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Sep 7, 1999 - The decorrelator results contain no MAI components and, what is more important, can be achieved without phase/amplitude knowledge. The.
An Iterative CDMA Multiuser Detector with Embedded Phase/Amplitude Estimation Zhiqiang Feng, Christian Schlegel Department of Electrical Engineering University of Utah Salt Lake City, UT 84112 Tel: (801) 581 5561 Email: [email protected] September 7, 1999 Abstract: This paper proposes an iterative multi-user detector for a direct-

sequence code division multiple access (DS-CDMA) system based on the iterative technique of Gauss-Seidel for solving linear algebraic equations. The detector starts with soft iterations, that is, no hard decisions are made until (near) signal decorrelation is achieved. The signals at this point are passed to an amplitude/phase estimator, and the iterative detector continues using hard decisions of the estimated symbols to achieve improved reliability. Only small blocks of data are required for estimation and no phase tracking loop and power control loops are needed. It is shown that this detection method achieves substantial gains over the decorrelator in systems with high load at a computational complexity of O(NK ) per bit rather than O(NK ) per bit for decorrelation (K is the number of users and N is the processing gain). Keyword index: Multistage cancellation, DS-CDMA system, GaussSeidel iterative detection, decorrelator, phase and amplitude estimation. 2

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1 Introduction A DS/CDMA system enables multiple access by using the spreading and despreading processes to suppress multiple access interference. Each user is typically detected independently of the other users by passing the received signals through a lter matched to the spreading code of that user. This is called conventional matched lter detection. The conventional matched lter receiver, however, is optimal only in the single user case. When the number of users accessing the channel increases, degradation becomes severe such that for moderate to high system loads, the conventional receiver becomes useless (see Section 6). Furthermore, the near-far problem can cause failure of the system using the conventional receiver even for small loads. In order to overcome these problems, Verdu [1] proposed an optimal multiuser detector, the maximum-likelihood (ML) detector. ML detection of interference-limited CDMA signals is near-far resistant and provides near single user performance [1], but its computational complexity is O(2K ) per bit, where K is the number of users. To circumvent the complexity problem, much e ort has been devoted to developing suboptimum receivers. Lupas and Verdu [2] considered decorrelation to reject MAI. The decorrelator has a complexity of O(NK )/bit and is near-far resistant. Decorrelation is the benchmark for suboptimum detector development. A new suboptimum detector is commonly compared with the decorrelator in complexity and performance to determine its promise. Varanasi and Aazhang proposed a multi-stage (iterative) detector using tentative hard decisions in [3]. Their detector has a computational complexity of O(MK )/bit (M being the number of iterations, or stages). The initial estimate of the information bits used in the rst iteration are provided by the outputs of the matched lters. Noticing that the initial guess of the information bits plays an important role in multi-stage detection , Varanasi and Aazhang proposed to use a decorrelator to provide the initial estimation of the bit values to achieve better performance [5]. However, the decorrelator adds a signi cant amount of extra complexity, defeating some of the purpose of multistage detection. The knowledge of the relationships between Gauss-Seidel iteration, decorrelation, and multi-stage detection leads us to develop a detection scheme which can achieve improved performance while 2

1

1

This is not the case in iterative implementations of linear detectors as discussed in [4].

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maintaining low computational complexity. Besides [3] [5], a series of papers on non-linear iterative detection have been published recently [6]{[12]. In [6] and [7], null-zone multistage detection and partial cancellation were proposed. Null-zone multistage shows slight improvement over the multistage detection proposed in [5]. Partial cancellation [7] was said to have dramatic improvement over the other iterative procedures at high initial bit error rate. In [4], the convergence rate of di erent linear detectors was discussed. It was shown that the parallel cancellation detector (Jacobi iteration, both hard and soft) diverges when the system load reaches 17 percent of its maximum. Serial soft cancellation however does not have this problem. All the nonlinear detectors need the phase and amplitude knowledge of each user, while the linear detectors do not. We will take advantage of this property to realize our phase and amplitude estimator. We call the detector we propose in this paper Gauss-Seidel Soft/Hard Detector (GS/SH) since it combines soft and hard iterations. Like the multistage detectors proposed in [3, 5], the hard decision part of the GS/SH detector needs knowledge of the signal energy and carrier phases of each symbol. Traditionally, this problem can be solved through power control and phase tracking loops. Assuming slowly changing systems, we can use simple adaptive lters to estimate the power and the carrier phase shifts. However, in a multiple user system, this approach becomes ine ective because the presence of MAI makes it much more dicult to suppress the noise and interference through averaging. However, using preprocessing which eliminates the MAI at the cost of enhancing the Gaussian noise (decorrelation), it is possible to obtain good phase/amplitude estimates by averaging. The rest of this paper is organized as follows. The CDMA model used in this paper is described in Section 2. Although the detection methods discussed in this paper are applicable to higher order modulation, we will restrict ourselves to BPSK. In Section 3, we review multi-stage detection and draw the parellels with the Gauss-Seidel iterative procedure for solving linear algebraic equations [4]. The GSSH detector is developed in Section 4. In Section 5, the method of estimating the signal power and carrier phase shift is described and incorporated in the receiver. Simulation results are presented in Section 6. 3

