Two Transmit Antennas, One Receive Antenna

Two Transmit Antennas, One Receive Antenna

When two antennas are employed at the transmitter, we must first specify how the information bits are transmitted across the two antennas. Here we adopt the well-known orthogonal space-time block coding scheme [12, 475]. Specifically, for user k, two information symbols, b_k,1 and b_k,2, are transmitted over two symbol intervals. At the first time interval, the symbol pair (b_k,1, b_k,2) is transmitted across the two transmit antennas; and at the second time interval, the symbol pair (–b_k,2, b_k,1) is transmitted. The received signals corresponding to these two time intervals are given by

Equation 5.130

Equation 5.131

where g_1,k (g_2,k,) is the complex channel response between the first (second) transmit antenna and the receive antenna; n₁ and n₂ are independent received N_c (0, I_N) noise vectors at the two time intervals.

Linear Diversity Multiuser Detector

We first consider the linear diversity multiuser detection scheme, which first applies the linear multiuser detector w₁ in (5.96) to the received signals r₁ and r₂ during the two time intervals, and then performs a space-time decoding. Specifically, denote

Equation 5.132

Equation 5.133

with

Equation 5.134

where ||w₁||² = [R^–1]_1,1.

Denote

It is easily seen that . Then (5.132)–(5.134) can be written as

Equation 5.135

with

Equation 5.136

As before, denote . Note that

Equation 5.137

The ML decision rule for b_1,1 and b_2,1 based on z in (5.135) is then given by

Equation 5.138

Using (5.135), it is easily seen that the decision statistic in (5.138) is distributed according to

Equation 5.139

Equation 5.140

Hence the probability of error is given by

Equation 5.141

This is the same expression as (5.117) for the linear diversity receiver with one transmit antenna and two receive antennas.

Linear Space-Time Multiuser Detector

Denote and . Then (5.130) and (5.131) can be written as

Equation 5.142

On denoting

the decorrelating detector for detecting the bit b_1,1 based on in (5.142) is given by

Equation 5.143

where is the first unit vector in . We have the following result.

Proposition 5.7: The decorrelating detector in (5.143) is given by

Equation 5.144

where w₁ is given by (5.96).

Proof: We need to verify that

Equation 5.145

We have

Equation 5.146

Equation 5.147

Equation 5.148

Equation 5.149

This verifies (5.145), so that (5.144) is indeed the decorrelating detector given by (5.143).

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

Equation 5.150

with

Equation 5.151

where using (5.99) and (5.144), we have

Equation 5.152

Therefore, the probability of detection error is given by

Equation 5.153

On comparing (5.141) with (5.153) we see that for the case of two transmit antennas and one receive antenna, the linear diversity receiver and the linear space-time receiver have the same performance. Hence the multiple transmit antennas with space-time block coding provide only diversity gain and no signal separation capability.