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Blind intersymbol decorrelating detector for asynchronous multicarrier CDMA system
ARTICLE IN PRESS
Signal Processing 85 (2005) 1511–1522 www.elsevier.com/locate/sigpro
Blind intersymbol decorrelating detector for asynchronous multicarrier CDMA system Gaonan Zhanga,b, Guoan Bia,b,, Qian Yua,b a Block S1, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore Block S2, Information System Research Lab, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
b
Received 6 June 2003; received in revised form 7 February 2005
Abstract This paper presents a blind decorrelating detector for asynchronous multicarrier CDMA systems with Rayleigh fading channel. The detector is derived by making use of the cross correlation matrix between the consecutively received signals. The main attraction of the detection algorithm is its simplicity since the detector can be implemented blindly without any channel estimation except for the synchronization of the desired user. Simulation results are provided to demonstrate the significant gains in performance and simplicity achieved by the proposed detector. r 2005 Elsevier B.V. All rights reserved. Keywords: Multicarrier CDMA; Decorrelating detector; Intersymbol information
1. Introduction Recently, various multicarrier code-division multiple-access (MC-CDMA) systems have been proposed [1,2]. The MC-CDMA is basically the combination of orthogonal frequency division multiplexing (OFDM) and CDMA [3,4]. The main advantage of OFDM is that the chip duration can be significantly increased to reduce frequencyCorresponding author. Tel.: +65 6790 4823; fax: +65 6793 3318. E-mail addresses: [email protected] (G. Zhang), [email protected] (G. Bi), [email protected] (Q. Yu).
selective fading. In MC-CDMA systems, each user is transmitted simultaneously over multiple subcarriers without any spreading per subcarrier. The resultant signal is a combination of multiple narrowband signals with regularly spaced subcarrier frequencies in the frequency domain. Hence, with the narrowband of the subcarriers, the frequency-selective fading can be reduced to the limit that the signal associated with each subcarrier behaves like one with flat fading. In contrast, the direct-sequence CDMA (DS-CDMA) systems need a complex RAKE receiver to equalize the channel fading. Another advantage of MC-CDMA is that the signal in each subcarrier
0165-1684/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2005.02.006
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can be dealt with by low-speed parallel-type processing, while the DS-CDMA systems require a high-speed serial-type processing. Because of the desirable advantages, MCCDMA systems have received significant attention and been considered as a promising candidate for future wideband wireless communications. Meanwhile, multiuser detections for MC-CDMA systems were also proposed in the literature. For example, several decorrelating detectors for synchronous MC-CDMA systems were developed in [5,6]. For asynchronous reception, the orthogonality among different subcarriers and the spreading codes of different users are destroyed. In [7], the receivers for asynchronous MC-CDMA systems using equal-gain combining (EGC) and maximum-ratio combining (MRC) were reported. The MRC detector achieves an optimum performance in the signal user case, while a perfect estimation of the timing and the fading channel of the desired user is required [7]. The EGC detector achieves a limited performance with a low complexity for implementation. In this paper, a new detection scheme is presented for asynchronous MC-CDMA systems. The proposed method avoids the channel estimation by exploiting the intersymbol information. The key idea of the proposed detector is the utilization of a cross correlation matrix constructed by the consecutively received signals. It is shown that the proposed detector is equivalent to the decorrelating detector [8]. Unlike the previous MRC detector, our method does not require the
channel estimation of the desired user, which leads to a simpler implementation. The simulation results show that the proposed detector significantly outperforms the EGC and MRC receivers. The paper is organized as follows. Section 2 presents the signal model of the asynchronous MC-CDMA system, and Section 3 introduces the proposed decorrelating detector. In Section 4, simulation examples are provided to demonstrate the performance of the proposed detector. Section 5 contains the conclusion.
2. Signal model Let us consider an asynchronous MC-CDMA system with K simultaneous users and N subcarriers in a frequency-selective Rayleigh fading channel. The system uses the Gold codes of length N followed by an OFDM modulator to construct the spreading codes of different users. The transmitter block diagram of the asynchronous MC-CDMA system for user k is shown in Fig. 1. In this figure, the transmitter spreads the ith data symbol of the kth user, i.e., bk ðiÞ 2 f1; 1g; in the frequency domain by using the kth user’s spreading code fck;n jn¼0;1;...;N1 g: The transmitted signal for the kth user can be written as sk ðtÞ ¼ ak
1 X i¼1
bk ðiÞ
N 1 X
ck;n
n¼0
Pðt iT s tk Þej2pnðttk Þ=T s ,
Fig. 1. Transmitter block diagram for the kth user of the asynchronous MC-CDMA system.
ð1Þ
ARTICLE IN PRESS G. Zhang et al. / Signal Processing 85 (2005) 1511–1522
where ak is the amplitude of the kth user, the delay of the kth user tk is a random variable and uniformly distributed in ½0; T s ; PðtÞ is the rectangular pulse waveform normalized uniformly in the interval ½0; T s and T s is the symbol duration. After passing through the frequency-selective fading channel, sk ðtÞ is affected by multipath fading and the additive white Gaussian noise (AWGN). By suitably choosing N and PðtÞ [9], we can assume that each subcarrier undergoes independent frequency non-selective slow Rayleigh fading and bk;n has a Rayleigh distribution and a uniform distribution for the phase between 0 and 2p: The final received signal of all active users is given by yðtÞ ¼
K X
1 X
ak
k¼1
bk ðiÞ
i¼1
N 1 X
Pðt iT s tk Þe
þ zðtÞ,
ð2Þ
where zðtÞ is the AWGN process with zero mean and variance s2 ; bk;n accounts for the overall effects of phase shift and fading for the nth subcarrier of the kth user with a Rayleigh distribution. At the receiver end, the received signal in (2) is correlated at each subcarrier frequency as shown in Fig. 2. After sampling the output of each subcarrier correlator at the symbol rate, the output of the nth correlator at the ith symbol is given by 1 rn ðiÞ ¼ pffiffiffiffiffiffi Ts
Z
K X
ðiþ1ÞT s
½hk;n bk ðiÞ þ h¯ k;n bk ði 1Þ þ vn ðiÞ,
where 1 X 1 N hk;n ¼ pffiffiffiffiffiffi ak ck;l bk;l ej2pltk =T s T s l¼0 Z Ts
ej2pðlnÞt=T s dt,
ð4Þ
tk 1 X 1 N h¯ k;n ¼ pffiffiffiffiffiffi ak ck;l bk;l ej2pltk =T s T s l¼0 Z tk
ej2pðlnÞt=T s dt,
ð5Þ
0
Z
ðiþ1ÞT s
zðtÞej2pnt=T s dt
(6)
iT s
for n ¼ 0; 1; . . . ; N 1: Without loss of generality, we assume that user 1 is the desired user which has been synchronized, i.e., t1 ¼ 0: According to (5), it is easy to prove h¯ 1;n ¼ 0; n ¼ 0; 1; . . . ; N 1 (7) which implies that the intersymbol interference (ISI) of the desired user is completely removed by synchronization operation, while the ISI of the other users still exist because of their delays. By using (4)–(6) and stacking the N outputs of the subcarrier correlators into a vector, rðiÞ9½r0 ðiÞ; r1 ðiÞ; . . . ; rN1 ðiÞ T ; where ½ T is the matrix transpose operator, the final output of the receiver can be given by
yðtÞej2pnt=T s dt
iT s
ð3Þ
k¼1
1 vn ðiÞ ¼ pffiffiffiffiffiffi Ts
ck;n bk;n
n¼0 j2pnðttk Þ=T s
¼
1513
rðiÞ ¼ h1 b1 ðiÞ þ
K X
½hk bk ðiÞ þ h¯ k bk ði 1Þ þ vðiÞ
k¼2
¼ h1 b1 ðiÞ þ HI bI ðiÞ þ vðiÞ, where hk 9½hk;0 ; hk;1 ; . . . ; hk;N1 T h¯ k 9½h¯ k;0 ; h¯ k;1 ; . . . ; h¯ k;N1 T HI 9½h2 ; h3 ; . . . ; hK ; h¯ 2 ; h¯ 3 ; . . . ; h¯ K bI ðiÞ9½b2 ðiÞ; b3 ðiÞ; . . . ; bK ðiÞ; b2 ði 1Þ, b3 ði 1Þ; . . . ; bK ði 1Þ T Fig. 2. Receiver block diagram of the asynchronous MCCDMA system.
vðiÞ9½v0 ðiÞ; v1 ðiÞ; . . . ; vN1 ðiÞ T .
