MINIMUM BIT-ERROR-RATE DECISION FUSION RECEIVERS FOR ROBUST DS SI'READ-SPECTRUM COMMUNICATIONS * Stella N. Batalama Dept. of Electrical and Computer Eng. 201 Bell Hall State Univ. of New York at Buffalo Buffalo, N Y 14260 E-Mail:
[email protected]
Dimitris A. Pados Dept. of Electrical and Computer Eng. and Center for Telecommunication Studies Universit:y of Southwestern Louisiana Laft3yette, LA 70504-3890 E-Mail:
[email protected]
Ioannis N. Psaromiligkos Dept. of Electrical and Computer Eng. 201 Bell Hall State Univ. of New York at Buffalo Buffalo, NY 14260 E-Mail:
[email protected]
ABSTRACT The conventional Matched Filter (MF) correlation receiver for Direct Sequence Spread Spectrum transmissions is reconsidered from the theoretical point of view of robust distributed detection (decision fusion). Separate chip- by-chip decisions on the transmitted information bit of interest are generated. The individual decisions are then optimally fused to produce the final decision on the transmitted bit. N is proven that the majority-vote strategy is the statistically optimal fusion rule for additive white Gaussian noise channels. A n exact closed form expression is given f o r the probability of error induced b y the majority-vote receiver. It is shown that the performance is independent of the identity of the active interfering user population, that is signatures and signature cross-correlations, in DS-CDMA environments. This perfect robustness with respect to the identity of the multiuser interference component is coupled b y significant resistance toward occasional high-power jamming (equivalently occasional natural severe channel imperfections) or outlier-prone background n,oise, or both. Some numerical studies illustrate these theoretical findings.
I. INTRODUCTION In this work we direct our attention to the design and performance evaluation of simple, low-cost, robust DS-CDMA receivers that are resistant to statistical variations of the operating environment. In their pioneering studies of spreadspectrum transmissions over impulsive channels in the late 'This work was supported in part by the LEQSF contract RD-A-35
0-7803-3682-8196 $5.00 0 1996 IEEE
OS, Aazhang and Poor [8],[9] considered hard-limiting receivers and chip-by-chip decision. From the point of view of distributed (decentralized) detection, this approach corresponds essentially to a majority-vote decision fusion rule. In this present work we prove that the majority-vote receiver is in fact the statistically optimal decision fusion rule on the test vector of the individual chip-based decisions. This is true under the assumption of equally probable information bit transmissions that are independent to each other and to the channel :noise. The latter is assumed to be either white Gaussian or a white Gaussian mixture. Our analysis leads to a simple exact expression for the probability of error induced by the majority-vote structure. It shows theoretically that the performance (average Bit Error Rate - B'ER) is completely independent of the identity of the interfering user population (signatures and signature cross-correlations). This perfect robustness toward the multiuser interference (MUI) component is coupled by significant resistan.ce with respect to statistical variations of the underlying channel noise. Indeed, as a consequence of the well documented statistical robustness of sum-of-sign tests [GI ,[7], the majority-vote receiver exhibits superior performance stability in the presence of occasional either intentional (single-user DS/SS military communications) or natural (civilian DS/CDMA transmissions) interference. The interference inay manifest itself as occasional high-power perfect chip-matching jamming, or impulsive outlier-prone background noise, or both. The rest of this paper is organized as follows. In Section I1 a brief presentation of the spread-spectrum channel model facilitates the introduction of our notation. Section I11 presents the main results for the analysis of the majority-
959
the only known quantity in ( 3 ) . All other signatures and all energies (including Eo) are treated as unknown deterministic variables.
vote decision fusion receiver. Robustness studies are carried out in Section IV. The theoretical analysis is coupled by numerical comparisons of the majority-vote receiver with the conventional MF and the decorrelating detector [1],[2],[3] for various statistical environments. The popular decorrelator assumes known interfering spreading codes and it is included merely as a reference point.
111. MAJORITY-VOTE DECISION FUSION In pursue of simple, low-cost, robust alternatives we reexamine the conventional M F correlator. As in the original studies of DS-CDMA transmissions over impulsive channels of Aazhang and Poor [SI, [ 9 ] , we consider individual chip/signature matched demodulation t o obtain, after sampling at the chip rate l/Tc, the following vector decision statistic: DT = [ ~ ( l ) S o ( .l .).,, r ( L ) S o ( L ) ] . (4)
11. SPREAD-SPECTRUM MODEL Let us consider a CDMA system where K users transmit synchronously over a channel that is nominally additive white Gaussian (AWG). The continuous-time received signal is modeled as follows: K-1 T(t)
=
JEI,bk(i) i
Sk(t
+ n(t)
- iT)
(1)
We note that according to the nota.tion introduced in the previous section the operation in (4) corresponds to the outer product of the vectors r and S o . Next, we proceed with individual chip-based decisions for the information bit bo of the form
k=O
where, with respect to the k-th user, Ek is the received energy, b k ( i ) E {-1, 1 ) is the 2-th information bit, s k ( t ) is the individual spreading code or signature, T is the duration of a single information bit, and n ( t ) is the additive white Gaussian noise component, say of variance u 2 .The signature S k ( t ) is of the form
Ui
= s g n [ D ( i ) ] = sgn[r(i)So(i)],i = 1 , .. ., L.
