IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 2, FEBRUARY 2005 103
Performance Analysis and Improvement of
Decorrelating Detection for Multi-Rate DS/CDMA
Huang Lee, Member, IEEE, and Kwang-Cheng Chen, Senior Member, IEEE
Abstract— We study access strategies for decorrelating
detec-tion applied in multi-rate direct-sequence code-division multiple-access (DS/CDMA) systems, including multi-modulation (MM), multi-code (MC), and variable-spreading-length (VSL) schemes by jointly considering signal constellations and multiple-access interference. The mathematical analysis shows that when the number of active users is large, the MM scheme outperforms MC and VSL schemes especially for high-rate transmission. We also conclude that the design of modulation is important in MC and VSL schemes. Numerical analysis demonstrates that applying 4-PSK instead of 2-4-PSK in MC and VSL schemes can improve about9 dB performance gain. In addition, by considering cross-correlation of noise components, we propose a detector that minimizes the symbol error probability under the constraint that the complexity grows linearly with the number of active users as decorrelating detectors. Simulations show that about 4 dB performance gain over conventional decorrelating detectors can be achieved for multi-rate DS/CDMA communications.
Index Terms— Decorrelating multiuser detection, code-division
multiple-access, multi-rate, multi-code, variable-spreading-length, multi-modulation.
I. INTRODUCTION
I
N THE past few years, there has been much researchon multi-rate direct-sequence code-division multiple-access (DS/CDMA) systems. According to previous research, multi-code (MC) [1] and variable-spreading-length (VSL) [2] are two widely considered access strategies in multi-rate systems. With conventional detection, it was shown in [3] [4] that MC and VSL schemes offer similar bit-error-rate (BER) perfor-mance and always outperform the multi-modulation (MM) scheme [3] for high level modulation (e.g., 16-QAM), where the MM strategy realizes multi-rate transmission by varying the number of elements in signal constellations. However, this result does not necessarily apply to systems considering multiuser detection such as decorrelating detection.
In this letter, we revisited decorrelating detection for MC, VSL, and MM schemes over synchronous additive white Gaussian noise channels. In [4] [5] [6], MC and VSL schemes were analyzed only by evaluating the multiple-access interfer-ence (MAI) introduced by spreading sequinterfer-ences, which ignored the fact that multi-rate can also be realized by designing signal constellations as the MM scheme. To thoroughly understand the design of multi-rate communications, a complete analysis of different multi-rate strategies is very much desired.
In addition, previous studies [4] [5] [6] on the low-rate decorrelator (LRD), which detects information bits in a low-rate interval, ignored the cross-correlation of noise compo-nents so that the performance of LRD is degraded. By taking Manuscript received May 20, 2004. The associate editor coordinating the review of this letter and approving it for publication was Prof. M. Saquib.
The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan (e-mail: [email protected]).
Digital Object Identifier 10.1109/LCOMM.2005.02019.
the cross-correlation into consideration, we propose a novel detector to improve the performance over LRD.
II. SYSTEMMODEL
For simplicity, we consider a dual rate DS/CDMA system
where the high data rater2is an integer multipleM of the low
data rate r1, and all low-rate users are assigned a spreading
sequence with length N and antipodal modulation. Assume
that (a) there areK1low-rate users (LRUs) andK2high-rate
users (HRUs) whereK K1+K2; (b) the average bit energy
Eb of all users are identical; and (c) the spreading sequences
used in the system are linearly independent.
After chip-matched filtering, the generalized form of the received signal in low-rate symbol interval can be written as:
rK
k=1ak
Dk
d=1xk,dsk,d+ n = SAx + n (1)
where |ak| and ak are the received amplitude and phase of
user k; Dk is the number of spreading sequences assigned
to user k; xk,d and sk,d represent the received symbols
and spreading sequences of user k; n is the N × 1
zero-mean Gaussian random vector (GRV) with covariance matrix
σ2IN×N; S [s1,1 · · · sk,1 · · · sk,Dk · · · sK,DK]N× K;
A diag {a1, · · · , ak, · · · , aK} is a K× K diagonal matrix;
x [x1,1 · · · xk,1· · · xk,Dk · · · xK,DK]
T is a K × 1 vector;
K Kk=1Dk; and the superscript (·)
T
denotes transpose. We specify the parameters of (1) for different schemes:
• MM scheme: Dk=1 for all users; sk,1=√1N[ck,1,1 · · ·
ck,1,N]T;xk,1∈ {PAM/PSK/QAM signal constellation}.