2 System Description Throughout this paper, bold face letters represent matrices and underlined letters represent vectors. In a CDMA system, let us assume that K users access the transmission channel simultaneously, each using a unique code (spreading sequence, signature waveform). The information bits of each user are modulated by its spreading sequences (random sequences are used in this paper). The received signal can be written as y (t) =

?

K L X1 X j =0 k =1

dj

(k )

r

wj(k) c(jk) (t ? jT ?  (k)) + nw (t):

(1)

The waveform cjk (t) is the wide band signature waveform of user k during the transmission of the j th binary symbol djk (dj 2 ?1; 1). The duration of the signature waveform is T . wjk is the energy of the j th symbol sent by the k th user, and  k is the delay for the k th user with respect to some arbitrary timing reference, usually the symbol boundaries of user 1. For convenience, we assume cjk (t) to be unit-energy and generated from a discrete spreading sequence as N ? X jNk n p(t ? jT ? nTc ); (2) cjk (t) = ( )

( )

( )

( )

( )

1

( )

( )

+

n=0

where jNk n 2 f?1; +1g are the components of the discrete spreading sequences, and p(t) is the chip waveform. The processing gain is assumed identical for all users and equal to N , and each user transmits a sequence of L symbols called a block. Given di erent delays  k , chip synchronous and symbol asynchronous cases can be realized if more than one sample is taken during one chip period. If  k = 0; 8k, the synchronous case is modeled. The CDMA system block diagram is shown in Figure 1, where `' denotes multiplication, and ` ' denotes convolution. In our simulation, pseudorandom signature sequences are used for each user, also called long codes because of their long periods. The spreading sequence a ecting symbol j is denoted by h i jk = jNk ;    ; kj N ? : (3) ( )

+

( )

( )

( )

( )

( ) ( +1)

4

1

d1 dk dK

P

matched filter bank





p(t)

 k

p(t)

p(?t)  k ; P

p(t)

 K ; P

(1)

( )



(K )

(1)

n(t)

;

( )

(

)

Fig. 1: The CDMA system block diagram used in this paper. The conventional receiver consists of a chip-matched lter and a bank of despreaders. The outputs of the matched lter are sampled and then despread. We assume that samples are taken chip synchronously. This model can be described by the linear algebraic equation y = AWd + n;

(4)

where the vector y = [y ; y ;    ; yL? ] , and yj = [yjN ;    ; y J N ? ] is the N -vector of samples during the j th interval of duration T = NTc, d = [d ; d ;    ; dL? ] , and dj = [dj ;    ; djK ] is the symbol vector at time j, containing the j -th symbols for all users. n is the additive white Gaussian sampled noise. W is a diagonal matrix of dimension LK determined by (the square root of) the user energies, i.e., T 0

T 0

T 1

T

1

T

diag(W) =

q

T

T

( +1)

1

(1)

T

T

h

T 1

(

q

)

q



(w );    ; (w k );    ; (wLK? ) : (1) 0

1

( ) 0

(

i

) 1

(5)

The matrix A = a ;    ; a K ;    ; aL? ;    ; aLK? , whose columns are the spreading sequences shifted appropriately such that jNk starts at position (1) 0

( ) 0

(1)

(

1

) 1

( )

5

jN + j + 1, i.e., a(jk)

2 6 ) = 640| ; {z  ; 0}; (jNk) ;    ; ((kj+1) ;;0 N ?1 ; 0 | {z }

??

jN +j

(L 1 j )N

?j

3T 7 7 : 5

(6)

The chip waveform matched lter is the front end of the receiver. Despreaders that follow are a bank of spreading sequences, multipliers and summers which correlate the sampled outputs of the matched lters. The outputs after correlation, denoted by yM F , can be written as y M F = AT y:

(7)

As is well known [13], these outputs provide sucient information (a sucient statistics).