ð8Þ
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In (8), h1 and b1 ðiÞ contain the information of the desired user. The multiple access interference (MAI) and ISI of the interfering users are represented by HI and bI ðiÞ: It is easy to see from (6) that vðiÞ is a white Gaussian noise vector with an N N covariance matrix 3
2
s2 6 pffiffiffiffiffiffi 6 Ts 6 6 0 6 Corv ¼ 6 6 . 6 .. 6 6 4 0
0 2
psffiffiffiffi Ts
.. .
..
0
.
0 7 7 7 0 7 7 7. .. 7 . 7 7 s2 7 pffiffiffiffiffiffi 5 Ts
2 d1 9argmin kdH 1 ½h1 ; HI k
subject to : dH 1 h1 ¼ 1, (11)
where d1 is the decorrelating detector for user 1 having the following properties: dH 1 h1 ¼ 1, dH 1 HI ¼ 0.
(9)
ð12Þ
3.2. Classical blind implementation of the decorrelating detector Let us now consider the conventional blind subspace implementation of the decorrelating detector in (12). Denote the autocorrelation matrix of the received signals as
3. Blind intersymbol decorrelating detector Let us consider the blind symbol-by-symbol detection for the desired user. The canonical form of a linear detector for the desired user can be represented by a correlating vector w1 2 RN followed by the received signal rðiÞ; such that the detection of user 1 is b^1 ðiÞ ¼ sgnfRðwH 1 rðiÞÞg,
weight vector d1 ¼ w1 ; given by
(10)
H
where ½ denotes conjugate transpose operator. A straightforward single user criterion for optimization of (10) is the maximization of the power of the desired user, i.e., w1 ¼ ½c1;0 b1;0 ; c1;1 b1;1 ; . . . ; c1;N1 b1;N1 T . This is referred to as MRC detector. A simpler implementation of the detector in (10) is the EGC detector, which is given by w1 ¼ c1 ; where c1 ¼ ½c1;0 ; c1;1 ; . . . ; c1;N1 T : 3.1. Canonical representation of linear decorrelating detector Let us focus on the linear decorrelating detection method, which is designed to completely eliminate the MAI and ISI from all interfering users at the expense of enhancing the ambient noise [8]. It has a canonical form of (10) with the
s2 R ¼ EfrðiÞrðiÞH g ¼ ½h1 ; HI ½h1 ; HI H þ pffiffiffiffiffiffi IN , Ts (13) where E½ is the expectation operator. By performing the eigendecomposition of R; we obtain 2 3 0 Ls 5½Us ; Un H , 2 R ¼ ½Us ; Un 4 ð14Þ 0 psffiffiffiffi I T N2Kþ1 s
where Us is an N ð2K 1Þ dimensional orthonormal basis of the signal subspace spanned by ½h1 ; HI and Un is the basis for the white noise subspace orthogonal to Us : The diagonal matrix Ls contains pffiffiffiffiffiffi the 2K 1 largest eigenvalues of R and s2 = T s is the eigenvalue of white noise. Thus, the canonical expression of the decorrelating detector in subspace form can be given by [10] d1 ¼
1 Us pffiffiffiffiffiffi T s ÞI2K1 Þ1 UH s h1
1 s2
Ls pffiffiffiffiffiffi I2K1 UH s h1 . Ts hH 1 Us ðLs
ðs2 =
ð15Þ
In (15), Un and Ls can be estimated from the received signal. In addition, the detector also requires to know the channel information of the desired user h1ffi and the variance of the background pffiffiffiffiffi noise s2 = T s : For blind multiuser detection,
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however, the detector is assumed to have the only prior knowledge of timing information and the spreading sequence of the desired user. Hence, parameter estimation is required for the method described in (15). When the background pffiffiffiffiffiffi noise is white as discussed in Section 2, s2 = T s can be estimated through an eigendecomposition of R by calculating the average eigenvalue of noise subspaces Un : The desired user’s channel information h1 can be estimated by exploiting the orthogonality between the signal subspace and the noise subspace and by oversampling the received signal [11]. However, channel estimation substantially increases the complexity of the detector and often leads to serious deterioration of the performance due to the errors of the estimation.
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the decorrelating detector. By performing the ¯ we obtain eigendecomposition of R; " ¯ n ¯ ¼ ½U ¯ s; U R
#
¯s L 0
¯ n H , ¯ s; U ½U
¯ sL ¯ sU ¯H ¼U s
ð17Þ
¯ s ¼ diagðl¯ 1 ; . . . ; l¯ 2K2 Þ contains 2K 2 where L ¯ s are the corresponding non-zero eigenvalues and U ¯ eigenvectors. With the eigendecomposition of R; we have the following lemma: ¯ s is equal to the null Lemma 1. The null space of U space of HI ; i.e., ¯ s Þ ¼ rangeðIN U ¯ sU ¯H nullðHI Þ ¼ nullðU s Þ.
3.3. Intersymbol blind implementation of the decorrelating detector
Proof. See Appendix A.
In this subsection, let us consider a new blind decorrelating detector for asynchronous MCCDMA systems by using the intersymbol information of the received signals. It is known that as the existence of ISI, the current received signal has certain relation with its previous and following ones. The basic idea of our proposed detector is to utilize such correlation to avoid the channel estimation. ¯ We first construct a cross correlation matrix R; which is made up of the cross correlation of the consecutively received signals, i.e., ¯ ¼ EfrðiÞrði 1ÞH g þ Efrði 1ÞrðiÞH g R " # 0 IK1 ¼ HI HH I . IK1 0
ð16Þ
¯ Unlike the autocorrelation matrix R in (13), R only contains the subspace spanned by the MAI and ISI of all interfering users and does not contain the information of the desired user and the background noise. Recalling that the decorrelating detector must belong to the signal subspace and be orthogonal to the subspace of all interfering users except for the desired user, it is natural to ¯ to develop exploit the useful property of R
(18)
&
Lemma 1 shows that the orthogonal subspace of interfering users can be acquired by computing the ¯ A necessary condition eigendecomposition of R: for the validity of Lemma 1 is 2KoN þ 2; where K is the number of users and N is the length of ¯ s to spreading code, which necessitates the matrix U be tall. By projecting the signal in (8) onto IN ¯H ¯ sU U s directly, we obtain ¯H ¯ sU r~ðiÞ ¼ ðIN U s ÞrðiÞ H
¯ sU ¯ s Þ½h1 b1 ðiÞ þ HI bI ðiÞ þ vðiÞ ¼ ðIN U ¯ sU ¯H ¼ ðIN U s Þ½h1 b1 ðiÞ þ vðiÞ ,
ð19Þ
where the third equality follows from Lemma 1, ¯ sU ¯H i.e., ðIN U s ÞHI ¼ 0: It is clear from (19) that the MAI and ISI of all interfering users are removed from r~ðiÞ: Then, we consider the following optimization problem: wopt 9 arg min Efjb1 ½i wH r~½i j2 g. w
(20)
H ~ ¯ sU ¯H Denoting R9Ef~ rðiÞ~rH ðiÞg ¼ EfðIN U s ÞrðiÞr H ¯ ¯ ðiÞðIN Us Us Þg and performing the derivative of (20) with respect to w; we obtain
~ ¼ ðIN U ¯ sU ¯H Rw s Þh1 .
(21)
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~ ¯ sU ¯H We next premultiply ðIN U s Þh1 by R to obtain H ~ N U ¯ sU ¯ ¯H ¯ ¯H ¯H RðI s Þh1 ¼ ½ðIN Us Us Þh1 h1 ðIN Us Us Þ
s2 ¯H ¯ sU ¯ ¯H þ pffiffiffiffiffiffi ðIN U s Þ ðIN Us Us Þh1 Ts
¯ ¯H ¼ ½hH 1 ðIN Us Us Þh1 s2 ¯H ¯ sU þ pffiffiffiffiffiffi ðIN U s Þh1 Ts
ð22Þ
1 s2 H ¯ ¯H ¯ ¯ sU p ffiffiffiffiffi ffi U wopt ¼ hH ðI U Þh þ ðIN U N s 1 s s Þh1 . 1 Ts (23) pffiffiffiffiffiffi 1 2 ¯ ¯H is only a Since ½hH 1 ðIN Us Us Þh1 þ ðs = T s Þ positive real scalar factor which is irrelevant to the performance of the detector, wopt can be rewritten as H
(24)
Lemma 2. wopt is the decorrelating detector for user 1. Proof. See Appendix B.
