7.8 Maximum-Likelihood Code-Aided Method

7.8 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 t ₁ =0, in which the NBI signal is given by

Equation 7.143

where A _I > 0 is the received amplitude of the NBI signal, m is the number of digital NBI symbols transmitted per spread-spectrum data symbol, d _I [ j ] is the j th (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 { s ₁ (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 { b ₁ [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 b ₁ [0], ... d _I [0], ..., d _I [ m “1]. Note that with the NBI symbols d _I [0], ... d _I [ m “1] fixed, the maximum-likelihood choice of the spread-data symbol b ₁ [0] is easily seen to be given by

Equation 7.147

so that the maximum over b ₁ [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 2 ^m possible values for these m binary symbols. However, in a practical overlay system, the parameters A _I , A ₁ , and r _j 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.

Figure 7.19. BER comparison of maximum-likelihood and MMSE code-aided suppression of digital NBI: N = 15, m = 3, and for several values of presuppression interference-to-signal ratio (ISR).

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.