Prerequisites & Notation

Before You Begin

Earlier chapters modeled the OTFS transmit signal as a delta-train on the DD grid, followed by an idealized pulse. That abstraction is mathematically clean but physically impossible — real pulses have finite duration, finite bandwidth, and complex time-frequency structure. This chapter confronts the practical pulse-shaping problem for OTFS: what transmit and receive filters are tolerable, what bi-orthogonality condition must they satisfy, and what is the tradeoff between ISI, ICI, and spectral efficiency.

Zak transform and symplectic Fourier(Review OTFS Ch. 2, 3)
Self-check: Can you state the Zak transform and its relation to the SFT?
OTFS transceiver chain(Review OTFS Ch. 6)
Self-check: Do you recall the ISFFT/SFFT + Heisenberg/Wigner-Ville mapping?
OTFS detection(Review OTFS Ch. 8)
Self-check: Are you familiar with the structure of the DD input-output relation?
Pulse shaping in OFDM(Review Telecom Ch. 14)
Self-check: Do you know what the raised-cosine filter does for OFDM?

Notation for This Chapter

Pulse-shaping-specific symbols.

Symbol	Meaning	Introduced
$g_{\mathrm{tx}}(t)$	Transmit pulse shape (time domain)	s01
$g_{\mathrm{rx}}(t)$	Receive pulse shape	s01
$A_{g_{\mathrm{rx}}, g_{\mathrm{tx}}}(\tau, \nu)$	Cross-ambiguity function of receive and transmit pulses	s02
$\mathrm{ENBW}$	Equivalent noise bandwidth of a window	s03
$\mathrm{SLL}$	Sidelobe level (dB)	s03
$\gamma$	Pulse roll-off factor (0: brickwall; 1: wide)	s03

← Ch 19 Ideal vs. Practical Pulses