis of various signalparameters to discriminate between amplitude shift keying (ASK),FSK, and PSK. Other techniques using signal parameters have

been reported in [3], [4], [5], [6], and [7]. A combination of tech-niques including pattern recognition are used in [8] and [9]. Sev-eral authors have applied techniques from higher-order statistics

that exploit cyclostationarity to identify modulation [10]. Still oth-ers have applied neural networks to the problem [11, 12]. A recent

This work was supported by the Advisory Committee for Researchand Development at Southwest Research Institute.

book by Azzouz and Nandi [13] gives more details on these andother recent techniques for modulation identification.

Another group of authors have applied techniques from max-

imum-likelihood (ML) decision theory to modulation identifica-tion. Kim et al. use a truncated series approximation of the like-

lihood ratio function for distinguishing a BPSK from an MPSK(M 4 ), but these results apply in low SNR only[14]. Extensionsto high SNR and 16-state QAM are presented in [15]. Sapiano

presents a PSK classification technique with improved sensitivityto parametric degradation [16]. Most recently, Boiteau presenteda comprehensive review of the literature on signal classification

and provided a generalized framework that does not require anyrestriction on the baseband pulse [17].

In this paper we extend previous results on maximum-likeli-hood classification for PSK/QAM by developing general solutions

for coherent and noncoherent classification of PSK/QAM signals

with an arbitrary number of signal states. Performance curves arepresented for both the coherent and noncoherent cases for variousmodulation types including 64-state QAM.

The paper begins with an introduction in Section 1. Section 2presents the signal modelfor PSK/QAMcommunications. The co-

herent classifier is presented in Section 3 along with performancecurves showing error rates and false alarm rates. The noncoherent

case is treated in Section 4. Performance curves are presented andthen noncoherent performance is compared with coherent perfor-mance. Section 5 contains conclusions and recommendations.

2. SIGNAL MODEL FOR PSK/QAM

We receive a signal r t = s t + n t , 0 t T where s t

is a signal emitted from a non-cooperative transmitter and n t

is additive white gaussian noise (AWGN) with a two-sided powerspectral density (PSD) of N

0

2

. The signal s t is represented by

s t = A t c o s !

c

t +

c

+ t = R e

A t e

j t

e

j !

c

t +

c

where A t and t are the modulated amplitude and phase, !

c

is the carrier frequency, and

c

is an unknown phase offset.

The received signal is one of N

candidate modulation types.Let the integers i = 0 1 ; : : : ; N , 1 enumerate the candidatemodulation types, such that m for i = 0 1 ; : : : ; N , 1 denotes

the event that the intercepted signal belongs to the i

t h modulationtype. We will assume equal a priori probabilities P m .

Let ~ s t = A t e

j t denote the complex envelope of s t .The complex envelope of a PSK/QAM signal can be expressed in

terms of

~ s t =

r

2 E

s

E

a

E

p

X

n

a

n

p t , n T

s

, t

d

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where a

n

is a sequenceof symbols taken from a set of M i com-plex numbers I m = f

1

2

; : : : ;

M

g , R

s

=

1

T

s

is thesymbol rate, and t

d

is an unknown timing offset. The pulse shape

p t is any of the standard symmetric types such as a square-rootraised cosine or a square pulse. The symbol energy is E

s

providedthat (1) we define

E

p

=

Z

1

1

p

2

t d t

and (2) wemodel a

n

as independentrandom variables with equally

likely assignment from the set I m such that

E

a

= E a

n

2

=

1

M i

M

X

n = 1

n

2

where E denotes the expected value operator. It is customary to

normalize the set f

n

g such that E

a

= 1 . The input signal-to-noise ratio (SNR) is defined by =

E

s

N

0

,

Each modulation type is characterized by its symbol configu-ration in the complex plane, which definesthe amplitude and phase

values for the set I m .

