Prerequisites & Notation

Before You Begin

This is the pivotal transceiver chapter. It assembles the mathematical tools of Chapters 1-3, the channel model of Chapter 4, and the OFDM contrast of Chapter 5 into a complete OTFS transmitter and receiver. All subsequent chapters (detection, estimation, ISAC, cell-free) build on the transceiver defined here.

ISFFT and SFFT — the 2D symplectic DFT pair(Review OTFS Ch. 3)
Self-check: Can you state the ISFFT formula and confirm it is the exact inverse of the SFFT?
Zak transform and its covariance(Review OTFS Ch. 2)
Self-check: Can you recall why the Zak transform intertwines delay-Doppler shifts with DD-plane translations?
OFDM modulation, CP, pulse shaping(Review Telecom Ch. 14)
Self-check: Can you write the OFDM transmit waveform and identify the standard pulse shapes?
DD input-output relation(Review OTFS Ch. 4)
Self-check: Can you state the 2D-convolution form of the DD channel?
OFDM failure under Doppler(Review OTFS Ch. 5)
Self-check: Do you know the ICI-to-signal ratio formula and when OFDM breaks?

Notation for This Chapter

Symbols introduced in this chapter.

Symbol	Meaning	Introduced
$X_{DD}[\ell, k]$	OTFS data symbol on the DD grid	s01
$X_{TF}[n, m]$	TF-grid symbol after ISFFT precoding	s01
$s(t)$	OTFS time-domain transmit waveform	s02
$g_{tx}(t), g_{rx}(t)$	Transmit and receive prototype pulses	s02
$L_{CP}$	Cyclic-prefix length in samples	s03
$\text{Heis}[\cdot]$	Heisenberg transform: $\{X_{TF}[n, m]\} \mapsto s(t)$	s02
$\text{Wig}[\cdot]$	Wigner transform: $s(t) \mapsto \{Y_{TF}[n, m]\}$	s04
$\hat{X}_{DD}[\ell, k]$	Estimated DD-grid data after SFFT demodulation	s04

← Ch 5 The OTFS Transmitter: Three Stages