Steady-state analysis of constrained normalized adaptive filters for MAI reduction by energy conservation arguments

ARTICLE IN PRESS

Signal Processing 88 (2008) 326–338 www.elsevier.com/locate/sigpro

Steady-state analysis of constrained normalized adaptive filters for MAI reduction by energy conservation arguments R.L.G. Cavalcante, I. Yamada Department of Communications and Integrated Systems, Tokyo Institute of Technology, 2-12-1-S3-60 Ookayama, Meguro-ku, Tokyo 152-8550, Japan Received 9 April 2007; received in revised form 18 July 2007; accepted 20 July 2007 Available online 29 August 2007

Abstract We derive the steady-state performance of a common class of adaptive filters for multiple access interference (MAI) reduction in code division multiple access (CDMA) systems. The adaptive filters under study utilize estimates of the desired user’s amplitude and are divided into three groups of least-mean-square (LMS) algorithms differing by the choice of the normalization factor. The steady-state performance is deduced from energy-conservation relations that include a possibly erroneous estimate of the desired user’s amplitude. The analyses show that blind algorithms using information about the desired user’s amplitude achieve similar performance to that of nonblind algorithms. In addition, geometric considerations reveal the conditions under which the choice of the normalization factor is expected to have great impact on the convergence properties of the algorithms. Numerical simulations show good agreement with theory. r 2007 Elsevier B.V. All rights reserved. Keywords: Multiuser interference; Energy conservation relations; Steady-state analysis

1. Introduction In direct sequence/code division multiple access systems (DS/CDMA), multiple users share the same channel at the same time. Hence, at the receiver the detection of the transmitted symbols of a desired user is strongly affected by the interference originated from the other users in the system. This interference is known as multiple access interference (MAI). Without proper mitigation, this MAI causes severe performance degradation in the detection of the transmitted information. Corresponding authors. Tel.: +81 3 5734 2503;

fax: +81 3 5734 2905. E-mail addresses: [email protected] (R.L.G. Cavalcante), [email protected] (I. Yamada).

In multiuser systems, a receiver that achieves the minimum bit error rate (BER) generally demands high computational complexity [1]. Therefore, linear adaptive filters with low computational complexity have been introduced to mitigate MAI [2]. Linear adaptive filters try to approximate a filter that minimizes (or maximizes) a mathematically tractable cost function closely related to the BER. Usually, these adaptive filters track the filter maximizing the signal-to-interference-noise ratio (SINR) [2–13], and the main difference between them lies in the choice of the adaptive algorithm. In particular, a great deal of effort has been devoted to the study of blind linear receivers based on the recursive least-squares (RLS) and leastmean-square (LMS) algorithms. RLS-based receivers show good convergence speed [10], but they

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ARTICLE IN PRESS R.L.G. Cavalcante, I. Yamada / Signal Processing 88 (2008) 326–338

demand higher computational complexity than LMS-based receivers and often suffer from numerical instabilities [8] caused by the inherent matrix inversion and possible model mismatch. Moreover, for this particular application, conventional fast versions with linear complexity of the RLS algorithm are hard to be applied because there is no time-index-shifting property of the input data [10]. Furthermore, in relatively fast time-varying scenarios, LMS-based receivers can outperform RLS-based receivers [13] because the tracking behavior of the LMS algorithm is usually superior to that of the RLS algorithm in nonstationary environments [14]. In this study, the focus is on LMS-based receivers. When a training sequence is necessary for the filter adaptation in a LMS-based receiver, the LMS algorithm is said to be nonblind; otherwise, it is said to be blind. Blind LMS algorithms are preferable because the overhead imposed by training sequences is not present, so the throughput of the system is potentially higher. They are usually obtained by restricting the adaptive filter to satisfy a constraint determined by the desired user’s signature. A celebrated blind LMS algorithm for MAI mitigation has been proposed in [2]. Unfortunately, the steady-state performance achieved with this approach is much worse than the performance achieved with nonblind algorithms, especially when the level of background noise is small. This performance difference between blind and nonblind algorithms can be reduced if estimates of the desired user’s amplitude and transmitted symbol are utilized effectively in the blind algorithms [4,9,12,15].

327

We can further divide LMS algorithms into nonnormalized algorithms [2,9] and normalized algorithms [4,5,9,11–13]. Normalized LMS algorithms are frequently used because they are potentially faster than non-normalized LMS algorithms for both uncorrelated and correlated input data [16]. The work in [9] introduces a normalization factor that reduces the sensitivity of LMS algorithms to changes in the number of users and/or the signal-tonoise ratio (SNR). In [4] the normalization factor has been derived from projections onto closed nonconvex sets. By considering geometric properties of projections onto closed convex sets, [5,12,15] have proposed a normalization factor aiming at fast convergence speed. In this study, we deduce closed-form expressions for the steady-state performance of constrained normalized LMS algorithms. The closed-form expressions are derived by including the possible information about the desired user’s amplitude into energy-conservation relations [17–19]. More precisely, we focus on the steady-state performance of the algorithms introduced in [4,5,9,12]. These algorithms are examples of a more general class of receivers presented in [13], which in turn is based on the adaptive projected subgradient method [20]. For simplicity, we divide the algorithms into three different groups according to their normalization factors (cf. Table 1):



Group I [4]: the OPM-based gradient projection (OPM-GP) algorithm, the generalized projection (GP) algorithm, and the space alternating generalized projection (SAGP) algorithm;

Table 1 Characterization of the adaptive filters under study Group

Algorithm

a

b~1 ½i

gðr½iÞ

I I I

GP [4] SAGP [4] OPM-GP [4]

