Combined Decision-Feedback Multiuser Detection/Soft ... - CiteSeerX

Combined Decision-Feedback Multiuser Detection/Soft-Decision Decoding for CDMA Channels Abdulrauf Hafeez and Wayne E. Stark Dept. of Elect. Engineering and Computer Science University of Michigan, Ann Arbor, MI-48109 Abstract{This paper evaluates the performance of decision-feedback multiuser detectors with coding for CDMA channels. A combined multiuser detection/softdecision decoding scheme is proposed which allows total control over the reliability of decision-feedback mechanism. Its performance is compared to that of conventional detection and linear-decorrelation detection with soft-decision decoding. Simulation and analysis is undertaken for synchronous CDMA channels. It is found that the scheme works very well especially for weak users and when multiuser interference is large. However, strong users do not gain much with this scheme over a linear-decorrelation detector.

decisions for the symbols causing the interference are available or not. A DFMD eliminates anti-causal MAI using zero-forcing ltering and attempts to cancel causal MAI using decision feedback. It is shown in [2] that the improvement in performance with a DFMD is appreciable over the LDD, especially when MAI is large. The detectors have a nearly ideal near-far resistance. Another desirable feature of these detectors is that their serial structure allows combining multiuser detection with soft-decision decoding for maximum bene t. Note that many of the low-complexity detectors proposed in recent years can not easily be combined with true soft-decision decoding. Examples include the multistage detector proposed by Varanasi and Aazhang[3] which employs successive interference cancellation and the post-processing likelihood detector of Seshadri and Hoeher (see [1] for references). In this paper we investigate the performance of singlestage DFMD with convolutional coding and soft-decision Viterbi decoding (SDD) over synchronous CDMA channels in additive white Gaussian noise (AWGN). Varying the depth of trellis truncation of the Viterbi decoder allows control over the reliability of decisions fed back. We compare the performance of combined DFMD/SDD with conventional and LDD with coding using simulation and analysis. Performance bounds are derived for a two-user coded synchronous case.

1 Introduction

Conventional detection for a direct sequence codedivision multiple-access (DS-CDMA) system employs a bank of matched lters, matched to the signature waveforms of individual users, followed by decision devices. This scheme suers from severe degradation if the signal strength of some interfering users is dominant. The linear-decorrelation detector (LDD) does a multidimensional linear ltering operation on the matched lter outputs, which completely decorrelates the signal component of each user with those of the interferering users. This detector performs signi cantly better than the conventional detector. However, since it employs channel inversion to get rid of MAI, it enhances noise. The maximum likelihood sequence detector minimizes the sequence error probability and has complexity that grows exponentially with the number of users, making it infeasible for implementation. Several low-complexity, sub-optimum multiuser detectors have been proposed in recent years but most of the research done so far on multiuser detection has considered uncoded transmissions, with a few exceptions [1]. Error control coding, however is important in a noisy and interference limited environment to obtain satisfactory performance. A family of single-stage decision-feedback multiuser detectors (DFMD) has been proposed for asynchronous CDMA channels in [2]. In these detectors, users make decisions in the order of decreasing signal strength. The MAI aecting a given symbol of a given user is classi ed as causal and anticausal depending on whether

2 Combined DFMD/SDD: Model

Consider a coded asynchronous CDMA system with K users. The baseband signal of the k th user is: 1 p X ck rw s (t ? iT) xk (t) = i=?1

i

k k

where cki are antipodal channel symbols of duration T obtained by encoding binary information symbols of energy wk with a rate r code and then mapping into f+1; ?1g. sk (t) is the normalized signature waveform of the k th user, sk (t) = 0 for t 62 [0; T]. Using the D transform, the input symbol stream of the k th user is written as 1 X cki Di : ck (D) = i=?1

The input symbol vector of K users is given by c(D) = (c1 (D); c2 (D); : : :; cK (D))T :

 This

research was supported by NSF under Grant NCR9115969 and by the Army Research Oce under contract number DAAH04-95-I-0246.

