LOCALLY OPTIMAL DETECTION FOR WEAK

SPREAD SPECTRUM SIGNAL IN THE

PRESENCE OF POWERFUL NARROWBAND

M-ARY ASK INTERFERENCE

S. Mohammad SaberaliAmirkabir University of Technology

Tehran, IranEmail: sm saberali@aut.ac.ir

Abstract- This paper provides a new spread spectrum signaldetection in the presence of digital narrowband interference usingnonlinear detectors by utilizing several moments in a frequencyband. The proposed procedure amounts to a nonlinear processingon the received data and matched filtering. The performanceassessments indicate that this receiver is able to suppress theeffect of narrowband interference, which is assumed as an M-ary amplitude shift keying (ASK) signal. The new detector usesthe information available in a bit duration of the spread spectrumsignal and requires no training data.

I. INTRODUCTION

To transmit over communication channels with preexistingnarrowband signals, spread spectrum techniques can be used.This is due to the inherent immunity of the spread spectrumsignal against narrowband interference. Such spread spectrumsignals would be constrained in power so as not to interferewith preexisting narrowband users. The optimal receiver forthe spread spectrum signal detection in additive white Gaus-sian noise (AWGN) is a filter matched to the signature sequenceof the spread spectrum signal. Even so, it has been shownthat the capability of the spread spectrum system in reject-ing narrowband interference can be significantly improvedif an appropriate narrowband interference suppressing filteris employed in the receiver. The presence of narrowbandinterference makes the observation noise density in the spreadspectrum receiver non-Gaussian especially for powerful nar-rowband interference. Hence, the matched filter is not optimalin such a case. It is shown that the optimal detector in thepresence of additive non-Gaussian noise contains nonlinearityin its structure, the approximate conditional mean (ACM) filter[1], [2] is used to suppress the effect of the narrowband signal.This filter is a nonlinear filter which is an approximation of theoptimal filter in non-Gaussian observation noise. This filter andsimilar others estimate the narrowband interference initially,then, the obtained estimation is subtracted from the receivedsignal to suppress the narrowband interference effect [1], [2],[4]. The main problem of these filters, ACM filter and one in[3] is their slow convergence rate using adaptive algorithms

Hamidreza AmindavarAmirkabir University of Technology

Tehran, IranEmail: hamidami@ aut.ac.ir

for implementation purposes. These algorithms require longtraining sequences to work properly; therefore, they suffer aloss in the bandwidth efficiency. In order to compensate thisdrawback, traditionally, blind algorithms are used requiringno training sequences but long times for convergence [5], [6].In this paper we propose a new nonlinear detector for theweak spread spectrum signal that does not require trainingdata, each decision in this new detector is made in onebit duration containing several spread spectrum chips, hence,long convergence times are avoided. We also provide somesimplified nonlinearities to enhance the spread spectrum signaldetection capability with low complexity and no need of anyestimations. This paper is organized as follows. In section II,the problem formulation is presented. In section III, for thenew detector, the locally optimal (LO) detector is obtainedand the suboptimal detectors are derived and performancecalculation is assessed. Section IV presents the simulationresults. Finally, conclusions are drawn in section V.

II. PROBLEM FORMULATION

We assume the following model for the samples of thereceived spread spectrum signal in the presence of narrowbandinterference, after chip matched filtering [1], [2], [4]

Yk = Sk +ik +vk, (1)

where sk, 'k, and Vk are samples of the spread spectrum,narrowband signal, and white Gaussian noise respectively.We assume sk is a sequence of independently identicallydistributed (IID) random variables[2]. Previously three basicmodels have been used for the narrowband signal; tonal signal,narrowband autoregressive process and digital narrowbandsignal [1]. We use digital narrowband signal; an M-ary ASK,which is a more realistic assumption. The sequences sk, vk and'k are assumed to be mutually independent [2]. In detectingthe spread spectrum signal the observation noise is the sum ofthe narrowband interference and white Gaussian noise

Wk = ik + vk. (2)

1-4244-1190-4/07/$25.00 C2007 IEEE. 182 PACRIM'07

Since wk is the sum of two independent random variables, thePDF of wk is the convolution of the PDF's of ik and vk. vk is aGaussian random variable and ik is a random variable takingon values Im = (2m -1-M)d with m = 1, 2, , M withequal probabilities. The PDF of ik is

(3) xfi(ik) = My , [ (ik - 1m)] -

m=1

Hence, the samples of the observation noise have the followingPDF,

fw(wk) = M EZV2(Wk -Im),m=l

where JV'2(X) is defined as exp(-x2/2u')/ 2wu2 and u2

is the variance of each Gaussian shape component. Thusthe observation noise is a non-Gaussian noise which can bemultimodal based on the relative values for or2 and in. ThePDF of this noise is shown in Fig. 1 for M = 2, 4. Thecharacteristic function of this noise is as follows

7w(Wi) =exp(2U2 ) E exp [j(2m -1 -M)d].m=

(5)After some simplification, (5) looks as

~1/$w(jW)2 2) MI 2w(M 5=2exp(-2 Cos[( -M)w]. (6)

m=

Hence, the underlying binary spread spectrum signal detectionis formulated in the following hypothesis testing problem

