Multiuser Detection
To finish this section we turn finally to the full
multiple-access model of (1.9), within which data
detection is referred to as multiuser detection.
This situation is very similar to the ISI channel described above. In
particular, we now consider the likelihood function of the observations r(·) conditioned on all symbols of all users. Sorting
these symbols first by symbol number and then by user number, we can collect
them in a column vector b given as
Equation 1.50
so that the nth element of b is given by
Equation 1.51
where [·]K denotes reduction of the argument modulo K and 
. 
denotes the integer part of the argument.
Analogously with (1.38) we can write the
corresponding likelihood function as
Equation 1.52
where y is a column vector
that collects the set of observables
Equation 1.53
indexed conformally with b, and where H denotes the KM x
KM Hermitian cross-correlation matrix of the
composite waveforms associated with the symbols in b, again with conformal indexing:
Equation 1.54
with
Equation 1.55
Comparing (1.52), (1.53), and (1.54) with their single-user counterparts (1.38), (1.39), and (1.40),
we see that y is a sufficient statistic for
making inferences about b, and moreover
that such inferences can be made in a manner very similar to that for the
single-user ISI channel. The principal difference is one of dimensionality:
Decisions in the single-user ISI channel involve simultaneous sequence detection
with M symbols, whereas decisions in the
multiple-access channel involve simultaneous sequence detection with KM symbols. This, of course, can increase the
complexity considerably. For example, the complexity of exhaustive search in ML
detection, or exhaustive summation in MAP detection, is now on the order of 
.
However, as in the single-user case, this complexity can be mitigated
considerably if the delay spread of the channel is small. In particular, if the
duration of the composite signaling waveforms is D, the matrix H
will be a banded matrix with
Equation 1.56
where, as before, D = 
D/T
. This bandedness allows the complexity of both
ML and MAP detection to be reduced to the order of 
via
dynamic programming.
Although further complexity reduction can be obtained in this
problem within additional structural constraints on H (see, e.g., [380]), the 
complexity
of ML and MAP multiuser detection is not generally reducible. Consequently, as
with the equalization of single-user channels, a number of lower-complexity
suboptimal multiuser detectors have been developed. For example, analogously
with (1.47), linear multiuser detectors
can be written in the form
Equation 1.57
where M is a KM x KM matrix, [My]n
denotes the nth component of My, and where, as before, q(·) denotes a quantizer mapping the complex numbers to
the symbol alphabet A. The choice M = H-
1 forces both MAI and ISI to zero and is known as the decorrelating
detector, or decorrelator. Similarly, the
choice
Equation 1.58
where Sb denotes
the covariance matrix of the symbol vector b, is known as the linear
MMSE multiuser detector. Linear and nonlinear iterative versions of these
detectors have also been developed, both to avoid the complexity of inverting
KM x KM matrices and to exploit the
finite-alphabet property of the symbols (see, e.g., [520]).
As a final issue here we note that all of the discussion above
has involved direct processing of continuous-time observations to obtain a
sufficient statistic (in practice, this corresponds to hardware front-end
processing), followed by algorithmic processing to obtain symbol decisions.
Increasingly, an intermediate step is of interest. In particular, it is often of
interest to project continuous-time observations onto a large but finite set of
orthonormal functions to obtain a set of observables. These observables can then
be processed further using digital signal processing (DSP) to determine symbol
decisions (perhaps with intermediate calculation of the sufficient statistic),
which is the principal advantage of this approach. A tacit assumption in this
process is that the orthonormal set spans all of the composite signaling
waveforms of interest, although this will often be only an approximation. A
prime example of this kind of processing arises in direct-sequence
spread-spectrum systems [see (1.6)], in which the received
signal can be passed through a filter matched to the chip waveform and then
sampled at the chip rate to produce N samples per
symbol interval. These N samples can then be
combined in various ways (usually, linearly) for data detection. In this way,
for example, the linear equalizer and multiuser detectors discussed above are
particularly simple to implement. A significant advantage of this approach is
that this combining can often be done adaptively when some aspects of the
signaling waveforms are unknown. For example, the channel impulse response may
be unknown to the receiver, as may the waveforms of some interfering signals.
This kind of processing is a basic element of many of the results discussed in
this book and will be revisited in more detail in
Sections