(5)
The final decision 60 may be based only on the binary vector A UT = [ul, . . . , U L ] of the intermediate chip-by-chip decisions. This scheme effectively translates the whole process into the familiar domain of dist,ributed detection (also known as statistically optimal “decision fusion” [ l o ] ,[ l l ] [7]). , We refrain to comment on the final decision fusion rule at this moment. Instead, we proceed with general performance evaluation studies that will eventually dictate the optimum fusion strategy. Assuming equally probable, independent information bit transmissions in white Gaussian noise of variance u 2 ,we may take advantage of the symmetricity Pr(ui = - l i b 0 = +1) = Pr(ui = + l i b 0 = - 1 ) and the equal priors to write
L j=1
where L is the so called system processing gain, c k ( j ) E { - I , l } , j = 1, . . . , L are the assigned signature bits, and &,(t) is the chip waveform with duration T, = T / L . Without loss of generality, we assume that all signatures are T normalized to unit energy, that is s i ( t ) d t = 1, V k = 0 , . . . , A-- 1 . Next, we drop the informatmionindex i in (1) and we focus our attention on a single bit interval (single-bit one-shot detection). After chip-matched demodulation and sampling
so
A
at the chip rate 1/T, we obtain the samples r [ n ] = r(nT,), n = 1, . . . , L . The discrete-time version of the received signal r ( t ) in ( 1 ) assumes the form
Pe(u1) =
i.(
= Eb,>j=i2...,~ = cr21L,L. Equation (3) offers the opportunity for a laconic review of the objectives of this work. The user of interest is assumed to be user 0 and the objective is the detect,ion of the information bit bo. All information bits b,, i = 0,. . . , I< - 1 , are assumed to be random, independent to each other and to the channel noise, and equally probable. The available decision statistic is the vector P and SO(the signature of user 0) is
11) (6)
where E b J , j = l , . , , , ~ - ldenotes {.} the statistical expectation with respect to the interfering bits b j , j = 1, . . . , K - 1, and Q(z) = A +m -&e-ya/zdy. The following theorem presents
s,
a core result of this work. T h e proof is
omitted due to lack
of space.
Theorem 1 Let r in (3) be t h e received signal vector with bi, i = 0,. . . , K - 1, equally probable, independent to each other and to the iioise vector n information bats. Let also the noise vector n be white Gaussian dzstributed. Then, t h e optimum detection rule f o r the information bat of interest bo on the data (test statistic) UT = [ . I I ~ , .. . , U L ] with U + , i = 1 , . . . , L , given by (5) is
960
bo = s g n ( U T W ) ,
(7)
where W is the FIR filter with coeficients (tap-weights)
In essence, the above theorem classifies as optimum and quantifies the common sense approach of weighting the output of each individual chip-based detector according to its performance. Indeed, the weighting coefficients wi are strictly monotonically decreasing in P,(ui) E (0,1), i = 1 , . . . , L. We also notice that for P,(u;)= 1/2 the decision U ; is disregarded as usless (wi= 0). Since the optimal FIR filter W is determined solely by P , ( u ~ )i, = 1,.. . , L , a closer look on (6) appears wise. With s k ( t ) and c k ( j ) as in (2), (6) can be rewritten as follows: =
Pe(.i)
2
= 1 , .. . , L ,
(9)
We recall that the signature bits co(i), i = 1,.. . , L , are assumed to be known, while c j ( i ) , j = 1 , .. . , I< - 1, i = 1,. . . , L , are treated as unknown deterministic f l binary quantities. Then, the newly defined random variables
remain binary, independent to each other, and equally probable. Substitution of (10) into (9) gives an equivalent expression for P,(u;), where the expectation is now taken with respect to the set b5, j = 1 , .. .,I< - 1. But the two sets { bl , . . . , b k - 1 ) and { b;, . . . , bE-l} are identically distributed. Hence, we have shown that for binary information bits on top of binary modulated signatures
P,(.i)
= P,(.)
A
=
v i = 1, . . . , L.