H1 Yk =A+wk, k =1,2, ,N (7)Ho Yk =-A +wk, k = 1, 2, , N

where Yk and wk represent the samples of the received signaland observation non-Gaussian noise respectively. A representsthe level of the binary spread spectrum signal and N isthe number of chips per bit in the spread spectrum signal.When the observation noise is non-Gaussian there are alwaysnonlinear detectors with better performance with respect toconventional optimum linear detectors [1], [2], [4], [7]. In thefollowing sections we propose a new nonlinear detector forspread spectrum signal that requires no training data and hasbetter performance than the matched filter.

III. LOCALLY OPTIMAL DETECTOR

We can use the noise PDF (4) in a likelihood ratio test, toobtain the optimum Bayes' detector. Since the noise samplesare IID variates, the test statistics for maximum likelihooddetection is

T(y) = IIf (Yk HO)

where y shows the received vector of length N. Havingobtained likelihood ratio, hypothesis H1 or Ho are to be

Fig. 1. Observation noise PDF in the presence of an M-ary ASK forM = 2,4 and d= 2.

chosen according to the following rule

Nzm A12 (Wk

k 1 Em= 1 2Wk

Im -A) H1> 0.

Im +A) Ho

(8)

We note that the test statistics depend on the level of the spreadspectrum, which its estimation is difficult in the presence ofa powerful interference. Therefore, we invoke LO rule whichis asymptotically optimal for weak signal situations [9]. Thenonlinearity in LO detector is

9LO = -fw (x)lfw (x) (9)

[7], [9]. For the observation noise PDF in the underlying spreadspectrum problem, after some algebra, it is straightforward toshow that LO test is

T(y)

EM exp (-(YKI)2)

H1

> O.Ho

After some mathematical simplification we have

=1] exp(4t ) sin (1)Yk

This nonlinearity is shown in Fig. 2 for M = 2 and M = 4.Replacing cos (.) and sin (.) by their Taylor series expansionwe have a Pade like approximation of YLO(Yk) Ykl/2, thishas an odd function Taylor series approximation. as seem from(10). For the case with M = 2 and I1 = -12 = I it isstraightforward to show

N

T(y) = Y2k=i

I(I

Hi

-tan I > ~0.Ho

The nonlinearity in the above test statistics is similar to the one

in ACM filter [1], [2]. Using the Taylor series approximation

183

0.2

0.18

0.16

0.14

0.12

0 M=2x- M=4

(4)

IV. SIMULATION RESULTS

105

,~~~~~~~~~~---M=20 M=4

-15-10 -5 0 5 10

x

Fig. 2. Nonlinearities of the LO detector for an M-ASK narrowbandinterference with M = 2, 4 and d = 3.

for the tan (.), the nonlinearity is expressed in the followingform

L(Y) ((J-22(,2 ),+ ( 4Y3 2 (6Y5 + 17 (8Y79LO (Y)=( Yk k Yk &~kYi 15 315

where ( I/or2. For high level interference, the higher degreeterms are dominant. Hence, we approximate the obtainednonlinear polynomial by the term with the highest degreeto obtain the suboptimal detector. Thus the approximatednonlinearity is g(x) = xm in which m is an odd integer. Weuse the Asymptotic Relative Efficiency (ARE) with respect tosome reference detector [8], [9], [10], [11] for performanceevaluation. This is a convenient measure of performance fornonlinear detectors that is defined as [8], [9], [10], [11]

ARE2,1 lrn Ni(Pe,~A) (1NAON2 (Re, A)when N1 -> o0, N2 o00,. Ni (Pe, A) denotes the numberof samples which detector i requires to achieve a Probabilityof Error (Pe) with signal strength A. The expression for AREfor the constant signal with respect to the linear detector isobtained as [8]

AREndld=f7° 2f ()dx f g'(x)fw(x)dx (12)

nd,l- °g(tf(t - [J _00 ()w )d

For the approximate nonlinearity g(x) = xm, m odd, asimpler expression can be obtain as follows

ARE,d,ld(m,,o,I) =mE[x2 ][E[x ]]2 (13)E[X2mn](3

The computation of the above equations; (12,13), to determineARE is accomplished by having the moments from (5), (6) andthe fact that E(xm) j)mn(m) (0) We can show that

lim ARE(m, (u, I) = m. (14)

Hence, this proves that the performance of the proposednonlinear detector is better than the linear detector with afactor of m for powerful interference.