H
¯ sU ¯ s ÞrðiÞrH ðiÞ wopt ¼ Principal-eigenvector EfðIN U H
where the first equality follows from (19). It is seen ¯ sU ¯H from (22) that ðIN U s Þh1 is an eigenvector of ~ with the corresponding eigenvalue hH ðIN R 1 pffiffiffiffiffiffi 2 ¯H ¯ sU U s Þh1 þ ðs = T s Þ: Hence, the solution to (21) is given by
¯ sU ¯ s Þh1 . wopt ¼ ðIN U
where the second equality follows from uH ðIN H ¯H ¯ sU U s Þh1 ¼ 0: It is clear from (25) that h1 ðIN ffiffiffiffiffi ffi p 2 ~ ¯H ¯ sU U s Þh1 þ ðs = T s Þ is the largest eigenvalue of R; ~ are smaller and the remaining eigenvalues of R pffiffiffiffiffiffi 2 than s = T s : It implies that the proposed detector wopt is equivalent to the principal eigenvector ~ i.e., of R;
¯ sU ¯ s Þg
ðIN U ¯ sU ¯H ¼ Principal-eigenvector ðIN U s Þ H
¯ sU ¯ s Þ.
RðIN U
ð26Þ
Hence, the proposed decorrelating detector wopt can be implemented by calculating the maximum ~ Table 1 summarizes the algoeigenvector of R: rithm for the proposed decorrelating detector. It is seen that the main task of the algorithm is the ¯ This algorithm is totally eigendecomposition of R: blind and simple since channel estimation of the desired user is not required. Remark 1. The eigendecomposition of the cross ¯ plays a key role in the correlation matrix R algorithm of Table 1 because it is utilized to null out the ISI and MAI of all interfering users. It ¯ is should be pointed out that the acquirement of R based on a perfect synchronization of the desired user. Although the prior time information of the
&
Lemma 2 shows that wopt is equivalent to the decorrelating detector for the desired user. Recal~ with the ling that wopt is an eigenvector of R H ¯ sU ¯H corresponding eigenvalue h1 ðIN U s Þh1 þ p ffiffiffiffiffi ffi ~ ðs2 = T s Þ; we consider the rest eigenvectors of R: ~ Denoting u as an eigenvector of R which is ¯ sU ¯H orthogonal to wopt ; i.e., uH ðIN U s Þh1 ¼ 0; we have H ~ ¼ ½ðIN U ¯ sU ¯ ¯H ¯H Ru s Þh1 h1 ðIN Us Us Þ
s2 ¯ sU ¯H þ pffiffiffiffiffiffi ðIN U s Þ u Ts s2 ¯H ¯ sU ¼ pffiffiffiffiffiffi ðIN U s Þu, Ts
Table 1 Intersymbol decorrelating detector for asynchronous MCCDMA system Signal frame: rð0Þ; rð1Þ; . . . ; rðPÞ Step 1: Compute autocorrelation and cross correlation 1 XP rðiÞrðiÞH R¼ i¼0 Pþ1 XP 1 XP ¯ ¼1 rðiÞrði 1ÞH þ rði 1ÞrðiÞH R i¼1 i¼1 P P ¯ Step 2: Compute eigendecomposition of R ¯ ¼U ¯ sL ¯ sU ¯H R s
ð25Þ
Step 3: Form the detector ¯ sU ¯ ¯H ¯H wopt ¼ Max-eigenvectorðIN U s ÞRðIN Us Us
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desired user is one of the most relaxed requirements for multiuser detection, our proposed detector, like other multiuser detectors [5,11], suffers performance deterioration when the desired user is not synchronized. The reason is that when the synchronization error occurs, the cross corre¯ also contains a small amount of lation matrix R desired user’s information in addition to the ISI and MAI of interfering users. As a consequence, ¯ s in (17) also removes the orthogonal subspace of U a small part of the useful channel information of the desired user as the synchronization errors occur. Hence, the accuracy of the detector would be affected since only the subspace spanned by the remaining information of the desired user and the background noise is considered in (26). Fortunately, satisfying synchronization accuracy has been guaranteed in practical CDMA communication systems, such as CDMA 2000 and WCDMA networks. In these systems, special synchronization channel is allocated to acquire the active user’s time information.
¯ sU ¯H users. By premultiplying ðIN U s Þ with the received signal, we obtain a new constructed signal which only contains the MAI of the synchronous users. Hence, simple synchronous multiuser detection techniques can be used for the newly constructed signal, since the interference of asynchronous users has been nulled out in advance.
Remark 2. The proposed detector in this paper focuses on the multiuser detection for asynchronous MC-CDMA communication which is valid in CDMA uplink. In the downlink communication, the received signal at mobile end includes both the intracell synchronous users and intercell asynchronous users. In this case, the algorithm in Table 1 will suffer deterioration because it is impossible to guarantee that the maximum eigenvector of ¯ sU ¯ ¯H ¯H ðIN U s ÞRðIN Us Us ) is orthogonal to other synchronous users except the desired user. In order to avoid this ambiguity, a simple and effective way to make the proposed detector valid for downlink communication is to assign a unique delay for each transmitted user in the base station, i.e., the signals of different users are sent consecutively with regular time interval. In fact, the intersymbol ¯ is still useful even for the autocorrelation matrix R synchronous CDMA downlink communication. For example, we assume that there are n intracell synchronous users and K n intercell asynchro¯ contains nous users in a cell. Then, the matrix R the subspace spanned by the K n intercell ¯ sU ¯H asynchronous users, and ðIN U s Þ removes the MAI and ISI generated by all asynchronous
r1 ðiÞ ¼ h11 b1 ðiÞ þ H1I bI ðiÞ þ v1 ðiÞ,
3.4. Intersymbol multiuser detection in correlated noise Although the proposed intersymbol detector primarily focuses on the channel with white ambient noise, the intersymbol information can be exploited as well when the noise is not white. Let us briefly discuss the intersymbol multiuser detection in the presence of correlated ambient noise. The key assumption here is that the signal is received by two well separated antennas so that the noise is spatially uncorrelated [12]. Then, the two augmented received signal vectors at the two antennas can be written, respectively, as
r2 ðiÞ ¼ h21 b1 ðiÞ þ H2I bI ðiÞ þ v2 ðiÞ,
ð27Þ
where ½h11 ; H1I and ½h21 ; H2I contain the channel information corresponding to the respective antennas. Since the two antennas are well separated, the ambient noise is spatially uncorrelated, i.e., Efv1 ðiÞv2 ðiÞH g ¼ 0: Without loss of generality, let us consider the decorrelating detector for the first receiver, i.e., r1 ðiÞ: Denote further R12 9Efr1 ðiÞr2 ðiÞH g ¼ ½h11 ; H1I ½h21 ; H2I H , R12I 9Efr1 ðiÞr2 ði 1ÞH g þ Efr2 ði 1Þr1 ðiÞH g " # 0 IK1 1 ¼ HI H2H I . IK1 0
ð28Þ
Here ½h11 ; H1I H ½h11 ; H1I and ½h21 ; H2I H ½h21 ; H2I are assumed to be of full rank. By performing the H eigendecompositions of R12 RH 12 and R12I R12I ; we obtain 1 1 2 2 H 2 2 1 1 H R12 RH 12 ¼ ½h1 ; HI ½h1 ; HI ½h1 ; HI ½h1 ; HI
¼ U112 L112 U1H 12 ,
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" R12I RH 12I
¼
H1I
0 IK1
H1H I
¼
IK1 0
#
" 2 H2H I HI
U112I L112I U1H 12I ,
0
IK1
IK1
0
#
ð29Þ
where L112 is a positive diagonal matrix containing 1 2K 1 eigenvalues of R12 RH 12 and U12 are the 1 corresponding eigenvectors, and L12I contains 2K 2 positive eigenvalues of R12I RH 12I with the corresponding eigenvectors U112I : With (28), (29) and the full rank assumption of ½h21 ; H2I H ½h21 ; H2I ; it is easy to show that range(U112 ) ¼ range(½h11 ; H1I ) contains the signal subspace at receiver 1, and range(U112I ) ¼ range(H1I ) only contains the subspace spanned by all interfering users except the desired user. Recalling that the decorrelating detector must be in the range of user subspace and orthogonal to all interfering users except the desired user, the detector can be expressed as dcor ¼ U112 g1 ; where dcor must be orthogonal to the subspace spanned by interfering users, i.e., 1 H 1H 1 dH cor U12I ¼ g1 U12 U12I ¼ 0.
(30)
Hence, an estimation of g1 can be obtained by computing the minimum eigenvector of the matrix 1 1H 1 U1H 12 U12I U12I U12 ; and the decorrelating detector is derived accordingly.