A1 A~ 1

sgnðhTi1 r½iÞ sgnðhTi1 r½iÞ -

kr½ik2 kr½ik2 kr½ik2

II II II

Modified GP Modified SAGP [12] Modified OPM-GP [5]

A1 A~ 1 0

sgnðhTi1 r½iÞ sgnðhTi1 r½iÞ –

r½iT Pr½i r½iT Pr½i r½iT Pr½i

III

BLMS (normalized)[9]

0

–

III

CLMS-AE (normalized)[9]

A~ 1

b1 ½i

III

DD-CLMS-AE (normalized)[9]

A~ 1

sgnðhTi1 r½iÞ

b½i ¼ ð1  kÞb½i  1 þ kkr½ik2 , b½0 ¼ kr½0k2 ; 0oko1 b½i ¼ ð1  kÞb½i  1 þ kkr½ik2 , b½0 ¼ kr½0k2 ; 0oko1 b½i ¼ ð1  kÞb½i  1 þ kkr½ik2 , b½0 ¼ kr½0k2 ; 0oko1

0

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Group II [5,12]: the modified OPM-GP algorithm, the modified GP algorithm, and the modified SAGP algorithm; Group III [9]1: the constrained (normalized) LMS with amplitude estimation (CLMS-AE) algorithm, the decision directed CLMS-AE (DD-CLMS-AE) algorithm, and the blind LMS (BLMS) algorithm.

The steady-state performance analysis reveals that (i) the algorithms of Group II have better convergence rate than those of Group I; (ii) the difference in steady-state performance between Groups I and II (when the same step size is used) increases as the angle between the desired user’s signature and the input signal decreases; (iii) the algorithms of Groups I and III have similar performance when the same amount of information is used; and (iv) proper use of the estimate of the desired user’s amplitude is highly expected to improve the steady-state performance. (NOTE: These results can be immediately extended to adaptive array antenna systems if the sources transmit symbols with constant modulus.) A short conference version of this study was presented in [21]. Here, we present a fully detailed derivation of the steady-state performance and show extensive numerical simulations to confirm the expressions.

where Ak 2 ½0; 1Þ, bk ½i 2 f1; þ1g, sk 2 RM (s1 ; . . . ; sK linearly independent), and n½i are the kth user’s amplitude, kth user’s transmitted bit, kth user’s received signature [2], and noise. Hereinafter we assume that the channel is a slowly fading channel, i.e., the channel is approximately constant for a sufficiently large block of transmitted symbols. Without loss of generality, the first user is the desired user and ksk k2 ¼ sTk sk ¼ 1 (k ¼ 1; . . . ; K). We also use the following common assumptions [9,22]. Assumption 1. The noise vector n½i is a zero-mean Gaussian random vector with Efn½in½iT g ¼ s2n I M , where I M is the M-dimensional identity matrix. Assumption 2. Efbk ½ig ¼ 0, k ¼ 1; . . . ; K and ( 1; q ¼ r; Efbq ½ibr ½ig ¼ 0; otherwise: Assumption 3. bk ½i, k ¼ 1; . . . ; K, and n½i are mutually independent. The detector estimates the transmitted bits with a properly designed filter h 2 RM by b~1 ½i ¼ sgnðhT r½iÞ,

(2)

where  sgnðxÞ ¼

þ1 for xX0; 1 otherwise:

2. Preliminaries

2.2. Optimal constrained filters

2.1. System model

The linear minimum mean squared error (MMSE) filter is defined by

For simplicity, we consider a synchronous BPSK DS-CDMA system with K active users and M chips per symbol over a flat fading channel model. Note that an extension of the analysis to asynchronous systems with constant modulus symbols over a frequency-selective channel model (see, for example, [3, Section II], [22]) is straightforward. If the receiver is synchronized with the desired user, the received vector r½i 2 RM is given by r½i ¼ A1 b1 ½is1 þ

K X

Ak bk ½isk þ n½i.

(1)

k¼2

hMMSE 2 arg min EfjhT r½i  b1 ½ij2 g. opt

Under Assumptions 1–3, we readily verifyPthat hMMSE ¼ A1 R1 s1 , where R ¼ E½r½ir½iT  ¼ K opt k¼1 2 T Ak sk sk þ s2n I M . In addition, from (2), we conclude that the BER is the same for hMMSE and any filter of opt the form h ¼ nR1 s1 , provided that n40. In blind receivers, a commonly used filter is hopt :¼ ðR1 s1 Þ=ðsT1 R1 s1 Þ, which is uniquely given as the solution to the optimization problem hopt 2 arg min EfjhT r½i  ab1 ½ij2 g; h2C s

1 A sound performance analysis for the non-normalized algorithms of this group is also provided in [9]. However, the work in [9] neither discusses the algorithms in [4,5,12] nor derives the steady-state performance by using energy conservation arguments.

