Interference rejection using time-dependent constant ...

Interference Rejection Using Time-Dependent Constant Modulus A l g o r i t h m ‘

R. Mendoza, .I.H. Reed, T. C. Hsia, B. G. Agee Departmcnt of Electrical Engineering and Computer Science University of California, Davis Davis, CA 95616 Abstract Traditional adaptive filters are implemented using timeindependent training-sequence directed techniques. In contrast, the technique prsentcd and evaluated in this paper is based on time-dependent blind weight adaptation. The corresponding adaptive filter is able t o exploit properties found in cyclostationary and constant-modulus signals. The steepest-descent CMAs, originally applied to time-independent adaptive processing, arc generalized and applied here t o timedependent adaptive filtering for interference rejection. By filtering frcquency-shifted versions of the received signal, CMA-based time-dependent adaptive filters make use of additional spectral information to improve the estimate of the signal-&interest. Cpmputer simulations show the potential of the proposed techntque.

GCMAs. Finally, Section V summarizes the main results of this paper.

II B A C K G R O U N D A. Time-Independent Adaptive Filters The mast common adaptive filter is the transversal finite impulse-rsponse (FIR) TIAF. The output of this filter can be expressed as

where q is the input signal a t time k; r denotes the delay time associated with the N+l filter coefficients wk(O,r). C o n s t a n t Modulus Algorithms A TIAF has so f a r been used t o implement constant modulus algorithms (CMAs) 11) ?hick have proven successful in a variety of applications ine “ding interference suppression, polarization combining, and equalization. CMAs exploit the constant envelope property, which many radar and communication signals have, to adaptively minimize any modulusvariation imposed by interference and/or multipath prapagation. As derived in 131, CMAs are based on minimizing a cost function of the form

I INTRODUCTION The problems of cc-channel interference, adjacentchannel interference, and jamming in civil and military communications call for the development of effective interference rejection techniques. Research in this area has produced several categories of adaptive filters, which can be classified in terms of filter stiucwre, cost function, and adaptation algorithm. Two classes of adaptive filters are considered here: time-independent adaptive filters (TIAFs),whose parameters are timeinvariant, and timedependent adaptive filters (TDAFs), whose parameters are periodically time-varying. TDAFs are capable of outperforming TIAFs when applied to eyelostationary signals such as PCM communication signals. Consequently, timedependent adaptive filtering is a promising approach for the implementation of interference rejection techniques. The classical method for implementing an adaptive filter optimizes quality-measure between the processor output and a known training signal. However, in many applications the inclusion of a training signal in the adaptation scheme is not possible. The need for more versatile adaptive filters has promptcd the development of blind adaptation techniques that use other types of performance measures t o adapt the filter parameters. In particular, constant modulus algorithms (ChMs) [I], which minimize the modulus variation of the observed signal, have been shown t o be useful in a variety of applications due t o their equalization and interference suppression effects. However, CMAs have also been observed to capture undesired signals. This paper addresses this limitation by advancing and evaluating new filters that can compensate for the effects of interference on constant-envelope signals: the generalized CMA (GCMA) TDAFs. These adaptive filters attempt t o outperform TIAFs and other TDAFs such as the spectral correlation discriminator (SCD) 121. Section 11 provides the background necessary for the development of the new technique. Section 111 presents the GCMA TDAFs. Via computer simulations, Section N tests and evaluates the performance of the proposed technique in suppressing interference as compared to other techniques. Our experimental results show the potential of

F = < IA’-l& 1’ I ‘ > M , (2) where < . > M denotes time averaging over M points; A represents the modulus of the signal; and p and q are pqsitive integers. The optimization i s accomplished in 131 usmg a steepcst-descent approach with a stochastic gradient estimate given by

a

I

This work

was aupporlrd b y

bFk =

(3)

- IL IP)q-llsgn(AP - l i k l P ) l g , where the symbol * d e n o t s the complex conjugate operation and the integers p and q are limited to 1 and 2 for practical reasons 131. Thus, four p-q combinations are possible, each of which yields a corresponding steepest-descent “p-q CMA”. Other rapidly converging techniques based on orthogonalized steepestdescent methods, and property-mapping approaches have also been developed [ 4 , 5 , 6 ] . The steepestdescent CMAs used in this paper are summarized by --4p=;sk

I& I ’ - * ( A ’

Wk+,(o,r)= w k ( o , r )

where

c

+c

,

(4)

i r ~ z L

and ck are selected from the following: 2-1:

c=2,

ik =

&8gn(A2-lGk

1’)

(54

These algorithms, when written in this form, resemble the complex version of the LMS algorithm. CMAs were originally developed to suppress correlated interference, most commonly produced by multipath propaga tion, wherein CMAs, acting a9 equalizers, try to counteract

Advanced CounterMeosurc S&rms.