1

The receiver observes the sum of the transmitted signals in additive white Gaussian noise (AWGN). r(t) =

K X

k=1

~1

1

c (D)

2

~2

v (D) v (D)

xk (t ? k ) + n(t)

G(D)

c (D)

-L

D

Viterbi Decoder 1 Viterbi Decoder 2

Forward Filter K

~K

v (D)

where k 2 [0; T] is the relative delay between users and n(t) is WGN with power spectral density N0 =2. The received signal is ltered by a bank of matched lters, each matched to the corresponding signature waveform. The sampled output of the matched lter v(D), where v(D) = (v1 (D); v2 (D); : : :; vK (D))T is given by v(D) = S(D)Wc(D) + z(D) where W = diag(prw1 ; prw2; : : :; prwK ) and S(D) is the channel spectrum S(D) = R?1D?1 + R0 + R1D R 1 s (t? )s (t? +iT)dt. z(D) is colwhere Rkli = ?1 k k l l ored Gaussian noise with spectrum Sz (D) = (N0 =2)S(D). S(D) is real-symmetric and non-singular on the unit circle. It can be factored as S(D) = F T (D?1 )F(D) F(D) and F(D)?1 are both causal and stable matrix lters. F(D) has the following form F(D) = F0 + F1 D + F2D2 +


(1) where F0 is lower triangular (See [4] for references). Note that the de nition of causality implied here de nes the \past " for each component k of a vector process at time i as given by all components at times i -1,i -2,... and the components 1, 2,... , k -1 at time i , and the \present " as the component itself. The single-stage decision-feedback multiuser detector proposed in [2] consists of an anticausal forward lter G(D) as a zero-forcing lter that eliminates multiuser interference from \future " symbols and a strictly causal feedback lter B(D) that attempts to cancel multiuser interference from \past " symbols. The optimum forward and feedback lters that would maximize the ideal signal-to-noise ratio (SNR) at the input to the decision devices are shown to be[2] G(D) = (F T (D?1 ))?1 ; B(D) = (F(D)?diag(Fo ))W (2) Note that G(D) is the whitening lter for the process v(D) and y(D) is the resulting whitened process y(D) = G(D)v(D) = F(D)Wc(D) + n(D) where n(D) is WGN. The detector proposed in [2] orders the users in decreasing symbol energy. In the i -th sampling time, user 1 makes its decision rst based1 on the corresponding output of the forward lter (yi ) and the decisions 2

D

c (D)

-KL

Viterbi Decoder K

^1

c (D)

Encoder 1

^2

B(D)

c (D)

Encoder 2

Feedback Filter ^K

c (D)

Encoder K

Figure 1: Combined Decision-Feedback Multiuser Detector/Soft-Decision Decoder (L=Truncation depth of the feedback device). of all users made at sampling times2 i -1,i -2,... . User 2 makes its decision next based on yi , the decision made by user 1 for the i -th sampling time and the past decisions of all users, and so on. The multiuser receiver structure we propose is shown in Figure 1. The decision-feedback mechanism feeds back tentative decisions for convolutionally encoded symbols (^ck (D); k = 1; 2; : : :) obtained from the trellis of corresponding decoders at depth L, where L is less than or equal to the truncation depth of the Viterbi decoder. Since the decision of user 2 at the i -th sampling time depends on the decision of user 1 at the same sampling time, user 2's decoder runs at a lag of L units behind the decoder of user 1, and so on for other decoders. Thus the total detection delay of a K user system is KL.

