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Signature Optimization for DS-CDMA with Limited Feedback Wiroonsak Santipach and Michael L. Honig Department of Electrical and Computer Engineering Northwestern University 2145 Sheridan Road Evanston, IL 60208 fsak,[email protected] Abstract— We examine a Direct Sequence (DS)-Code Division Multiple Access (CDMA) system with signature optimization for interference avoidance. Reduced-rank signature optimization, in which each signature is constrained to lie in a lower-dimensional subspace, is analyzed with a matched filter receiver. Performance is studied assuming that the signature coefficients are quantized for transmission over a feedback channel with limited bandwidth. Assuming scalar quantization of each reduced-rank coefficient, the dimension which maximizes the output Signal-to-Interference Plus Noise Ratio depends on the load and number of feedback bits, and can be much less than the number of signature coefficients. The performance of quantized reduced-rank signature optimization is compared with a random vector quantization scheme, which gives an upper bound on performance.

I. I NTRODUCTION Interference poses a major limitation on the performance of Direct-Sequence (DS)-Code Division Multiple Acess. It has been recently recognized that in addition to interference suppression at the receiver, it is also possible to avoid interference by optimizing the signatures through a feedback channel [1], [2], [3], [4]. Here we study signature optimization with limited feedback, i.e., with a constraint on the number of bits allowed to specify the signature. To reduce the amount of feedback needed for signature optimization, reduced-rank signature optimization is proposed in [4]. Specifically, the signature is constrained to lie in a lower-dimensional subspace, which reduces the number of coefficients to estimate and quantize. This is analogous to reduced-rank interference suppression at the receiver [5]. As the dimension of the subspace in which the signature resides decreases, there are fewer coefficients to estimate and quantize. This is traded off against fewer degrees of freedom for avoiding interference. We analyze the performance of reduced-rank signature optimization for synchronous Code-Division Multiple Access (CDMA) with a matched filter receiver. The subspace for each optimized signature is chosen randomly, and is fixed. The optimal reduced-rank signature for a given user is computed at the receiver, and the filter coefficients are quantized individually, according to a constraint on the total number of feedback bits B . The B bits are then relayed to the transmitter over an error-free feedback channel. This work was supported by the U.S. Army Research Office under grant DAAD19-99-1-0288.

To evaluate performance, we compute the large system Signal-to-Interference plus Noise Ratio (SINR) at the output of the matched filter receiver as B , the number of dimensions D, the number of users K , and the processing  = K=N , gain N all tend to infinity with fixed ratios K   D = D=K , and B = B=N . Both single-user and group optimization are considered. The former refers to the scenario in which a single user optimizes his/her signature in the presence of fixed interference, and the latter refers to the scenario in which all users optimize signatures. Numerical results are presented assuming that the quantization coefficient is Gaussian. Although the SINR expression for group optimization is approximate, it accurately predicts the corresponding simulation results.  can be selected to maxThe normalized dimension D  decreases imize the output SINR, and the optimal D  with load K and increases with bits/dimension B=D. (It is a less sensitive function of background noise level.) We compare the performance of the optimized reducedrank scheme for single-user signature optimization with a Random Vector Quantization (RVQ) scheme in which the B feedback bits choose one of 2B random signatures. Our results show that the optimized reduced-rank scheme performs nearly as well as RVQ, and is substantially better than scalar quantization of the optimized full-rank signature coefficients. In the next section, we present the system model, and in Section III we present the quantization scheme. The evaluation of reduced-rank SINR is presented in Section IV, and the comparison with RVQ is presented in Section V. II. S YSTEM M ODEL We consider the synchronous CDMA model in which the N  1 received vector is given by

r=

K X k=1

pk bk + n

where pk is the N  1 signature for user k , bk is the corresponding transmitted symbol, n is an N  1 noise vector with covariance n2 I, and there are K users. For reducedrank signature optimization, we constrain the signature vector to lie in a D-dimensional subspace (D  N ). Namely, pk = F k k

