Chapter 5: Zak transform analysis for delay-Doppler communications
The butterfly counts not months but moments, and has time enough.
Rabindranath Tagore
Abstract
The delay-Doppler modulation of a finite-length sequence of information symbols is mathematically equivalent to the well-known discrete Zak transform, a tool which makes OTFS simpler to understand, by applying its known properties. This chapter will introduce to the readers the basic concepts of the Zak transform. First, we present the properties of the continuous Zak transform and model the continuous-time wireless channel input–output relation in the continuous Zak (delay-Doppler) domain. We then introduce the discrete Zak transform, which is later used to analyze the discrete-time model ...
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