Automatic modulation recognition using time domain parameters

Signal Processing 13 (1987) 323-328 North-Holland

323

SHORT COMMUNICATION

AUTOMATIC M O D U L A T I O N RECOGNITION USING TIME D O M A I N PARAMETERS Janet AISBETT Electronics Research Laboratory, Defence Science and Technology Organisation, Department of Defence, GPO Box 2151, Adelaide, South Australia 5001

Received 19 November 1986 Revised 30 March 1987

Abstract. One of the important variables to be determined when monitoring unknown radio transmissions is the modulation type. Automatic recognition procedures based on time-domain parameters additional to the standard envelope and instantaneous frequency are proposed and results of initial simulations reported. Zusammenfassung. Wenn eine unbekannte Radiosendung angemessen wird, ist die Art der Modulation eine der wichtigen zu bestimmenden Grrl3en. Vorgeschlagen und anhand von Simulationen vorgestellt werden automatische Erkennungsverfahren, die fiber die Bestimmung der Einhiillenden und der Augenblicksfrequenz hinaus Parameter im Zeitbereich verwenden. R6sum~. Une des variables importantes qui doit ~tre drterminre dans la surveillance des transmissions radio inconnues est le type de modulation. Des procrdures automatiques de reconnaissance, basres sur des param~tres du domaine temporel en plus de l'enveloppe standard et de la frrquence instantanre, sont proposres et les rrsultats des simulations initiales sont reportrs. Keywords. Modulation recognition, signal classification, signal analysis.

1. Introduction

When unknown radio signals are monitored, either for military purposes or to enforce civilian compliance, one of the important variables of the transmission is the modulation type [2]. The problem of replacing operator intervention by machine recognition has recently received attention. Pattern recognition procedures based on time-averaged behaviour of instantaneous envelope, frequency and zero-phase have been successfully applied to strong signals [1, 3, 4]. Existing recognition procedures are less successful at low signal-to-noise ratios (SNR), in part because these methods depend on parameters which, in the presence of bandlimited Gaussian noise, are biased estimators of the true signal parameters. Moreover, the standard approach to modulation recognition does not explicitly deal with SNR, and so ignores the fact that noisy signals are more similar to each other, r e g a r d l e s s o f m o d u l a t i o n t y p e , t h a n t h e y a r e to s t r o n g signals o f the s a m e m o d u l a t i o n type. W e p r o p o s e t h e use o f a r e c o g n i t i o n p r o c e d u r e b a s e d o n t i m e - d o m a i n signal p a r a m e t e r s w h i c h are u n b i a s e d e s t i m a t o r s o f t h e t r u e signal p a r a m e t e r s in t h e p r e s e n c e o f a d d i t i v e G a u s s i a n n o i s e w i t h s y m m e t r i c s p e c t r a l d e n s i t y . T i m e a v e r a g e s a n d d i g i t i z e d d i s t r i b u t i o n s o f t h e s e p a r a m e t e r s w o u l d be u s e d to f o r m f e a t u r e v e c t o r s , as u s u a l .

However, we propose

a s s i g n i n g to e a c h m o d u l a t i o n

0165-1684/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)

t y p e m u l t i p l e classes,

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corresponding to various SNR ranges. Results of limited simulations support the theoretical indications that the new approach enhances recognition of modulation type.

2. Signal characteristics and SNR Suppose a received signal r(t) with envelope A(t) and phase O(t) is the sum of a modulated signal s(t) with envelope B(t) and phase th(t) plus narrow-band Gaussian noise n(t) of power o.2. It is well known that the expected value, E(A(t)), of the envelope at any instant t is

E(A(t))= (~ar) ' ,/20-M(-~, 1,-B2(t)/20-2),

(1)

where M is the confluent hypergeometric function (see, for example, [6, p. 105]). E(A(t)) increases monotonically with o.2, having asymptotic expressions B(t)+O(2o.2/B(t)) when o.2<>B2(t). Rice's classic analysis of FM modulation showed that the instantaneous frequency is a biased estimator of the true signal parameter in the presence of narrowband Gaussian noise with power spectrum symmetrically distributed about the centre frequency f [5]:

E(O'(t)-2wf) = (1 -exp(B2(t)/2o.z))(¢'(t)- 2rrf).