• MC scheme: Dk=1 for LRUs, and Dk=M for HRUs.
sk,d=√1N[ck,d,1 · · · ck,d,N]T.xk,d ∈ {+ √ Eb,− √ Eb}. • VSL scheme: Dk=1 and sk,d=√1N [ck,d,1 · · · ck,d,N]T
for LRUs. Dk=M and sk,d=
M N[0
T
(d−1)N/M ck,d,1 · · · ck,d,N/M 0T(M−d)N/M]T for HRUs where 0n represents
the n × 1 zero vector. xk,d∈ {+
√ Eb,−
√ Eb}.
ck,d,n∈ {+1, − 1} are the chip codes.
III. DECORRELATINGDETECTION
For any user k, the received signal r can be decorrelated
by yk 1 ak SH kSk− SHkTk TH kTk −1 TH kSk −1 · SH k I − Tk TH kTk −1 TH k r (2) where Sk sk,1 sk,2 · · · sk,Dk N×Dk, Tk is the N × K − Dk
matrix by striking out all columns related
to the user k from the matrix S, and the superscript (·)H
denotes Hermitian operator. After decorrelating, we define
104 IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 2, FEBRUARY 2005
nk the noise vector corresponding to user k’s signal xk
xk,1 xk,2 · · · xk,Dk
T
, and its Dk × Dk covariance
matrix is given by E nknHk = σ2|ak|−2Rk where Rk = SH kSk− SHkTk TH kTk −1 TH kSk −1 .
The LRD [4] [5] [6] ignores the fact that Rk is not a
diagonal matrix, yielding the following decision for userk:
xLRD k arg maxx k Re 2yHkxk− xHkxk . (3)
However, since the noise components innk are correlated,
the matrixRkshould be considered in the detector to improve
the performance. We propose the modified detector
xk arg minx k yk− xk H R−1 k yk− xk = arg max xk Re 2yHk xk− xHk xk (4)
wherexk =R−1k xk. In Appendix, we prove that the modified
detector is optimal in the sense of minimizing the symbol
error probability while the receiver of userk is only allowed
to utilize statisticYk as LRD. Therefore, we call the
modi-fied decorrelating detector the optimum decorrelating detector (ODD). Note that the performance of ODD is still worse than that of the jointly optimum multiuser detector [7] because the
receiver of userk observes only Yk not Yk for k= 1, . . . ,
K, and suffers the noise enhancement effect. Besides, because
the ODD improves the performance by considering the
cross-correlation ofnk, it is easy to show that LRD and ODD have
the same performance whenDk is equal to one.
IV. SYSTEMANALYSIS
We prove in Appendix that when the noise power is small enough, the performance can be evaluated by the normalized minimum distance d2min min xk=xk 1 Eb (x k− xk) H R−1k (xk− xk) . (5)
In addition to the minimum distance, MAI is also character-ized in (5) since MAI can be regarded as the noise
enhance-ment effect represented byR−1k while applying decorrelating
detector. Moreover, because R−1k is composed of spreading
sequences, the performance is often dependent on the use of spreading sequences [6] [8], thus making global comparison of access schemes difficult. To remove the influence of spreading sequences in the comparison, we employ the random spreading sequence approach to analyze performance as [3] [4] [8]. The expectation value of the normalized minimum distance is
d2min min xk=xkE 1 Eb(x k− xk) H R−1 k (xk− xk) . (6)
The d2min of LRUs can be obtained from the first LRU
without loss of generality and is given by
d2min= γlowE
1− sH1,1T1T1HT1−1TH1 s1,1
= γlowηlow
whereγlow= min
x1=x1
1
Eb|x
1− x1|2is the normalized minimum
distance of the signal constellation without MAI. By the similar procedure, the expectation value of the normalized
8 10 12 14 16 18 20 22 24 26 10−5 10− 4 10− 3 10− 2 10− 1 100 Eb/N0 (dB) BER
Low−Rate Users; R1=1; MC 2−PSK (ODD) High−Rate Users; R2=4; MC 2−PSK (ODD) High−Rate Users; R2=4; MC 2−PSK (LRD) High−Rate Users; R2=4; MC 4−PSK (ODD)
8 10 12 14 16 18 20 22 24 26 10−5 10−4 10−3 10−2 10−1 10 0 Eb/N0 (dB) BER
Low−Rate Users; R1=1; VSL 2−PSK (ODD) High−Rate Users; R2=4; VSL 2−PSK (ODD) High−Rate Users; R2=4; VSL 2−PSK (LRD) High−Rate Users; R2=4; VSL 4−PSK (ODD)
(a) MC scheme. (b) VSL scheme.