273 23ACSSC-12/89/0273 $1.00 Q 1989 MAPLE PRESS

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multipath channel effects [1,7,8l. However, they are also effective in suppressing uncorreiated interference since the cast function ~ensesthe presence of any additive interferer that causes a deviation in the amplitude of a constantmodulus signal. A CMA-based adaptive filter acts as a frequencydependent attenuator that suppresses bauds where the interfeerence spectrally overlaps the signal-of-interest (SOI).U no overlap exists, an adaptive filter with sufficient spectral resolution can suppress the interferer. If significant overlap eriste, however, a CMA-based TIAF may have problems in trying to suppress undesired signals 191. One 8ource of concern when using a CMA ia inherent in singlechannel time-invariant filters: in trying t o remove the interference, the processor decreases it? gain in the portion of the band where the additive interference is located, which may result in signal degradation. CMA-based filters in particular may degrade the SO1 by forcing it t o be constant modulus and the resulting signal may not accurately resemble the shaping of the original signal. In general, the compromise between interference suppression and signal distortion depends on the type of algorithm used. For instance, LMS-based filters are known t o converge to the Wiener solution. CMA-based filters, on the other hand, can exhibit Wiener-like behavior near convergence 111, but can also introduce residual distortion in QPSK signals in order to equalize the modulus variation over the entire interval 171. A second source of concern is the so-called “capture” phenomenon, which is a result of the nonlinear nature of the constant modulus cost function. One version of the problem is called interference capture and can occur when a constant modulus (or even noneonatant-modulus) interferer is stronqer than the SO1 [1,4,8,101. The second version, known as nome capture [5,11,12], can occur when the SO1 ia received in the presence of broadband noise and is due to tranaient convergence of the adaptive algorithm to a saddle point of the constant modulus cost function.

(9)

I

where eit = yk-& and a. is assumed here to be zero, forming a time-independent branch.

The Spectral Correlation Discriminator Unlike the FSR TDAF, the spectral correlation discriminator (SCD 12 does not require a training signal for adaptation. This blind algorithm can be used to estimate wideband or narrowband signals regardless of the bandwidth of the interferer. Signals t h a t possess different statistical pciiadicities are decorrelated by frequency-shifting the observed data; in this way, noise and interference, not correlated with the SOI, are filtered out. The SCD is obtained by replacing yk by y in Fig. 1. The cwfficient update equation is

+

wk+l(n,r) = wh(n,r) 2,,(it2;_,cizno”ik-r’ 3 (10) where t = zk-& and en# 0. The frequency shifts are chosen such t h a t the frequencyshifted data exhibit spectral correlation with the SO1 but not with the SNOI. Therefore, 0 %is, in general, a nonzero multiple of the baud rate of the SOI. This restriction degrades SCD performance, unless the energy corresponding to the missing time-independent term is obtained by other means. Unfortunately, if the SO1 is phasemodulated (e.g. BPSK, QPSK), exploiting multiples of the SO1 baud rate [ails to estimate the missing component because the SO1 lacks energy at these f r a quenciw. In 1171 an innovative technique solves this problem by combining the SCD with a CMA TIAF, which provides the required energy. Ul TEE PROPOSED T D A I Since many communication signals have a low modulus variation and are eyelostationary or almost eyelostationary, there exists high potential for joint exploitation constant modulus and spectral correlation properties. Therefore, taking advantaga of blind adaptive algorithms such as CMAS and time-dependent adaptive filtering, this section presents a new technique that promises to he successful.