(3)

h2RM

8a 2 ½0; 1Þ, (4)

M

T

where C s :¼fh 2 R jh s1 ¼ 1g. Note that, by imposing the constraint C s , the solution does not change irrespective of the choice

ARTICLE IN PRESS R.L.G. Cavalcante, I. Yamada / Signal Processing 88 (2008) 326–338

of a [9,12]. By setting a ¼ 0, hopt can be realized without knowledge of b1 ½i. However, as shown in Section 3, practical blind adaptive algorithms can be realized even with aa0 by replacing b1 ½i by its estimate b~1 ½i, and proper choice of a40 gives better performance than a ¼ 0. 3. Steady-state analysis of adaptive filters for MAI suppression 3.1. Classification of the adaptive filters under study The adaptive filters of interest track the solution hopt in (4) and are expressed by2   r½i hi ¼ P hi1  mðhTi1 r½i  b~1 ½iaÞ (5) þ s1 , gðr½iÞ where h0 ¼ s1 and m is the step size. The parameters a, b~1 ½i, and the normalization factor gðr½iÞ are summarized in Table 1. In this table, the parameter A~ 1 is an estimate of A1 and k is a forgetting factor used to track E½kr½ik2 , the estimation of which is given by b½i. For the algorithms of Groups I and II, the step size m belongs to the range 0omo2; for the algorithms of Group III, m is a sufficiently small positive constant. The algorithms of Group I and their modified versions (algorithms of Group II) differ in the choice of the normalization factor gðr½iÞ. The matrix P:¼ðI M  s1 sT1 Þ is the orthogonal projection onto ðspanfs1 gÞ? , hence P satisfies: (a) P2 ¼ P, (b) Ps1 ¼ 0, and (c) kPxkpkxk, x 2 RM . We introduce for the first time the modified GP algorithm so that the GP algorithm also has its modified counterpart. Note that Px þ s1 is the metric projection of x onto C s , hence hi 2 C s , i.e., hTi s1 ¼ hTopt s1 ¼ 1, i ¼ 0; 1; 2; . . . . For mathematical tractability, we assume in our analyses that b~1 ½i ¼ b1 ½i for the algorithms using aa0. In Section 4, we show that the proposed analysis is in very good agreement with numerical simulations not imposing this assumption. For the algorithms requiring an amplitude estimate, we study the performance by fixing a to a possibly erroneous estimate of A1 . With this approach we see how well the amplitude estimation algorithm should perform in order to achieve a desired SINR level (an efficient amplitude estimation algorithm has been independently proposed in [4] and [9]). 2 See [12] for the details of the derivation of this equation for the algorithms of Groups I and II; for the details of the algorithms of Group III, see [9].

329

For notational simplicity, we define the following: h~ i :¼hi  hopt 2 ðspanfs1 gÞ? , ep ½i:¼r½iT Ph~ i ¼ r½iT h~ i ,

ð7Þ

ea ½i:¼r½i Ph~ i1 ¼ r½i hi1 ,

ð8Þ

vðaÞ½i:¼hTopt r½i  b1 ½ia, uðaÞ½i ¼ hTi1 r½i  b1 ½ia.

ð9Þ

T~

T

ð6Þ

ð10Þ

The above variables can be interpreted as follows:

   

h~ i : weight error distribution ep ½i : a posteriori estimation error ea ½i : a priori estimation error vðaÞ½i : vðaÞ½i ¼

ðA1  aÞb1 ½i |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} desired user’ s residual symbol K X þ Ak bk ½ihTopt sk k¼2

|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} residual interference þ hTopt n½i |fflfflffl{zfflfflffl} filtered noise



[vðaÞ½i is the output of the filter hopt minus the scaled version of the transmitted symbol (ab1 ½i).] uðaÞ½i : output of the filter hi1 minus the scaled version of the transmitted symbol ðab1 ½iÞ.

Note that, unlike the discussion in [17] and [18, Chapter 6], the above definitions include the available information about the desired user’s amplitude. We also adopt the following assumptions to simplify the analysis in Section 3.2. Although these assumptions do not hold in general, they are good approximations due to the reasons given below. Assumption 4. At steady-state, ea ½i and the considered normalization factors gðr½iÞ are independent. The justification of this assumption is given in Appendix A. Assumption 5. v2 ðaÞ½i is independent of r½i. This assumption holds when the noise variance approaches zero because in such a case hTopt r½i ¼ A1 b1 ½i (see [6]). Hence v2 ðaÞ½i is a constant and therefore independent of r½i. Even in more general situations, Assumption 5 simplifies the steady-state

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analysis and shows good agreement with numerical simulations. 3.2. Steady-state performance We introduce one more assumption before proceeding with the steady-state analysis.

3.2.1. Algorithms based on gðr½iÞ ¼ r½iT Pr½i This case is the simplest one because a ¼ b. Hence, from Proposition 1, the steady-state SINR for the algorithms of Group II is given by SINRGroup

II

¼

A21 s2v ðaÞm 2m

þ

hTopt Rhopt

. 

(13)

A21

For fixed step size m, the global maximum of (13) is clearly reached at a ¼ A1 . Thus we conclude that a good estimate of the amplitude A1 is expected to improve the steady-state performance.

Assumption 6.     r½ir½iT Efr½ir½iT g b1 ½ir½i E and E ¼ Efgðr½iÞg gðr½iÞ gðr½iÞ Efb1 ½ir½ig A 1 s1 ¼ ¼ . Efgðr½iÞg Efgðr½iÞg The above relations are reasonable approximations because gðr½iÞ  constant for the normalization factors used in this paper. The justification is similar to the justification of Assumption 4 in Appendix A. Now we are ready to analyze the steady-state performance of the adaptive filters in (5). Proposition 1. Under Assumptions 1–6, the adaptive filters in (5) with b~1 ½i ¼ b1 ½i satisfy (1) (unbiasedness) lim Efhi g ¼ hopt for 0omo2 i!1 and A21 (2) lim SINRðh Þ ¼ , i 2 i!1 sv ðaÞm T 2 þ hopt Rhopt  A1 2b=a  m (11)

3.2.2. Algorithms based on gðr½iÞ ¼ kr½ik2 For these algorithms, by the Cauchy–Schwartz’s inequality, we have  T    r½i Pr½i kr½ik kPr½ik a¼E pE g2 ðr½iÞ g2 ðr½iÞ     2 kr½ik 1 pE 2 ¼ b. ¼E g ðr½iÞ kr½ik2 Hence, the steady-state performance is given by SINRGroup I ¼

X

A21 s2v ðaÞm þ hTopt Rhopt  A21 2b=a  m A21

s2v ðaÞm þ hTopt Rhopt  A21 2m ¼ SINRGroup II .

