A matrix-algebraic approach to successive interference cancellation in ...

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 1, JANUARY 2000

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A Matrix-Algebraic Approach to Successive Interference Cancellation in CDMA Lars K. Rasmussen, Member, IEEE, Teng J. Lim, Member, IEEE, and Ann-Louise Johansson, Member, IEEE

Abstract—In this paper, we describe linear successive interference cancellation (SIC) based on matrix-algebra. We show that linear SIC schemes (single-stage and multistage) correspond to linear matrix filtering that can be performed directly on the received chip-matched filtered signal vector without explicitly performing the interference cancellation. This leads to an analytical expression for calculating the resulting bit-error rate which is of particular use for short-code systems. Convergence issues are discussed, and the concept of -convergence is introduced to determine the number of stages required for practical convergence for both short and long codes. Index Terms—Code-division multiaccess, linear algebra, multiuser channels, signal detection.

I. INTRODUCTION

I

N A MOBILE communications system, multiple access to the common channel resources is vital. In a system based on spread-spectrum transmission techniques, code division provides simultaneous access for multiple users. By selecting mutually orthogonal codes for all users, they each achieve interference-free single-user performance. It is however not possible to maintain orthogonal spreading codes at the receiver in a mobile environment, and thus multiple-access interference (MAI) arises. Conventional single-user detection techniques are severely affected by MAI, making such systems interference-limited [1]. Traditional matched filter receivers for code-division multiple-access (CDMA) also require strict power control in order to alleviate the near–far problem where a high-powered user creates significant MAI for low-powered users. More advanced detection strategies can be adopted to improve performance. In [2], Verdú developed the optimal (0,1)constrained maximum-likelihood (ML) detector for multiuser CDMA. The inherent complexity however increases exponentially with the number of users, rendering the optimal ML detector impractical.

Paper approved by B. Aazhang, the Editor for Spread Spectrum Networks of the IEEE Communications Society. Manuscript received June 9, 1997; revised July 1, 1998. This paper was presented in part at the IEEE Vehicular Technology Conference, Phoenix, AZ, May 1997, and in part at the Multiaccess, Mobility, and Teletraffic Workshop, Melbourne, Australia, December 1997. L. K. Rasmussen is with the Telecommunication Theory Group, Department of Computer Engineering, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden (e-mail: [email protected]). T. J. Lim is with the Centre for Wireless Communications, Teletech Park, Singapore 117674 (e-mail: [email protected]). A.-L. Johansson is with Nokia Svenska AB, SE-164 25 Kista, Sweden (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(00)00517-1.

For practical implementation, parallel and successive interference cancellation (SIC) schemes have been subject to most attention. These techniques rely on simple processing elements constructed around the matched filter concept. The first structure based on the principle of interference cancellation was the multistage detector in [3]. Here, the cancellation is decision-directed (i.e., nonlinear) and is done in parallel. In [4], Dent et al. proposed a serial approach, a single-stage nonlinear SIC scheme, while Kawabe et al.suggested a multistage nonlinear SIC technique in [5]. A closely related scheme was suggested by Sawahashi et al. in [6]. Linear SIC detectors have been considered in detail for both single-stage and multistage cases in [7] and [8], while Jamal and Dahlman have compared the performance of the linear and the nonlinear SIC approaches in [9]. An algebraic approach to SIC was initially introduced in [10] and further developed in [11]. Closely related work by Elders-Boll et al. was presented in [12] and [13], where they suggest linear detectors based on the application of classic iterative techniques for solving linear systems. The Gauss–Seidel iteration was here identified as SIC. Iterative methods for linear detector design have also been proposed by Juntti et al. in [14]. The equivalence to interference cancellation was however not recognized. In this paper, we describe the linear SIC scheme based on matrix-algebra. We show that the linear SIC schemes (single-stage and multistage) correspond to linear matrix filtering that can be performed directly on the received chip-matched filtered signal vector without explicitly performing the interference cancellation. This leads to an analytical expression for calculating the resulting bit-error rate (BER), which is of particular use for short-code systems. Convergence issues are discussed, and the concept of -convergence is introduced to determine the number of stages required for practical convergence for both short and long codes. The paper is organized as follows. In Section II, the uplink model is described, and the techniques for SIC are briefly summarized. In Section III, we introduce a matrix-algebraic approach for describing linear SIC, which allows for new insight into the behavior of the schemes. The equivalent matrix filters for linear SIC are derived, and convergence issues are discussed for multistage schemes, including the concept of -convergence for both short and long codes. Numerical examples are presented in Section IV, and conclusions are drawn in Section V. The following notation is used for the product of matrices