2

where Fk is an N  D matrix of basis vectors, and k is a D  1 vector of combining coefficients, both for user k . The basis matrix is fixed and known to both the transmitter and receiver. We assume that the transmitter has no a priori knowledge about the distribution of interferers, so that the subspace spanned by the columns of Fk is chosen randomly. In what follows, we assume that the columns of Fk are nonoverlapping segments of a normalized signature vector with i.i.d. elements, so that Fyk Fk = I. Note that D = 1 corresponds to conventional power control and D = N corresponds to full-rank signature optimization. Varying the subspace dimension D allows a trade-off between achievable performance and the number of coefficients to be estimated and fed back to the transmitter. Our objective is to select the signature and receiver for user k to maximize the received SINR subject to the transmitted power constraint kpk k2 = 1. A linear receiver is assumed, which produces soft outputs cyk r, where ck is the filter for user k . Here we consider the matched filter receiver given by

^ k and  k be the quantized coefsame quantizer. Let ficient vector and the quantization error vector for user k , respectively. Hence,

^ k = k +  k

To construct the quantizer which minimizes the distortion, we must have the probability density function (pdf) for k;i [7]. Simulation results indicate that the Gaussian pdf is a good approximation of the actual pdf. This is illustrated in Figure 1, which compares the measured pdf of a combining coefficient for N = 16 and D = 6 with the Gaussian pdf with zero mean and variance 1=6. Since the D elements of k are assumed to be i.i.d. and k k k = 1, the variance of each element is 1=D. With B bits per signature update, there are B=D bits available for quantizing each coefficient. PDF of α for D = 6 and N = 16 1 Simulated Gaussian 0.9

0.8

0.7

ck = pk f(α)

0.6

Although more sophisticated receivers (e.g., Minimum Mean Squared Error) can also be considered, the matched filter is relatively simple to analyze, and is robust in an adaptive mode with limited training for signature estimation [4]. The SINR for user k is

0.5

0.4

0.3

0.2

0.1

y 2 ck pk

0 −1

k = y ck Rk ck

−0.8

−0.6

−0.4

−0.2

(1)

where y denotes Hermitian transpose, the interference plus noise covariance matrix is

Rk = Pk Py + 2 I; k

(3)

n

(2)

and Pk is the N  (K ? 1) signature matrix excluding the signature for user k . In what follows, we will assume that the signatures are real-valued. To maximize the SINR for user k , given by (1), we choose k to be the eigenvector of Fyk Rk Fk corresponding to the minimum eigenvalue [4]. In an adaptive mode, Rk can be replaced with a sample covariance matrix. When D = N , the matched filter corresponding to the optimized signature is the same as the MMSE receiver [2], [6]. However, this is not true when D < N . III. S IGNATURE Q UANTIZATION Here we assume that the receiver quantizes the D coefficients with a scalar quantizer, and transmits B bits back to the transmitter via an error-free feedback channel. We assume that each element in the vector is i.i.d. so that each coefficient k;i , 1  i  D, is quantized with the

0 α

0.2

0.4

0.6

0.8

1

Fig. 1. The pdf of a reduced-rank signature coefficient for N = 16 and D = 6.

We can compute the quantization noise power as M ?1 Z xj+1 i h X (^ j ? y)2 f (y) dy (4) E k k k2 = D x j j =1

where M is the number of the quantization levels, fxj g,

1  j  M , are the decision thresholds, ^j is the quantized value, and f (y ) is the pdf of the combining coefficient. For large M = 2B=D and Gaussian f , the asymptotic quantization noise power is given by [8] h

E k k

k2

i

=

p

3 + O(1=M 2 )

22B=D+1

(5)

IV. L ARGE S YSTEM A NALYSIS In this section, we analyze the effect of the previous quantization scheme on system performance by computing the large system SINR. Specifically, we let (D; K; N; B ) tend to infinity with fixed normalized di = D=K , normalized load K = K=N , and mension D  = B=N . We first consider normalized feedback bits B single-user optimization followed by group optimization.

