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260 CHAPTER 6 Equalization of Time-Varying Channels With a TV channel, Q will not be circulant, and thus Q will not be diagonal. In this case, the off- diagonal terms of Q will be nonzero, complicating the estimation of a from y. In fact, the interference power profile of Q with CP-SCM is identical to that of Q with CP-OFDM, which (as shown in Fig. 6.4) decays quite slowly with distance from the diagonal. However, through the application of time-domain windowing 17 at the demodulator (Schniter & Liu, 2003), it is possible to give Q the narrowly quasi- banded support of Fig. 6.2(b), in which case any of the fast linear equalization techniques described in Section 6.3.3 can be used to estimate a from y. For example, Tang & Leus (2008) proposed a method to equalize a single-carrier system using the OFDM fast LMMSE technique (Rugini et al., 2005). Because a does not have a finite-alphabet structure, the trellis, DFE, and tree-search-based tech- niques discussed in Section 6.3.3 are not directly applicable to the estimation of a. Iterative soft equalization, however, is applicable. We now summarize the approach proposed by Schniter & Liu (2003). First, the fast serial iterative soft equalization technique of Schniter (2004) is used to compute ^ the LMMSE interference-canceled estimate a from the frequency-domain windowed observations y ^ (given current estimates of the time-domain symbol means and variances). Next, the estimates a are ^ ^ transformed to the time domain through a = W H a , from which posterior LLRs are calculated for each of the bits c[ j]. The posterior LLR computation is more complicated than (6.54)(6.55), though, due to the correlation that results from the time-frequency transformation. Finally, the posterior LLRs are used as priors in the next iteration, which begins by recalculating the time-domain symbol means and variances. In Schniter & Liu (2003), a fast algorithm for the entire procedure was derived that con- sumes only O(D 2 K log K) operations per block iteration. Ng & Falconer (2004) later extended the technique of Schniter & Liu (2003) to include widely linear estimation (although they neglected the receiver windowing step). Although the windowed FDE method above focuses on CP-SCM, similar techniques can be applied to ZP-SCM under appropriate processing of the received guard samples. For single-carrier modu- lation without a prefix, the use of IBI-cancellation and cyclic-prefix reconstruction (Kim & St uber, ¨ 1998) enables the application of the CP-SCM-based windowed FDE methods discussed earlier, as demonstrated by Schniter & Liu (2004). 6.3.4.2 Other Approaches to Equalization for Single-Carrier Schemes Barhumi, Leus, & Moonen (2005) proposed a CP-SCM equalization technique based on linear TV filters whose time-variations were constrained to obey a (possibly oversampled) complex-exponential basis expansion model of order I - 1. Under these constraints, LZF and LMMSE equalizers, requiring O(KI 3 M 3 ) operations per block, were designed. However, because of the cubic complexity in M, these schemes are much more expensive than frequency-domain equalization when M is large. 6.4 NONCOHERENT EQUALIZATION In Section 6.3, we discussed the coherent approach to equalization, i.e., estimation of a from y in (6.2), where the channel H, and hence the effective channel Q, was assumed to be known. Here, we discuss noncoherent equalization, where the channel realization H is unknown but its statistics may be known. 17 With time-domain windowing, y = W y and Q = W QW H for suitably chosen diagonal .