ABSTRACT
The waveform design for doubly dispersive channels has been investigated extensively since 1946. It is vital to look back on these design principles. This chapter presents an extensive survey of these waveform design principles. It presents a short summary of legacy waveforms. The chapter provides mathematical preliminaries such as linear operator on Hilbert space, frames on Hilbert space and Gabor transform. It then presents time-frequency analysis tools. The chapter also presents the mathematical models of time varying wireless channel. It discusses the waveform design framework using Gabor transforms and properties of pulse shapes which are relevant for time-varying channel. The chapter then discusses a survey of legacy waveforms such as orthogonal frequency division multiplexing (OFDM), windowed OFDM, filtered OFDM, variable sub-carrier OFDM, filter bank multi-carrier, universal filtered multi-carrier and generalized frequency division multiplexing. OFDM is a multi-carrier modulation technique.