B. Time-Depcndcnt A d a p t i v e Filters The principle behind timedependent sdaptive filterin conaista in usin spectrally correlated portions of the SO8 orland the signa? notof-interest (SNOI)to improve the partion of the SO1 spectrum corrupted by interference. When aignal statistice are rapidly changing, a TDAF is upacted to o u t perform B TIAF in approximating the o timal timevaryin solution. However, if the SO1 and the SdOI are band-limitaa such that no portion of the available spectrum exhibits coirclsr tion, or if the spectrally correlated portions are sufficiently corrupted by noiso or interference, a TDAF may not provida any si nificant bansfit over TIAF. The output of a transvaraai F b TDAF is given by .EO

-jZrm.[h-r]

+ 286kzk-,c I

w k + l ( n d = wk(nlr)

A. The Qsneraliaed CMA TDAFa Until now ChlAe have been implemented with

a TIAF. However, reseamh has shown that the use of a linear eombinatien of frequsney-shifted vereions of I usua~iy provides improved signal extraction perfocmance 1151. Hence, CMAbased filters that take advantage of timedependent adaptive filteringi are tdvaneed: the generalized CMA (GCMA TDAFs A8 de cted m Flg. 2, the basic structure of these filters con: silts o! a time-independent path, which filters a n unshifted “ersion of z,, and a number of time-dependent paths, which filter frequency-shifted versions of z k . GCMA TDAF coefficientsare updated here by any of the steepest-descent C C U s given by,

r--N/P

-j2res(k-r]

Wk+dnlr)

where as represents the n t h filter periodicity and L Is the number of Bitor psriadicities.

= wk(n,r)

+ C~Lfkz&-rE

I

~

(13)

where ao=o I which r?u!ts in the time-independept spctipn; c and f k are given by any of the pairs in Eq. (5); and is obtained by

The FSR T D A F One method of Implementing a TDAF Is tha Fourier series representation (FSR) TDAF [ l a , 141. Although capable ef optimal performance, this adaptive filter must ha provided with a tralning signal. The original FSR TDAF has been generalized to handle multiple signals in the observed data, as well as certain modulated slgnals whose statistical periodicitlea are not multiple of a single fundamental harmonic. As derived in 15,181, the generalised version of the FSR TDAF, depicted in Ig. 1, is b a e d on minimixing n eoet runctlon of the form

The non!ipear nature of the constant modulus cost functiop results in CMA- adaptation problems and analysi_s difficulties. Extending the “se of CMAs t o timedependent adaptive filtering introduces further theoretical e&plicdti&s. Fortunately, our experinenta! investigation h a s shown promis, ing results regarding this extension. G C h N TDAFs are expected to out erforin CMA TIAFs due t o the spectral information provide{ by t h e r&&,cy-shirted versions of z k . Versions t h a t result from shifting zk by periodicities associated with the SO1 help t o estimate the SOI, while versions that are the outcome of shifting zk by periodicities associated with the SNOI help to reduce the interference. However, since G C h S s are just generalimed versions of CMAs, they are prone to suffer

k.

The optimiiation is acromplished in [15] using a steepest descent approach with a stochastic gradient estimate given by resulting in the coefficientupdate equation

274

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To illustrate difference in performance, the output signals of the proposed and competing techniques are displayed in Fig. 6 t o 10. As expected, the SCD output spectrum shows distortion at its center frequency. In trying t o suppress the interference, the CMA has also produced a distorted power spectrum. On the other hand, the timedomain plot in Fig. 8 highlights the superior performance achieved by the GCMA; the sharp baud transitions reflect high-frequency content provided by time-dependent filtering. In contrast, as seen in the timedomain plots of Fig. 7 and 9, a TlAF is incapable of obtaining an output with such features. Finally, Fig. 10 shows t h e best SO1 estimate: the TSD TDAF output. The MSE and BER computations shown in Fig. 11, 12, and 13 substantiate the information provided by the signal plots. As expected, the highest MSE and BERs are given by the SCD, while the lowest are produced by the TSD TDAF. Both the 2-1 and the 1-2 GCMAs attain lower MSEs than t h e corresponding CMAs. However, since the constant modulus approach is not based on restoring signal quality but rather on reducing modulus variation, MSE may not be the best performance measure for CMA-based adaptive filters. Thus, BER computations may provide a more appropriate performance measure. For instance, although MSE results indicates t h a t t h e TIAF is superior to both the GCMAs and the CMAs, matched filtering BER and baud sampling BER indicate otherwise. In this case, a reduction in modulus variations implies an increase in the number of signal bauds that have the correct polarity at the sampling instants. Therefore matched filtering and baud sampling detect few, il any, ~ F I O T Sin the output of the CMAs and the GCMAs.

from pitfalls imposed by the constant modulus cost function and modified by time-dependent adaptive filtering.