3 Multi-Stage Detection In this section, we review the multistage detector proposed by [3] and relate it to Gauss-Seidel procedure and decorrelation. The multi-stage detector works directly with the outputs of the matched lter. From equation (4) and (7), we have yM F = AT y = AT AWd + n0 = HWd + n0 ;

(8)

where H is block diagonal, and is illustrated in Figure 2. Decomposing yM F into sections of size N , i.e., yM F = (yM F; ;    ; yM F;L), we can write y ; as 1

MF r

y MF;r = HL;r?1 Wdr?1 + Hr Wdr + HTL;r Wdr+1 + n0r ;

(9)

where dr = 0 for r < 0 and r > L. In (9), we have broken H into three components (see Fig. 2), where HL;r is the transpose of HL;r. Writing Hr as Hr ? I + I, (9) can then be rearranged into: T

Wd + n0 = y r

r

MF;r

? HL;r Wdr? ? (Hr ? I)Wdr ? HL;r Wdr T

1

6

+1

;

(10)

0

0

T

H= A A =

0 (0)

0 (0)

h 11 h 12 (0) (0) h 21 h(0)2 2 h(0)31 h 3 2 h(0)12 =

0

(0)

h 13 (0) (0) 0 h(0)2 3 h 21 (0) (0) T h 3 3 h 31h 3 2 (1) H0 HL,0 (1) (1) (0) h13 h 11 h 12 h 13 (1) T h2(0)3 h(1)21 h(1)2 2h2(1)3 h21 = HL,0 H1 HL,1 (1) (1) (1) (1) (1) h 31 h3 2 h 3 3 h 31 h3 2 (1) (1) (2) (2) h12 h 13 h 11 h12 h(2)13 HL,1 H2 h2(1)3 h(2)21 h2(2)2 h2(2)3 (2) h 31 h3(2)2 h(2)3 3

Fig. 2: Graphical illustration of H for K=3 asynchronous users. If we have the matched lter outputs as an initial estimate of the symbols Wdr? , Wdr , Wdr , we can use the above equation to compute iteratively: 1

+1

dr (i + 1) = sgn[(Wdr + n0 )] = sgn[yMF;r ? HL;r Wdr?1(i) ? (Hr ? I)Wdr (i) ?HTL;r Wdr+1(i)]:

(11)

where sgn[] is the signum function. The iteration (11) is called a simple iteration, also known as parallel interference cancellation [3]. We can actually realize an improved form of this iterative procedure, which converges faster: dr (i + 1) =

sgn[y ; ? HL;r Wdr? (i) ? (Hr ? I) Wdr (i) ?(Hr ? I) Wdr (i + 1) ? HL;r Wdr (i + 1)]; up

1

MF r

T

low

+1

(12)

where (Hr ? I) is the strictly upper triangular part of (Hr ? I), and (Hr ? I) is the strictly lower part. In (12), we reuse updates of d as soon as they become available, this is known as successive interference cancellation. up

low

7

Equation (11) represents the multistage detector proposed in [3]. As is well known [4], the iterations

Wd (i + 1) = r

y MF;r ? HL;r Wdr?1 (i) ? (Hr ? I)Wdr (i) ?HTL;rWdr+1(i):

(13)

and

Wd (i + 1) = r

y MF;r ? HL;r Wdr?1 (i) ? (Hr ? I)upWdr (i) ?(Hr ? I)low Wdr (i + 1) ? HTL;rWdr+1(i + 1); (14)

are the Jacobi and Gauss-Seidel iterative solutions for the linear equation y M F = HWd, i.e., they are convergent approximations of the solution Wd = H? yM F . Studies in [4] show, however, that the Jacobi iteration diverges for K=N  0:17, but that Gauss-Seidel always converges. The Gauss-Seidel iteration also converges faster than the Jacobi iteration. Equations (11) and (13), (12) and (14) are more di erent than they look. The most important distinction is that equations (11) and (12) form the estimation of the interference components using hard decisions, and subtract the interference components from the matched lter outputs, while (13) and (14) use soft-decision iteratively to compute decorrelation. The quality of the initial estimate of the information bits is very important to the results of (11) and (12), while the soft iterations (13) and (14), converge to H? yM F irrespective of the starting value. Furthermore, iterations (13) and (14) are insensitive to the amplitude and phase of the received signals. The results from iterations (13) and (14) contain complete information and can be used to obtain amplitude, phase and information bits of each user. Iterations (11) and (13), on the contrary, need this knowledge to operate. 1