ð14Þ

(12)

As in the case of the algorithms based on gðr½iÞ ¼ r½iT Pr½i, algorithms of Group I using an estimate of A1 have better performance than those relying on a ¼ 0 (if the step size is the same). In addition, the algorithms of Group I achieve higher SINR than their modified versions (algorithms of Group II) when the same step size is used. Therefore, for a fair comparison,3 different step sizes should be employed. The steady-state performance of the algorithms of Group I depends on the ratio b=a, of which the exact value may not be easily obtained. However, if the bound in (14) is tight, the performance of the

Next, we study the influence of the ratio a=b and the parameter a on the steady-state performance of each group of adaptive filters.

3 For a fair comparison, the step sizes should be selected in such a way that all algorithms present the same convergence rate and then we compare the steady-state performance. A second approach is to set the step sizes in such a way that all algorithms present the same steady-state performance and then we compare the convergence rate [9]. We derived the steady-state performance of all algorithms, so the later approach is used in this study.

where 

 r½iT Pr½i , g2 ðr½iÞ   1 , b¼E gðr½iÞ

a¼E

s2v ðaÞ ¼ Efv2 ðaÞ½ig ¼ a2  2aA1 þ hTopt Rhopt , and SINRðhi1 Þ ¼

A21 . EfjhTi1 r½i  A1 b1 ½ij2 g

Proof. The proof is given in Appendix B.

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r½iT s1 and y½i is the angle between kr½ik r½i and s1 (recall that ks1 k ¼ 1). From (15), if cos y½i is small on average, the ratio a=b is expected to be close to one and thus the bound in (14) is tight. A small cos y½i has the equivalent geometric interpretation of r½i being nearly orthogonal to s1 . From the model in (1), when good spreading codes are used (jsTk sj j5ksk k2 ; kaj) and the power of the interfering users is high, the ratio a=b is expected to be close to one, and we conclude that the bound is tight in such a situation. The ratio a=b for different scenarios is illustrated in Fig. 1.

where cos y½i ¼

Remark 1. The algorithms of Group II have been specially devised to outperform those of Group I when the angle y½i is small. Unfortunately, in such a situation the bound (14) is not tight. Hence, for a fair comparison, the relation between the step size of the algorithms of Group I (denoted by m1 ) and the step size of the algorithms of Group II (denoted by m2 ) has to be set in such a way that SINRGroup I ¼ SINRGroup II . If the same parameter a is the same, SINRGroup I ¼ SINRGroup II is satisfied when m2 a ¼ . m1 b

(16)

3.2.3. Algorithms based on gðr½iÞ ¼ b½i Making use of the fact that kr½ik  constant for well designed signatures, we have that b½i  kr½ik2 (see Appendix A). Therefore, we expect that the algorithms with gðr½iÞ ¼ b½i and those with gðr½iÞ ¼ kr½ik2 perform similarly if the parameter a is the same.

1 0.98 0.96 0.94 a/b

algorithms of Group II is also a good approximation of the performance of the algorithms of Group I. By Pr½i ¼ r½i  ðr½iT s1 Þs1 , we deduce  T  r½i Pr½i E a kr½ik4   ¼ 1 b E kr½ik2   kr½ik2 ks1 k2 cos2 y½i E kr½ik4 n o ¼1 1 E kr½ik 2  2  cos y½i E kr½ik2  , ¼1  ð15Þ 1 E kr½ik2

331

0.92 0.9 0.88 0.86

Gold sequences, K=5 Random sequences, K=5 Gold sequences, K=20 Random sequences, K=20

0.84 0.82 1

2

3

4

5

6

7

8

9

Ak/A1, k=2,…,K

Fig. 1. Influence of the number of users (K), the choice of the signatures, and power relations on a=b. The results were obtained by simulations. Length-31 codes and SNR 15 dB. Note that, for well-designed signatures, such as those based on Gold sequences, and/or for highly interfered channels, the steady-state performance of the algorithms of Group I and their counterparts of Group II is expected to be very close.

Corollary 1. The steady-state SINR achieved by the modified OPM-GP algorithm is bounded by SINRmodified

OPMGP p

2m . m

(17)

Proof. The presence of noise cannot increase the SINR, so the best performance is achieved when sn ! 0. In this situation, we have s2v ðaÞja¼0 ¼ ðhTopt r½iÞ2 ¼ A21

(18)

and hTopt Rhopt ¼ A21

(19)

(see the discussion after Assumption 5). Substitution of (18) and (19) into (13) yields SINRmodified

OPMGP

¼ ð2  mÞ=m,

which concludes the proof.