4. Simulation results In this section, simulation results are presented to illustrate the performance of the proposed decorrelating detector for asynchronous MCCDMA system under Rayleigh fading channel. The system uses BPSK modulation. It is assumed that user 1 is the desired user which has been synchronized. We use a 15-subcarrier MC-CDMA system with Gold code of length N ¼ 15: A normalized Rayleigh channel bk;n is generated with the probability density function (pdf) defined as ! b2k;n bk;n pðbk;n Þ ¼ 2 exp 2 , s 2s where k ¼ 1; 2; . . . ; K; n ¼ 0; 2; . . . ; N 1 and E½b2k;n ¼ 2s2 ¼ 1: The delay of each interfering user, i.e, tk is randomly selected from [0; T s ], and the signal frame in Table 1 is P ¼ 500: In order to check the proposed detector in Table 1, it is assumed throughout this section that the background noise is white with the same variance. Figs. 3 and 4 compare the signal to interference and noise ratios (SINR) produced by four detectors, i.e., the optimal decorrelating detector which is presumed to know the channels information
Fig. 3. SINRs of the optimal decorrelating, the proposed, EGC and MRC detectors (SNR ¼ 2 dB). SINR comparison of various detectors: (a) k ¼ 4 and (b) k ¼ 7:
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Fig. 4. SINRs of the optimal decorrelating, the proposed, EGC and MRC detectors (SNR ¼ 6 dB). SINR comparison of various detectors: (a) k ¼ 4 and (b) k ¼ 7:
of all users, the proposed decorrelating detector, the EGC detector and the MRC detector. The performance measure SINR is defined as SINRðiÞ ¼
jwH h1 j2 , jwH ½rðiÞ h1 b1 ðiÞ j2
(31)
where w is the detector, rðiÞ and b1 ðiÞ are the ith received signal and the ith symbol of the desired user in the signal frame, respectively. In these two figures, we simulate a severe near-far case in which the power of each interfering user is 10 dB more than that of the desired user. In order to improve the legibility and the accuracy of the simulations results, all data in Figs. 3 and 4 were plotted by the average of 100 random runs. It is seen from Fig. 3(a) that the performance of the proposed detector is significantly superior to both EGC and MRC detectors, and is close to the optimal decorrelating detector. Similar result is also observed in Fig. 3(b), where the number of active users increases from 4 to 7. The difference between Fig. 3(a) and 3(b) is that the performance gap between the proposed detector and its optimal in Fig. 3(b) is larger than that in Fig. 3(a). Fig. 4 illustrates the SINR comparison of the previously mentioned detectors when the SNR increases from
2 to 6 dB. It is observed that both the proposed detector and the optimal decorrelating detector have significant performance gain as SNR increases, while the improvement of the EGC and MRC detectors are negligible. It can be explained that the performance of the EGC and MRC detectors is limited by the multiuser interference introduced by the successive symbols of the interfering users due to the timing offset between different users, while the proposed detector and the optimal decorrelating detector have more freedom than EGC and MRC detectors to suppress such interference. It is also observed from Figs. 3 and 4 that the proposed detector suffers more deterioration compared with the optimal decorrelating detector when the system has more interfering users. This is because that the optimal decorrelating detector can always exactly remove MAI and ISI of all interfering users, while the proposed detector is based on the signal subspace estimation of Table 1. For the limited length of the signal frame, there exists an estima¯ When tion error in the eigendecomposition of R: the system has more interfering users, more eigenvectors should be estimated. The estimation error is then increased to deteriorate the accuracy of the proposed detector.
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Fig. 5 compares the near-far resistance of the optimal decorrelating detector, the proposed detector, the EGC detector and the MRC detector.
The number of users in this figure is K ¼ 7: We assume that all interfering users have the same signal power, and the interference to signal ratio
Fig. 5. Bit error rate versus ISR (SNR ¼ 5 dB).
Fig. 6. Performance comparison among various detectors.
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(ISR) is defined as ISR¼ ak =a1 for k ¼ 2; 3; . . . ; K: It is shown that the proposed detector significantly outperforms EGC and MRC detectors. Meanwhile, the proposed detector achieves a performance similar to that of its optimal in low interference region. It shows that the proposed detector possesses a satisfying capability of interference suppression. Fig. 6 compares the BER performance of the above-mentioned detectors. We again assume that there are seven active users in the system and the power of each interfering user is 10 dB more than that of the desired user. The comparison is based on the same channel condition and the system parameters were given at the beginning of this section. It is again observed that the proposed decorrelating detector significantly outperforms both EGC and MRC detectors, and its performance is close to that of the optimal decorrelating detector. It reveals the fact that the proposed detector is more capable of rejecting the MAI and ISI generated from interfering users, compared with the EGC and MRC detectors.
1521
we obtain " ¯R ¯ H ¼ HI R
0
IK1
IK1
0
#
" HH I HI
This paper presents a decorrelating detector for asynchronous MC-CDMA systems over a frequency selective Rayleigh fading channel. The basic idea of the proposed method is the construction of a useful intersymbol cross correlation matrix. By utilizing this matrix, the proposed detector can be implemented conveniently since no channel estimation is required except the synchronization of the desired user.
Appendix A Denote an N 1 vector x: The proof is equivalent to the proof of the following items. ¯ s ¼ 0: 1. If xH HI ¼ 0¼)xH U ¯ s ¼ 0¼)xH HI ¼ 0: 2. If xH U ¯R ¯ H : Denoting Let us now consider R " # " # 0 IK1 0 IK1 H W¼ HI HI , IK1 0 IK1 0
IK1
IK1
0
# HH I
¼ HI WHH I ¯ sL ¯ sU ¯H ¯ ¯ ¯H ¼U s Us Ls Us 2
H
¯ sL ¯ sU ¯s ¼U
ð32Þ
where the third equality follows from (17). To prove the first item, we have xH HI ¼ 0¼)xH HI WHH I x ¼ 0, ¯ sL ¯ 2s U ¯H ¼)xH U s x ¼ 0, H¯ ¼)x Us ¼ 0,
ð33Þ
¯ 2s is where the last step follows from the fact that L a positive diagonal matrix. For the second item, the proof is as follows 2
H
¯ s ¼ 0¼)xH U ¯ sL ¯ sU ¯ s x ¼ 0, xH U ¼)xH HI WHH I x ¼ 0, ¼)xH HI ¼ 0.
5. Conclusions
0
ð34Þ
Similarly, the last step in (34) is achieved because W is also positive definite and of full rank. Combining items 1 and 2, we arrive at the desired result.
Appendix B According to the definition in [10], the decorrelating detector for the desired user must be in the range of user subspace and orthogonal to all interfering users except the desired one. Therefore, based on (24) and the eigendecomposition of (14), we need to verify that H
¯ sU ¯ s Þh1 2 rangeðUs Þ, d1 ¼ ðIN U ( H d1 h1 40; and dH 1 HI ¼ 0:
ð35Þ
According to (14), the proof of the first condition is equivalent to the proof of UH n d1 ¼ 0 since rangeðUs Þ ¼ nullðUn Þ; which is easily
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achieved by UH n d1
¼ ¼
¯ sU ¯H U s Þh1 H H¯ ¯H Un h1 Un Us Us h1 UH n ðIN
¼ 0,
[4]
ð36Þ
where the third equality follows from the fact that h1 2 rangeðUs Þ is orthogonal to Un ; and rangeðUn Þ ¼ nullðh1 ; HI Þ belongs to the subspace ¯ s Þ; i.e., UH ¯ of nullðHI Þ ¼ nullðU n Us ¼ 0: Now, we consider the second condition. Based on Lemma 1, it is easy to obtain H ¯ ¯H dH 1 h1 ¼ h1 ðIN Us Us Þh1 40,
dH 1 HI
¼
hH 1 ðIN
¯ sU ¯H U s ÞHI
¼ 0.
[5]
[6]
[7]
ð37Þ
Therefore, d1 is the decorrelating detector for user 1.
[8] [9]
References [10] [1] L. Vandendorpe, Multitone spread spectrum multiple access communications system in a multipath Rican fading channel, IEEE Trans. Veh. Technol. 44 (2) (February 1995) 327–337. [2] S. Hara, R. Prasad, Overview of multicarrier CDMA, IEEE Commun. Mag. 35 (12) (December 1997) 126–133. [3] N. Yee, J.P. Linnartz, G. Fettweis, Multi-carrier CDMA indoor wireless radio networks, in: Proceedings of the
[11]
[12]
International Symposium on Personal and Mobile Radio Communications, September 1993, pp. 109–113. K. Fazel, L. Papke, On the performance of convolutionally-coded CDMA/OFDM for mobile communication system, in: Proceedings of the International Symposium Personal and Mobile Radio Communications, September 1993, pp. 468–472. M.A. Visser, Y. Bar-Ness, Adaptive reduced complexity multi-carrier CDMA (MC-CDMA) structure for downlink PCS, European Trans. Telecommun. 10 (4) (1999) 437–444. M. A. Visser, Y. Bar-Ness, Joint multiuser detection and frequency offset correction for downlink MC-CDMA, Global Telecommunications Conference, 1999, pp. 2400–2404. X. Cui, T.S. Ng, Performance of asynchronous orthogonal multicarrier CDMA system in frequency selective fading channel, IEEE Trans. Commun. 47 (7) (July 1999) 1084–1091. S. Verdu´, Multiuser Detection, Cambridge University Press, Cambridge, UK, 1998. S. Kondo, B. Milstein, Performance of multicarrier DS CDMA systems, IEEE Trans. Commun. 44 (2) (February 1996) 238–246. X. Wang, H.V. Poor, Blind multiuser detection: a subspace approach, IEEE Trans. Inform. Theory 44 (2) (February 1998) 677–690. J. Namgoong, T.F. Wong, J.S. Lehnert, Subspace MMSE receiver for multicarrier CDMA, IEEE Trans. Commun. 48 (11) (November 2000) 1897–1908. X. Wang, A. Host-Madsen, Group-Blind Multiuser Detection for Uplink CDMA, IEEE J. Selected Area Commun. 17 (11) (November 1999) 1971–1984.
