Prerequisites & Notation

Before You Begin

This chapter closes the single-link OTFS analysis by proving the main performance theorem (full DD diversity) and quantifying the OTFS-vs-OFDM gap. It rests on the channel model of Ch. 4 and the detectors of Ch. 8.

DD channel matrix structure(Review OTFS Ch. 4)
Self-check: Can you identify the rank of $\mathbf{H}_{DD}$ under a $P$ -path channel?
OTFS detection algorithms (ML, MP, LCD)(Review OTFS Ch. 8)
Self-check: Do you know that MP achieves BER slope $P$ asymptotically?
Diversity order of classical fading channels(Review Telecom Ch. 10)
Self-check: Can you derive the diversity gain of an $L$ -tap Rayleigh fading channel?
Diversity-multiplexing tradeoff (DMT)(Review Telecom Ch. 15)
Self-check: Can you state the Zheng-Tse DMT curve for an $n_t \times n_r$ MIMO Rayleigh channel?
Ergodic capacity of fading channels(Review ITA Ch. 13)
Self-check: Can you write the ergodic capacity of a Rayleigh channel in closed form?

Notation for This Chapter

Symbols introduced in this chapter.

Symbol	Meaning	Introduced
$d$	Diversity order — slope of BER curve at high SNR	s01
$r$	Multiplexing gain — rate scaling in $\log \mathrm{SNR}$	s04
$\mathrm{PEP}(\mathbf{x} \to \hat{\mathbf{x}})$	Pairwise error probability from transmitted $\mathbf{x}$ to $\hat{\mathbf{x}}$	s01
$\Delta\mathbf{x}$	Error vector $\hat{\mathbf{x}} - \mathbf{x}$	s01
$d^*(r)$	Optimal diversity at multiplexing gain $r$	s04
$\mathrm{BER}_{\text{OTFS}}(\mathrm{SNR}, P)$	OTFS BER as a function of SNR and path count	s03

← Ch 8 Pairwise Error Probability and Diversity