&

Remark 2 (On Corollary 1). All studied algorithms with a ¼ 0 are expected to achieve similar steadystate performance when good spreading codes are used [see discution after (14)]. In such a situation, the bound in (17) is also a good approximation of the best performance achieved by these algorithms, and we conclude that the choice of the step size implicitly defines the maximum achievable SINR. In contrast, algorithms using the desired user’s amplitude (a ¼ A1 ) satisfy limsn !0;i!1 SINRðhi Þ ¼ 1,

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SINR achieved by the modified OPM-GP algorithm is calculated according to Corollary 1. Hereafter, we define SNR ¼ A21 =s2n . From Fig. 2(a), we see that, when MAI is high and Gold sequences are employed, the algorithms of Group I and their counterparts of Group II show very close performance, as predicted (see also Fig. 1 and (14)). In Fig. 2(b) we decrease the number of users to five. In addition, the signatures are changed to random sequences—i.e., element of sk belongs pffiffiffiffiffiffi each pffiffiffiffiffi ffi to the set f1= M ; þ1= M g and the possible values have the same probability—, and all users have the same power. These changes decrease the angle between the desired user’s signature and the received signal, so the bound in (14) is not expected to be tight because the signatures lose their good cross-correlation properties. Note that Assumption 4 is less reliable under such a scenario. Other parameters are the same as in Fig. 2(a). From the numerical simulations, we see that the gap in terms of steady-state performance between the algorithms of Group I and their modified counterparts of Group II (with the same step size) is more pronounced for the OPM-GP algorithm. This fact is also predicted by (14) because s2v ðaÞ is larger for the OPM-GP algorithm. In Fig. 3 we show that the algorithms of Groups I and III achieve the same steady-state performance when the parameters a and m are the same. To verify the relative performance of blind and nonblind algorithms, we fix the amplitude estimate to A1 for

8m 2 ð0; 2Þ, which can be easily verified by substituting s2v ðaÞja¼A1 ¼ 0 and hTopt Rhopt ¼ A21 into (11). 4. Simulation results Fig. 2(a) shows theoretical and simulated curves for the OPM-GP algorithm, the GP algorithm, and their modified counterparts, which belong to Group II. The algorithms of Group III perform similarly to the algorithms of Group I, so the performance curves of the former group are omitted for visual clarity. The system has 20 users, the signatures are chosen from length-31 Gold spreading codes, the desired user’s amplitude is A1 ¼ 1, and each interferer has 9.54 dB power advantage over the desired user’s power. All algorithms have the same step size m ¼ 0:3, so comparing their relative performance is not possible because such a comparison would not be fair (see discussion after Eq. (14)). However, by fixing all step sizes to the same value, we can verify the following: (i) the tightness of the bound in (14) according to the angle y½i, (ii) the bound given by Corollary 1 for the algorithms that employ a ¼ 0, and (iii) the validity of Assumptions 4 and 5. The simulated values are obtained by averaging the last 1000 out of 3000 iterations of the ensemble-average curve, which in turn is obtained by averaging the SINR curve over 1000 realizations. The ratio b=a, which is necessary to calculate the SINR of the algorithms of Group I, is obtained through simulations. The maximum

30

30 GP (theory) GP (simulation) Modified GP (theory) Modified GP (simulation) Modified OPM−GP (limit) OPM−GP (theory) OPM−GP (simulation) Modified OPM−GP (theory) Modified OPM−GP (simulation)

SINR [dB]

20 15

25 20 SINR [dB]

25

GP (theory) GP (simulation) Modified GP (theory) Modified GP (simulation) Modified OPM−GP (limit) OPM−GP (theory) OPM−GP (simulation) Modified OPM−GP (theory) Modified OPM−GP (simulation)

10 5

15 10 5

−OPM−GP (theory and simulation)

−OPM−GP (theory and simulation)

0

0

−Modified OPM−GP (theory and simulation)

−Modified OPM−GP (theory and simulation)

-5

-5 0

5

10

15 SNR [dB]

20

25

30

0

5

10

15

20

25

30

SNR [dB]

Fig. 2. Simulated and theoretical SINR curves as a function of SNR. (a) High MAI and Gold sequences. (b) Low MAI and Gold sequences.

ARTICLE IN PRESS R.L.G. Cavalcante, I. Yamada / Signal Processing 88 (2008) 326–338 30

SINR [dB]

20 15 10

Modified OPM-GP

12 10

OPM-GP

8 SINR [dB]

25

Steady-state performance (theory)

14

GP (theory) GP (simulation) OPMGP (theory) OPMGP (simulation) BLMS (theory) BLMS (simulation) DDCLMSAE (theory) DDCLMSAE (simulation) CLMS (theory) CLMS (simulation)

333

GP (Theory and simulation) DDCLMSAE (Theory and simulation) CLMS (theory and simulation)

6

BLMS

4

GP

2

5 OPMGP (theory and simulation) BLMS (theory and simulation)

DD-CLMS-AE

0

0

Modified GP

-2

5

200

0

5

10

15 SNR [dB]

20

25

400

600

800 1000 1200 1400 1600 1800 2000

30

Fig. 3. Simulated and theoretical curves as a function of SNR.

Iteration

Fig. 5. Tracking characteristic of the algorithms under study. SNR 15 dB.

16

100

14

10-1

12

10-2

8

BER

SINR [dB]

10

6 4

μ=0.1 (theory) μ=0.1 (simulation) μ=0.3 (theory) μ=0.3 (simulation) μ=0.5 (theory) μ=0.5 (simulation)

2 0 -2

0.5

GP OPM-GP Modified GP Modified OPM-GP DD-CLMS-AE BLMS Optimal

10-4 10-5 10-6

-4 0

10-3

1 α

1.5

2

Fig. 4. SINR as a function of a for different values of m. SNR 15 dB, A1 ¼ 1.

the DD-CLMS-AE and CLMS-AE algorithms. Others parameters are the same as in Fig. 2(b). The parameter k is set to 0.01. As predicted by the derivations, the DD-CLMS-AE, CLMS-AE, and GP algorithms present very close performance, and the performance of the OPM-GP and BLMS algorithms are similar. The numerical results also show that the DD-CLMS-AE and GP algorithms, which are blind, can achieve similar performance to that of the nonblind algorithm CLMS-AE. This confirms the assumption that the estimation of the symbols are reliable over a wide range of SNR at steady state. In Fig. 4, we consider many versions of the modified GP algorithm with amplitude mismatches

2

4

6

8

10

12

14

16

SNR [dB]

Fig. 6. BER as a function of SNR.

