5.6 Adaptive Space-Time Multiuser Detection in Multipath CDMA

5.6.1 Signal Model

In this section we develop adaptive space-time multiuser detectors for asynchronous CDMA systems with two transmit and two receive antennas. The continuous-time signal transmitted from antennas 1 and 2 due to the k th user for time interval i {0,1, . . .} is given by

Equation 5.227

Equation 5.228

where M denotes the length of the data frame, T denotes the information symbol interval, and { b _k [i] } _i is the symbol stream of user k . Although this is an asynchronous system, we have, for notational simplicity, suppressed the delay associated with each user's signal and incorporated it into the path delays in (5.230). We assume that for each k the symbol stream, { b _k [i] } _i , is a collection of independent random variables that take on values of +1 and -1 with equal probability. Furthermore, we assume that the symbol streams of different users are independent. As discussed in Chapter 2, for the direct-sequence spread-spectrum (DS-SS) format, the user signaling waveforms have the form

Equation 5.229

where N is the processing gain, { c _q,k [ j ]}, q {1, 2} is a signature sequence of ±1's assigned to the k th user for bit b _q,k [ i ], and y ( t ) is a normalized chip waveform of duration T _c = T/N . The k th user's signals, and , propagate from transmit antenna a to receive antenna b through a multipath fading channel whose impulse response is given by

Equation 5.230

where is the complex path gain from antenna a to antenna b associated with the l th path for the k th user, and is the sum of the corresponding path delay and initial transmission delay of user k . It is assumed that the channel is slowly varying, so that the path gains and delays remain constant over the duration of one signal frame ( MT ).

The received signal component due to the transmission of and through the channel at receive antennas 1 and 2 is given by

Equation 5.231

Equation 5.232

Substituting (5.228) and (5.230) into (5.232), we have for receive antenna b {1,2},

Equation 5.233

For a, b, q {1, 2}, we define

Equation 5.234

In (5.234), ( t ) is the composite channel response for the channel between transmit antenna a and receive antenna b , taking into account the effects of the chip pulse waveform and the multipath channel. Then we have

Equation 5.235

The total received signal at receive antenna b {1,2} is given by

Equation 5.236

At the receiver, the received signal is match-filtered to the chip waveform and sampled at the chip rate (i.e., the sampling interval is T _c , N is the total number of samples per symbol interval, and 2 N is the total number of samples per time slot). The n th matched-filter output during the i th time slot is given by

Equation 5.237

Denote the maximum delay (in symbol intervals) as

Equation 5.238

Substituting (5.235) into (5.237), we obtain

Equation 5.239

Further substitution of (5.234) into (5.239) shows that

Equation 5.240

We may write more compactly as

Equation 5.241

where

Equation 5.242

Equation 5.243

For j = 0,1, . . . ,[ I /2], define the following notation:

Equation 5.244

Equation 5.245

Equation 5.246

Then we have

To exploit both temporal and spatial diversity, we stack the vectors received from both receive antennas,

and observe that

Equation 5.247

where

By stacking m successive received sample vectors, we create the following quantities :

where . We can write (5.247) in matrix form as

Equation 5.248

We will see in Section 5.6.3 that the smoothing factor, m , is chosen such that

Equation 5.249

for channel identifiability. Note that the columns of H (the composite signature vectors) contain information about both the timings and the complex path gains of the multipath channel of each user. Hence an estimate of these waveforms eliminates the need for separate estimates of the timing information .

5.6.2 Blind MMSE Space-Time Multiuser Detection

Since the ambient noise is white (i.e., E { v [ i ] v [ i ] ^H } = s ² I _{4 Nm} ), the autocorrelation matrix of the received signal in (5.248) is

Equation 5.250

Equation 5.251

where (5.251) is the eigendecomposition of R . The matrix U _s has dimensions 4 Nm x r and U _n has dimensions 4 Nm x (4 N _{m “ r} ).

The linear MMSE space-time multiuser detector and corresponding bit estimate for b _a,k [ i ] ,a {1, 2} are given, respectively, by

Equation 5.252

Equation 5.253

The solution to (5.252) can be written in terms of the signal subspace components as

Equation 5.254

where is the composite signature waveform of user k for bit a {1,2}. This detector is termed blind since it requires knowledge only of the signature sequence of the user of interest. Of course, we also require estimates of the signal subspace components and of the channel. We address the issue of channel estimation next .

5.6.3 Blind Adaptive Channel Estimation

In this section we extend the blind adaptive channel estimation technique described in Section 5.6.2 to the asynchronous multipath case. First, however, we describe the discrete-time channel model in order to formulate an analog to the optimization problem in (5.208).

Discrete-Time Channel Model

Using (5.244), it is easy to see that

Equation 5.255

From (5.241) we have for j = 0, . . . , [ I /2]; n = 0, . . . , 2 N “ 1; b = 1,2,

Equation 5.256

Equation 5.257

We will develop the discrete-time channel model for . The development for follows similarly. From (5.240) we see that

Equation 5.258

From (5.240) we can also see that the sequences and are zero whenever i < 0 or i > I N . With this in mind we define the following vectors:

Then we have

Equation 5.259

where

A similar development for produces the result

Equation 5.260

where

The final task in the section is to form expressions for the composite signature waveforms h _{1, k} and h _{2, k} in terms of the signature matrices C _{1, k} , C _{2, k} and the channel response vectors and . Denote by C _a,k [ j ], j = 0, 1, . . . , [ I /2], a {1,2} the submatrix of C _a,k consisting of rows 2 Nj + 1 through 2( j + 1) N . Then it is easy to show that

Equation 5.261

where

Blind Adaptive Channel Estimation

The blind channel estimation problem for the asynchronous multipath case involves the estimation of f _a,k from the received signal r [ i ]. As we did for the synchronous case, we exploit the orthogonality between the signal subspace and noise subspace. Specifically, since U _n is orthogonal to the column space of H , we have

Equation 5.262

Denote by z [ i ] the projection of the received signal r [ i ] onto the noise subspace:

Equation 5.263

Equation 5.264

Using (5.262), we have

Equation 5.265

Our channel estimation problem, then, involves solution of the optimization problem

Equation 5.266

subject to the constraint f = 1. If we denote , we can use the Kalman-type algorithm described in (5.209) “(5.211), where h ₁ [ i ] is replaced with f _a,k [ i ].

