5.3 Optimal Space-Time Multiuser Detection

In Section 5.2, we considered (linear) spatial processing as a mechanism for separating multiple users sharing identical temporal signatures. In the remainder of this chapter we examine the situation in which both temporal and spatial signatures of the users differ and consider the joint exploitation of these differences to separate users. Such joint processing is known as space-time processing . In this and the following sections, we discuss such processing in the context of multiuser detection in a CDMA system with multipath channel distortion and multiple receive antennas. We begin in this section with consideration of optimal (nonlinear) processing, turning in subsequent sections to linear and adaptive linear methods . The materials in this and the next sections first appeared in [554].

5.3.1 Signal Model

Consider a DS-CDMA mobile radio network with K users, employing normalized spreading waveforms s ₁ , s ₂ , . . . s _K , and transmitting sequences of BPSK symbols through their respective multipath channels. The transmitted baseband signal due to the k th user is given by

Equation 5.42

where M is the number of data symbols per user per frame, T is the symbol interval, b _k [ i ] {+1, “1} is the i th symbol transmitted by the k th user, and A _k and s _k ( t ) denote, respectively, the amplitude and normalized signaling waveform of the k th user. It is assumed that s _k ( t ) is supported only on the interval [0, T ] and has unit energy. (Here, for simplicity, we assume that periodic spreading sequences are employed in the system. The generalization to the aperiodic spreading case is straightforward.) It is also assumed that each user transmits independent equiprobable symbols and that the symbol sequences from different users are independent. Recall that in the direct-sequence spread-spectrum multiple-access format, the user signaling waveforms are of the form

Equation 5.43

where N is the processing gain, { c _j,k : j = 0, . . ., N “ 1} is a signature sequence of ±1's assigned to the k th user, and y is a normalized chip waveform of duration T _c = T/N.

At the receiver an antenna array of P elements is employed. Assuming that each transmitter is equipped with a single antenna, the baseband multipath channel between the k th user's transmitter and the base station receiver can be modeled as a single-input/multiple-output channel with the following vector impulse response:

Equation 5.44

where L is the number of paths in each user's channel, a _,k and t _l,k are, respectively, the complex gain and delay of the l th path of the k th user's signal, and a _l,k = [ a _{l,k ,1} · · · a _l,k,P ] ^T is the array response vector corresponding to the l th path of the k th user's signal. The total received signal at the receiver is then the superposition of the signals from the K users plus the additive ambient noise, given by

Equation 5.45

where * denotes convolution; n (t) = [ n ₁ ( t ) · · · n _P ( t )] ^T is a vector of independent zero-mean complex white Gaussian noise processes, each with power spectral density s ² .

5.3.2 Sufficient Statistic

We next derive a sufficient statistic for demodulating the multiuser symbols from the space-time signal model (5.45). To do so, we first denote the useful signal in (5.45) by

Equation 5.46

where and . Using the Cameron “Martin formula [377], the likelihood function of the received waveform r ( t ) in (5.45) conditioned on all the transmitted symbols b of all users can be written as

Equation 5.47

where

Equation 5.48

The first integral in (5.48) can be expressed as

Equation 5.49

Since the second integral in (5.48) does not depend on the received signal r ( t ), by (5.49) we see that { y _k [ i ]} is a sufficient statistic for detecting the multiuser symbols b . From (5.49) it is seen that this sufficient statistic is obtained by passing the received signal vector r ( t ) through KL beamformers directed at each path of each user's signal, followed by a bank of K maximum-ratio multipath combiners (i.e., RAKE receivers). Since this beamformer is a spatial matched filter for the array antenna receiver, and a RAKE receiver is a temporal matched filter for multipath channels, the sufficient statistic { y _k [ i ]} _i;k is simply the output of a space-time matched filter . Next we derive an explicit expression for this sufficient statistic in terms of the multiuser channel parameters and transmitted symbols, which is instrumental to developing various space-time multiuser receivers in subsequent sections.

