Maximum-Likelihood Code-Aided Method
In Section 7.7, we saw that
the linear MMSE detector is a very useful tool for NBI suppression in DS-CDMA
systems. A natural question to ask is whether its favorable performance
properties can be improved upon. Within the context of linear code-aided
methods, an optimal method was proposed and analyzed in [426], although without comparison to
the linear MMSE technique (which had not yet been explored in this context at
that time). In this section we look briefly at a more generally optimal,
nonlinear, code-aided NBI suppression technique.
We saw in Section 7.2 and 7.3 that in
the context of predictive suppression, performance gains can be obtained by
going from a linear to a nonlinear method to exploit signal structure. In the
code-aided context, this suggests that it could be of use to progress from
linear methods to optimal methods. One such method is maximum-likelihood
detection, which is known to offer the ultimate performance improvement against
MAI.
In the context of NBI suppression, we can examine a
maximum-likelihood detector in the setting of digital NBI discussed in Section
7.4.4. To examine this situation, let us look at the signal model of (7.1)–(7.2) with a
single spread user (i.e., K = 1) and with t1 =0, in which the
NBI signal is given by
Equation 7.143
where AI > 0 is
the received amplitude of the NBI signal, m is
the number of digital NBI symbols transmitted per spread-spectrum data symbol,
dI[j]
is the jth (binary) symbol of the NBI, and {p(t)} is the basic pulse shape (having unit energy and
duration T/m) used by the digital NBI. To
simplify the discussion, we assume that {p(t)} is
synchronous with {s1(t)} so that
exactly m symbols of the NBI interfere with each
symbol of the spread data signal. Similar to the situation in Fig. 7.11, we can think of the
signal (7.143) as adding a set of m additional users to the channel, so that we have a
multiple-access channel with m + 1 total
users.
We now consider the maximum-likelihood detection of the symbol
stream {b1[i]} of the overlaid spread
signal {S(t)}. Due to the synchrony and the
assumption of white Gaussian noise, we can restrict attention to a single symbol
interval. Examining the i = 0 spread-data symbol
interval, the log-likelihood function of the received waveform {r(t)} can be shown straightforwardly to be proportional
to (see, e.g., [381])
Equation 7.144
where 
Equation 7.145
Equation 7.146
Let us examine the likelihood function (7.144) for a maximum over the unknown symbols b1[0], ... dI[0], ..., dI[m–1]. Note
that with the NBI symbols dI[0], ...
dI[m–1]
fixed, the maximum-likelihood choice of the spread-data symbol b1[0] is easily seen to be given by
Equation 7.147
so that the maximum over b1[0] can we written as
Equation 7.148
To find the global maximum-likelihood solution, we must
maximize the quantity in (7.148) over
the NBI data symbols, which generally requires direct search over the
2m possible values for these m binary symbols. However, in a practical overlay
system, the parameters AI, A1, and rj should be such that the narrowband
symbols can be detected by conventional methods with a relatively low
probability of error. Thus, (7.148) is
dominated by the first term on its right-hand side and so should be
approximately maximized by the choice
Equation 7.149
which maximizes this first term 
. So an
approximate maximum-likelihood detector for the spread user's data symbol is
Equation 7.150
This detector is essentially an "onion peeling" detector, in
which the layer of NBI symbols is peeled off (i.e., detected and subtracted)
using a conventional narrowband detector, and then the residual left after
peeling is used for conventional detection of the spread user's symbol. Note
that this detector fits the general mode of NBI suppression systems, in which a
replica of the NBI is formed and then subtracted from the spread-spectrum signal
before it is detected. A distinction is that here, this process takes place
after despreading, so it fits within the code-aided framework. Note that
multiple spread users can also be handled in this way, by first peeling off the
NBI and then applying a standard multiuser detector on the residual. Similar
ideas have been proposed in the context of multirate systems in [220, 299, 371, 509, 567].
Whether the detector of (7.150) offers general performance improvements over the
linear code-aided methods of Section 7.7 is an
interesting open question. Results from a simulation example comparing the
maximum-likelihood and linear MMSE code-aided detectors for digital interferers
with N = 15 and m
= 3 are shown in Fig. 7.19. In this
example it is seen that for a presuppression interference-to-signal ratio (ISR)
of 0 dB the linear MMSE detector is better than the ML detector, but at ISR = 5
dB (and, of course, for larger values of ISR), the opposite behavior is
observed. Also observe that for increasing ISR, the linear MMSE performance
degrades (even though very slightly, in view of the near–far resistant feature
of the linear MMSE receiver), whereas for increasing ISR, the performance of the
ML detector improves. This matches with the intuition that for large ISR, the
NBI can be better canceled with such a receiver.
There are many other techniques and aspects of the NBI
suppression problem that we have not discussed in this chapter. Such
contributions include a variety of other adaptive techniques [58, 72, 135, 153, 173, 285, 286, 352, 451, 480]; subspace-based methods [16, 118, 182]; Markov chain Monte Carlo
(MCMC)-based Bayesian methods [594]; results for higher-order
signaling [254, 539, 556]; other types of interference,
such as chirp signals [151]; the effects of NBI suppression
on tasks such as acquisition and tracking and on the correlation properties of
spreading sequences (and vice versa) [155, 242, 328, 469]; and the explicit exploitation
of cyclostationarity in this context [57]. The interested reader is
referred to these sources for further details.
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