(11)
The implication of expression (11) is twofold. First, we see from (8) that wi = logV i = 1 , .. . , L , and the optimal decision fusion rule in (7) degenerates to L i=l
This is known as the "majority-vote" (or majority-selector) rule in the context of the pertinent distributed detection literature [Ill. In addition, since Vi = 1 , .. . , L
P,(.;)
= = n
=
P r ( q = + l l b o = -1) P r ( u ; = - 1 p o = $1)
P(U)
1
(13) 961
the random variables ui, i = 1,. . . , L become conditionally i.i.d. Bernoulli and the fusion test U ; becomes conditionally binomially distributed. Therefore, due to the symmetricity of the two hypotheses, the final probability of error induced by the optimum majority-vote receiver is
where 1.r denotes the smallest integer that is equal to or greater thain 2 (we recall that L is odd of the form 2" - 1). So far, the derivation of expression (11) has yielded a significant payoff. It single-handedly proved the statistical optimality of the majority-vote fusion rule and it led to the closed form probability of error expression (average BER) in (14). In addition, a close inspection of (11) indicates that the probabiility of error for every chip-based decision P,(u) is completely independent of the signatures and signature cross-corre1,ations of the interfering users. We conclude that the majority-vote receiver is immune to the exact identity of the active MU1 population and its performance is not affected by signatures and signature cross-correlations (cf. (14) and (11)). This perfect robustness toward the MU1 component is cxpected to be coupled by significant resistance toward occasional natural or malicious man-made perfect chip-matched high-power interference and/or heavy-tailed, outlier-prone background noise. It is well known [6], [7] that these are properties shared by all sum-of-sign tests such as in (7) or (12).
I V . ROBUSTNESS STUDIES To acquire a certain practical feeling for the majority-vote detector and the probability of error expressions in (11) and (14), we begin with the setup of a DS-CDMA communication system case study. This example will be used throughout this section and it will be modified and adapted according to our special needs and interests. We consider a 4-user DS-CDMA scenario where each user is equipped1 with a signature of length L = 15. The user of interest is assumed to be user 0. To magnify the effects of the multi-user interference component (MUI), the signature of the user of interest is significantly correlated with the signatures (of all interferers (< S o , S i >= 7/15, i = 1 , 2 , 3 ) while the interfering signatures are nearly uncorrelated to each other (< S;, Sj >= -1/15, i # j # 0). We would like to see the probability of error induced by the majority-vote receiver (expression (14)) as a function of the SNR of the user of interest, say SNRo. We fix the SNRs of the interferers to S N R l = 5dB, S N R 2 = 6dB, SNR3 = 7dB and in Fig. 1 we plot P,(&) as a function of SNRo for the convent,ional M F , the decorrelator, and the majority-vote receiver. A11 expressions allow exact numerical evaluation. Fig. 1 indicates that the majority-vote structure outperforms the compet,itors over a whole range of practical interest (0 14dB).
Next, we intent to exploit the perfect robustness of the majority-vote structure with respect to the identity of the interferers as proved in the previous section. We fix SNRo to 8dB and we maintain S N R i , i = 1 , 2 , 3as in Fig. 1. We vary < SO,!$ >, i = 1 , 2 , 3from 1/15 to 13/15 in increments of 2/15 and we plot Pe(&o) as before. As expected, Fig. 2 shows that the majority-vote detector is totally immune to variations of the cross-correlations < SO,S i >, i = 1 , . . . , K - 1, while this is not the case for the conventional MF or the decorrelating receiver. Next, we proceed with mainstream robustness studies for the optimum majority-vote decision fusion rule. We consider scenaria where, in addition to the typical multiple-access interference components, we may have occasional high-power perfect chip-matching interference. The latter may be due to malicious jamming attempts or due to occasional severe natural channel imperfections. Maintaining the assumptions of statistical independence across channel noise samples, both phenomena can be described by a first order €-contamination model [e]:
fn(z) = (1 - E)N(O,a)
+~h(z).
(15)
This identifies the univariate probability density function (pdf) of the background noise random variable n as nominally U-mean a’-variance Gaussian and with probability t distributed according t o some contamination pdf h ( z ) . Worst case robustness studies call for deterministic symmetric “contamination” distribution at +8:
h ( z )=
-21q z
-
1 8) 3- - q z 2
+e),
and U-mean, X2ai-variance AWG with probability E . Thus, the noise random variable n follows a white Gaussian mixture distribution and the heavy-tailed (impulsive) effect is achieved by choosing some X >> 1. To maintain consistency with our previous studies, the total noise power must remain the same: a2 = (1 - t)ai + ~X’ai. This implies that the relationship between F O , a , E , and X is F O = a/J1 - E €A2. With background noise as in (18), the probability that any individual chip-based decision ut, i = 1, . . . , L is in error becomes
+
As with the earlier jamming studies, Pe(bolc,A) maintains the binomial form of (14) with Pe(Ul€,X) in place of Pe(.). Fig. 5 replicates the studies of Fig. 1. The only difference is that Fig. 5 accounts for a 10% impulsive noise contribution ( E = 0.1) with A’ = 25. The robustness of the majority-vote detector is clearly captured. In terms of further studies and future research, it would be appealing to apply the optimal decision fusion rule of Theorem 1 for the robustification of a recently proposed line of blindly optimized FIR filters [4],[5]. In this domain the majority-vote rule loses its optimum status.