In this section we examine the performance of the proposeddetectors. The performance measure is Pe obtained via MonteCarlo simulation. We simulate for a system with o2 = 1,d = 10 and different number of chips per bit N. The resultsare depicted in Fig. 3, Fig. 4, Fig. 5 and Fig. 6. In Fig. 3performance of the LO detector with respect to matched filterand the lower bound is shown. This simulation is achievedfor two different processing gain N = 5 and N = 11, andbinary versus 4-ASK interference signal. We note, the LOdetector has better performance than matched filter and itsperformance is near the lower bound when spread spectrumsignal is weak; i.e, low SNR case. Lower bound is obtained forthe case in when no interferences is available, i.e. I = 0. It isalso shown that nonlinear detector can use the processing gainmore efficiently than the matched filter, because increasingthe processing gain in the matched filter case doesn't haveconsiderable improving effect, this is in contrast to the LOdetector. As wee see for N = 5 the performance of the LOdetector degrades in high SNR's, which is due to the Taylorseries approximation about zero used in obtaining LO detector.Although this approximation is true only for weak signal case,but by increasing the processing gain N we can get goodperformance for larger range of SNR's as we see from Fig. 3.In Fig. 4, the performance of the suboptimal detector withnonlinearity g(x) = xm is shown for different values of min the binary ASK case. This illustrates that by increasingthe degree of the nonlinearity g(x) = xm, the performanceof the detector is enhanced. Fig. 5 shows the performanceof the suboptimal detector with m = 11 for binary and 4-ASK. As we see the detector has also superior performancewith respect to matched filter in the 4-ASK case. As we seein the previous section the only constraint for the suboptimaldetector is the even symmetry of the observation noise PDF.Since by increasing the number of the M-ary ASK narrowbandinterferers this condition will remain true, suboptimal detectorsare appropriate in such a case. This situation is considered inFig. 6 where the nonlinear detector has better performancewith respect to matched filter in an environment with threeinterferers.

V. CONCLUSION

In this paper we introduce new blind nonlinear detectorsto improve the performance of spread spectrum receiver inthe presence of narrowband interference. The optimal detectordoesn't require training data and the time required for eachdecision is one bit avoiding prohibitive convergence times.Simulation results indicate that this nonlinear detector outper-forms conventional optimal linear detector. Then suboptimaldetectors are proposed which requires no training data and noestimations. This detector is especially appropriate when theenvironment is dynamic with different number of interferersand unknown parameters.

184

100

10-2 t \vlaOLM rlLN11

O LO,M=4,N= 11g. 3. .B p.e..f.o..r.. o. te .d n d. a

-N

l A

spread spectrum processing gain N.

10 : X0

o-. .-.,

Low er Bound (1=0)3 Matched Filter

10 BinaryASK.......0-4............ \ X ary ASK

:.-. -----:\------:-----

0 5 10 15 20Eb/No (dB)

Fig. 5. Suboptimal detector performance for Binary and 4-ary ASKnarrowband interference and N =17.

00

..L, . ....

o 10 51, ,-

10Eb/NoEb/No

20

Fig. 4. Suboptimal detector performance with nonlinearity g(x)N= 17.

Xm and Fig. 6. Suboptimal detector performance with mASK narrowband interference and N = 17.

= 11 for multiple Binary

REFERENCES

[1] H. V. Poor, "Active interference suppression in CDMA overlay systems,"IEEE Select. Areas Commun., vol. 19, no. 1, pp. 4-18, Jan. 2001.

[2] R. Vijayan and H. V. Poor, "Nonlinear techniques for interferencesuppression in spread-spectrum systems," IEEE Trans. Commun., vol.38, no. 7, pp. 1060-1065, July 1990.

[3] J. Wang and L. Milstein, "Adaptive LMS filters for cellular CDMAoverlay situations," IEEE J Select. Areas Commun., vol. 14, no. 8, pp.1548-1559, Oct. 1996.

[4] C. L. Wang and K. M. Wu, "A new narrowband interference suppres-sion scheme for spread-spectrum CDMA communications," IEEE Trans.Signal Processing, vol. 49, no. 11, pp. 2832-2838, Nov. 2001.

[5] M. Lops and A. M. Tulino, "Automatic suppression of narrow-bandinterference in direct sequence spread spectrum systems," IEEE Trans.Commun., vol. 47, no. 8, pp. 1113-1136, August 1999.

[6] H. V. Poor and X. Wang, " Code-aided interference suppression forDS/CDMA communications-Part II: parallel blind adaptive implemen-tation,"IEEE Trans. Commun., vol. 45, no. 9, pp. 1112-1122, Sep. 1997.

[7] S. Kay, Fundamentals of Statistical Signal Processing-Detection Theory,Englewood Cliffs, NJ: PTR Prentice-Hall, 1998.

[8] J. H. Miller, J. B. Thomas,"Detectors for Discrete-Time Signals in Non-Gaussian Noise," IEEE Trans. Information Theory, vol. IT 18, no. 2,pp.241-250, Mar. 1972.

[9] S. A. Kassam, Signal Detection in Non-Gaussian Noise, NY: Springer-Verlag, 1988

[10] H. V. Poor, An Introduction to Signal Detection and Estimation, NY:Springer-Verlag, 1994.

[11] X. Yang, H. V. Poor,"Memoryless DicreteTime Signal Detection inLong-Range Dependent noise," IEEE Trans. signal processing., vol. 52,no. 6, pp.1607-1619, Jun. 2004.

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