Sufficient Statistic
We next derive a sufficient statistic for demodulating the
multiuser symbols from the space-time signal model (5.45). To do so, we first denote the useful signal in (5.45) by
Equation 5.46
where 
and 
. Using
the Cameron–Martin formula [377], the likelihood function of the
received waveform r(t) in (5.45) conditioned on all the transmitted
symbols b of all users can be written
as
Equation 5.47
where
Equation 5.48
The first integral in (5.48) can be expressed as
Equation 5.49
Since the second integral in (5.48) does not depend on the received signal r(t), by (5.49) we
see that {yk[i]} is a sufficient statistic for detecting the
multiuser symbols b. From (5.49) it is seen that this sufficient statistic is
obtained by passing the received signal vector r(t) through
KL beamformers directed at each path of each
user's signal, followed by a bank of K
maximum-ratio multipath combiners (i.e., RAKE receivers). Since this beamformer
is a spatial matched filter for the array antenna receiver, and a RAKE receiver
is a temporal matched filter for multipath channels, the sufficient statistic
{yk[i]}i;k is
simply the output of a space-time matched filter.
Next we derive an explicit expression for this sufficient statistic in terms of
the multiuser channel parameters and transmitted symbols, which is instrumental
to developing various space-time multiuser receivers in subsequent sections.
Assume that the multipath delay spread of any user signal is
limited to at most D symbol intervals, where D is a positive integer. That is,
Equation 5.50
Define the following cross-correlations of the delayed user
signaling waveforms:
Equation 5.51
Since tl,k 
D T and sk(t) is
nonzero only for t 
[0, T], it then follows that 
for |j| > D. Now
substituting (5.45) into (5.49), we have
Equation 5.52
where {ul,k[i]} are zero-mean complex Gaussian random sequences
with the following covariance:
Equation 5.53
where Ip denotes a p x p identity matrix
and d(t) is the Dirac delta function. Define the following
quantities:
We can then write (5.52)
in the following vector form:
Equation 5.54
where ° denotes the Schur matrix product (i.e., elementwise
product), and from (5.53) the covariance
matrix of the complex Gaussian vector sequence {u[i]} is
Equation 5.55
Substituting (5.54) into
(5.49), we obtain a useful expression for
the sufficient statistic y[i], given by
Equation 5.56
where {v[i]} is a
sequence of zero-mean complex Gaussian vectors with covariance matrix
Equation 5.57
Note that by definition (5.51) we have 
. From this it follows that R[-j] = R[j]T,
and therefore H[-j] =
H[j]H .
Sections