REFERENCES
as 8 --+ CO [6]. In this setup, the probability that a single chip-based decision ui,i = 1, . . . , L in (5) is in error becomes
Pe(UI€) = (1 - €)
S.Verdu, “Minimum probability of error for asynchronous Gaussian multiple-access channels,” IEEE Trans. Inform. Theory, vol. 32, pp. 85-96, Jan. 1986.
’
The overall majority-vote probability of error continues to follow the binomial formula in (14), with Pe(uI€)in place of pe(u). As a numerical example we return to our 4-user case study and we set SNRo = 6dB and S N R i = 5,6,and 7dB for i = 1 , 2 and 3, respectively. In Fig. 3 we plot the BER of the majority-vote, the MF, and the decorrelating receiver as a function of the cole‘w?inationlevel E E (0,l). We note the superior robustness and performance stability of the majority-vote detector. In Fig. 4 we fix the contamination level at 10% ( E = 0.1) and we plot the BERs as a function of the SNR of the user of interest. We conclude this section with performance studies for transmissions over outlier-prone channels. We use the classic first-order impulsive noise model [e],[8],[9] fn(.)
= (1 - r)N(O,CO)
+ tN(0, AFO),
(18)
where the channel noise samples are assumed to be nominally U-mean, ai-variance AWG with probability 1 - E 962
K. S. Schneider, “Optimum detection of code division multiplexed signals,” IEEE Trans. Aerospace Electron. Syst., vol. 15, pp. 181-185, Jan. 1979. R. Lupas and S. Verdu, “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 35, pp. 123-136, Jan. 1989.
M. L. Honig, U. Madhow, and S. Verdu, “Blind adaptive multiuser detection,” IEEE Trans. Inform. Theory, vol. 41, pp. 944-960, July 1995. S. N . Batalama and D. A . Pados, “Blind real-time lowcomplexity receivers for DS-CDMA mobile users,” in Proc. IEEE CTMC/GLOBECOM ’95,pp. 122-125, Singapore, Nov. 1995. P. J . Huber, Robust Statastacs, Wiley, 1981. D. A. Pados, E(. W . Halford, D. Iiazakos, and P. Papantoni-Kazakos, “Distributed binary hypothesis testing with feedback,” IEEE Trans. Syst. Man and Cyberii., vol. 25, pp. 21-42, Jan. 1995.
[8] B. Aazhang and H. V. Poor, “Performance of DS/SSMA communications in impulsive channels-Part I: Linear correlation Receivers,” IEEE Trans. Commun., vol. 35, pp. 1179-1188, NOV. 1987.
[9] B. Aazhang and H. V. Poor, “Performance of DS/SSMA communications in impulsive channels-Part 11: HardLimiting correlation Receivers,” IEEE Trans. Commun., vol. 36, pp. 88-97, Jan. 1988.
[lo] D. A. Pados, P. Papantoni-Kazakos, D. Kazakos, and A. G. Koyiantis, “On-line threshold learning for NeymanPearson distributed detection,” IEEE Trans. Syst. Man and Cybern., vol. 24, pp. 1519-1531, Oct. 1994.
0.1
L-
0
I
0.1
0.2
0.3
0.4 0.5 0.6 ContaminationLevel
0.7
0.8
0.9
Fig. 3. F’robability of error for worst-case contamination with probability E .
[11] B. V. Dasarathy, Decision Fusion, Los Alamitos, CA: IEEE Computer Society Press, 1994.
I
0.5,
0 0001
Y
Y
0
2
4
8
6
12
10
14
16
Fig. 4. 13ER versus SNR with MU1 SNRs at 5, 6, 7dB and 10% worst case contamination ( E = 0.1).
SNR (dB)
Fig. 1. Probability of error versus SNR for the user of interest. The SNRs of the three interferers are fixed to 5dB, 6dB, and 7dB.
0.5,
x: Matched Filler Decorrelalor *: Majority-Vote Detector
0:
10P
Fig. 5. IBER versus SNR with MU1 SNRs a t 5, 6, 7dB and 10% impulsive noise contribution at X2 = 25.
I 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
signature Cross-correlation
Fig. 2. Probability of error as a function of the signature cross-correlation coefficient < So, Si >, i = 1,2 , 3 (SNRo = 8dB, S N R l = 5dB, S N R z = GdB, and SNR3 = 7dB). 963