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if if (1)

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II. SYSTEM MODELS A specific user in this -user communication system by transmits a binary information symbol multiplying with a spreading code of length chips and then transmitting over an additive white Gaussian noise (AWGN) channel using binary phase-shift keying (BPSK).1 The spreading codes transmitted by each user in any given symbol interval are assumed to be symbol synchronous, and the channel imposes no phase rotation on the transmitted signal. Symbol synchronism is assumed for clarity. Similar arguments hold for the symbol-asynchronous case as demonstrated in [13]. Each user is received at a user-specific that is assumed constant over one bit interval. energy level . The output of a Note that we have assumed that chip-matched filter is then expressed as a linear combination of spreading codes, specifically, the chip matched filtered received vector is a column vector of length , encompassing the transmissions for all users. The received vector is hence , described through matrix-algebra as where

Fig. 1. Linear SIC unit.

(2) (3) (4) and has To avoid rank deficiencies, we assume that full rank, i.e., the spreading codes for all users are linearly independent. The noise vector consists of independently, identically distributed additive white Gaussian distributed samples . with zero mean and variance The received vector is contained in a vector space of dimen. It is however only the part of residing in the sion , signal space2 that is affecting the detector decision. The signal . If , . space is determined by with and In general however, , where denotes the null space or equivalently the orthogonal complement of . of and by orthogonal projections, We can determine . SIC schemes are best described by defining an interference cancelling unit (ICU) as shown in Fig. 1. This unit is then used as a building block in the multistage SIC scheme shown in Fig. 2. The single-stage scheme is obtained by omitting all but the first stage. It is assumed that the users are ordered according to their received signal power. The input residual signal vector to . For the first user in the first an ICU of user at stage is . The contribution to be cancelled in the ICU stage, . In geometrical terms, this is a of user at stage is onto the relevant projection of the residual received vector for all ICU spreading code . For the first stage, blocks.

Fig. 2. Multistage linear SIC structure with four users and three stages.

III. MATRIX-ALGEBRAIC APPROACHES A. Linear SIC , The first-stage, first-unit output is leading to the resulting input vector for the next unit as , where is an identity ma, and trix. The next step is then . In general, , and we get . A single-stage linear SIC scheme can thereof the received fore be represented as a linear filtering . signal vector , where represents the residual received The output vector vector at the end of the first stage after all users have been processed. It is determined as (5) For notational purposes in what is to follow, we define

(6) . Furthermore, Equation (5) then becomes , so the decision statistic is then (7)

1Binary

data and chip formats are assumed for clarity. All the presented concepts generalize to -ary formats. 2The irrelevance theorem allows for the portion of the noise that lies outside of fSg to be ignored.

m

and the residual received vector for the next ICU is (8)

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In general, the terms in the second stage are described by and , where . Continuing this way, we obtain the general expressions for user in stage (9) and (10) where

and

For the linear SIC however, for . At intermediate stages where the residual received vector gets close to estimating the noise vector correctly, it is therefore possible that the linear SIC performs better in terms of BER than at convergence. This was first observed in [14]. The potential improvement is however relatively insignificant and of theoretical interest only. Since the detector is described by linear matrix filtering, a single-user matched filter approach with each matched filter is possible. The complexity is then inherent in the given by filter computation. It should be noted however that it is not necessary to perform a matrix inverse to obtain the filter. It is also possible to analytically evaluate the BER performance using techniques similar to those used for the conventional detector. The BER for user after stages is thus

(11)

(16)

So far, we have considered users successively as suggested by the detector structure, deriving equivalent user specific matched . It is also possible to directly derive filters, represented by . Writing out the the matrix filter decision statistic for user as a function of the most recent decision statistics for the other users, we get

when using short codes. For long codes, the above expression must be averaged over all segments of length .