W SIMULATION RESULTS Two sets of simulations illustrate the strengths and limitations of the proposed TDAFs, and compare their performance to that of the competing blind SCD and CMA-based TIAFs, as well as t o the training-sequence directed (TSD) TIAF and TDAF in extracting signals in noise and interference. For conciseness, the CMA TIAFs and GCMA TDAFs are referred t o as CMAs and GCMAs, respectively. The TSD TIAF and TDAF are implemented using the structure shown in Fig. 1, where the TSD TIAF is the degenerate form of the TSD TDAF, i.e., only a time-independent branch (ao= 0) is used. The TSD TIAF and the TSD TDAF represent the optimum TIAF and the optimum TDAF, respectively. The frequency shifts are chosen here t o be multiples of t h e baud rate of the SO1 or/and the baud rate of the SNOI. Each filter has all of its start-up weights initialized t o O.O+jO.O except the center tap of its time-independent section (the only section in TIAFs), which is set to l . O i j 0 . 0 ; this prevents Cr from remaining a t zero when using a TIAF solely updated by a CMA. The value of the step size p is arbitrarily set far one filter; for other filters p i s found by sealing the original value in inverse proportion to the number of periodieities used by the filter. Mean-square error (MSE) and bit-error rate (BER) graphs are used t o compare GCMAs t o the competing methods. In this paper the MSE between the output signal and the desired signal y ~ .is computed using all of the signal points in a specified interval. Using the same data interval, the BER of Gk is calculated by matched filtering (matched filtering BER) and by baud sampling (baud sampling BER). Whereas matched filtering BER is computed using all of the samples of a baud, baud sampling BER is computed using only the middle sample (if the number of samples per baud is odd) or the average of t h e two middle samples (if the number of samples per baud is even).

B.

P e r f o r m a n c e as a F u n c t i o n of SIR with a QPSK SO1 and a QPSK SNOI This set of simulations evaluates the influence of SIR on the performance of the 1-2 GCMA, the 1-2 CMA, and the training-sequence directed filters. By adding a SO1 with a n amplitude of 1.0 to each one of ten SNOI, ten corrupted signals are generated, each consisting of 410,000 points. The resulting SIR^ are -4.0, -2.0, 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, and 8.0 dB. Table 3 lists the SO1 and the SNOI parameters. Each filter operates on the corrupted signal over 410,000 iterations; then, the output signals are compared to the SO1 over a data interval or in,ooo points (400,ono to 4119,999) to generate MSE and BER data. Table 3. The Signal Parameters

A. P e r f o r m a n c e with a BPSK SO1 and a Q P S K SNOI Table 1 lists the signal parameters for the first set of simulations. The SIR is 2.0 dB. Both signala possess the same baud rate. In addition, the SNOI, unlike the SOI, has a startu p delay of one second. Fig. 3 through 5 depict the the input signals, each of which is 20,000 points long. All of the adaptive filters, whose parameters are given in Table 2, operate over 20,000 iterations. The last 5,000 points of the SO1 and the

output signal are used to compute MSE and BER. Table 1. The Signal Parameters

Table 4. The Filter Parameters

T y p e of Filter

GCMA

TSD TD4F

CMA & TSDTIAF

Table 2. The Filter Parameters No. of Taps

I

Adaptation TSD TIAF

Filter Periodicities EIS)

0.0, 1 1 . 0

0.0, H.O 2~1.4, 11.4 or 0.0, ~ . 4 +

er Periodicit

Filter 10.25

!75

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0.0

SO1 are eliminated t o reduce adaptation errors. Fig. 14, 15, and 16 present the results. By 6.0 dB, all the techniques have achieved zero-valued matched filtering BE&; by 8.0 dB, all the TDAFs have attained zero-valued baud sampling BERs. Therefore, these points are not plotted in Fig. 15 and 16. As seen in these graphs, the filter parameters given in Table 4 maintain t h e superiority of the GCMA over the CMA for positive SIRS. However, the nonlinear nature of the constant modulus fuunetion causes the GCMA to abruptly break away from the desired goal in this e a ~ enear 0.0 dB.

