Extension to Dispersive Channels
So far we have assumed that the channels are nondispersive
[i.e., there is no intersymbol interference (ISI)]. We next extend the
techniques considered in previous subsections to dispersive channels and develop
space-time processing techniques for suppressing both co-channel interference
and intersymbol interference.
Let D be the delay spread of the
channel (in units of symbol intervals). Then the received signal at the antenna
array during the ith symbol interval can be
expressed as
Equation 5.40
where gl,k is the array steering vector for
the kth user's lth
symbol delay and 
. Denote 
. By
stacking m successive data samples, we define the
following quantities:
Then from (5.40) we can
write
Equation 5.41
where P is a matrix of the
form
with
Here, as before, m is the
smoothing factor and is chosen such that the matrix G is a "tall" matrix [i.e., Pm 
K(m + D – 1)]. Hence 
. We assume that G has full column rank. From the signal model (5.41), it is evident that the techniques
discussed in previous subsections can be applied straightforwardly to dispersive
channels, with signal processing carried out on signal vectors of higher
dimension. For example, the linear MMSE combining method for estimating the
transmitted symbol b1[i] is based on quantizing the correlator output wHr[i], where w = C-1g1, with
with
To apply the subspace-based adaptive array algorithm, we first
estimate the signal subspace (Us,Ls) of C by forming the sample autocorrelation matrix of
r[i] and
then performing an eigendecomposition. Notice that the rank of the signal
subspace is K x
(m + D – 1). Once the
signal subspace is estimated, it is straightforward to apply the algorithms
listed in Section 5.2.3 to estimate
the data symbols.
Simulation Examples
In what follows we provide some simulation examples to
demonstrate the performance of the subspace-based adaptive array algorithm
discussed above. In the following simulations, it is assumed that an array of
P = 10 antenna elements is employed at the base
station. The number of symbols in each time slot is M = 162 with mt = 14 training symbols, as in IS-54/136
systems. The modulation scheme is binary PSK (BPSK). The channel is subject to
Rayleigh fading, so that the steering vectors {gk,
k = 1, . . . , K}
are i.i.d. complex Gaussian vectors, 
, where 
is the
received power of the kth user. The desired user
is user 1. The interfering signal powers are assumed to be 6 dB below the
desired signal power (i.e., Ak = A1/2, for k =
2, . . . , K). The ambient noise process {n[i]} is a
sequence of i.i.d. complex Gaussian vectors, n[i] ~ Nc(0, s2IP).
In the first example we compare the performance of the two
steering vector estimators 
in (5.31) and in (5.7). The number of users is six (i.e., K = 6) and the channels have no dispersion. For each
SNR value, the normalized root-mean-square error (MSE) is computed for each
estimator. For the subspace estimator, we consider its performance under both
the exact signal subspace parameters (Us,
Ls) and the estimated signal subspace parameters
(
, 
). The results are plotted in Fig. 5.2. It is seen that the subspace-based steering
vector estimator offers significant performance improvement over the
conventional correlation estimator, especially in the high-SNR region. Notice
that although both estimators tend to exhibit error floors at high SNR values,
their causes are different. The floor of the sample correlation estimator is due
to the finite length of the training preamble mt, whereas the floor of the subspace
estimator is due to the finite length of the time slot M. It is also seen that the performance loss due to
inexact signal subspace parameters is not significant in this case.
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