Fig. 1. BER vs SNR.K1= 6, K2= 6, r1= 1, and r2= 4.
minimum distance of HRUs is given by d2min = γhighηhigh
whereγhigh= min
xk=x k
1
Eb|x
k− xk|2andk indicates any HRU.
The noise enhancement effect and the pattern of signal
constellations can be characterized by ηlow (ηhigh) and γlow
(γhigh), respectively. Further, the coefficient ηlow (ηhigh) is
equal to the optimum near-far resistance [7] and can be regarded as a coefficient that shortens the normalized mini-mum distance. Therefore, the performance analysis of different schemes is achieved by jointly considering MAI and signal constellations.
The d2min of the MM scheme is given by
d2min=
γlow(1− (K1+ K2− 1) /N) LRU
γhigh(1− (K1+ K2− 1) /N) HRU. (7)
The γlow and γhigh of MC and VSL schemes are equal to
4, and the ηlow and ηhigh of MC and VSL schemes can be
obtained by Propositions in [8]. For the MC scheme, we get
d2min=
4 (1− (K1+ M K2− 1) /N) LRU
4 (1− (K1+ M K2− 1) /N) HRU. (8)
For VSL (with general random code [8] applied), we obtain
d2min=
4 (1− (K1+ M K2− 1) /N) LRU
4 (1− (K1+ M K2− M) /N) HRU. (9)
In the MM scheme, the coefficientsγlowandγhighare equal to
4 for 2-PSK and 4-PSK modulation, and γhigh is smaller than
4 for higher level modulation (e.g., γhigh= 85 for 16-QAM).
Hence, for high level modulation, when the number of active users is small, MC and VSL schemes have better performance than MM. However, once the number of active users increases,
d2minof MC and VSL decreases more rapidly than that of MM
scheme so that the MM scheme can outperform MC and VSL.
V. SIMULATIONRESULTS
In all simulations, we apply random spreading sequences
[3] [4] [8] with length N equal to 32.
We demonstrate that the proposed ODD achieves better performance than LRD. In our simulation, ODD improves the performance about 5 and 4 dB over LRD for high-rate MC and VSL users, respectively, as shown in Fig. 1.
We investigate the BER performance for different access
strategies and transmission rates as the number of HRUs K2
increases in Fig. 2. We apply ODD in simulations where the
signal-to-noise ratio (SNR =Eb
2σ2) is fixed to be 10 dB. The
numerical analysis indicates that when the number of active users increases, the performance of MC and VSL schemes degrades more rapidly than that of MM scheme.
LEE and CHEN: PERFORMANCE ANALYSIS AND IMPROVEMENT OF DECORRELATING DETECTION FOR MULTI-RATE DS/CDMA 105 2 3 4 5 6 7 8 9 10 11 12 10−5 10−4 10−3 10−2 10−1 10 0
Number of High-rate Users
BER
Low−Rate Users; r1=1; 2−PSK High−Rate Users; r2=2; 4−PSK High−Rate Users; r2=2; 4−PAM Low−Rate Users; r1=1; MC 2−PSK High−Rate Users; r2=2; MC 2−PSK Low−Rate Users; r1=1; VSL 2−PSK High−Rate Users; r2=2; VSL 2−PSK 2 3 4 5 6 10−5 10−4 10−3 10−2 10−1 100
Number of High−rate Users
BER
Low−Rate Users; r1=1; 2−PSK High−Rate Users; r2=4; 16−QAM Low−Rate Users; r1=1; MC 2− PSK High−Rate Users; r2=4; MC 2− PSK Low−Rate Users; r1=1; VSL 2− PSK High−Rate Users; r2=4; VSL 2− PSK High−Rate Users; r2=4; MC 4− PSK High−Rate Users; r2=4; VSL 4− PSK M = 2 M = 4
Fig. 2. BER vs number of high-rate usersK2.K1= 6.