Assume that the multipath delay spread of any user signal is limited to at most D symbol intervals, where D is a positive integer. That is,

Equation 5.50

Define the following cross- correlations of the delayed user signaling waveforms:

Equation 5.51

Since t _l,k D T and s _k ( t ) is nonzero only for t [0, T ], it then follows that for j > D . Now substituting (5.45) into (5.49), we have

Equation 5.52

where { u _l,k [ i ]} are zero-mean complex Gaussian random sequences with the following covariance:

Equation 5.53

where I _p denotes a p x p identity matrix and d ( t ) is the Dirac delta function. Define the following quantities :

We can then write (5.52) in the following vector form:

Equation 5.54

where ° denotes the Schur matrix product (i.e., elementwise product), and from (5.53) the covariance matrix of the complex Gaussian vector sequence { u [ i ]} is

Equation 5.55

Substituting (5.54) into (5.49), we obtain a useful expression for the sufficient statistic y [ i ], given by

Equation 5.56

where { v [ i ]} is a sequence of zero-mean complex Gaussian vectors with covariance matrix

Equation 5.57

Note that by definition (5.51) we have . From this it follows that R ^[-j] = R ^[j]T , and therefore H ^[-j] = H ^[j]H .

5.3.3 Maximum-Likelihood Multiuser Sequence Detector

We now use the sufficient statistic above to derive the maximum-likelihood detector for symbols in b . The maximum-likelihood sequence decision rule chooses b that maximizes the log-likelihood function (5.48). Using (5.46), the second integral in (5.48) can be computed as

Equation 5.58

where H denotes the following MK x MK block Jacobi matrix:

Equation 5.59

( denotes the Kronecker matrix product), , and recall that .

Substituting (5.49) and (5.58) into (5.48), the log-likelihood function W ( b ) can then be written as

Equation 5.60

For any integer n satisfying 1 n MK , denote its modulo- K decomposition with remainder k ( n ) = 1, . . . , K , by n = h ( n ) K + k ( n ) [518]. Then we can write ^[2]

^[2] Notation: A [ i,j ] denotes the ( i,j )th element of matrix A ; b [ i ] denotes the i th element of vector b .

Equation 5.61

Equation 5.62

where in (5.62) the vectors x _n and f _n have dimensions D K + K “1, given, respectively, by

where denotes the k th column of matrix H ^[j] , and , denotes the ( k , k ')th entry of H ^[j] . Substituting (5.61) and (5.62) into (5.60), the log-likelihood function can then be decomposed as follows:

Equation 5.63

where and

Equation 5.64

with the state vector recursively defined according to x _{n +1} = [ x _n [2], . . . , x _n [ D K + K “ 1], x _n ] ^T and x ₁ = ( D K + K “1), where _m denotes a zero vector of dimension m .

Given the additive decomposition (5.63) of the log-likelihood function, it is straightforward to apply the dynamic programming to compute the sequence that maximizes W ( b ) (i.e., the maximum-likelihood estimate of the transmitted multiuser symbol sequences). Since the dimensionality of the state vector is D K + K “ 1, the computational complexity of the maximum-likelihood sequence detector is on the order of (2 ^{( D +1) K} ). Note that in the absence of multipath (i.e., L = 1 and D = 1), if the users are numbered according to their relative delays in ascending order (i.e., 0 t _1,1 · · · t _{1, K} < T ), the matrix H ^{[ “1]} becomes strictly upper triangular . In this case, the dimension of the state vector is reduced to K “ 1 and the computational complexity of the corresponding maximum likelihood sequence detection algorithm is (2 ^K ) [324, 518]. However, in the presence of multipath, even if the multipath delays are still within one symbol interval (i.e., D = 1), the matrix H ^{[ “1]} no longer has an upper triangular structure in general. Hence the dimension of the state vector in this case is 2 K “ 1 and the complexity of the dynamic programming is (2 ^{2 K} ). Even though the (2 ^{( D +1) K)} complexity is generally much lower than the (2 ^KM ) complexity of brute-force maximization of (5.60) ( D is typically only a few symbols, while the frame length M can be hundreds or even thousands of symbols), this complexity is still prohibitively high if the number of users is even moderate (say a dozen ). Thus, it is of interest to find low-complexity alternatives.