V CONCLUSIONS In this paper, the performance of the steepest-descent GCMAs is tested under diRerent conditions and then compared t o the performance oRered by other blind and nonblind adaptive techniques. The simulation results indicate t h a t the use of CMAs in conjunction with time-dependent adaptive filtering usually results in improved performance. Even though GCMAs can also capture undesired signals, they are in most cases supriot to CMAs, which take advantage of only one signal property. Furthermore, as the SIR increases, GCMAs are better able t o perform well relative to training-sequence. directed adaptive filters. 1. J. R. Treiehler and E. G. Agee, “A New Approach t o Multipath Correction of Constant Modulus Signals,” IEEE Tinnsoctions on deoustics, Specch, and Signal Processing, vol. ASSP-31, pp. 459-472, April 1983. 2. J. H. Reed and T. C. Hsia, “A Technique for Sorting and Detecting Signals in Interference,’’ 1988 IEEE Military Communications Conference, Son Diego C A , 1988. 3. M. G.. Larimore and .I.R. Treiehler, “Convergence Behavior of the Constant Modulus Algorithm,” Pioc. ICASSP~BS,Boston MA, pp. 13-16, April 1983. 4. R. P. Gooch and J. Lundell, “The CMA Array: An Adaptive Beamformer for Constant Modulus Signals,” Proc. ICASSP, Tokyo Japan, April 1986. 5. B. G. Agee, “The Least-Squares CMA: A New Technique for Rapid Correction of Constant Modulus Signals,” ICASSP-86, Tokyo Japan, pp. 953-956, April 1986. 6. R. &eh, M. Ready, and J. Svoboda, “A Latticc-Based Constant Modulus Adaptive Filter,” Proc. Twentieth A86 lomor Conference on Signals, Systems ond Computers, Pacific Grove CA. November 1986. 7. M. G. Larimare and J. R. Treichler, “Data Equalization Based on the Constant Modulus Adaptive Filter,” Proc. ICASSP, Tokyo Japan, pp. 949-952, April 1986. 8. D. N. Godard, “Self-Recovering Equalisation and Carrier Tracking in Two-Dimensional Data Communication Systems,“ IEEE Transactions on Communicntions, vol. COM-28, pp. 1867-1875, November 1980. J. R. Treiehler and M. G. Larimore, “CMA-based Tech9. niques for Adaptive Interference Rejection,” IEEE M E COM Proceedings, pp. 47.3.1-47.3.5, 1986. 10 J. R. Trekhler and M. G. Larimore, “The Tone Capture Properties of C U - b a s e d Interference Suppressors,“ IEEE Trans. ASSP, pp. 946-958, August 1985. 11 M. G. Larimare and J. R. Treiehler, “Noise Capture Prcperties of the Constant Modulus Algorithm,” Proc. ICASSP-85, Tompa FL, pp. 11651168, March 1985. 12 B. G. A g e , “Convergent Behavior of Modulus-Restoring Adaptive Arrays in Gaussian Interference Environments,” Proc. Twenty-Second Asilomor Conf. an Signals, Systems and Computers, Pacific G o v a CA, November 1988. 13 E. R. Feriara, “Frequency-Domain Implementations of Periodically Time-Varying Adaptive Filters,” IEEE Tronsoctions on Acoustics, Specch, ond Signal Processing, vol. ASSP-33, pp. 883-892, August 1985. 14 W. A. Gardner, “Optimization and Adaptation of Linear Periodically Time-Variant Digital Systems,” Signal and Image Processing Lab Report SIPL-85-15, University of California, Davis, CAJ98.5.

J. H. Reed, “Time-Dependent Adaptive Filters for Interference Rejection,” Ph.D. Dissertation, Department of Electrical Engineering and Computer Science, University of California, Davis CA, December 1987. 16. W. A. Gardner, Statistical Spectral Analysis: A Nonprobabilistic Thcary, Prentiee-Hall, Englewwd Cliffs, New Jersey, 1987. 17. R. Mendoza, J. H. Reed, and T. C. Hsia, “Interference Rejection Using a Hybrid of a Constant Modulus Algcrithm and the Spectral Correlation Discriminator,” IEEE Military Communications Confercncc, Boston MA, 1989.

15.

~~

g1 ~

niliuu

7_

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"I

1

Fig. 7. The CMA TIAF Ourput Signal

Fig. 4. The SNOI.

Fig 9. The TSD T I M Outpul Signal.

il

Fig. 10. The TSD TDAFOutpul Signal

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D

-

ui=

Y

--C

-12

-

5

-

3

TSOTHF

TSDTDAF -

1

,

3

s

7

5

SIR. dB

Fig. 14. Comparison of MSE YS. SIR.

Fig. 11. Comparison of O u p a MSE.

I o.m,

0.m

Fig. 12. Comparisonof Ma(ched Filtering BER.

SCOBER*O.iS

a

I

Fig. IS. Malched Filtering BER vs. SIR

3

0.m

-

6

.

4

-

2

0

SIR,

Fig. 13. Comparison of Baud Sampling BER.

2

4

dB

Fig. 16. Baud Sampling BER YS. SIR

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6

8