One Transmit Antenna, Two Receive Antennas
Consider the following discrete-time K-user synchronous CDMA channel with one transmit antenna and two receive antennas. The received baseband signal at the pth antenna can be modeled as
where sk is the N-vector
of the discrete-time signature waveform of the kth user with unit norm (i.e., ||sk|| =
1), bk 
{+1, –1} is the data bit of the kth
user, gp,k is the complex channel
response of the pth receive antenna element to
the kth user's signal, and np ~
Nc (0, s2 IN) is
the ambient noise vector at antenna p. It is
assumed that n1 and n2 are independent.
Linear Diversity Multiuser Detector
Denote
Suppose that user 1 is the user of interest. We first consider the linear diversity multiuser detection scheme, which first applies a linear multiuser detector to the received signal rp in (5.95) at each antenna p = 1, 2, and then combines the outputs of these linear detectors to make a decision. For example, a linear decorrelating detector for user 1 based on the signal in (5.95) is simply
where e1
denotes the first unit vector in 
. This detector is applied to the received
signal at each antenna p = 1, 2, to obtain z = [z1
z2]T, where
with
where ||w1||2 = [R–1]1,1. Denote
and 
. Since the noise vectors from
different antennas are independent, we can write
with
The maximum-likelihood (ML) decision rule for b1 based on z in (5.100) is then
Let 
be the total received desired user's
signal energy. The decision statistic in (5.102) can be expressed as
with
The probability of detection error is computed as
Linear Space-Time Multiuser Detector
Denote
Then, by augmenting the received signals at two antennas, (5.95) can be written as
with 
. A linear space-time multiuser
detector operates on the augmented received signal 
directly. For
example, the linear decorrelating detector for user 1 in this case is given
by
This detector is applied to the augmented received signal to
obtain
with
where 
. Denote
An expression for 
can be found as follows. Note that
where (5.111) and (5.112) follow, respectively, from the following two matrix identities:
Hence
where ° denotes the Schur matrix product (i.e., elementwise product).
The ML decision rule for b1 based on 
in (5.108) is then
The probability of detection error is computed as
Performance Comparison
From the discussion above it is seen that the linear space-time multiuser detector exploits the signal structure in both the time domain (i.e., induced by the signature waveform sk) and the spatial domain (i.e., induced by the channel response gk) for interference rejection; whereas for the linear diversity multiuser detector, interference rejection is performed only in the time domain, and the spatial domain is used only for diversity combining. The next result, which first appeared in [324], shows that the linear space-time multiuser detector always outperforms the linear diversity multiuser detector.
Proposition 5.6: Let 
(e)
given by (5.105)
and 
(e) given by (5.117) be,
respectively, the probability of detection error of the linear diversity
detector and the linear space-time detector. Then
Proof: By (5.105) and (5.117) it suffices to show that
We make use of the following facts. Denote by Ai,j the submatrix of A obtained by striking out the ith row and the jth column. Then it is known that
It is also known that
Assuming that 
and 
, and
using the two results above, we have
where (5.120) follows
from the fact that 
and 
; (5.121) follows from the matrix identity
and (5.122) follows from
Hence we have
We next consider a simple example to demonstrate the performance difference between the two receivers discussed above. Consider a two-user system with
where r is the correlation of the
signature waveforms of the two users and q1
and q2 are the directions of arrival of the
two users' signals. Define 
. Then we have E1 = E2 = 1 and
These expressions are plotted in Fig. 5.12. It is seen that while the multiuser space-time receiver can exploit both the temporal signal separation (along the r-axis) and the spatial signal separation (along the a-axis), the multiuser diversity receiver can exploit only the temporal signal separation. For example, for large r, the performance of the multiuser diversity receiver is poor, no matter what value a takes; but the performance of the multiuser space-time receiver can be quite good as long as a is large.
Figure 5.12. Performance comparison between a multiuser diversity receiver (top) and multiuser space-time receiver (bottom).