Signal Processing 85 (2005) 1511–1522 www.elsevier.com/locate/sigpro
Blind intersymbol decorrelating detector for asynchronous multicarrier CDMA system Gaonan Zhanga,b, Guoan Bia,b,, Qian Yua,b a Block S1, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore Block S2, Information System Research Lab, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
b
Received 6 June 2003; received in revised form 7 February 2005
Abstract This paper presents a blind decorrelating detector for asynchronous multicarrier CDMA systems with Rayleigh fading channel. The detector is derived by making use of the cross correlation matrix between the consecutively received signals. The main attraction of the detection algorithm is its simplicity since the detector can be implemented blindly without any channel estimation except for the synchronization of the desired user. Simulation results are provided to demonstrate the significant gains in performance and simplicity achieved by the proposed detector. r 2005 Elsevier B.V. All rights reserved. Keywords: Multicarrier CDMA; Decorrelating detector; Intersymbol information
1. Introduction Recently, various multicarrier code-division multiple-access (MC-CDMA) systems have been proposed [1,2]. The MC-CDMA is basically the combination of orthogonal frequency division multiplexing (OFDM) and CDMA [3,4]. The main advantage of OFDM is that the chip duration can be significantly increased to reduce frequencyCorresponding author. Tel.: +65 6790 4823; fax: +65 6793 3318. E-mail addresses: [email protected] (G. Zhang), [email protected] (G. Bi), [email protected] (Q. Yu).
selective fading. In MC-CDMA systems, each user is transmitted simultaneously over multiple subcarriers without any spreading per subcarrier. The resultant signal is a combination of multiple narrowband signals with regularly spaced subcarrier frequencies in the frequency domain. Hence, with the narrowband of the subcarriers, the frequency-selective fading can be reduced to the limit that the signal associated with each subcarrier behaves like one with flat fading. In contrast, the direct-sequence CDMA (DS-CDMA) systems need a complex RAKE receiver to equalize the channel fading. Another advantage of MC-CDMA is that the signal in each subcarrier
0165-1684/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2005.02.006
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can be dealt with by low-speed parallel-type processing, while the DS-CDMA systems require a high-speed serial-type processing. Because of the desirable advantages, MCCDMA systems have received significant attention and been considered as a promising candidate for future wideband wireless communications. Meanwhile, multiuser detections for MC-CDMA systems were also proposed in the literature. For example, several decorrelating detectors for synchronous MC-CDMA systems were developed in [5,6]. For asynchronous reception, the orthogonality among different subcarriers and the spreading codes of different users are destroyed. In [7], the receivers for asynchronous MC-CDMA systems using equal-gain combining (EGC) and maximum-ratio combining (MRC) were reported. The MRC detector achieves an optimum performance in the signal user case, while a perfect estimation of the timing and the fading channel of the desired user is required [7]. The EGC detector achieves a limited performance with a low complexity for implementation. In this paper, a new detection scheme is presented for asynchronous MC-CDMA systems. The proposed method avoids the channel estimation by exploiting the intersymbol information. The key idea of the proposed detector is the utilization of a cross correlation matrix constructed by the consecutively received signals. It is shown that the proposed detector is equivalent to the decorrelating detector [8]. Unlike the previous MRC detector, our method does not require the
channel estimation of the desired user, which leads to a simpler implementation. The simulation results show that the proposed detector significantly outperforms the EGC and MRC receivers. The paper is organized as follows. Section 2 presents the signal model of the asynchronous MC-CDMA system, and Section 3 introduces the proposed decorrelating detector. In Section 4, simulation examples are provided to demonstrate the performance of the proposed detector. Section 5 contains the conclusion.
2. Signal model Let us consider an asynchronous MC-CDMA system with K simultaneous users and N subcarriers in a frequency-selective Rayleigh fading channel. The system uses the Gold codes of length N followed by an OFDM modulator to construct the spreading codes of different users. The transmitter block diagram of the asynchronous MC-CDMA system for user k is shown in Fig. 1. In this figure, the transmitter spreads the ith data symbol of the kth user, i.e., bk ðiÞ 2 f1; 1g; in the frequency domain by using the kth user’s spreading code fck;n jn¼0;1;...;N1 g: The transmitted signal for the kth user can be written as sk ðtÞ ¼ ak
1 X i¼1
bk ðiÞ
N 1 X
ck;n
n¼0
Pðt iT s tk Þej2pnðttk Þ=T s ,
Fig. 1. Transmitter block diagram for the kth user of the asynchronous MC-CDMA system.
ð1Þ
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where ak is the amplitude of the kth user, the delay of the kth user tk is a random variable and uniformly distributed in ½0; T s ; PðtÞ is the rectangular pulse waveform normalized uniformly in the interval ½0; T s and T s is the symbol duration. After passing through the frequency-selective fading channel, sk ðtÞ is affected by multipath fading and the additive white Gaussian noise (AWGN). By suitably choosing N and PðtÞ [9], we can assume that each subcarrier undergoes independent frequency non-selective slow Rayleigh fading and bk;n has a Rayleigh distribution and a uniform distribution for the phase between 0 and 2p: The final received signal of all active users is given by yðtÞ ¼
K X
1 X
ak
k¼1
bk ðiÞ
i¼1
N 1 X
Pðt iT s tk Þe
þ zðtÞ,
ð2Þ
where zðtÞ is the AWGN process with zero mean and variance s2 ; bk;n accounts for the overall effects of phase shift and fading for the nth subcarrier of the kth user with a Rayleigh distribution. At the receiver end, the received signal in (2) is correlated at each subcarrier frequency as shown in Fig. 2. After sampling the output of each subcarrier correlator at the symbol rate, the output of the nth correlator at the ith symbol is given by 1 rn ðiÞ ¼ pffiffiffiffiffiffi Ts
Z
K X
ðiþ1ÞT s
½hk;n bk ðiÞ þ h¯ k;n bk ði 1Þ þ vn ðiÞ,
where 1 X 1 N hk;n ¼ pffiffiffiffiffiffi ak ck;l bk;l ej2pltk =T s T s l¼0 Z Ts
ej2pðlnÞt=T s dt,
ð4Þ
tk 1 X 1 N h¯ k;n ¼ pffiffiffiffiffiffi ak ck;l bk;l ej2pltk =T s T s l¼0 Z tk
ej2pðlnÞt=T s dt,
ð5Þ
0
Z
ðiþ1ÞT s
zðtÞej2pnt=T s dt
(6)
iT s
for n ¼ 0; 1; . . . ; N 1: Without loss of generality, we assume that user 1 is the desired user which has been synchronized, i.e., t1 ¼ 0: According to (5), it is easy to prove h¯ 1;n ¼ 0; n ¼ 0; 1; . . . ; N 1 (7) which implies that the intersymbol interference (ISI) of the desired user is completely removed by synchronization operation, while the ISI of the other users still exist because of their delays. By using (4)–(6) and stacking the N outputs of the subcarrier correlators into a vector, rðiÞ9½r0 ðiÞ; r1 ðiÞ; . . . ; rN1 ðiÞ T ; where ½ T is the matrix transpose operator, the final output of the receiver can be given by
yðtÞej2pnt=T s dt
iT s
ð3Þ
k¼1
1 vn ðiÞ ¼ pffiffiffiffiffiffi Ts
ck;n bk;n
n¼0 j2pnðttk Þ=T s
¼
1513
rðiÞ ¼ h1 b1 ðiÞ þ
K X
½hk bk ðiÞ þ h¯ k bk ði 1Þ þ vðiÞ
k¼2
¼ h1 b1 ðiÞ þ HI bI ðiÞ þ vðiÞ, where hk 9½hk;0 ; hk;1 ; . . . ; hk;N1 T h¯ k 9½h¯ k;0 ; h¯ k;1 ; . . . ; h¯ k;N1 T HI 9½h2 ; h3 ; . . . ; hK ; h¯ 2 ; h¯ 3 ; . . . ; h¯ K bI ðiÞ9½b2 ðiÞ; b3 ðiÞ; . . . ; bK ðiÞ; b2 ði 1Þ, b3 ði 1Þ; . . . ; bK ði 1Þ T Fig. 2. Receiver block diagram of the asynchronous MCCDMA system.
vðiÞ9½v0 ðiÞ; v1 ðiÞ; . . . ; vN1 ðiÞ T .