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4 Gauss-Seidel Soft/Hard Detector Noticing that the initial estimate of the information bits plays an important role in multistage detector performance, Varanasi and Aazhang [5] proposed to use a decorrelator as the front end of the multi-stage detector to achieve better performance. This proposition has several drawbacks, however. First, 8

the computational complexity of the decorrelator is too high to serve as a front end for a less complex multi-stage detector. Second, the decorrelator does not work for all spreading codes and system loads since the matrix inverse does not necessarily exist. Thus, the system might fail in some situations. As we have stated above, the Gauss-Seidel iteration can do di erent jobs for di erent types of feedback: Soft iteration approximates decorrelation (equation (14)), and hard iteration performs interference cancellation (equation (12)). Thus, it is natural to use the Gauss-Seidel iteration to complete a dual mode iterative detection scheme. The rst mode performs soft iterations for a number of steps until decorrelation is achieved, after which the iterations switch to hard decisions. A few hard iterations are then performed. Based upon the better initialization of the second mode improved performance and convergence is accomplished. This dual mode detector is illustrated in Figure 3.

y

Matched lter bank

yM F

GaussSeidel iteration

~d(i + 1)

Unit Delay

W^d(i)

Sign()

Fig. 3: Gauss-Seidel soft/hard joint detection/interference cancellation scheme. 9

As we will show in Section 6, this scheme can have substantial gain over the decorrelator but with much less complexity. Let us analyze the computational complexity of the GSSH detector now. Since the elements in A are 2 f?1; +1g, computation of H = A A involves only addition, and in computational complexity analysis, we are more concerned with multiplication, so we ignore the computation of A A. H is a block diagonal matrix with KL rows and at most 2K ? 1 elements per row (Only K elements in synchronous CDMA). If the matrix is normalized, i.e., the diagonal elements are all unity, each iteration in the Gauss-Seidel method involves at most 2(K ? 1)KL multiplications. If the total number of iterations is M , then the total complexity in term of multiplications is fewer than 2M (K ? 1)KL to decode KL bits, or 2M (K ? 1)/bit (for the decorrelator, O(K M )/bit, where in this case M denotes the window size of the approximation of the inverse of the block-band-diagonal matrix H). T

T

2

5 Amplitude and Phase Estimation The detectors described so far all assume phase and amplitude knowledge for each user. That is, we need to provide such knowledge in some way. In GSSH detection, decorrelation can be achieved before switching to hard iterations. The decorrelator results contain no MAI components and, what is more important, can be achieved without phase/amplitude knowledge. The decorrelated values can then be used to estimate phase and amplitude. Averaging over a block of data is a powerful tool in suppressing Gaussian noise. Assume the signal amplitudes and phase shifts of each user keep constant during a short block and the phase shifts lie in (?=2; =2]. To precise the model developed in Section 2, a phase term is added to each symbol in (1): y (t) =

If we de ne

?

K L X1 X j =0 k =1

dj

(k )

then equation (4) becomes

r

wj(k) eij c(jk)(t ? jT ?  (k) ) + nw (t): (k )

bj = dj (k )

(k )

r

y = Ab + n;

10

(k )

wj(k) eij ;

(15) (16) (17)

where b has element bjk in its (j  K + k)-th position. Now, we have a new variable b, whose maximum likelihood solution, ^b = arg min ky ? Abk = (A A)? A y (18) b ( )

