6.5 Turbo Multiuser Detection in CDMA with Multipath Fading

In this section we generalize the low-complexity SISO multiuser detector developed in Section 6.3.3 for synchronous CDMA systems to general asynchronous CDMA systems with multipath fading channels. The method discussed in this section was first developed in [257].

6.5.1 Signal Model and Sufficient Statistics

We consider a K - user asynchronous CDMA system employing a periodic spreading waveforms and signaling over multipath fading channels. The transmitted signal due to the k th user is given by

Equation 6.131

where M is the number of data symbols per user per frame, T is the symbol interval, and A _k , b _k [ i ], and { s _i,k (t) ; 0 t T } denote, respectively, the amplitude, i th transmitted bit, and normalized signature waveform during the i th symbol interval of the k th user. It is assumed that s _i,k (t) is supported only on the interval [0, T ] and has unit energy. Note here that we allow the possibility that aperiodic spreading waveforms are employed in the system, and hence the spreading waveforms vary with symbol index i .

The k th user's signal x _k ( t ) propagates through a multipath channel with impulse response

Equation 6.132

where L is the number of paths in the k th user's channel, and where a _l,k (t) and t _l,k are, respectively, the complex fading process and the delay of the l th path of the k th user's signal. It is assumed that the fading is slow:

which is a reasonable assumption in many practical situations. At the receiver, the received signal due to the k th user is then given by

Equation 6.133

where * denotes convolution. The received signal at the receiver is the superposition of the K users' signals plus the additive white Gaussian noise, given by

Equation 6.134

where n(t) is a zero-mean complex white Gaussian noise process with power spectral density s ^2.

Denote and . Define

Equation 6.135

Similar to the situations in earlier chapters, using the Cameron “Martin formula [377], the likelihood function of the received waveform r(t) in (6.134) conditioned on all the transmitted symbols b of all users can be written as

Equation 6.136

where C is some positive scalar constant, and

Equation 6.137

The first integral in (6.137) can be expressed as

Equation 6.138

Since the second integral in (6.137) does not depend on the received signal r(t) , by (6.138) a sufficient statistic for detecting the multiuser symbols b is { y _k [ i ]} _i;k . From (6.138) it is seen that this sufficient statistic is obtained by passing the received signal r(t) through a bank of K maximal-ratio multipath combiners (i.e., RAKE receivers). 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 the SISO multiuser detector. Note that the derivations below are similar to those in Section 5.3.1 for space-time CDMA systems.

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

Equation 6.139

Define the following correlation of the delayed signaling waveforms:

Equation 6.140

Since t _,k D T and s _i,k ( t ) is nonzero only for t [0, T ], it then follows that for j D . Now substituting (6.134) into (6.138), we have

Equation 6.141

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

Equation 6.142

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 (6.141) in the following vector form:

Equation 6.143

and from (6.142), the covariance matrix of the complex Gaussian vector sequence { u [ i ]} is

Equation 6.144

Substituting (6.143) into (6.138), we obtain an expression for the sufficient statistic y [ i ], given by

Equation 6.145

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

Equation 6.146

Note that by definition (6.140), we have . It then follows that R ^{[- j ]} [ i ] = R ^{[ j ]} [ i ] ^T , and therefore, H ^{[- j ]} [ i ] = H ^{[ j ]} [ i ] ^H .

6.5.2 SISO Multiuser Detector in Multipath Fading Channels

In what follows we assume that the multipath spread is within one symbol interval (i.e., D = 1). Define the following quantities:

We can then write (6.145) in matrix form as

Equation 6.147

where by (6.146), v (i) ~ N _c ( , s ² H [ i ]).

Based on the a priori LLR of the code bits of all users, { l ₂ ( b _k [ i ])} _i;k , provided by the MAP channel decoder, we first form soft estimates of the user code bits:

Equation 6.148

Denote

Equation 6.149

Equation 6.150

Equation 6.151

where e _k denotes a 3 K -vector of all zeros, except for the ( K + k )th element, which is 1.

At symbol time i , for each user, a soft interference cancellation is performed on the received discrete-time signal y [ i ] in (6.147), to obtain

Equation 6.152

Equation 6.153

An instantaneous linear MMSE filter is then applied to y _k [ i ] , to obtain

Equation 6.154

where the filter is chosen to minimize the mean-square error between the code bit b _k [ i ] and the filter output z _k [ i ]:

Equation 6.155

where

Equation 6.156

Equation 6.157

with

The solution to (6.155) is given by

Equation 6.158

As before, in order to form the LLR of the code bit b _k [ i ], we approximate the instantaneous linear MMSE filter output z _k [ i ] in (6.154) as being Gaussian [i.e., z _k [ i ] ~ N _c ( m _k [ i ] b _k [ i ], [ i ])]. Conditioned on the code bit b _k [ i ], the mean and variance of z _k [ i ] are given, respectively, by

Equation 6.159

Equation 6.160

Therefore, the extrinsic information l ₁ ( b _k [ i ]) delivered by the instantaneous linear MMSE filter is given by

Equation 6.161

The SINR at the instantaneous linear MMSE filter output is given by

Equation 6.162

Recursive Algorithm for Computing Soft Output

Similar to our earlier discussion, computation of the extrinsic information can be implemented efficiently . In particular, the major computation involved is the following K x K matrix inversion:

Equation 6.163

Note that D _k [ i ] can be written as

Equation 6.164

where

Equation 6.165

Substituting (6.164) into (6.163), we have

Equation 6.166

where H _{(:,K+ k )} [ i ] denotes the ( K + k )th column of H [ i ]. Define

Equation 6.167

Then by the matrix inversion lemma, (6.166) can be written as

Equation 6.168

Equations (6.167) and (6.168) constitute a recursive procedure for computing Y _k [ i ] in (6.163).

Next, we summarize the SISO multiuser detection algorithm in multipath fading channels ( D = 1) as follows.

Algorithm 6.4: [SISO multiuser detector ”multipath fading channel]

• Form the soft bit estimates using (6.148) “(6.151) .

• Compute the matrix inversions using (6.167) and (6.168).

• For i = 0, ..., M “ 1 and for k = 1, ..., K, compute z _k [ i ] using (6.152), (6.154), and (6.158); compute m _k [ i ] using (6.159); and compute l ₁ (b _k [ i ]) using (6.161).

Finally, we examine the computational complexity of the SISO multiuser detector in multipath fading channels. By (6.167), it takes ( K ³ ) multiplications to obtain Y [ i ] using direct matrix inversion. After Y [ i ] is obtained, by (6.168), it takes ( K ² ) more multiplications to get Y _k [ i ] for each k . Since at each time i , Y [ i ] is computed only once for all K users, it takes ( K ² ) multiplications per user per code bit to obtain Y _k [ i ]. After Y _k [ i ] is computed, by (6.154), (6.159), and (6.161), it takes ( K ² ) multiplications to obtain l ₁ ( b _k [ i ]). Therefore, the total time complexity of the SISO multiuser detector is ( K ² ) per user per code bit.