3 Performance analysis

3.1 Bounds: two-user synchronous system

The input to the Viterbi decoders in Figure 1 is c~(D) = G(D)v(D) ? B(D)^c(D): Substituting G(D) and B(D) from (2), we get c~(D) = diag(F0 )Wc(D) + n(D) +(F(D) ? diag(F0))W(c(D) ? c^(D)):(3) Note that F(D) = F0 for a synchronous CDMA system. Thus B(D) is lower-triangular with zero diagonal elements. For a two-user system, (3) simpli es to (4) c~1(D) = F011prw1c1(D) + n1 (D) 2 22 p 2 2 c~ (D) = F0 rw2c (D) + n (D) +2F012prw1
c1(D) (5)





where
c1(D) = c (D)?2 c^ (D) . Since user 1 sees WGN only, an upper bound on its bit error probability 1

1

Thus,

can be found [5] by substituting Es = (F011)2 rw1 in r 2d E !  d E  f s exp f s A(x) Pb1  Q x=e?Es=N0 N0 N0 (6) where df is the free distance of the code and A(x) is obtained as @ : (7) A(x) = @y G(x; y; z)

l=1

Zl2 = F022prw2d +

d X

k

l=1

n2l + 2F012prw1

d X l=1








c1l

where N is arbitrary.

3.2 DFMD, LDD and the conventional detector: multiuser synchronous system

We consider a system with K equal energy users. For each detector, we nd the rst-order statistic for the l -th code symbol of the k -th user: DFMD Assuming that ckl = +1 is sent, the statistic is given by

p

p

Zlk = F0kk rw + nkl + 2 rwIlk P where F0kk  1 8 k and Ilk = km?=11 F0mk
cml . Ilk is the residual multiuser interference that aects the l -th symbol of the k -th user and comes from the decision errors of the lower-indexed interfering users. We model
cml ; m = 1; 2; : : :; k ? 1 as i.i.d. random variables with distribution 8 < +1 121 Pc1 m
cl = : ?1 2 Pc1 0 1 ? Pc1 where Pc1 is the probability of code symbol error of user 1. We note that if a decision-error in the l -th symbol of user i forces user j (j  i) to make an error when it is fed back in the absence of noise,
cil and
cjl would have opposite signs. This means that the correlation among
cml ; m = 1; : : :; k ? 1 is such that they tend to cancel each other out when added. Thus our assumption about residual multiuser interference (that it is the sum of independent r.v.'s
cml ) should be pessimistic.

Note that the superscripts in Z 2 and n2 identify user 2. Note from Figure 1 that it's the code symbols that are fed back not the information symbols. l

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 N +X df ?1  Ad P2 (d) ? 21 Z d (11) Pb2  21 T(x) + x=Z d=df

is less than zero1 . (Code symbol 0 is mapped to +1). n2l ; l = 1; 2; : : : are independent Gaussian random variables with mean 0 and variance N0 =2. We model
c1l ; l = 1; 2; : : :, decision errors of user 1 fed back to user 2, as i.i.d. random variables with distribution 8 < +1 121 Pc1 1
cl = : ?1 2 Pc1 0 1 ? Pc1 where Pc1 is the probability of code symbol error of user 12 derived in [6]. Since at the output of a Viterbi decoder, we usually get errors in bursts,
c1l are not independent. But we can assume independence if we assume an interleaver which randomizes all error bursts at the output of decoder 1 before feedback. However, for all practical purposes an interleaver is not required. Using the above model, we get the pairwise error probability as  d  1 k1+k2 d X P2(d) = k1 ; k 2 2 Pc1 k1 =k2 =0;k1+k2 d (1 ? Pc1 )qd?k1 ?k2 q 0 22 w2 1 F0 2r N0 d + 2F012 2r Nw10 (k1 ? k2) A p Q@ d 1 2

Ad P2(d)



where gijk is the number of paths on the trellis of length k with Hamming weight i, corresponding to information vectors of Hamming weight j, that diverge from the all-zero path at a node and remerge with it. For user 2, we assume that the all-zero codeword is sent. We nd the probability that a path having Hamming weight d that diverges from the all-zero path at a node and then remerges with it, accumulates higher metric over the unmerged segment or equivalently the decision variable d X

d=df

for w1  w2 F022 2 (9) Pb2  21 T(x) N0 N0 2F012 x=Z where  (F 22)2rw  Z = exp ? 0 N 2 0   F 22jF 12jpw1w2  2 1 + 2Pc1 sinh 0 0N :(10) 0 Combining (8) and (9), we get a tight closed form upper bound, which we call a compound union bound