2

T

1

T

can be calculated by the soft iteration procedure discussed above: ^br (i + 1) = y ; ? HL;r br? (i) ? (Hr ? I) br (i) ?(Hr ? I) br (i + 1) ? HL;r br (i + 1): (19) Assuming that the signal amplitudes and phase shifts of each user keep constant during a small frame, and the phase shifts lie in (?=2; =2], the amplitudes and phases of each user can be estimated in the q following way: q k i k k k Since dj 2 f1; ?1g, then E (kbj k) = E [kdj wj e k] = wjk . Without Gaussian noise, the above estimate is exact. In order to suppress the noise, kaverages should be taken for each user over a block of data. Since the bit dj is not known beforehand, steps must be taken to eliminate the e ect of djk . At moderately high signal to noise ratio, we assume sgn[real(bjk )] = sgn[real(^bjk )] for ?=4  jk  =4, with a small probability of error, and sgn[imag(bjk )] = sgn[imag(^bjk )] for ?=2  jk  ?=4 or =4  jk  =2. We describe the ways of averaging for the two cases above. Case 1: ?=4  jk  =4, such that djk = sgn[real(bjk )] = sgn[real(^bjk )] with a small probability of error. up

1

MF r

T

low

( )

( )

( )

( )

+1

(k )

( )

j

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

s

(k ) 1

( )

( )

( )

L ? X 1 = L sgn[real(^bjk )]  ^bjk 1

( )

j =0

?

= L1 =

( )

L X1

p

j =0

sgn[real(bjk )]  [real(bjk ) + nj;rk + i(imag(bjk ) + nj;ik )] ( )

w(k) cos((k) ) 

p

( )

( )

( )

( )

? ? p 1 LX 1 LX nj;rk + if w k sin( k )  nj;ik g L L 1

( )

j =0

( )

( )

1

( )

j =0

+ n k ; (20) where nj;rk and nj;ik are the real and imaginary parts of the enhanced complex Gaussian noise and n k) is complex Gaussian noise (A precise consideration =

( )

w(k) e

i(k )

( )

( )

(

11

shows that E [n k n k ]  N ). Now the amplitude of the signals of user k can be estimated as: pdk w = ks k k: (21) The phase of each user can be estimated as   ^ k = arctan imag(s k )=real(s k ) : (22) ( )

( )

0

( ) 1

( )

( ) 1

( )

( ) 1

Case 2: (?=2  jk  ?=4 or =4  jk  =2), such that sgn[imag(bjk )] = sgn[imag(^bjk )] with a small probability of error. ( )

( )

( )

( )

? L? 1 LX ^bjk )]  ^bjk = 1 X sgn[imag(bjk )]  ^bjk sgn[imag( Lj Lj r L ? L ? X X 1 k k k = L dj sgn[sin(j )]  dj wjk cos(jk )  L1 nj;rk 1

s(2k) =

( )

1

( )

=0

1

( )

?

L X1

L j =0

p

( )

=0

( )

( )

j =0

+1

( )

( )

1

( )

( )

j =0

r

d(jk) sgn[sin((jk) )]  d(jk) wj(k) sin((jk)) 

? 1 LX nk 1

L j =0

( )

j;i

L ? p X 1 = f w cos( )sgn[sin( )]  L nj;rk g + if w k j sin( k )j

 L1

(k )

(k )

?

L X1 j =0

1

(k )

( )

( )

( )

j =0

n(j;ik)g:

(23)

From s k a form analogous to s k can be derived: ( ) 2

( ) 1

s(3k) = sgn[real(s(2k) )]  s(2k)

L ? L ? p k X X 1 1 k k = f w cos( )  L nj;r g + if w sin( )  L nj;ik g

p

=

p

(k )

1

(k )

( )

j =0

w e

(k ) (k ) i

( )

1

( )

( )

j =0

+ n :

(24)

(k )

Now the amplitude of the signals of user k can be estimated as:

pdk

w( ) = ks(3k) k:

12

(25)

The phase of each user can be estimated as   ^ k = arctan imag(s k )=real(s k ) : ( ) 3

( )

(26)

( ) 3

If the range of jk is not known beforehand, both s k and s k need to be calculated. Then choose the one with the larger norm ( )

( ) 1

s

(k )

=

(

( ) 3

s(1k) if ks(1k)k  ks(3k)k s(3k) otherwise:

After the amplitudes and phases have been estimated, the information bits can be initially estimated as: h  i d^jk = sgn real ^bjk conj(s k ) : (27) ( )

( )

( )

d^(jk) and s(k) are then fed back into the multistage detector to complete

the detection. In this estimator, the operations only involve two divisions, two norm calculations and L (frame length) additions. The detector with the estimator is shown in Figure 4. So far, we have assumed  2 (?=2; =2], in fact, like all the other phase estimation schemes, our method has a  ambiguity. This problem can be easily solved using NRZI (Non-return to Zero Inverse) coding, which costs only one bit for a block of information bits of any size.

