For coherent detection in a wireless communication system, channel state information (CSI) is indispensable. Channel estimation has drawn tremendous attention in the literature (see [TSD04] and references therein), where the pilot-aided method is one of the most intensively studied approaches. This method is especially attractive for time-varying channels because of their short coherence time. In this chapter, we will address pilot-aided channel estimation for both orthogonal frequency division multiplexing (OFDM) and single-carrier systems, where pilots are inserted in the frequency domain and time domain, respectively. We study these two systems under one framework because in the context of channel estimation, both systems can be characterized by data models of the same form. More specifically, the received samples can be expressed as the joint effect of the information part (due to the pilots), the interference part (due to the unknown data symbols), and the noise. Consequently, our task is to design a channel estimator that can combat both the interference and the noise. Such a data model is typical for OFDM over time-varying channels, where due to the Doppler effect, the orthogonality between the subcarriers is destroyed and the channel matrix in the frequency domain becomes effectively a diagonally-dominant yet full matrix instead of a diagonal matrix. As a result, the received frequency-domain samples depend on both the pilots and the unknown data symbols. For single-carrier systems, the channel matrix in the time domain is a strictly banded matrix if a finite impulse response (FIR) assumption for the channel is applied, and therefore, we can in practice find some received samples that solely depend on the pilots. However, it is sometimes beneficial to also consider received samples that depend on the unknown data symbols as well, to better suppress the interference and the noise. In any case, the resulting data model for single-carrier systems looks very similar to the data model for OFDM systems, and similar channel estimation techniques can be applied. Note that the considered data model can also account for superimposed pilot schemes [GS06, HT07a], where the pilots and the data symbols co-exist on the same subcarriers or time instants. Whether we are dealing with OFDM or single-carrier systems, estimating a time-varying channel implies estimating a large number of unknowns, making the channel estimation problem much more difficult than in the time-invariant case. As a remedy, we adopt in this chapter a parsimonious model, referred to as the basis expansion model (BEM), to approximate the time variation of the channel (see Section 1.3 of Chapter 1). If the BEM is accurate with negligible approximation error, channel estimation can be achieved by just estimating the BEM coefficients, which are much smaller in number than the actual unknowns, i.e., the channel tap values at different time instants. In the remainder of the chapter, we first discuss the system and channel model in Section 1.2. In Section 1.3, we then present channel estimation algorithms within a single OFDM symbol/time block. We indicate how to position the pilots, where to select observation samples, and what is the best channel estimation strategy. In Section 1.4, we extend these methods to situations where multiple OFDM symbols/time blocks are utilized simultaneously. In this case, the position of the pilots plays an important role in the performance. Extensions to multiple antenna systems are considered in Section 1.5. We conclude this chapter in Section 1.7.
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