3

A. Single-User Optimization The quantized signature for user k is

p^ k = Fk ^ k

(6)

which must be renormalized to satisfy the power constraint. The SINR for user k is

k =

p^ yk p^ k 1 = (^pk =kp^ k k)y Rk (^pk =kp^ k k) p^ yk Rk p^ k

Combining with (3), (6), and (2), and using the fact that Fyk Fk = I, we can write the averaged SINR as i

h

E k ^ k k2 i i h h

k = 2 E k ^ k k2 + E ^ y Fy Pk Py Fk ^ k n

k k

Figure 2 compares the large system expression (11)  = 1, and normalized loads with simulated results with B   K = 1=2 and K = 2=3. For the simulations, the processing gain N = 96 and the background Signal to Noise Ratio (SNR) = 8 dB. The SINR is plotted versus nor . The trade-off between the quantimalized dimension D zation error and available degrees of freedom D to avoid interference is apparent. As the dimension increases from D = 0, the SINR increases because the degrees of free , increasing D degrades perdom increase. For large D formance because of the quantization error. The large system SINR is within a fraction of a dB of the simulated  = 0:5 for results. The maximum SINR is achieved at D K = 2=3, and at D = 0:65 for K = 1=2.

(7)

One user adapts, N = 96, SNR = 8 dB, feedback Bit = 96 8

k

7

i

h

h

E k ^k k2 = 1 ? E k k k2

i

(8)



(9)

  1. Since the elements of Fk  k are i.i.d., we for K can evaluate y y

y

lim  k Fk Pk Pk Fk  k (K;N;D;B )!1 i h  k   k2 = KE

E



(10)

k

 k = lim(N;D;B)!1  k . We assume that the where  limiting distribution for each coefficient k;i is Gaussian with mean zero and variance 1=D, so that this term can   be computed from (4), where M = 2B=D = 2B=(DK ) , or from the asymptotic approximation (5) for large M . Finally, if the random variables k;i and  k;i are uncorrelated, then the expected value of the two cross-terms containing k and  k evaluates to zero. Combining (7)-(10) gives the large system SINR for user k , h

4

2 Simulated; load = 1/2 Large System; load = 1/2 Simulated; load = 2/3 Large System; load = 2/3

1

0

0

0.1

0.2

0.3

0.4

0.5 D/K

0.6

0.7

0.8

0.9

1

p 

y Fy Pk Py Fk k = 1 ? D 2 K k (K;N;D)!1 k k



5

3

Substituting (3) into (7), and expanding the interference term in the denominator gives four terms, which we evaluate as follows. Using the fact that k is the eigenvector of Fyk Pk Pyk Fk corresponding to the minimum eigenvalue, which we denote as min , we have [4]

lim

6

SINR (dB)

where the expectation is over the distribution of the quantization noise. When the quantizer is optimal in the mean squared error sense, the quantized output power is given by [9]

i

1 ? E k  k k2 1

k = i  p 2  h ? n2 + 1 ? D K + K ? n2 E k  k k2 (11)

Fig. 2. Average SINR in dB versus normalized dimension with singleuser signature optimization.

B. Group Optimization We now assume that all users optimize their signatures. ^ k be the quantized signature matrix for all users Let P except user k . The performance metric is average SINR per user, given by PK

k ^ k4

k k=1

 = 2 PK P ^ yk Fyk P^ k P^ yk Fk ^ k n k=1 k ^ k k2 + K k=1

(12)

^ k = P^ k P^ yk + n2 I, and we have used (6). where R Computing the large system SINR appears to be quite difficult, so that we resort to an approximation presented in [4]. Specifically, we assume the users are added sequentially, and each signature is optimized given the users present, and is then fixed. The performance of this scheme lower bounds the performance of the system with jointly optimized signatures. We make the further approximation that the interference appears i.i.d. to each new user, which leads to an accurate prediction of the performance of the sequential optimization scheme.