13

r

Matched lter banks

yM F

GaussSeidel iteration Switch

~d(i + 1)

Unit Delay

W Amplitude/phase Estimator

Sign()

Fig. 4: Gauss-Seidel soft/hard detection scheme with proposed amplitude/phase estimator.

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6 Numerical Results In this section, simulations are presented based on a synchronous BPSK CDMA system. The number of samples per chip is assumed to be one. Each user is given an identical carrier phase shift, which is the worst situation (for BPSK situation), and identical energy. The spreading codes are pseudo random sequences. Figure 5 shows the performance of the Gauss-Seidel iterative detector using soft-decision feedback only, for a set of random spreading codes of length N = 31 for K = 21 users. We can see that Gauss-Seidel iteration converges to decorrelation as expected. Figure 6 shows the performance of the same system if we switch the iterations to hard-decisions after 9 soft iterations, i.e., after decorrelator performance is approached. At this point the bit error rate drops dramatically to a lower error oor. −1

10

−2

Bit Error Rate

10

−3

10

:Gauss−Seidel iterations :Decorrelator N=30,K=20,SNR=11 −4

10

0

5

10

15 Number of iterations

20

25

30

Fig. 5: Performance of a Gauss-Seidel soft iteration detector illustrating convergence to decorrelation. Figure 7 compares the performance of the decorrelator, the multi-stage detector, and the GSSH detector for di erent system loads, where the system load is de ned as K=N . The SNR is chosen to be 7dB . In the simulation, all 15

users are at an identical energy level. The bit error probability of the matched lter receiver rises rapidly to an unacceptable level when the system load reaches one third. The multistage detection (parallel IC) method proposed in [3] shows lower BER but still is not useful at high system load. Decorrelation [2] is better than parallel IC, but has almost the same BER as the matched lter when the load is  0:8. The GSSH detection has the lowest BER and works even for full load. Its BER rises slowly with increasing system load. −1

10

:Decorrelator :Gauss−Seidel iteration with hard decision after 9th iteration N=30, K=20, SNR=11

−2

Bit Error Rate

10

−3

10

−4

10

0

5

10

15 Number of users

20

25

30

Fig. 6: Performance of the Gauss-Seidel soft/hard iteration (GSSH) detector. The performance degradation of the GSSH detector and the decorrelator w.r.t to single user performance is shown in Fig. 8 as a function of the system load for the BER level of 10? . The degradation is measured in degradation factor de ned as the ratio (in dB) of the Eb =N reqiured to achieve BER=10? , in the presence of K users, to that which is required by a single user. Figure 9 shows the performance of the GSSH and the decorrelator as a function of the SNR for the two cases of K=10, and K=21. The processing gain N is 31. With 10 users, the Gauss-Seidel soft/hard iteration has less than 1 dB degradation at BER=10? from single user performance, while the decorrelator su ers about 2 dB degradation. With 21 users, GSSH su ers less than 3 dB degradation while the decorrelator has almost a 6 dB degradation. 2

0

2

3

16

0

10

:GSSH :Single user :Decorrelator :Multistage (parallel IC) :Matched filter SNR=7

−1

Bit Error Rate

10

−2

10

−3

10

−4

10

0

0.1

0.2

0.3

0.4

0.5 System load

0.6

0.7

0.8

0.9

1

Fig. 7: Performance comparison of di erent detectors (processing gain N=31). 16 :Matched filter :Decorrelator :GSSH detector :Processing gain N=31

14

Performance degradation in dB

12

10

8

6

4

2

0

0

0.1

0.2

0.3

0.4

0.5 System load

0.6

0.7

0.8

0.9

1

Fig. 8: Performance degradation of the GSSH and decorrelator at BER=10? , equal power for all users. 2

