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
Equation 5.95
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
Equation 5.96
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
Equation 5.97
with
Equation 5.98
where ||w1||2 = [R–1]1,1. Denote
Equation 5.99
and  . Since the noise vectors from
different antennas are independent, we can write
Equation 5.100
with
Equation 5.101
The maximum-likelihood (ML) decision rule for b1 based on z in (5.100) is then
Equation 5.102
Let  be the total received desired user's
signal energy. The decision statistic in (5.102) can be expressed as
Equation 5.103
with
Equation 5.104
The probability of detection error is computed as
Equation 5.105
Linear Space-Time Multiuser Detector
Denote
Then, by augmenting the received signals at two antennas, (5.95) can be written as
Equation 5.106
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
Equation 5.107
This detector is applied to the augmented received signal to
obtain
Equation 5.108
with
Equation 5.109
where  . Denote
Equation 5.110
An expression for  can be found as follows. Note that
Equation 5.111
Equation 5.112
where (5.111) and (5.112) follow, respectively, from the
following two matrix identities:
Equation 5.113
Equation 5.114
Hence
Equation 5.115
where ° denotes the Schur matrix product (i.e., elementwise
product).
The ML decision rule for b1 based on  in (5.108) is then
Equation 5.116
The probability of detection error is computed as
Equation 5.117
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
Equation 5.118
It is also known that
Equation 5.119
Assuming that  and  , and
using the two results above, we have
Equation 5.120
Equation 5.121
Equation 5.122
where (5.120) follows
from the fact that  and  ; (5.121) follows from the matrix identity
Equation 5.123
and (5.122) follows
from
Equation 5.124
Hence we have
Equation 5.125
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
Equation 5.126
Equation 5.127
Equation 5.128
Equation 5.129
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.
| 
Sections
Syndication