4

Optimal D/K vs B/N; SNR = 8 dB; K/N = 0.5 1

0.9

0.8

0.7

0.6 Optimal D/K

In analogy with single-user case, the interference term in the denominator of (12) can be expanded into four terms. Following the approach in [4], and substituting into the SINR expression, we obtain (13) shown at the bottom of the page. Figure 3 compares (13) with simulation results. The  with figure shows SINR versus normalized dimension D  B = 1. Two sets of curves are shown corresponding  = 1=2 and K = 2=3. For the simulations, to loads K the processing gain N = 96. These results show that the large system approximation is within one dB of the  are 0.5 and 0.4 simulated curve. The optimal values of D   for loads K = 1=2 and K = 2=3, respectively.

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0.3

0.2

0.1 Single−user optimization Group optimization 0

0

0.5

1

1.5

2 B/N

All users adapt, N = 96, SNR = 8 dB, feedback BW = 96

2.5

3

3.5

4

8

Fig. 4. Optimal normalized dimension vs. normalized feedback bits.

7

Let Xj = i6=k (vjy pi )2 , and Imin = minfX1 ;    ; X2B g be the minimum interference. The corresponding expression for SINR for user k is P

6

SINR (dB)

5

4

1

k = 2 n + E [Imin ]

3

2 Simulated; load = 1/2 Large System; load = 1/2 Simulated; load = 2/3 Large System; load = 2/3

1

0

0

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0.9

1

D/K

Fig. 3. Average SINR in dB vs normalized dimension with group signature optimization.

Figure 4 shows the normalized dimension, which maximizes the SINR, as a function of normalized feedback bits for both single-user and group optimization. The  = 0:5 curves are generated from (11) and (13) with K  and SNR = 8 dB. The plots show that the optimal D for single-user optimization is always larger than that for group optimization.

We wish to select the set V to maximize the SINR. We first assume that the elements of the vectors fvj g are i.i.d. with variance 1=N , compute the associated SINR, and later show that this choice is indeed optimal. Note that as (K; N; B ) ! 1, the energy constraint kvj k  1 is satisfied with probability one. An optimized vector quantization scheme for representing transmitter antenna coefficients (spatial signatures) is discussed in [10]. As N ! 1, viy pj converges in distribution to a Gaussian random variable with zero mean and variance 1=N . Therefore, Xi converges in distribution to a Gamma random variable with mean K=N and variance 2K=N 2 , which has the pdf

fX (x) =

V. R ANDOM V ECTOR Q UANTIZATION In this section we compare the performance of scalarquantized reduced-rank signature optimization with a Random Vector Quantization (RVQ) scheme for choosing the signature. In RVQ, the user can select one of 2B randomly assigned signatures. Here we consider only single-user optimization. Let V = fvj g, 1  j  2B , denote the set of quantized signatures for user k where kvj k = 1 for each j . The receiver measures the SINR for each signature, and selects

pk = arg min v2V

X

i6=k

(vjy pi )2

(14)

(15)

N?

2?



K

2

Nx

2

 K2?2

e?

Nx

2

;x0

(16)

where ?(x) is the incomplete Gamma function. Let FX;N (x) denote the cumulative distribution function (cdf) of Xi where the subscript N denotes the finite processing gain. Since the vectors fvi g are independent, the Xi ’s are i.i.d., and the cumulative distribution function (cdf) for Imin;N is

FImin (x) = 1 ? (1 ? FX;N (x))2

B

;N

(17)

The average interference with RVQ is therefore

E [Imin;N ] =

1

Z

0

x2B (1 ? FX;N (x))2 ?1 fX;N (x) dx: B

5



i2

h

1 ? E k  k k2 1  i  i h h

   p  n2 + 1 ? D32 + 2D ? 83 D K + E k  k k2 K 1 ? E k  k k2

Single−user signature quantization, K/N =0.5 SNR =8 dB

Theorem 1: The large system interference power is given by

1 = Imin

lim

(K;N;B )!1

E [Imin;N ] =

F ?1

lim (K;N;B )!1 X;N



1 2B

(13)

