Prerequisites & Notation

Before You Begin

The Zak transform is a fundamental tool of time-frequency analysis. It is not usually taught in signal processing courses, but the machinery required is elementary: Fourier series, the Poisson summation formula, and the notion of a function as an element of an $L^2$ space.

Fourier series of periodic functions(Review Telecom Ch. 4)
Self-check: Can you write the Fourier series $f(t) = \sum_k c_k e^{j 2\pi k t / T_0}$ and the dual pair?
Poisson summation formula: $\sum_k f(kT) = (1/T)\sum_m \hat{f}(m/T)$ (Review Telecom Ch. 4)
Self-check: Can you state and prove the Poisson summation formula?
The Short-Time Fourier Transform (STFT) and windowed Fourier analysis(Review FSP Ch. 14)
Self-check: Can you write the STFT of a signal $x(t)$ with window $g(t)$ ?
Chapter 1 of this book(Review OTFS Ch. 1)
Self-check: Can you recall the spreading function $h(\tau, \nu)$ and why it is sparse?
$L^2(\mathbb{R})$ and Hilbert-space conventions(Review FSI Ch. 1)
Self-check: Can you state the Plancherel theorem for $L^2$ ?

Notation for This Chapter

Symbols introduced in this chapter.

Symbol	Meaning	Introduced
$Z_f(t, \nu)$	Zak transform of $f$ , evaluated at time $t$ and Doppler $\nu$	s01
$T_0, \nu_0 = 1/T_0$	Zak transform time period and dual frequency period	s01
$\pi_{\tau_0, \nu_0}$	Delay-and-Doppler-shift operator: $(\pi_{\tau_0, \nu_0} f)(t) = f(t - \tau_0)\,e^{j 2\pi \nu_0 t}$	s01
$g(t)$	Prototype pulse (window) on the DD plane	s03
$Z_f^{(d)}[n, k]$	Discrete Zak transform with delay index $n$ and Doppler index $k$	s04
$\mathbb{T}^2$	The fundamental domain (2D torus) $[0, T_0) \times [0, \nu_0)$	s02

← Ch 1 Definition and Motivation