ðaaA1 Þ and different step sizes under SNR 15 dB. Other parameters are the same as in Fig. 2(a). From Fig. 4, we see that the algorithm is unable to reduce MAI for a large value of the parameter a because the estimates of the received bits are not reliable. However, for a ¼ 0, which corresponds to the modified OPM-GP algorithm, and for a fairly close to the desired user’s amplitude, the mathematical analysis and the simulation results show remarkably good agreement. In addition, the numerical simulation shows that the algorithm is robust to amplitude mismatches when the step size is small. Other algorithms requiring an estimate of the amplitude perform similarly. In Figs. 5 and 6, we make a fair comparison of the algorithms. First, we fix the step size of the GP algorithm to m ¼ 0:3 and calculate its steady-state

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SINR. Then, the step sizes of all other algorithms are calculated in such a way that all algorithms achieve the same steady-state performance as the GP algorithm. Unless explicitly mentioned, Ak ¼ 3A1 , ka1, and other parameters of the simulations are the same as in Fig. 2(b). The ratio a=b for the algorithms of Group I is obtained by substitution of the expectations in (15) by temporal means in each realization. We assume that the ratio a=b is the same for the algorithms of Groups I and III because these algorithms have similar steadystate performance when the parameters a and m are the same. The BER curves are obtained by considering the last 1000 out of 3000 transmitted symbols. In Fig. 6 we also plot the BER of the optimal filter hopt ¼ R1 s1 =ðsT1 Rs1 Þ for comparison purposes. From Fig. 5, we verify that the algorithms that use estimates of the desired user’s transmitted bits and amplitude are faster than those that rely on a ¼ 0. Comparing algorithms with the same parameter a, we have a slight advantage in terms of convergence speed to the algorithms of Group II. This advantage is not surprising because the algorithms of Group II have been proposed to improve the tracking characteristics of those of Group I [5,15], especially when y½i is small on average. Regarding Fig. 6, we adjusted the step sizes so that all algorithms have the same SINR, therefore the BER performance is almost the same for all compared algorithms. This result is also expected because linear filters achieving the same SINR should have close BER performance (see [9] and references therein).

stead-state performance with good convergence speed. Finally, we have demonstrated that the maximum SINR achieved by the algorithms that do not use any information about the desired user’s amplitude is severely limited by the choice of the step size, and even a rough estimate of this amplitude can greatly improve the steady-state SINR. Acknowledgment The authors would like to thank Prof. Kohichi Sakaniwa of Tokyo Institute of Technology for helpful suggestions and comments. This work was partially supported by JSPS Grants-in-Aid (C-19500186). Appendix A. Justification of Assumption 4 We start by justifying Assumption 4 for gðr½iÞ ¼ r½iT Pr½i. If we rewrite (1) as r½i ¼ SAb½i þ n½i, where S ¼ ½s1 ; . . . ; sK , A ¼ diagðA1 ; . . . ; AK Þ, b½i ¼ ½b1 ½i; . . . ; bK ½iT , then gðr½iÞ ¼ r½iT Pr½i ¼ kr½ik2  ðr½iT s1 Þ2 ¼ b½iT AS T SAb½i þ n½iT n½i þ 2n½iT ASb½i  ðr½iT s1 Þ2 .

Noticing that in CDMA systems the signatures are designed to satisfy sTk sj  0, jak, and the symbols have constant modulus, we get

5. Conclusion b½iT AS T SAb½i  We have studied the steady-state performance of a common class of adaptive filters for MAI reduction. A closed-form steady-state expression for algorithms with different normalization factors has been derived in a unified way by including the possible information about the desired user’s amplitude into the energy conservation relation. This closed-form expression has been useful to compare different normalized algorithms in a fair way in many different scenarios without a time-consuming trial-and-error approach to adjust the step sizes. Even though the algorithms of Group II should use smaller step sizes than those of Groups II and III for a fair comparison, the algorithms of Group II have superior convergence speed. We have also shown that the algorithms (of any Group) that use information about the desired user’s amplitude achieve good

ðA:1Þ

K X

A2k

(A.2)

k¼1

and ðr½iT s1 Þ2  A21 .

(A.3)

Using the law of large numbers [23, pp. 191–194] and the fact that in CDMA systems the spreading factor M is large, we can replace sample means by expectations (a similar approximation is found in [24]), i.e., n½iT n½i M PM nk ½i2 ¼ M k¼1 M  MEfn1 ½i2 g

n½iT n½i ¼ M

¼ Ms2n

ðA:4Þ

ARTICLE IN PRESS R.L.G. Cavalcante, I. Yamada / Signal Processing 88 (2008) 326–338

(U T ¼ U 1 ). Hence

and n½iT ASb½i ¼ M

n½iT ASb½i  0, M

(A.5)

where we define n½i¼:½n1 ½i . . . nM ½iT . In the above derivation, we used the fact that the random variables nj ½i, j ¼ 1; . . . ; M, are i.i.d. and independent of bk ½i, k ¼ 1; . . . ; K. Substitution of (A.2)–(A.5) into (A.1) yields gðr½iÞ 

K X

A2k þ Ms2n ,

k¼2

which is a constant. Hence the independence of ea ½i and gðr½iÞ is a good approximation. Similarly, we can repeat the above derivation for gðr½iÞ ¼ kr½ik2 to show that gðr½iÞ 

335

K X

A2k þ Ms2n .

k¼1

For the algorithms of Group III, by substituting kr½ik2  constant (as shown above) into the definition of b½i in Table 1, we also have that b½i  constant.

