Prerequisites & Notation

Before You Begin

This chapter compares OFDM to OTFS by translating the former into the DD language. We assume OFDM at the level of Telecom Ch. 14 and the DD machinery of Chapters 1-4.

OFDM modulation: IFFT, cyclic prefix, per-subcarrier equalization(Review Telecom Ch. 14)
Self-check: Can you write the OFDM transmit signal $s(t) = \sum_k X_k g(t)e^{j 2\pi k \Delta f t}$ and state the orthogonality of subcarriers?
Inter-carrier interference (ICI) from Doppler(Review Telecom Ch. 14, §14.5)
Self-check: Do you recall that Doppler destroys subcarrier orthogonality, and the leaked energy follows a sinc-squared pattern?
DD input-output relation(Review OTFS Ch. 4)
Self-check: Can you state $Y_{DD}[\ell, k] = \sum_i h_i X_{DD}[(\ell-\ell_i)\bmod M, (k-k_i)\bmod N] + W_{DD}[\ell, k]$ ?
The ISFFT as a 2D DFT precoder(Review OTFS Ch. 3)
Self-check: Can you apply the ISFFT to a single DD-grid symbol and recognize the output as a TF-plane wave?
Bello's system functions(Review OTFS Ch. 1)
Self-check: Can you identify $H(f, t)$ as the TF-domain view of the channel?

Notation for This Chapter

Symbols used throughout the chapter.

Symbol	Meaning	Introduced
$X_{TF}[n, m]$	OFDM data symbol on the time-frequency grid (symbol $n$ , subcarrier $m$ )	s01
$g_{tx}(t), g_{rx}(t)$	OFDM transmit/receive pulse shapes	s01
$\Delta f$	OFDM subcarrier spacing	s01
$T_s$	OFDM symbol duration (per symbol, including CP)	s01
$H[n, m]$	TF-domain channel matrix entry at symbol $n$ , subcarrier $m$	s02
$\Phi_{mm'}(t)$	Inter-carrier leakage coefficient from subcarrier $m$ to $m'$	s02
$\rho_{\text{ICI}}$	ICI-to-signal ratio under Doppler	s02

← Ch 4 OFDM Signaling in the Time-Frequency Domain