ð8Þ
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In (8), h1 and b1 ðiÞ contain the information of the desired user. The multiple access interference (MAI) and ISI of the interfering users are represented by HI and bI ðiÞ: It is easy to see from (6) that vðiÞ is a white Gaussian noise vector with an N N covariance matrix 3
2
s2 6 pffiffiffiffiffiffi 6 Ts 6 6 0 6 Corv ¼ 6 6 . 6 .. 6 6 4 0
0 2
psffiffiffiffi Ts
.. .
..
0
.
0 7 7 7 0 7 7 7. .. 7 . 7 7 s2 7 pffiffiffiffiffiffi 5 Ts
2 d1 9argmin kdH 1 ½h1 ; HI k
subject to : dH 1 h1 ¼ 1, (11)
where d1 is the decorrelating detector for user 1 having the following properties: dH 1 h1 ¼ 1, dH 1 HI ¼ 0.
(9)
ð12Þ
3.2. Classical blind implementation of the decorrelating detector Let us now consider the conventional blind subspace implementation of the decorrelating detector in (12). Denote the autocorrelation matrix of the received signals as
3. Blind intersymbol decorrelating detector Let us consider the blind symbol-by-symbol detection for the desired user. The canonical form of a linear detector for the desired user can be represented by a correlating vector w1 2 RN followed by the received signal rðiÞ; such that the detection of user 1 is b^1 ðiÞ ¼ sgnfRðwH 1 rðiÞÞg,
weight vector d1 ¼ w1 ; given by
(10)
H
where ½ denotes conjugate transpose operator. A straightforward single user criterion for optimization of (10) is the maximization of the power of the desired user, i.e., w1 ¼ ½c1;0 b1;0 ; c1;1 b1;1 ; . . . ; c1;N1 b1;N1 T . This is referred to as MRC detector. A simpler implementation of the detector in (10) is the EGC detector, which is given by w1 ¼ c1 ; where c1 ¼ ½c1;0 ; c1;1 ; . . . ; c1;N1 T : 3.1. Canonical representation of linear decorrelating detector Let us focus on the linear decorrelating detection method, which is designed to completely eliminate the MAI and ISI from all interfering users at the expense of enhancing the ambient noise [8]. It has a canonical form of (10) with the
s2 R ¼ EfrðiÞrðiÞH g ¼ ½h1 ; HI ½h1 ; HI H þ pffiffiffiffiffiffi IN , Ts (13) where E½ is the expectation operator. By performing the eigendecomposition of R; we obtain 2 3 0 Ls 5½Us ; Un H , 2 R ¼ ½Us ; Un 4 ð14Þ 0 psffiffiffiffi I T N2Kþ1 s
where Us is an N ð2K 1Þ dimensional orthonormal basis of the signal subspace spanned by ½h1 ; HI and Un is the basis for the white noise subspace orthogonal to Us : The diagonal matrix Ls contains pffiffiffiffiffiffi the 2K 1 largest eigenvalues of R and s2 = T s is the eigenvalue of white noise. Thus, the canonical expression of the decorrelating detector in subspace form can be given by [10] d1 ¼
1 Us pffiffiffiffiffiffi T s ÞI2K1 Þ1 UH s h1
1 s2
Ls pffiffiffiffiffiffi I2K1 UH s h1 . Ts hH 1 Us ðLs
ðs2 =
ð15Þ
In (15), Un and Ls can be estimated from the received signal. In addition, the detector also requires to know the channel information of the desired user h1ffi and the variance of the background pffiffiffiffiffi noise s2 = T s : For blind multiuser detection,
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however, the detector is assumed to have the only prior knowledge of timing information and the spreading sequence of the desired user. Hence, parameter estimation is required for the method described in (15). When the background pffiffiffiffiffiffi noise is white as discussed in Section 2, s2 = T s can be estimated through an eigendecomposition of R by calculating the average eigenvalue of noise subspaces Un : The desired user’s channel information h1 can be estimated by exploiting the orthogonality between the signal subspace and the noise subspace and by oversampling the received signal [11]. However, channel estimation substantially increases the complexity of the detector and often leads to serious deterioration of the performance due to the errors of the estimation.
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the decorrelating detector. By performing the ¯ we obtain eigendecomposition of R; " ¯ n ¯ ¼ ½U ¯ s; U R
#
¯s L 0
¯ n H , ¯ s; U ½U
¯ sL ¯ sU ¯H ¼U s
ð17Þ
¯ s ¼ diagðl¯ 1 ; . . . ; l¯ 2K2 Þ contains 2K 2 where L ¯ s are the corresponding non-zero eigenvalues and U ¯ eigenvectors. With the eigendecomposition of R; we have the following lemma: ¯ s is equal to the null Lemma 1. The null space of U space of HI ; i.e., ¯ s Þ ¼ rangeðIN U ¯ sU ¯H nullðHI Þ ¼ nullðU s Þ.
3.3. Intersymbol blind implementation of the decorrelating detector
Proof. See Appendix A.
In this subsection, let us consider a new blind decorrelating detector for asynchronous MCCDMA systems by using the intersymbol information of the received signals. It is known that as the existence of ISI, the current received signal has certain relation with its previous and following ones. The basic idea of our proposed detector is to utilize such correlation to avoid the channel estimation. ¯ We first construct a cross correlation matrix R; which is made up of the cross correlation of the consecutively received signals, i.e., ¯ ¼ EfrðiÞrði 1ÞH g þ Efrði 1ÞrðiÞH g R " # 0 IK1 ¼ HI HH I . IK1 0
ð16Þ
¯ Unlike the autocorrelation matrix R in (13), R only contains the subspace spanned by the MAI and ISI of all interfering users and does not contain the information of the desired user and the background noise. Recalling that the decorrelating detector must belong to the signal subspace and be orthogonal to the subspace of all interfering users except for the desired user, it is natural to ¯ to develop exploit the useful property of R
(18)
&
Lemma 1 shows that the orthogonal subspace of interfering users can be acquired by computing the ¯ A necessary condition eigendecomposition of R: for the validity of Lemma 1 is 2KoN þ 2; where K is the number of users and N is the length of ¯ s to spreading code, which necessitates the matrix U be tall. By projecting the signal in (8) onto IN ¯H ¯ sU U s directly, we obtain ¯H ¯ sU r~ðiÞ ¼ ðIN U s ÞrðiÞ H
¯ sU ¯ s Þ½h1 b1 ðiÞ þ HI bI ðiÞ þ vðiÞ ¼ ðIN U ¯ sU ¯H ¼ ðIN U s Þ½h1 b1 ðiÞ þ vðiÞ ,
ð19Þ
where the third equality follows from Lemma 1, ¯ sU ¯H i.e., ðIN U s ÞHI ¼ 0: It is clear from (19) that the MAI and ISI of all interfering users are removed from r~ðiÞ: Then, we consider the following optimization problem: wopt 9 arg min Efjb1 ½i wH r~½i j2 g. w
(20)
H ~ ¯ sU ¯H Denoting R9Ef~ rðiÞ~rH ðiÞg ¼ EfðIN U s ÞrðiÞr H ¯ ¯ ðiÞðIN Us Us Þg and performing the derivative of (20) with respect to w; we obtain
~ ¼ ðIN U ¯ sU ¯H Rw s Þh1 .
(21)
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~ ¯ sU ¯H We next premultiply ðIN U s Þh1 by R to obtain H ~ N U ¯ sU ¯ ¯H ¯ ¯H ¯H RðI s Þh1 ¼ ½ðIN Us Us Þh1 h1 ðIN Us Us Þ
s2 ¯H ¯ sU ¯ ¯H þ pffiffiffiffiffiffi ðIN U s Þ ðIN Us Us Þh1 Ts
¯ ¯H ¼ ½hH 1 ðIN Us Us Þh1 s2 ¯H ¯ sU þ pffiffiffiffiffiffi ðIN U s Þh1 Ts
ð22Þ
1 s2 H ¯ ¯H ¯ ¯ sU p ffiffiffiffiffi ffi U wopt ¼ hH ðI U Þh þ ðIN U N s 1 s s Þh1 . 1 Ts (23) pffiffiffiffiffiffi 1 2 ¯ ¯H is only a Since ½hH 1 ðIN Us Us Þh1 þ ðs = T s Þ positive real scalar factor which is irrelevant to the performance of the detector, wopt can be rewritten as H
(24)
Lemma 2. wopt is the decorrelating detector for user 1. Proof. See Appendix B.