Appendix B. Proof of Proposition 1 B.1. Proof of Part 1 An extension of the proof of Part 1 to OSTBCMIMO systems with imperfect channel state information has been given in [25,26]. Here, we show a slightly simplified version of the proof shown in [26] for completeness. We start with the following claim.

PR ¼ R1=2 R1=2 PR1=2 R1=2 ¼ R1=2 USU T R1=2 ¼ R1=2 USU 1 R1=2 . The matrix R has full rank and the positive semidefinite matrix P has rank M  1, so we have that USU T (or, equivalently, USU 1 ) is a positive semidefinite matrix with rank M  1. Thus we can arrange S and U in such a way that S ¼ diagðl1 ;    ; lM Þ with l1 X . . . XlM1 40 and lM ¼ 0. Upon defining Q ¼ R1=2 U 2 RMM , we conclude that PR ¼ QSQ1 .4 To prove that qTM P ¼ 01M , we only need to observe that the left eigenvectors of PR are the columns of QT [27, p. 316], and qM is the eigenvector associated with the eigenvalue 0. Thus we verify that qTM PR ¼ 01M ¼ qTM PRR1 ¼ qTM P. The vector qM is a scaled version of s1 because rankðPRÞ ¼ M  1 and sT1 PR ¼ 01M . Considering that the filter provides b~1 ¼ b1 , we can rewrite (5) as   r½i T þ s1 hi ¼ P hi1  mðhi1 r½i  b1 ½iaÞ gðr½iÞ r½i r½i þ maPb1 ½i ¼ Phi1 þ s1  mPhTi1 r½i gðr½iÞ gðr½iÞ r½i r½i r½iT hi1 þ maPb1 ½i ¼ hi1  mP gðr½iÞ gðr½iÞ  T r½ir½i r½ib1 ½i , ¼ I M  mP hi1 þ maP gðr½iÞ gðr½iÞ where we used the fact that hi1 ¼ Phi1 þ s1 . Using (6) in the above equation and taking expectation on both sides, we arrive at    r½ir½iT ~ Efhi g ¼ I M  mPE Efh~ i1 g gðr½iÞ     r½ib1 ½i r½ir½iT þ maPE  mPE hopt . gðr½iÞ gðr½iÞ

Claim 1. If R has full rank, the matrix PR can be decomposed into PR ¼ QSQ1 , where Q 2 RMM , S ¼ diagðl1 ; . . . ; lM Þ 2 RMM (lk , k ¼ 1; . . . ; M, are the eigenvalues of PR), l1 X    XlM1 40, and lM ¼ 0. In addition, by further decomposing QT into QT ¼:½q1 . . . qM , we have that qTM P ¼ 01M , where 01M is the M-dimensional row vector of zeros and qM is a scaled version of s1 .

To obtain the last identity, we also used the fact that r½i and h~ i1 are independent [see the model in (1)].5

Proof. As R1=2 PR1=2 2 RMM is symmetric, there always exist two matrices U 2 RMM and S 2 RMM such that R1=2 PR1=2 ¼ USU T , where S ¼ diagðl1 ; . . . ; lM Þ, and U is an orthogonal matrix

4 The existence of such a decomposition has been mentioned in [9]. 5 Even if r½i and the filter h~ i1 are not independent, which can happen in frequency selective fading channels, this assumption is frequently used in the analysis of adaptive filters [16,22].

ðB:1Þ

ARTICLE IN PRESS R.L.G. Cavalcante, I. Yamada / Signal Processing 88 (2008) 326–338

336

The last two terms of (B.1) are equal to zero because (i) Ps1 ¼ 0 and (ii) PRhopt ¼ Ps1 =ðsT1 R1 s1 Þ ¼ 0 (see also Assumption 6). Hence,  PR Efh~ i g ¼ I M  m (B.2) Efh~ i1 g. Efgðr½iÞg

reduces to lim Q1 Efh~ i g ¼ ½0 . . . 0 qTM ðPx þ s1  Phopt  s1 ÞT

i!1

¼ 0M1 ,

Decomposing PR into PR ¼ QSQ1 according to Claim 1, we can rewrite (B.2) as

which completes the proof.

&

 m S Q1 Efh~ i1 g IM  Efgðr½iÞg   m m S IM  S Q1 Efh~ i2 g ¼ IM  Efgðr½iÞg Efgðr½i  1Þg   m m S . . . IM  S Q1 Efh~ 1 g ¼ IM  Efgðr½iÞg Efgðr½2Þg  i1 m S ¼ IM  Q1 Efh~ 1 g Efgðr½iÞg 3 2 i1 l1 1  m    0 0 7 6 Efgðr½iÞg 7 6 7 6 6 .. 7 .. .. .. 6 .7 . . . 7Q1 Efh~ 1 g, ¼6 7 6  i1 7 6 lM1 6 0  1m 07 7 6 Efgðr½iÞg 5 4