3. COHERENT ML CLASSIFICATION

Automatic signal classification is a rather difficult problem in com-

posite hypothesis testing since so many parameters are unknown:symbol rate R

s

; carrier frequency !

c

; carrier phase

c

; pulseshape p t ; SNR ; and timing offset t

d

. A common approach is to first

estimate the unknown parameters and then attempt to classify the

signal according to modulation type. Although estimating theseparameters is nontrivial, it is not impractical. There are a wide

variety of techniques for estimating the signal parameters some of which are given in [18].

In this section we evaluate the performance of coherent MLclassification in which all of the signal parameters are known. In

this case, the signal is classified by forming likelihood ratios fromthe demodulated matched-filter output

~ r

n

=

Z

1

1

r t e

j !

c

t +

c

r

2

E

p

p t , n T

s

, t

d

d t

= r

I n

+ j r

Q n

It follows that r

I n

=

p

E

s

a

I n

+ n

I n

and r

Q n

=

p

E

s

a

Q n

+

n

Q n

wherea

I n

= R e f a

n

g

anda

Q n

= I m f a

n

g

. The noisecomponents

n

I n

andn

Q n

are independent, zero mean, with vari-ance N

0

2

.

Given that the modulation type is m , the probability density

function (PDF) governing the demodulated symbols ~ r

n

can be ex-pressed in the form

p

~ r

r

I n

r

Q n

m = m =

1

M i

M

X

k = 1

1

p

2

e

r

I n

p

E

s

I k

2

r

Q n

p

E

s

Q k

2

2

2

where

I k

= R e f

k

g

and

Q k

= I m f

k

g

and =

N

0

2

. It is

worth noting that r

I n

and r

Q n

are not necessarily independent—for example consider 8-PSK and QAM-32 from the V.32 standard[19].

We represent N demodulated symbols in vector form:

~r = r

I

+ j r

Q

=

2

6

6

4

~ r

1

~ r

2

...~ r

N

3

7

7

5

The PDF governing ~r is given by

p

~r

r

I

r

Q

m = m =

N

Y

n = 1

p

~ r

r

I n

r

Q n

m = m

The coherent maximum likelihood (ML) classifier is simply a rule

for choosing among the candidate modulation types given ~r .Choose

m = m

k

if and only if

p

~r

r

I

r

Q

m = m

is maximum for i = k .We investigate the performance of the coherent classifier by

evaluating its error rate as a function of SNR for the following

PSK/QAM modulation types: (m

1

) BPSK; (m

2

) QPSK; (m

3

) 8-PSK; (m

4

) QAM-16; (m

5

) QAM-32; and (m

6

) QAM-64. ForPSK, the symbol configurations are well known. For QAM how-ever, there are many possiblities including rectangular and circular

configurations. We consider the rectangular configurationsdefinedby the V.32 and V.33 standards [19] and shown in Figure 1.

(a) BPSK (b) QPSK. (c) PSK-8.

(d) QAM-16 (e) QAM-32. (f) QAM-64.

Figure 1: PSK/QAM symbol configurations.

Figure 2 shows the performance in terms of probability of er-ror and false-alarm rate for the coherent classifier resulting from

1000 Monte-Carlo simulations. The performance indicatesthat the

coherent ML classifier makes less than one error in ten across allsix modulation types provided the SNR is greater than or equal to

10 dB.This performance represents the best possibleerror rate thatcan be achieved. This level of performance is unlikely in practice

due primarily to phase incoherence between the transmitter andreceiver; that is, the parameter

c

is rarely known when the SNRis 10 dB. In fact coherent carrier acquisition for high-state QAM

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requires SNR levels much larger than 10 dB. Nevertheless, Figure2 provides a benchmarkfor classification performance from whichto compare noncoherent techniques.

4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Es/No (dB)

P r o b a b i l i t y o f E r r o r

BPSKQPSKPSK−8QAM−16QAM−32QAM−64

(a) Error rate.