H
¯ sU ¯ s ÞrðiÞrH ðiÞ wopt ¼ Principal-eigenvector EfðIN U H
where the first equality follows from (19). It is seen ¯ sU ¯H from (22) that ðIN U s Þh1 is an eigenvector of ~ with the corresponding eigenvalue hH ðIN R 1 pffiffiffiffiffiffi 2 ¯H ¯ sU U s Þh1 þ ðs = T s Þ: Hence, the solution to (21) is given by
¯ sU ¯ s Þh1 . wopt ¼ ðIN U
where the second equality follows from uH ðIN H ¯H ¯ sU U s Þh1 ¼ 0: It is clear from (25) that h1 ðIN ffiffiffiffiffi ffi p 2 ~ ¯H ¯ sU U s Þh1 þ ðs = T s Þ is the largest eigenvalue of R; ~ are smaller and the remaining eigenvalues of R pffiffiffiffiffiffi 2 than s = T s : It implies that the proposed detector wopt is equivalent to the principal eigenvector ~ i.e., of R;
¯ sU ¯ s Þg
ðIN U ¯ sU ¯H ¼ Principal-eigenvector ðIN U s Þ H
¯ sU ¯ s Þ.
RðIN U
ð26Þ
Hence, the proposed decorrelating detector wopt can be implemented by calculating the maximum ~ Table 1 summarizes the algoeigenvector of R: rithm for the proposed decorrelating detector. It is seen that the main task of the algorithm is the ¯ This algorithm is totally eigendecomposition of R: blind and simple since channel estimation of the desired user is not required. Remark 1. The eigendecomposition of the cross ¯ plays a key role in the correlation matrix R algorithm of Table 1 because it is utilized to null out the ISI and MAI of all interfering users. It ¯ is should be pointed out that the acquirement of R based on a perfect synchronization of the desired user. Although the prior time information of the
&
Lemma 2 shows that wopt is equivalent to the decorrelating detector for the desired user. Recal~ with the ling that wopt is an eigenvector of R H ¯ sU ¯H corresponding eigenvalue h1 ðIN U s Þh1 þ p ffiffiffiffiffi ffi ~ ðs2 = T s Þ; we consider the rest eigenvectors of R: ~ Denoting u as an eigenvector of R which is ¯ sU ¯H orthogonal to wopt ; i.e., uH ðIN U s Þh1 ¼ 0; we have H ~ ¼ ½ðIN U ¯ sU ¯ ¯H ¯H Ru s Þh1 h1 ðIN Us Us Þ
s2 ¯ sU ¯H þ pffiffiffiffiffiffi ðIN U s Þ u Ts s2 ¯H ¯ sU ¼ pffiffiffiffiffiffi ðIN U s Þu, Ts
Table 1 Intersymbol decorrelating detector for asynchronous MCCDMA system Signal frame: rð0Þ; rð1Þ; . . . ; rðPÞ Step 1: Compute autocorrelation and cross correlation 1 XP rðiÞrðiÞH R¼ i¼0 Pþ1 XP 1 XP ¯ ¼1 rðiÞrði 1ÞH þ rði 1ÞrðiÞH R i¼1 i¼1 P P ¯ Step 2: Compute eigendecomposition of R ¯ ¼U ¯ sL ¯ sU ¯H R s
ð25Þ
Step 3: Form the detector ¯ sU ¯ ¯H ¯H wopt ¼ Max-eigenvectorðIN U s ÞRðIN Us Us
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desired user is one of the most relaxed requirements for multiuser detection, our proposed detector, like other multiuser detectors [5,11], suffers performance deterioration when the desired user is not synchronized. The reason is that when the synchronization error occurs, the cross corre¯ also contains a small amount of lation matrix R desired user’s information in addition to the ISI and MAI of interfering users. As a consequence, ¯ s in (17) also removes the orthogonal subspace of U a small part of the useful channel information of the desired user as the synchronization errors occur. Hence, the accuracy of the detector would be affected since only the subspace spanned by the remaining information of the desired user and the background noise is considered in (26). Fortunately, satisfying synchronization accuracy has been guaranteed in practical CDMA communication systems, such as CDMA 2000 and WCDMA networks. In these systems, special synchronization channel is allocated to acquire the active user’s time information.
¯ sU ¯H users. By premultiplying ðIN U s Þ with the received signal, we obtain a new constructed signal which only contains the MAI of the synchronous users. Hence, simple synchronous multiuser detection techniques can be used for the newly constructed signal, since the interference of asynchronous users has been nulled out in advance.
Remark 2. The proposed detector in this paper focuses on the multiuser detection for asynchronous MC-CDMA communication which is valid in CDMA uplink. In the downlink communication, the received signal at mobile end includes both the intracell synchronous users and intercell asynchronous users. In this case, the algorithm in Table 1 will suffer deterioration because it is impossible to guarantee that the maximum eigenvector of ¯ sU ¯ ¯H ¯H ðIN U s ÞRðIN Us Us ) is orthogonal to other synchronous users except the desired user. In order to avoid this ambiguity, a simple and effective way to make the proposed detector valid for downlink communication is to assign a unique delay for each transmitted user in the base station, i.e., the signals of different users are sent consecutively with regular time interval. In fact, the intersymbol ¯ is still useful even for the autocorrelation matrix R synchronous CDMA downlink communication. For example, we assume that there are n intracell synchronous users and K n intercell asynchro¯ contains nous users in a cell. Then, the matrix R the subspace spanned by the K n intercell ¯ sU ¯H asynchronous users, and ðIN U s Þ removes the MAI and ISI generated by all asynchronous
r1 ðiÞ ¼ h11 b1 ðiÞ þ H1I bI ðiÞ þ v1 ðiÞ,
3.4. Intersymbol multiuser detection in correlated noise Although the proposed intersymbol detector primarily focuses on the channel with white ambient noise, the intersymbol information can be exploited as well when the noise is not white. Let us briefly discuss the intersymbol multiuser detection in the presence of correlated ambient noise. The key assumption here is that the signal is received by two well separated antennas so that the noise is spatially uncorrelated [12]. Then, the two augmented received signal vectors at the two antennas can be written, respectively, as
r2 ðiÞ ¼ h21 b1 ðiÞ þ H2I bI ðiÞ þ v2 ðiÞ,
ð27Þ
where ½h11 ; H1I and ½h21 ; H2I contain the channel information corresponding to the respective antennas. Since the two antennas are well separated, the ambient noise is spatially uncorrelated, i.e., Efv1 ðiÞv2 ðiÞH g ¼ 0: Without loss of generality, let us consider the decorrelating detector for the first receiver, i.e., r1 ðiÞ: Denote further R12 9Efr1 ðiÞr2 ðiÞH g ¼ ½h11 ; H1I ½h21 ; H2I H , R12I 9Efr1 ðiÞr2 ði 1ÞH g þ Efr2 ði 1Þr1 ðiÞH g " # 0 IK1 1 ¼ HI H2H I . IK1 0
ð28Þ
Here ½h11 ; H1I H ½h11 ; H1I and ½h21 ; H2I H ½h21 ; H2I are assumed to be of full rank. By performing the H eigendecompositions of R12 RH 12 and R12I R12I ; we obtain 1 1 2 2 H 2 2 1 1 H R12 RH 12 ¼ ½h1 ; HI ½h1 ; HI ½h1 ; HI ½h1 ; HI
¼ U112 L112 U1H 12 ,
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" R12I RH 12I
¼
H1I
0 IK1
H1H I
¼
IK1 0
#
" 2 H2H I HI
U112I L112I U1H 12I ,
0
IK1
IK1
0
#
ð29Þ
where L112 is a positive diagonal matrix containing 1 2K 1 eigenvalues of R12 RH 12 and U12 are the 1 corresponding eigenvectors, and L12I contains 2K 2 positive eigenvalues of R12I RH 12I with the corresponding eigenvectors U112I : With (28), (29) and the full rank assumption of ½h21 ; H2I H ½h21 ; H2I ; it is easy to show that range(U112 ) ¼ range(½h11 ; H1I ) contains the signal subspace at receiver 1, and range(U112I ) ¼ range(H1I ) only contains the subspace spanned by all interfering users except the desired user. Recalling that the decorrelating detector must be in the range of user subspace and orthogonal to all interfering users except the desired user, the detector can be expressed as dcor ¼ U112 g1 ; where dcor must be orthogonal to the subspace spanned by interfering users, i.e., 1 H 1H 1 dH cor U12I ¼ g1 U12 U12I ¼ 0.