(2)

As an example of the distortion involved, consider a signal received with a SNR of 0 dB, i.e., B(t) = (20 "2) 1/2. The expected value of O'(t) -2~rf in (2) is 63% of the true value and the expected value of A(t) in (1) is 128% of the true value. Signal modulation recognition uses time-averaged sampled characteristics such as the variance O.2(A) of the signal envelope in the observation period T:

(

o.2(A)=(t/T) ~ A2(jt) - (t/T) ~ A(jt) j=l

)2

where t = T / n .

j=l

If the signal strength falls so that B(t) is significantly less than o. for the entire observation period, it is easy to show that E(0-2(A)) is proportional to the average power received. It is independent of the true envelope's variance, and so carries no information on modulation type. The sampled estimator of ( 0 ' - 2 ~ r f ) 2 is a complicated function of the transmitted signal's envelope B and its time derivative B', and of the transmitted signal's instantaneous frequency ~b'. The second moment of 0'(t) is infinite if the narrow-band noise has power spectrum symmetrically distributed about the centre frequency (cf. [5]). The sensitivity of the estimator to transmitted signal parameters is manifested in another commonly used time-averaged characteristic, the sample variance of the instantaneous angular frequency. To illustrate this, various signals modulated by a sinusoid of frequency 5 kHz and corrupted by additive Gaussian noise were simulated at a sampling rate slightly greater than 10 kHz, with 1024 samples. The sample variance of the instantaneous angular frequency for various modulation types is recorded against SNR in Table 1. The theoretical zero variance of pure AM, DSB and CW signals is observed only at 100 dB. Again, the difficulty of using the variance as a modulation-type discriminator without reference to SNR is apparent. Signal Processing

325

J. Aisbett / Automatic modulation recognition Table 1 Standard deviation of instantaneous angular frequency by SNR by modulation type; (a) modulation depth 0.8, (b) modulation index 0.3, (c) modulation index 3 SNR (dB)

Standard deviation (kHz x 10) AM (a)

100 40 2o 10 0 -oo

0 1 9 23 29 42

DSB

0 127 132 72 82 42

SSB

0 0 3 9 42 42

CW

0 0 1 4 18 42

FM (b)

(c)

1 1 2 5 18 42

13 13 13 14 30 42

3. Some alternative signal characteristics The p r e c e d i n g c o n s i d e r a t i o n s m o t i v a t e the definition o f e s t i m a t o r s o f various signal characteristics which are u n b i a s e d in the p r e s e n c e o f n a r r o w - b a n d G a u s s i a n noise with p o w e r s p e c t r u m s y m m e t r i c a b o u t its centre f r e q u e n c y to/2rr. In the following, time d e p e n d e n c e is not m a d e explicit. A n o v e r b a r on a f u n c t i o n o f t i m e d e n o t e s the s a m p l e m e a n .

3.1. Lemma (i) E(AA')--- BB' and (ii) E ( A 2 ( O - ~ o ) ) = B2(A'-to).

Further, if the noise__is statio___naryover the sampling period, then (iii) E ( A 2 - A 2 ) = B e - B : and (iv) E(A20 ' - A 2 0 ')=B2& ' - B 2 & '. Proof. T h e p r o o f o f (iii) trivially follows f r o m the i d e n t i f i c a t i o n E ( A z) = B2+ 2o -2 [6]. A p p e n d i x A outlines a p r o o f to (i) a n d (ii), a n d (iv) follows from these. [] Since the i n s t a n t a n e o u s values are u n b i a s e d , so are the digitized d i s t r i b u t i o n s o f o c c u r r e n c e s o f values o f the s a m p l e d f u n c t i o n s A2(O ' - to) a n d AA' a b o u t their s a m p l e means. In o t h e r words, given a sufficiently long o b s e r v a t i o n p e r i o d the d i s t r i b u t i o n s built up from the received signal will reflect the c o r r e s p o n d i n g d i s t r i b u t i o n s o f the true signal c o m p o n e n t a n d will not be shifted by the noise c o m p o n e n t , Because the p a r a m e t e r s are s i m p l e f u n c t i o n s ( p o l y n o m i a l s o f o r d e r 2) o f the q u a d r a t u r e c o m p o n e n t s a n d their time derivatives, we e x p e c t that they will p e r f o r m b e t t e r t h a n the s t a n d a r d p a r a m e t e r s in any noise with a circularly s y m m e t r i c p r o b a b i l i t y d i s t r i b u t i o n function. 3.2. Lemma. In the presence of bandlimited Gaussian noise, the sample variance minus the square of the

sample mean of the squared envelope, A4-2(A2) 2

(3)

is an unbiased estimator, provided the sample size is much larger than 1 / S N R . Vol. 13, No. 3, October 1987