17

−1

10

−2

10

Bit Error Rate

−3

10

−4

10

−5

10

:GSSH detector :Decorrelator :Single User N=31, random codes −6

10

4

6

8

10 SNR E /N : in dB b

12

14

16

0

Fig. 9: Coherent Gauss-Seidel soft/hard detector performance. −1

10

−2

10

K=21

Bit Error Rate

−3

10

K=10

−4

10

−5

10

:GS soft/hard estimator :GS soft/hard coherent N=31, random codes −6

10

5

6

7

8

9 SNR E /N :in dB b

10

11

12

13

0

Fig. 10: Gauss-Seidel soft/hard detection performance with phase and amplitude estimation. The performance of GSSH detector incorporating the proposed estimator is shown in Figure 10. Amplitudes and phases are estimated over blocks of 50 symbols in this plot. There is no perceptible degradation to ideal 18

phase/amplitude knowledge. To see the e ect of the block length on the performance of the detector, block lengths of L = 5 and L = 10 are also plotted in Figure 11. Compared with L = 50, L = 10 degrades less than 0.1 dB and L = 5 degrades less than 0.3 dB for K=N = 10=31. The degradation for L = 10 is less than 0.4 dB, and for L = 5 is around 1.0 dB for K=N = 21=31. The degradation for larger system loads increases. −1

10

−2

10

K=21

Bit Error Rate

−3

10

K=10 −4

10

−5

10

:Block Length L=50 :L=10 :L=5 Random codes N=31 −6

10

5

6

7

8

9 10 SNR E /N : in dB b

11

12

13

0

Fig. 11: Block length e ect on Gauss-Seidel soft/hard detector performance with phase and amplitude estimation.

7 Conclusion In this paper, we have proposed an iterative scheme (Gauss-Seidel soft/hard) to improve the receiver performance for CDMA systems. This scheme has less complexity and better performance than the decorrelator. Based on the intermediate results from the GSSH detector, a simple estimator is used to obtain the amplitude and phase information necessary to complete detection.

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References [1] S. Verdu,\Minimum Probability of Error for Asynchronous Gaussian Multiple-Access Channels," IEEE Transaction on Information Theory vol. IT-32, no. 1, pp. 85-96, Jan, 1986. [2] R. Lupas and S. Verdu,\Linear multiuser detectors for synchronous code-division multiple-access channels," IEEE Transaction on Information Theory vol. 35, pp. 123-136, Jan 1989. [3] M.K. Varanasi and B. Aazhang, \Multistage detection in asynchronous code-division multiple access communications," IEEE Trans. Commun. vol. 38, pp. 509{519, April 1990. [4] A. Grant and C. Schlegel, \Iterative implementations for linear multiuser detectors," IEEE Trans. Commun., submitted February 1999. [5] M.K. Varanasi and B. Aazhang, \Near-optimum detection in synchronous code-division multiple-access systems," IEEE Trans. Commun. vol. 39, pp. 725{736, May 1991. [6] D. Divsalar and M. K. Simon, \Improved CDMA performance using parallel interference cancellation," JPL Publication 95-21, Oct. 1995. [7] D. Divsalar and M. K. Simon, \Improved parallel interference cancellation for CDMA," IEEE Trans. Commun. vol. 46, pp. 258{268, Feb 1998. [8] Y. C. Yoon, R. Kohno, and H. Imai, \Cascaded co-channel interference cancelling and diversity combining for spread-spectrum multiaccess over multipath fading channels," in Symp. Information Theory and its Applications (SITA 92), Minakami, Japan. Sept. 8-11, 1992. [9] Y. C. Yoon, R. Kohno, and H. Imai, \A spread-spectrum multiaccess system with a cascade of cochannel interference cancellers for multipath fading channels," in Symp. Information Theory and its Applications (ISSSTA 92), Yokohama, Japan. Nov. 29{Dec. 2, 1992.

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[10] Y. C. Yoon, R. Kohno, and H. Imai, \A spread-spectrum multi-access system with cochannel interference cancellation,"IEEE J. Select. Areas Commun. vol. 11, pp. 1067{1075, Sept. 1993. [11] M. Kawabe, T. Kato, A. Kawahashi, T. Sato, and A. Fukasawa, \Advanced CDMA scheme based on interference cancellation," in Proc. 43rd Annu. IEEE Vehicular Technology Conf., May 18{20, 1993. pp.448451. [12] P. Patel and J. Holtzman, \ Analysis of a simple successive interference cancellation scheme in a DS/CDMA system," IEEE J. Select. Areas Commun. vol. 12, pp. 796{807, June 1994. [13] J.G. Proakis,\ Digital Communications," Third edition, Mc-Graw Hill, New York, 1989. [14] S. Moshavi,\Multi-user detection for DS-CDMA communications," IEEE Commun. Mag., pp. 124-136. Oct. 1996.

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