Note that a necessary condition for the channel estimate to be unique is that the matrix is tall [i.e., 4 Nm “ 2 K ( m + I 2 ) 2 N ( I + 1)+ 2]. Therefore, we choose the smoothing factor m such that

Equation 5.267

Using the same constraint, we find that for a fixed m , the maximum number of users that can be supported is

Equation 5.268

Equation (5.268) illustrates one of the benefits of the proposed space-time receiver structure over the linear diversity structure. Notice that for reasonable choices of m and I , (5.268) is larger than the maximum number of users for the linear diversity receiver structure, given by

Equation 5.269

Another significant benefit of the space-time receiver is reduced complexity. The diversity structure requires four independent subspace trackers operating simultaneously since the receiver performs detection on each of the four received signals, and each has a different signal subspace. Since we stack received signal vectors before detection, the space-time structure requires only one subspace tracker.

Once an estimate of the channel state, , is obtained, the composite signature vector of the k th user for bit a is given by (5.261). Note that there is an arbitrary phase ambiguity in the estimated channel state, which necessitates differential encoding and decoding of the transmitted data. Finally, we summarize the blind adaptive linear space-time multiuser detection algorithm in multipath CDMA as follows.

Algorithm 5.6: [Blind adaptive linear space-time multiuser detector ”asynchronous multipath CDMA, two transmit antennas, and two receive antennas]

• Stack matched-filter outputs in (5.237) to form r [ i ] in (5.248).

• Form as discussed in Section 5.6.3.

• Using a suitable signal subspace tracking algorithm (e.g., NAHJ), update the signal subspace components U _s [ i ] and L _s [ i ] at each time slot i.

• Track the channel f _a,k according to the following

  Equation 5.270

  

  
  

  Equation 5.271

  

  
  

  Equation 5.272

  

  
  

  Equation 5.273

  

  
  

  Equation 5.274

  

  
  

• Form the detectors:

  Equation 5.275

  

  
  

• Perform differential detection:

  Equation 5.276

  

  
  

  Equation 5.277

  

  
  

Simulation Results

In what follows we present simulation results to illustrate the performance of blind adaptive space-time multiuser detection. We first look at the synchronous flat fading-case; then we consider the asynchronous multipath-fading scenario. For all simulations we use the two transmit/two receive antenna configuration. Gold codes of length 15 are used for each user. The chip pulse is a raised cosine with roll-off factor 0.5. For the multipath case, each user has L = 3 paths. The delay of each path is uniform on [0, T ]. Hence, the maximum delay spread is one symbol interval (i.e., I = 1). The fading gain for each user's channel is generated from a complex Gaussian distribution and is fixed for all simulations. The path gains in each user's channel are normalized so that each user's signal arrives at the receiver with the same power. The smoothing factor is m = 2. For SINR plots, the number of users for the first 1500 iterations is four. At iteration 1501, three users are added so that the system is fully loaded. At iteration 3001, five users are removed. For the steady-state bit-error-probability plots, the frame size is 200 bits and the system is allowed 1000 bits to reach steady state before errors are checked. The forgetting factor for the subspace tracking algorithm for all simulations is 0.995.

The performance measures are bit-error probability and the signal-to-interference-plus-noise ratio, defined by , where the expectation is with respect to the data bits of interfering users, ISI bits, and ambient noise. In the simulations, the expectation operation is replaced by time averaging. SINR is a particularly appropriate figure of merit for MMSE detectors since it has been shown [386] that the output of an MMSE detector is approximately Gaussian distributed. Hence, the SINR values translate directly and simply to bit-error probabilities [i.e., ]. The labeled horizontal lines on the SINR plots (Figs. 5.14 and 5.16) represent BER thresholds.

Figure 5.14. Adaptation performance of space-time multiuser detection for synchronous CDMA. The labeled horizontal lines represent bit-error-probability thresholds.

Figure 5.14 illustrates the adaptation performance for the synchronous case. The SNR is 8 dB. Notice that the BER does not drop below 10 ^-3 even when users enter or leave the system.

Figure 5.15 shows the steady-state performance for the synchronous case for different system loads. We see that the performance changes little as the system load changes.

Figure 5.15. Steady-state performance of space-time multiuser detection for synchronous CDMA.

Figure 5.16 shows the adaptation performance for the asynchronous multipath case. The SNR for this simulation is 11 dB. Again, notice that the BER does not drop significantly as users enter and leave the system.

Figure 5.16. Adaptation performance of space-time multiuser detection for asynchronous multipath CDMA. The labeled horizontal lines represent BER thresholds.

Figure 5.17 shows the steady-state performance for the asynchronous multipath case for different system loads. It is seen that system loading has a more significant effect on performance for the asynchronous multipath case than it does for the synchronous case.

Figure 5.17. Steady-state performance of space-time multiuser detection for asynchronous multipath CDMA.