(12) Let triangular part of

B. Convergence Rate When considering convergence rate, it is convenient to have the received vector completely contained in the signal space. , we will therefore consider rather than . The In case . starting vector is therefore In order to investigate the convergence rate, we reconsider (9)

, where is the strict lower left . Then, (12) leads to (17) (13)

, , , and . Equation (13) is in fact identical to the Gauss–Seidel iteration [15] that is known to converge has a spectral radius less than one, i.e., eigenvalues with if is symmetric modulus less than one. This is guaranteed if positive definite or, equivalently, has full rank, which we have assumed. Therefore, the linear SIC always converges. Furthermore, the Gauss–Seidel iteration was developed for matrix inversion so the linear SIC converges to the decorrelator. This was first pointed out in [12]. From (13), it follows that

where

in (6). For

where we have made use of the definition of convergence, we have

for That obviously corresponds to for

(18)

depends on the th power of , the conSince . From the vergence is dependent on the eigenvalues of and definition of eigenvectors, it is easy to show that have the same eigenvalues for all and . It is therefore sufficient to consider only one representative matrix, and we select . From (12) and (13), we have

(14) (the decorrelator) for where it is clear that since has a spectral radius less than one. Column of is obviously given by (11). Assuming perfect cancellation for all users after stages, then and

(19) i.e., we have that , we focus on

. Considering only

for (15)

(20)

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for convergence rate estimation. Following the approach for the derivation of the steepest descent algorithm, it is convenient to . Since is not symapply a similarity transformation to is not always dimetric and is occasionally defective, i.e., agonalizable, we use the Jordan decomposition [16], which is applicable to all square matrices. For notational simplicity, we in the sequel. So , where is a nonlet matrix, selected such that is a Jordan matrix singular of the form

..

Each block

(21)

.

is a Jordan block of the form

..

.

..

.

(22)

where is the eigenvalue associated with the Jordan block of order . The number of Jordan blocks is the number of linear independent eigenvectors of . The number of Jordan blocks corresponding to a given eigenis the geometric multiplicity of , while the value sum of the orders of all Jordan blocks corresponding to that of . If and eigenvalue is the algebraic multiplicity , then is diagonalizable and only if strictly diagonal. The Jordan matrix can be expressed as a sum of a diagonal and a nilpotent matrix , such that matrix of eigenvalues . The diagonal matrix of eigenvalues is defined , where the identical as are numbered consecutively. Assuming eigenvalues of and that , that the largest Jordan block is of order and then

consider the impact of a Jordan block. For a Jordan block of order at location to , we can combine (23) and to as (25) to express the elements

(28) can be divided into segments where each segThe vector ment is only affected by one specific Jordan block, i.e., all elecan be expressed as indicated above. As we know, ments of the linear SIC scheme converges when has full rank, and then all eigenvalues have modulus less than one. To make conclusive remarks about the convergence rate, however, we have to know the eigenvalues of the individual Jordan blocks. , Considering the eigenvalues of . The nonzero eigenvalues of we first examine are obviously the same as the eigenvalues of . We further have that so has nonzero eigenvalues zero eigenvalues corresponding to the orthogonal and complement of or, equivalently, corresponding to the noise space. These eigenvalues are not relevant since has no com, so the corresponding elements of are ponents in null eigenvalues of one that zero. It then follows that has relevant eigenvalues3 of are irrelevant for convergence. The are the same as the eigenvalues of . Thus, all the eigenvalues have modulus less than one. so has a zero It is possible to show that eigenvalue. It is not possible in general to analytically conclude anything further about the relevant eigenvalues with respect to the geometric and algebraic multiplicities and their relation to the individual Jordan blocks. We therefore resort to simulations in order to gain some useful insight. For all experiments conducted, all nonzero eigenvalues were simple, corresponding to Jordan blocks of order 1. Only the zero eigenvalue was observed to cause to occasionally become defective. Fortunately corresponding to the zero eigenvalues have the elements of obviously no impact on the convergence rate, so we only need nonzero eigenvalues. We to consider the remaining , where can thus express (25) as

(23) (29) For (24)

, correand is the set of indices of cardinality relevant for convergence. A sponding to the elements of is obtained from (27) and (29) similar expression for

(25) (30)

where Letting (26)

(31)

(27)

of

and denotes individual elements . To get a simple relationship between and , we

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