4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Es/No (dB)

P r o b a b i l i t y o f F a l s e A l a r m


(b) False alarm rate.

Figure 2: Coherent performance using N = 2 5 6 symbols.

4. NONCOHERENT PSUEDO-ML CLASSIFICATION

In this section we evaluate the performance of noncoherent MLclassification in which all of the signal parameters are known ex-cept the carrier phase

c

. In this case the demodulated symbol is

rotated by an unknown carrier phase

~ r

n

=

p

E

s

a

n

e

j

c

+ ~ n

n

In this case, the signal is classified by finding the amplitude ~ r

n

and the phase difference

n

=

n

,

n 1 m o d 2

where

n

= t a n

1

r

Q n

r

I n

Given the transmitted symbola

n

, the amplitude~ r

n

is a Ricean-

distributed random variable with PDF

p

~ r

n

r

n

a

n

=

r

n

2

e

r

2

n

+ E

s

a

n

2

2

2

I

0

r

n

p

E

s

a

n

2

where I

0

is the 0

t h -order modified Bessel function and

2

=

N

0

2

.The exactexpression for the PDF of the phase difference

n

is very complicated, but for sufficiently large SNR, it can be ap-proximated by a Gaussian:

p

n

n

a

n

a

n 1

1

p

2

n

e

n

2

2

2

n (1)

where

2

n

=

N

0

2

1

a

n

+

1

a

n 1

and = t a n

1

a

Q n

a

I n

,

t a n

1

a

Q n 1

a

I n 1

. The approximation given by (1) follows from the

approximation t a n

1

for small .For sufficiently large values of SNR, ~ r

n

and

n

are very

nearly independent, hence we can approximate their joint PDF by p

~ r

n

r a

n

a

n 1

p

~ r

n

r a

n

p

a

n

a

n 1

[20].We next apply the law of total probability to find the conditionaldensity function p

~ r

n

n

r

n

n

m = m . There are effi-cient ways to perform this calculation, but as a general expression

p

~ r

n

n

r

n

n

m =

M

X

n = 1

M

X

k = 1

p

~ r

n

r

n

a

n

m P a

n

p

n

n

a

n

a

k

m P a

n

a

k

Given N demodulated symbols, the PDF for the N , 1 -dimen-

sional decision vectors

R =

2

6

6

4

~ r

1

~ r

2

..

.~ r

N 1

3

7

7

5

P =

2

6

6

4

1

2

..

.

N 1

3

7

7

5

is

p

R P

R P m = m =

N 1

Y

n = 1

p

~ r

n

n

r

n

n

m

We select the modulation type that corresponds to the largest of the f p

R P

R P m = m g .

The performance of the psuedo-maximimum-likelihood mod-ulation classifier is illustrated in Figure 3. This figure shows theperformance in terms of probability of error and false-alarm rate

for the noncoherent classifier resulting from 1000 Monte-Carlosimulations. These results indicate that the noncoherent psuedo-

ML classifier makes less than one error in ten across the testedmodulation types provided the SNR is greater than or equal to 13dB. Comparing Figures 2 and 3, it is evident that the noncoherent

classifier exhibits a performance loss of approximately 3 dB.

A critical parameter in the design of a PSK/QAM recognizeristhe number of symbols that are used to decide between modulation

types. Using a large number of symbols in the likelihood-ratiotest reduces the probability of error and probability of false alarm;

hence, the one-error-in-ten performance can be sustained down tolower SNR. When fewer symbols are used,we require higher SNRfor this performance level.

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6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Es/No (dB)

P r o b a b i l i t y o

f E r r o r


(a) Error rate.

6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Es/No (dB)

P r o b a b i l i t y o f F a l s e A l a r m

BPSK

QPSKPSK−8

QAM−16QAM−32QAM−64

(b) False alarm rate.

Figure 3: Noncoherent performance (256 symbols).

