Two Transmit and Two Receive Antennas

Two Transmit and Two Receive Antennas

We combine the results from the two preceding sections to investigate an environment in which we use two transmit antennas and two receive antennas. We adopt the space-time block coding scheme used in the preceding section. The received signals at antenna 1 during the two symbol intervals are

Equation 5.154

Equation 5.155

and the corresponding signals received at antenna 2 are

Equation 5.156

Equation 5.157

where is the complex channel response between transmit antenna i and receive antenna j for user k. The noise vectors , and are independent and identically distributed with distribution N_c(0, s²I_N).

Linear Diversity Multiuser Detector

As before, we first consider the linear diversity multiuser detection scheme for user 1, which applies the linear multiuser detector w₁ in (5.96) to each of the four received signals , and and then performs a space-time decoding. Specifically, denote the filter outputs as

Equation 5.158

Equation 5.159

Equation 5.160

Equation 5.161

with

Equation 5.162

where, as before, .

We define the following quantities:

Then (5.158)–(5.162) can be written as

Equation 5.163

with

Equation 5.164

It is readily verified that

Equation 5.165

with

Equation 5.166

To form the ML decision statistic, we premultiply z by G₁ and obtain

Equation 5.167

with

Equation 5.168

The corresponding bit estimates are given by

Equation 5.169

The bit error probability is then given by

Equation 5.170

Linear Space-Time Multiuser Detector

We denote

Then (5.154)–(5.157) may be written as

Equation 5.171

Equation 5.172

where

Since and (5.171) has the same form as (5.142), it is easy to show that the decorrelating detector for detecting the bit b_1,1 based on is given by

Equation 5.173

Hence the output of the linear space-time detector in this case is given by

Equation 5.174

with

Equation 5.175

where

Equation 5.176

Therefore, the probability of detection error is given by

Equation 5.177

Comparing (5.177) with (5.170), it is seen that when two transmit antennas and two receive antennas are employed and the signals are transmitted in the form of a space-time block code, the linear diversity receiver and the linear space-time receiver have identical performance.

Remarks

We have seen that the performance of space-time multiuser detection (STMUD) and linear diversity multiuser detection (LDMUD) are similar for two transmit/one receive and two transmit/two receive antenna configurations. What, then, are the benefits of the space-time detection technique? They are as follows:

1. Although LDMUD and STMUD perform similarly for the 2 x 1 and 2 x 2 cases, the performance of STMUD is superior for configurations with one transmit antenna and P 2 receive antennas.

2. User capacity for CDMA systems is limited by correlations among composite signature waveforms. This multiple-access interference will tend to decrease as the dimension of the vector space in which the signature waveforms reside increases. The signature waveforms for linear diversity detection are of length N (i.e., they reside in ). Since the received signals are stacked for space-time detection, these signature waveforms reside in for two transmit and one receive antennas or for two transmit and two receive antennas. As a result, the space-time structure can support more users than linear diversity detection for a given performance threshold.

3. For adaptive configurations (Section 5.5.4 and Section 5.6.2), LDMUD 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. The space-time structure requires only one subspace tracker.