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326

(The lemma can obviously also be expressed in terms of the sample mean and variance of the signal power.) Proof. The proof is provided by the following equations [6]: E ( A 2) = B 2 + 2 o -2,

E ( A 4) = B4+ 8B2o-2 + 8o -4,

whenever the expected value of (A2) 2 can be equated to (E(A2)) 2. This is easily shown to be approximated under the stated assumption. [] For a fixed amplitude signal (FM, CW, etc.) the expected value of the parameter defined in (3) is proportional to the square of the transmitted signal power, regardless of the total noise-plus-signal power received. This parameter is potentially extremely useful even with amplitude-varying signals. For instance, if a quartenary AM signal is observed over sufficient interval T to record a more-or-less equal number of bit levels + b, +2b, then the parameter has expected value - 4 b 4 and thus gives the true signal amplitude.

4. Simulations

Various modulation types were simulated from a modulating signal which was either digitized male speech or a synthesized pulse stream generated from binary data carried on Gaussian pulses truncated to a 3-symbol duration. We supposed that the received signal was mixed down to baseband modulo a tuning offset of no more than 25% of the message bandwidth, and a constant phase offset. A total observation period of about 90 ms was allowed. The average power was approximately normalized to one after adding Gaussian noise to the in-phase and quadrature components of the simulated signal. Time derivatives of the noise quadrature components were sampled from Gaussian distributions with variance dependent on the type of bandpass filter modelled. Two signal strength ranges were considered, with SNR levels in the range 15 to 30 dB for 'strong signals' and - 1 to 3 dB for 'weak signals'. Within these ranges were the signal types: voice modulated AM, binary modulated AM, voice DSB, binary DSB, voice wideband FM, voice narrow-band FM, binary narrow-band FM and CW. As well, a pure Gaussian noise signal was analyzed. For each of the 17 signal type/strength combinations, 10 runs were performed using 3 or 4 different modulating signals at 1 to 3 different SNR levels in the relevant range. As yet, insufficient data have been collected to use a feature-vector-based classification scheme. Instead, the performance of the new parameters was compared against that of the standard parameters by applying Student's t-test to the peak and tail bins of the various distributions, for various modulation types. A total of six features was used in each case: the peak and tails values of the standard variables of envelope A and instantaneous frequency 0% and the peak and left tail values of A 2, A A ' , and A20 '. We ignored correlations between the six features since these seemed not to prejudice the results either way. Signal types i and j were deemed to be indistinguishable if all features in the set under consideration were the same for the ith and jth signal types (t-test, 99% confidence limit). Using the standard parameter set there were 14 unresolved cases out of the 136 signal pairs, whilst with the new parameter set there were only t w o - - w e a k binary modulated DSB and AM, and weak voice DSB and strong binary AM. Table 2 lists, for the various modulation types and for strong and weak signal groups, the mean and associated standard deviation over the 10 simulation runs of the parameter defined in Lemma 3.2. The third column gives the calculated values for pure signals, assuming also for AM and DSB that the values Signal Processing

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Table 2 The variance minus the squared mean of the squared envelope Sample mean ± sample standard deviation SNR 15 to 3 0 d B AM voice AM binary DSB voice DSB binary FM voice FM binary CW Noise

-3.1 ±0.3 - 1.0 ± 0.7 4.0± 1.4 1.0±2.1 -3.9±0.1 -3.9±0.1 -3.8±0.1 -0.11 ±0.16

Pure signal (see text)

SNR-1 to3dB -1.0±0.3 -0.3 ± 0.3 1.1 ±0.7 0.4+0.6 -1.2±0.4 -1.1 ±0.4 -1.1 ±0.4

2.0 4.0 -4.0 -4.0 0.0

of the modulating signal at the sampling instants are Gaussian distributed, and that half the AM power is in the carrier. The departure from these values for even the strong AM and DSB is because the Gaussian sampling is not well approximated, particularly by the regularly shaped pulses of the binary signal. As expected, the parameter appears to be a very good way of determining the true signal power, and hence the SNR, of constant envelope signals.