5. CONCLUSIONS

In this paper we investigated automatic modulation classificationfor PSK and QAM signals. We presented a maximum likelihoodframework for both the coherent and noncoherent cases. This gen-

eral approach treated PSK and QAM up to an arbitrary numberof signal states. The performance of both the coherent and non-

coherent classifier was investigated for various modulation typesincluding 64 state QAM. Noncoherent performance exhibited a 3dB loss compared to coherent performance.

6. REFERENCES

[1] C.S. Weaver, C.A. Cole, R.B. Krumland, and M.L. Miller,

The Automatic Classification of Modulation Types by Pattern

Recognition, Standford Electronics Laboratories, TechnicalReport No. 1829-2, April 1969.

[2] F.F. Liedtke, Computer Simulation of an Automatic Classi-

fication Procedure for Digitally Modulated Communication

Signals with Unknown Parameters, Signal Processing, vol. 6,

pp. 311-323, 1984

[3] P.H. Halpern and P.E. Mallory, A Simple Method for Distin-

guishing Modulation Types, IEEE Trans. ASSP, vol. 30, pp.97-99, Feb. 1982.

[4] N.F. Krasner, Optimum Detection of Digitally Modulated

Signals, IEEE Transactions on Communications, vol. Com-30, pp. 885-895, May 1982.

[5] J.E. Hipp, Modulation Classification Based on Statistical

Moments, Milcom-86, vol. 2, pp. 20.2.1-6, 1986.

[6] J. Aisbett, Automatic Modulation Recognition Using Time-

Domain Parameters, Signal Processing 13, pp. 323-328,1987

[7] Y.T. Chan and L.G. Gadbois, Identification of the Modulation

Type of a Signal, Signal Processing, vol. 16, pp. 149-154,1989.

[8] L.V. Dominguez, J.M.P. Borrallo, J.P. Garcia, and B.R.

Mezcua,A General Approach to the Automatic Classifica-

tion of Radiocommunication Signals, Signal Processing, vol.

22, pp. 239-250, 1991.

[9] F. Jondral, Automatic Classification of High Frequency Sig-

nals, Signal Processing, vol. 9, pp. 177-190, 1985.

[10] W.A. Gardner and C.M. Spooner, Signal Interception: Per-

formance Advantages of Cyclic-Feature Detectors, IEEETransactions on Communications, vol 40, No. 1, January1992.

[11] A.K. Nandi and E.E. Azzouz, Modulation Recognition Using

Artificial Neural Networks, Signal Processing, vol. 56, pp.165-175, 1997.

[12] A. Bernardini and S.D. Fina, Optimal Decision Boundaries

for M-QAM Signal Formats Using Neural Classifiers, IEEE

Trans. on Neural Networks, vol. 9, no. 2, March 1998.

[13] E. Azzouz and A.K. Nandi, Automatic Modulation Recogni-

tion of Communication Signals, Kluwer, 1997.

[14] K. Kim and A. Polydoros, Digital Modulation Classification:

the BPSK v.s. QPSK case, Milcom 1988.

[15] C.S. Long, K.M. Chugg, and A. Polydoros, Further Results

in Likelihood Classification of QAM Signals, Proceedings of MILCOM-94, pp. 57-61, Fort Momounth, NJ, October 2-5,

1994.

[16] P.C. Sapiano and J.D. Martin, Maximum Likelihood PSK

Classifier , Milcom-96, pp. 1010-1014, 1996.

[17] D. Boiteau and C. Le Martret, A Generalized Maximum Like-

lihood Framework for Modulation Classification, Interna-tional Conference on Acoustics, Speech, Signal Processing,1998.

[18] J.A. Sills, A QAM Demodulator for Digital Wideband Ar-

chitectures, Technical Report, Southwest Research Institute,

1998.

[19] CCITT Blue Book , Vol. 8, 1988.

[20] J.K. Patel and C.B. Read, Handbook of the Normal Distribu-

tion, 2nd Ed., Marcel Dekker Inc., New York, 1996.

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