Prerequisites & Notation

Before You Begin

This chapter lifts the single-antenna OTFS-ISAC framework of Chapter 12 into the MIMO setting. The scene now has directions as well as delays and Dopplers; the waveform carries a precoding matrix as well as DD symbols; and the receiver estimates angles along with ranges and velocities. The price of the extra dimension is the same kind of price one pays moving from SISO-OFDM to MIMO-OFDM — but the DD structure makes it more manageable than in the time-frequency domain.

OTFS modulation and detection(Review OTFS Ch. 6, Ch. 8)
Self-check: Can you state the DD input-output relation and describe MP detection?
OTFS radar and ambiguity theory(Review OTFS Ch. 11)
Self-check: Do you recall the thumbtack ambiguity and the range/velocity resolutions?
OTFS-ISAC fundamentals(Review OTFS Ch. 12)
Self-check: Can you explain the rate-CRB tradeoff and joint estimation-detection?
MIMO and beamforming basics(Review Telecom Ch. 18, Ch. 19)
Self-check: Are you comfortable with precoding, beam steering vectors, and MRT/ZF/MMSE beamforming?
Cramér-Rao bounds for parameter estimation(Review ITA Ch. 10; OTFS Ch. 11 §3)
Self-check: Can you compute the CRLB for a single parameter from the Fisher information?

Notation for This Chapter

Symbols introduced or specialized in this chapter.

Symbol	Meaning	Introduced
$N_t, N_r$	Number of transmit and receive antennas	s01
$\theta_i, \phi_i$	Angle of arrival (AoA) and angle of departure (AoD) of path $i$	s01
$\mathbf{a}(\theta) \in \mathbb{C}^{N_r}$	Array response vector at angle $\theta$	s01
$\mathbf{F} \in \mathbb{C}^{N_t \times N_s}$	Transmit beamforming (precoding) matrix, $N_s$ streams	s02
$\mathcal{H}_{DD} \in \mathbb{C}^{N_r \times N_t \times MN}$	MIMO-OTFS channel tensor: delay-Doppler-angle	s01
$\mathbf{R}_x = \mathbb{E}[\mathbf{x}\mathbf{x}^H]$	Transmit spatial covariance matrix (sensing handle)	s02
$\mathrm{CRB}(\Theta)$	Cramér-Rao bound for target parameter vector $\Theta$	s03
$R_{\text{sum}}$	Sum communication rate across $K$ users (bits/s/Hz)	s02

← Ch 12 MIMO-OTFS Channel: The Delay-Doppler-Angle Tensor