Above analyses show that modulation plays an important role in multi-rate systems, which leads us to modify mod-ulation schemes in MC and VSL strategies. We verify that utilizing 4-PSK instead of 2-PSK in MC and VSL schemes improves performance, even though applying 4-PSK and 2-PSK gives the same performance in single-rate [9] and MM multi-rate systems. Because 4-PSK MC and VSL schemes utilize less spreading sequences, they suffer less MAI than 2-PSK MC and VSL schemes. The performance therefore degrades slower than that of 2-PSK MC and VSL schemes as show in Fig. 2 (b). It is also shown in Fig. 1 that applying 4-PSK in MC and VSL schemes improves the performance about 9 dB performance gain.
VI. CONCLUSIONS
The performance of MM, MC, and VSL access strategies was analyzed by the normalized minimum distance such that the structure of signal constellations and MAI are jointly considered. Both the mathematical and numerical analyses demonstrated that the noise enhancement effect does more harm to MC and VSL strategies especially for high-rate users. Because the performance of MC and VSL degrades more rapidly as the number of active users increases, the MM scheme outperforms MC and VSL schemes when the number of active users is large.
In addition, a modified decorrelating detector for multi-rate systems was proposed. We proved that the proposed detector minimizes the symbol error probability while maintaining reasonable complexity, which grows linearly with the number of active users as LRD. The cross-correlation of noise com-ponents ignored by LRD was considered to achieve around 4 dB performance gain over LRD.
We showed that the design of modulation is critical in multi-rate systems. The MM scheme usually provides smaller minimum distance than MC and VSL schemes. However, the MM scheme also induces less MAI in systems. When applying MC and VSL schemes, we can design their signal constellations to improve the performance. In our simulations, we applied 4-PSK to obtain about 9 dB performance gain. The trade-off on the design of modulation schemes depends on the number of active users and spreading sequences. Since the normalized minimum distance reflects the effect of modulation schemes and MAI, we can use it to evaluate the performance when designing multi-rate communications.
APPENDIX
We prove that the proposed detector minimizes the symbol
error probability. The HRU k observes statistic Yk as LRD,
and makes decision among M = 2r2 possible transmitted
vectors {θ1,θ2, · · · , θM}. The statistic Yk is a GRV with
covariance R = σ2|ak|−2Rk. Define the observation space
Ω that contains all possible value Yk and is partitioned
into M disjoint subspace Ωn where Ω = ∪Mn=1Ωn. Denote
P (Yk ∈ Ωn| θm) the probability that the receiver chooses
θn when θm is transmitted. The symbol error probability is
given by Pe= 1 M M m=1 M n=1,n=mP (Yk ∈ Ωn| θm) = 1− 1 M M m=1 ···Ωmfθm yk dyk where fθm yk
is the probability density function of
GRV with mean θm and covariance R. The optimum
so-lution that minimizes Pe is given by choosing Ωm =
yk | fθm(yk)≥ fθn(yk), m = n
. Hence, it suffices to say
that the optimum receiver should be xk = arg max
xk fxk(yk)
equal to (4).
We derive the upper bound of Pe. Define the random
variable Tn,m= Re (xn− xm) HR−1 Yk− 1 2(xn+ xm) .
Since Tn,m(Yk) is the linear transformation of
GRV, Tn,m(Yk) is Gaussian random variable. We
can get its mean tn,m = E [Tn,m(Yk)|θm] =
−1 2 (xn− xm) HR−1 (xn− xm) and variance d2n,m = ETn,m(Yk)− tn,m 2|θ m = −2tn,m under θm
transmitted. The Peis upper bounded by
Pe≤ 1 M M m=1 M n=1,n=m ∞ 0 1 √ 2πdn,m e −(tn,m+tn,m)2 2d2n,m dt n,m = 1 M M m=1 M n=1,n=m Q ⎛ ⎝|ak| (xn− xm) HR−1 (xn− xm) σ ⎞ ⎠ . When the noise power is small enough, the performance is consequently dominated by the minimum distance (5).
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