Q1 Efh~ i g ¼

0



where Efgðr½iÞg¼trðEfr½iT Pr½igÞ¼trðEfPr½ir½iT gÞ ¼ trðPRÞ for the algorithms of Group II or Efgðr½iÞg ¼ trðEfr½ir½iT gÞ ¼ trðRÞ for the algorithms of Groups I and III. We also used the fact that the channel is slowly fading, which implies that Efgðr½iÞg ¼ Efgðr½jÞg, iaj (at least for a block that is large enough to guarantee the PM convergence of the algorithms). As trðPRÞ ¼ k¼1 lk , lk X0 (k ¼ 1; . . . ; M) and R is positive definite, we have that 0plk otrðPRÞ ¼ trðR  s1 sT1 RÞ ¼ trðRÞ  trðsT1 Rs1 ÞotrðRÞ, k ¼ 1; . . . ; M, where trð:Þ stands for trace. Then we conclude that 0olk =trðPRÞo1, for k ¼ 1; . . . ; M  1. Thus, for k ¼ 1; . . . ; M  1, any step size m 2 ð0; 2Þ is enough to guarantee that limi!1 ð1  mlk =Efgðr½iÞgÞi1 ¼ 0 because j1  mlk =Efgðr½iÞgjo1.

Eq. (5) shows that Efh1 g ¼ Pxþ s1 , where x ¼ E h0  mðhT0 r½i  b~1 ½iaÞr½i=gðr½iÞ and h0 is an initial deterministic filter. Hence, as hopt ¼ Phopt þ s1 and qTM P ¼ 0 (from Claim 1), for i ! 1, (B.3)

0

ðB:3Þ

1

B.2. Proof of Part 2 From Assumptions 2 and 3, we have EfjhTi1 r½i  A1 b1 ½ij2 g ¼ EfðhTi1 r½iÞ2 g  2EfhTi1 r½iA1 b1 ½ig þ A21 ¼ EfðhTi1 r½iÞ2 g  A21 .

ðB:4Þ

By noting that (i) hi1 is independent of r½i and that (ii) Efxg ¼ E y fE x fxjygg [17] [18, p. 296] for any two random variables x and y, we deduce an approximation of EfðhTi1 r½iÞ2 g for sufficiently large i as follows. First, we can check that Efe2a ½ig þ hTopt Rhopt ¼ EfðhTi1 r½iÞ2 g  2EfhTi1 r½ir½iT hopt g þ EfhTopt r½ir½iT hopt g þ hTopt Rhopt ¼ EfðhTi1 r½iÞ2 g  2EfEfhTi1 r½ir½iT hopt jhi1 gg þ 2hTopt Rhopt ¼ EfðhTi1 r½iÞ2 g  2ðEfhi1 g  hopt ÞT Rhopt ¼ EfðhTi1 r½iÞ2 g

ði ! 1Þ,

where, in the last line, we used Part 1 of the proposition (limi!1 Efhi g ¼ hopt ).

ARTICLE IN PRESS R.L.G. Cavalcante, I. Yamada / Signal Processing 88 (2008) 326–338

At steady state, (B.4) is equivalently rewritten as EfjhTi1 r½i  A1 b1 ½ij2 g ¼ Efe2a ½ig þ hTopt Rhopt  A21 , i ! 1.

ðB:5Þ

Hence, to obtain the steady-state SINR, we only need to evaluate Efe2a ½ig, i ! 1. For the adaptive filters in (5), by Ps1 ¼ 0, we have   r½i Phi ¼ P hi1  muðaÞ½i . gðr½iÞ With this relation, we follow [17] [18, Chapter 6] to find an approximation for Efe2a ½ig, i ! 1. Subtracting both sides from Phopt :   r½i Ph~ i ¼ P h~ i1  muðaÞ½i . (B.6) gðr½iÞ Thus, left multiplying both sides by r½iT , we have ep ½i ¼ ea ½i  muðaÞ½i

r½iT Pr . gðr½iÞ

(B.7)

Hence ea ½i  ep ½i uðaÞ½i . ¼m gðr½iÞ r½iT Pr½i

(B.8)

Substituting (B.8) into (B.6), we deduce Ph~ i þ

Pr½iea ½i Pr½iep ½i ¼ Ph~ i1 þ T . r½iT Pr½i r½i Pr½i

Taking the square norm of both sides with P2 ¼ P, we arrive at the energy-conservation relation (the cross terms cancel out due to (7) and (8)): T h~ i Ph~ i þ

e2p ½i e2a ½i ~ T Ph~ i1 þ ¼ h . i1 r½iT Pr½i r½iT Pr½i

(B.9)

Taking expectation on both sides and imposing the T T steady-state condition, Efh~ Ph~ i g ¼ Efh~ Ph~ i1 g, i

i1

we get 

e2a ½i E r½iT Pr½i

(

 ¼E

e2p ½i r½iT Pr½i

) .

(B.10)

Substitution of (B.7) into (B.10) yields the variance relation for steady-state performance,     r½iT Pr½i ea ½iuðaÞ½i E mu2 ðaÞ½i 2 . (B.11) ¼ 2E g ðr½iÞ gðr½iÞ Using the relation uðaÞ½i ¼ ea ½i þ vðaÞ½i in the last equation together with Assumptions 4 and 5, we can

337

show that 2Efe2a ½igE



1 gðr½iÞ



 r½iT Pr½i ¼ g2 ðr½iÞ  T  r½i Pr½i þ mEfv2 ðaÞ½igE . g2 ðr½iÞ mEfe2a ½igE



Isolating Efe2a ½ig, we finally get Efe2a ½ig ¼

s2v ðaÞm . 2b=a  m

(B.12)

Therefore, combining (12), (B.5) and (B.12), we arrive at the desired result (11).

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