(30)
Hence, an estimation of g1 can be obtained by computing the minimum eigenvector of the matrix 1 1H 1 U1H 12 U12I U12I U12 ; and the decorrelating detector is derived accordingly.
4. Simulation results In this section, simulation results are presented to illustrate the performance of the proposed decorrelating detector for asynchronous MCCDMA system under Rayleigh fading channel. The system uses BPSK modulation. It is assumed that user 1 is the desired user which has been synchronized. We use a 15-subcarrier MC-CDMA system with Gold code of length N ¼ 15: A normalized Rayleigh channel bk;n is generated with the probability density function (pdf) defined as ! b2k;n bk;n pðbk;n Þ ¼ 2 exp 2 , s 2s where k ¼ 1; 2; . . . ; K; n ¼ 0; 2; . . . ; N 1 and E½b2k;n ¼ 2s2 ¼ 1: The delay of each interfering user, i.e, tk is randomly selected from [0; T s ], and the signal frame in Table 1 is P ¼ 500: In order to check the proposed detector in Table 1, it is assumed throughout this section that the background noise is white with the same variance. Figs. 3 and 4 compare the signal to interference and noise ratios (SINR) produced by four detectors, i.e., the optimal decorrelating detector which is presumed to know the channels information
Fig. 3. SINRs of the optimal decorrelating, the proposed, EGC and MRC detectors (SNR ¼ 2 dB). SINR comparison of various detectors: (a) k ¼ 4 and (b) k ¼ 7:
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Fig. 4. SINRs of the optimal decorrelating, the proposed, EGC and MRC detectors (SNR ¼ 6 dB). SINR comparison of various detectors: (a) k ¼ 4 and (b) k ¼ 7:
of all users, the proposed decorrelating detector, the EGC detector and the MRC detector. The performance measure SINR is defined as SINRðiÞ ¼
jwH h1 j2 , jwH ½rðiÞ h1 b1 ðiÞ j2
(31)
where w is the detector, rðiÞ and b1 ðiÞ are the ith received signal and the ith symbol of the desired user in the signal frame, respectively. In these two figures, we simulate a severe near-far case in which the power of each interfering user is 10 dB more than that of the desired user. In order to improve the legibility and the accuracy of the simulations results, all data in Figs. 3 and 4 were plotted by the average of 100 random runs. It is seen from Fig. 3(a) that the performance of the proposed detector is significantly superior to both EGC and MRC detectors, and is close to the optimal decorrelating detector. Similar result is also observed in Fig. 3(b), where the number of active users increases from 4 to 7. The difference between Fig. 3(a) and 3(b) is that the performance gap between the proposed detector and its optimal in Fig. 3(b) is larger than that in Fig. 3(a). Fig. 4 illustrates the SINR comparison of the previously mentioned detectors when the SNR increases from
2 to 6 dB. It is observed that both the proposed detector and the optimal decorrelating detector have significant performance gain as SNR increases, while the improvement of the EGC and MRC detectors are negligible. It can be explained that the performance of the EGC and MRC detectors is limited by the multiuser interference introduced by the successive symbols of the interfering users due to the timing offset between different users, while the proposed detector and the optimal decorrelating detector have more freedom than EGC and MRC detectors to suppress such interference. It is also observed from Figs. 3 and 4 that the proposed detector suffers more deterioration compared with the optimal decorrelating detector when the system has more interfering users. This is because that the optimal decorrelating detector can always exactly remove MAI and ISI of all interfering users, while the proposed detector is based on the signal subspace estimation of Table 1. For the limited length of the signal frame, there exists an estima¯ When tion error in the eigendecomposition of R: the system has more interfering users, more eigenvectors should be estimated. The estimation error is then increased to deteriorate the accuracy of the proposed detector.
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Fig. 5 compares the near-far resistance of the optimal decorrelating detector, the proposed detector, the EGC detector and the MRC detector.
The number of users in this figure is K ¼ 7: We assume that all interfering users have the same signal power, and the interference to signal ratio
Fig. 5. Bit error rate versus ISR (SNR ¼ 5 dB).
Fig. 6. Performance comparison among various detectors.
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(ISR) is defined as ISR¼ ak =a1 for k ¼ 2; 3; . . . ; K: It is shown that the proposed detector significantly outperforms EGC and MRC detectors. Meanwhile, the proposed detector achieves a performance similar to that of its optimal in low interference region. It shows that the proposed detector possesses a satisfying capability of interference suppression. Fig. 6 compares the BER performance of the above-mentioned detectors. We again assume that there are seven active users in the system and the power of each interfering user is 10 dB more than that of the desired user. The comparison is based on the same channel condition and the system parameters were given at the beginning of this section. It is again observed that the proposed decorrelating detector significantly outperforms both EGC and MRC detectors, and its performance is close to that of the optimal decorrelating detector. It reveals the fact that the proposed detector is more capable of rejecting the MAI and ISI generated from interfering users, compared with the EGC and MRC detectors.
1521
we obtain " ¯R ¯ H ¼ HI R
0
IK1
IK1
0
#
" HH I HI
This paper presents a decorrelating detector for asynchronous MC-CDMA systems over a frequency selective Rayleigh fading channel. The basic idea of the proposed method is the construction of a useful intersymbol cross correlation matrix. By utilizing this matrix, the proposed detector can be implemented conveniently since no channel estimation is required except the synchronization of the desired user.
Appendix A Denote an N 1 vector x: The proof is equivalent to the proof of the following items. ¯ s ¼ 0: 1. If xH HI ¼ 0¼)xH U ¯ s ¼ 0¼)xH HI ¼ 0: 2. If xH U ¯R ¯ H : Denoting Let us now consider R " # " # 0 IK1 0 IK1 H W¼ HI HI , IK1 0 IK1 0
IK1
IK1
0
# HH I
¼ HI WHH I ¯ sL ¯ sU ¯H ¯ ¯ ¯H ¼U s Us Ls Us 2
H
¯ sL ¯ sU ¯s ¼U
ð32Þ
where the third equality follows from (17). To prove the first item, we have xH HI ¼ 0¼)xH HI WHH I x ¼ 0, ¯ sL ¯ 2s U ¯H ¼)xH U s x ¼ 0, H¯ ¼)x Us ¼ 0,
ð33Þ
¯ 2s is where the last step follows from the fact that L a positive diagonal matrix. For the second item, the proof is as follows 2
H
¯ s ¼ 0¼)xH U ¯ sL ¯ sU ¯ s x ¼ 0, xH U ¼)xH HI WHH I x ¼ 0, ¼)xH HI ¼ 0.
5. Conclusions
0
ð34Þ
Similarly, the last step in (34) is achieved because W is also positive definite and of full rank. Combining items 1 and 2, we arrive at the desired result.
Appendix B According to the definition in [10], the decorrelating detector for the desired user must be in the range of user subspace and orthogonal to all interfering users except the desired one. Therefore, based on (24) and the eigendecomposition of (14), we need to verify that H
¯ sU ¯ s Þh1 2 rangeðUs Þ, d1 ¼ ðIN U ( H d1 h1 40; and dH 1 HI ¼ 0:
ð35Þ
According to (14), the proof of the first condition is equivalent to the proof of UH n d1 ¼ 0 since rangeðUs Þ ¼ nullðUn Þ; which is easily
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achieved by UH n d1
¼ ¼
¯ sU ¯H U s Þh1 H H¯ ¯H Un h1 Un Us Us h1 UH n ðIN
¼ 0,
[4]
ð36Þ
where the third equality follows from the fact that h1 2 rangeðUs Þ is orthogonal to Un ; and rangeðUn Þ ¼ nullðh1 ; HI Þ belongs to the subspace ¯ s Þ; i.e., UH ¯ of nullðHI Þ ¼ nullðU n Us ¼ 0: Now, we consider the second condition. Based on Lemma 1, it is easy to obtain H ¯ ¯H dH 1 h1 ¼ h1 ðIN Us Us Þh1 40,
dH 1 HI
¼
hH 1 ðIN
¯ sU ¯H U s ÞHI
¼ 0.
[5]
[6]
[7]
ð37Þ
Therefore, d1 is the decorrelating detector for user 1.
[8] [9]
References [10] [1] L. Vandendorpe, Multitone spread spectrum multiple access communications system in a multipath Rican fading channel, IEEE Trans. Veh. Technol. 44 (2) (February 1995) 327–337. [2] S. Hara, R. Prasad, Overview of multicarrier CDMA, IEEE Commun. Mag. 35 (12) (December 1997) 126–133. [3] N. Yee, J.P. Linnartz, G. Fettweis, Multi-carrier CDMA indoor wireless radio networks, in: Proceedings of the
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