8



7

6

  1 and B  where the latter limit converges for 0  K 0. The proof relies on the asymptotic theory of extreme statistics [11]. As (K; N; B ) ! 1, the distribution of Imin;N converges to a Weibull distribution [11, Ch. 2].  = 0, which corresponds to no feedback, we When B 1 = K , and as B ! 1, Imin ! 0 for K < 1. have Imin The large system SINR for RVQ is therefore 1

1 =

RVQ 1 n2 + Imin

(19)

The following theorem states that choosing the signature vectors independently with i.i.d. elements maximizes the SINR. Theorem 2: Let V1 denote the large system SINR for RVQ, where the entire set of signatures V is chosen from 1 . an arbitrary distribution FV . Then V1  RVQ This Theorem implies that RVQ with i.i.d. signatures gives an upper bound on the performance of reduced-rank scalar quantization. Figure 5 compares the performance of scalar-quantized reduced-rank signature feedback with scalar-quantized full-rank feedback and RVQ. The figure shows SINR versus normalized feedback bits with single-user optimization. The large system SINR for RVQ is evaluated numerically from (18). The other two curves are evaluated  . These results show that the from (11) with the optimal D reduced-rank scheme with optimal rank performs within 1 dB of RVQ. Both schemes perform substantially better than full-rank optimization with scalar quantization. VI. C ONCLUSIONS We have analyzed the performance of quantized signature optimization for CDMA. The large system SINR has been evaluated for reduced-rank optimization assuming a single-user optimizes in the presence of fixed interference, and assuming all users optimize. In both situations, the dimension which maximizes the output SINR can be substantially less than N . The performance of the reduced-rank scheme with scalar quantization has also been compared with RVQ, which gives an upper bound

SINR (dB)

(18) 5

4

3

2 Reduced−rank with optimal D/K Full−rank RVQ 1

0

0.5

1

1.5

2 B/N

2.5

3

3.5

4

Fig. 5. SINR in dB vs B=N with single-user signature optimization.

on performance. For the case considered, the SINR for the optimized reduced-rank scheme is within 1 dB of the SINR for RVQ. R EFERENCES [1]

P. Rapajic and B. Vucetic. Linear adaptive transmitter-receiver structures for asynchronous CDMA systems. European Transactions on Telecommunications, 6(1):21–28, Jan-Feb 1995. [2] S. Ulukus and R. D. Yates. Iterative construction of optimum sequence sets in synchronous CDMA systems. IEEE Transactions on Information Theory, 47(5):1989–1998, July 2001. [3] T. F. Wong and T. M. Lok. Transmitter adaptation in multicode DS-CDMA systems. IEEE Journal on Selected Areas in Communications, 19(1):69–82, January 2001. [4] G. S. Rajappan and M. L. Honig. Signature sequence adaptation for ds-cdma with multipath. IEEE Journal on Selected Areas in Communications, 20(2):384–395, February 2002. [5] M. L. Honig and M. K. Tsatsanis. Adaptive techniques for multiuser CDMA receivers. IEEE Signal Processing Magazine, 17(9):49–61, May 2000. [6] P. Viswanath, V. Anantharam, and D. Tse. Optimal sequences, power control and capacity of spread-spectrum systems with multiuser linear receivers. ieeeit, 45(6):1968–1983, September 1999. [7] J. Max. Quantizing for minimum distortion. IEEE Transactions on Information Theory, 6:7–12, March 1960. [8] S. P. Lloyd. Least squares quantization in pcm. IEEE Transactions on Information Theory, 28(2):129–137, March 1982. [9] J. A. Bucklew and N. C. Gallagher. A note on optimal quantization. IEEE Transactions on Information Theory, 25(3):365–366, May 1979. [10] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell. Efficient use of side information in multiple antenna data transmission over fading channels. IEEE Journal on Selected Areas in Communications, 16(8):1423–1436, October 1998. [11] J. Galambos. The Asymptotic Theory of Extreme Order Statistics. Robert E. Krieger, 2nd edition, 1987.