Prerequisites & Notation

Before You Begin

This chapter builds the 2D Fourier transform that bridges the DD and TF planes. It is the last conceptual piece before we can derive the input-output relation of the DD channel in Chapter 4 and build the OTFS transceiver in Chapter 6.

Zak transform (continuous and discrete)(Review OTFS Ch. 2)
Self-check: Can you state the definition of $Z_f(t, \nu)$ and its quasi-periodicity?
Ordinary discrete Fourier transform and FFT(Review Telecom Ch. 4)
Self-check: Can you compute the $N$ -point DFT of a sequence by hand for $N = 4$ ?
2D Fourier transform on $\mathbb{R}^2$ (Review Telecom Ch. 4)
Self-check: Can you write the 2D Fourier pair $f(x, y) \leftrightarrow F(u, v)$ ?
OFDM basics: subcarriers, IFFT at the transmitter(Review Telecom Ch. 14)
Self-check: Can you write the OFDM transmit signal as $s(t) = \sum_k X_k\,g(t)\,e^{j 2\pi k \Delta f t}$ ?

Notation for This Chapter

Symbols introduced in this chapter.

Symbol	Meaning	Introduced
$\mathcal{F}_s$	Symplectic Fourier transform (SFT): 2D DFT on the DD plane	s01
$\mathcal{F}_s^{-1}$	Inverse symplectic Fourier transform (ISFFT)	s01
$X_{DD}[\ell, k]$	DD-domain data symbol at delay index $\ell$ , Doppler index $k$	s02
$X_{TF}[n, m]$	TF-domain symbol (OFDM grid) at symbol index $n$ , subcarrier index $m$	s02
$\Delta f, T_s$	OFDM subcarrier spacing and symbol duration: $T_s \Delta f = 1$	s03

← Ch 2 The Symplectic Fourier Transform