G(x; y; z) is the transfer function of the code, given by XXX i j k G(x; y; z) = gijkx y z j

1 X

where Ad is the coecient of the term xd in the expansion of A(x). We also obtain a relatively loose closedform union bound for Pb2 (see [6] for derivation)

y=z=1

i

Pb2 

l

3

"

#

kX ?1 w Var(Nk ) = (F kk)2 1 + 8r N Pc1 (F0mk )2 : (13) 0 0 m=1

1

LDD q The rst-order statistic for LDD, normallized

by N0 is given by (12) where Nk has variance 2

1 = Var(Nk ) = (H kk )2

Z

m=1;m6=k

0

2

Conventional, Uncoded

Decision−feedback, Uncoded Conventional, Coded −2

10

Linear decorrelator, Coded Decision−feedback, Coded Single user, Coded

[S(ej 2f )?1 ]kkdf  1 : 0 (14) In the synchronous case, if all users have equal crosscorrelations, then H kk = ((F0 )kk )11 8 k. The Conventional Detector The matched lter output for the l -th symbol of the k -thquser given that a +1 is sent can be normallized by N20 to get (12) where Nk has variance 0

o DF, Coded, L=32 x DF, Coded, L=16 + DF, Coded, L=08 * DF, Coded, L=04

1

2 3 K X w Var(Nk ) = 41 + 2r N (Rmk ) 5 :

Simulation results for various multiuser detectors

−1

10

Probability of error of user 2

Using this model for
cml , we approximate Ikl as a GaussianPrandom variable with mean zero and variance Pc1 km?=11 (F0mk )2 . Ilk is independent of theqnoise k component nkl that aects user k . Letting Dlk = N20 FZ0kkl , we get p Dlk = 2r(w=N0) + Nk (12) where Nk is a Gaussian r.v. with mean 0 and variance

−3

10

1

2

3

4 5 SNR1 (SNR2 = 3.0 dB)

6

7

Figure 2: Simulation Results.

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4 Simulation Results & Analysis

We simulated a two-user synchronous CDMA system with rate 21 , memory 4, convolutional encoding and Viterbi decoding with 8-level quantization. We employed length 31 spreading sequences for the coded system and length 63 spreading sequences for the uncoded system. Note that for a synchronous two-user system, decision-feedback multiuser detection involves zero-forcing detection for user 1 and complete decisionfeedback interference cancellation for user 2. Thus for user 1, DFMD is the same as LDD. Bit-error probability of user 2 with DFMD is compared to that with conventional and LDD in Fig. 2. Truncation depths L| 4,8,16 and 32 are used for the feedback mechanism in the coded system. With all truncation depths decisionfeedback detection outperforms linear-decorrelation for the normal operating range of the system (SNR1SNR2). When multiuser interference is large (SNR1SNR2), L = 4 is as good as L = 32, and both achieve the single-user bound. This happens because the tentative decisions of user 1 are reliable even when obtained from the trellis at small depth, owing to the high SNR of the interfering user. Figure 2 shows how the near-far resistance of DFMD with coding depends on L. For the code used, the minimum truncation depth (Lmin ) which does not result in appreciable performance loss due to truncation is known to be 15. It can be seen 4

that for L  Lmin , the eect on the near-far resistance of the detector is negligible. Thus if variable detection delay is allowable, weak users can virtually achieve single-user performance at all interference conditions by adjusting L according to the levels of multiuser interference. Performance bounds are shown in Fig. 3. By combining the loose union bound of (9) with an arbitrary number of exact union bound terms, we get a tighter upper bound. For nite truncation L, the bounds diverge for SNR1