5. Conclusions We have shown theoretically why features based on the standard time-domain parameters of signal envelope and instantaneous frequency are strongly influenced not only by the modulation type but also by the signal to noise ratio. Parameters which are unbiased in the presence of Gaussian noise have been introduced and limited simulations run to compare their performance with those of the standard parameters. Discrimination between transmissions of modulation type AM, FM, DSB, and CW on the basis of characteristics of the distribution of the new parameters appears to be at least as good on strong signals, while in a noisy environment modulation recognition is markedly enhanced. The further discrimination between voice and binary data using these techniques may also be feasible. More work on the statistics of the speech and multilevel digital modulating signals needs to be done. Performance in non-Gaussian noise also needs to be evaluated both theoretically and practically.

Appendix A Let f * denote the Hilbert transform of a real-valued function f Write

r(t)=s(t)+n(t),

r*(t)=s*(t)+n*(t),

(A.1)

where r = A cos 0 and s = B cos ~b and n is Gaussian noise with power cr:. If n~ and n2 are respectively the in-phase and quadrature components of n with respect to the centre frequency ½w/~r, then nl, n2 and their time derivatives n'l and n~ are (at any instant t) independent zero-mean Gaussian random variables. The variance of n~ and n2 is ~rz, and the variance of n'~ and n" is Vol 13, No. 3, October 1987

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J. Aisbett / Automatic modulation recognition

C ( B ) o "2, w h e r e the f o r m o f the f u n c t i o n C o f the noise b a n d w i d t h B is d e t e r m i n e d by the noise p o w e r s p e c t r u m shape. N o w n!

= n ~tc o s t o t - n 2 s i n! t o t - t o n * ,

n * ' = n ' 2 c o s t o t + n l s i nt t o t + t o n .

(A.2)

Using e q u a t i o n (A.2) a n d the i n d e p e n d e n c e o f the r a n d o m variables, it is s t r a i g h t f o r w a r d to c o m p u t e E(nn*) = E(nn') = E(n*n*') = 0 E ( n * n ' ) = - t o E (n .2) = -oJo "2 = - E ( n n * ' ) ,

(A.3)

E ( n '2) = C ( B ) + tour2 = E (n*'2). To p r o v e L e m m a 3.1(ii) viz. E ( A 2 ( O ' - t o ) ) = 0'= d/dt (tan-l(r(t)/r*(t)))

B 2 ( ( b ' - t o ) , use the definition

= ( r r * ' - r * r ' ) / ( r 2 + r .2)

to write A2( O' - to) = ( r r * ' - r* r') - to(r2+ r .2) = r( r * ' - tor) - r*( r' + wr*). R e w r i t i n g as in (A.1), this gives

(s + n)(s*'+ n*'-to(s + n)) - ( s * + n*)(s' + n'+ to(s*+ n*)) = s ( s * ' - tos) - s * ( s ' + tos*) + n n * ' - ton 2 - n * n ' - ton .2 + t e r m s l i n e a r in noise. Using e q u a t i o n (A.3), E ( n n * ' - n * n ' ) = t o E ( n 2 + n .2) a n d , o f course, the e x p e c t e d v a l u e o f any term l i n e a r in noise (i.e. in n, n*, n', or n*') is zero. Thus, E ( A 2 ( O' - to)) = s ( s * ' - tos) - s * ( s ' + tos*) =- B2(& ' - to). This p r o v e s L e m m a 3.1(ii). Part (i) is p r o v e d similarly.

References [1] T.G. Callaghan, J.L. Perry and J.K. Tjho, "Sampling and algorithms aid modulation recognition", Microwaves and RF, September 1985, pp. 117-121. [2] G.H. Hagn, D.M. Jansky and J.l. Dayharsh, "Definition of a measurement capability for spectrum managers", IEEE Trans. Elect. Compat., Vol. EMC-19, August 1977, pp. 216-224.

Signal Processing

[3] F. Jondral, "Automatic classification of high frequency signals", Signal Process,, Vol. 9, No. 3, 1985, pp. 177-190. [4] F.F. Liedtke, "Computer simulation of an automatic classification procedure for digitally modulated communication signals with unknown parameters", Signal Process., Vol. 6, No. 4, 1984, pp. 311-323. [5] S.O. Rice, "Noise in FM receivers", in: Time Series Analysis, Wiley, New York, 1963, pp. 395-424. [6] A.D. Whalen, Detection of Signals in Noise, Academic Press, New York/London, 1971, Chap. 4.