Prerequisites & Notation

Before You Begin

This chapter combines the tools of Chapters 1-3 to derive the DD-domain input-output relation — the formal statement that "the DD channel acts by 2D convolution with a sparse kernel." The derivation is the technical heart of OTFS; every subsequent chapter uses its output as a premise.

The spreading function $h(\tau, \nu)$ and its sparsity(Review OTFS Ch. 1)
Self-check: Can you write $h(\tau, \nu)$ for a $P$ -path channel as a sum of 2D Dirac impulses?
The Zak transform and its covariance under delay-Doppler shifts(Review OTFS Ch. 2)
Self-check: Can you state the identity $Z_{\pi_{\tau_0, \nu_0} f}(t, \nu) = e^{j 2\pi \nu_0 t}\,Z_f(t - \tau_0, \nu - \nu_0)$ ?
The symplectic Fourier transform and the ISFFT/SFFT(Review OTFS Ch. 3)
Self-check: Can you compute the ISFFT of a single DD-grid impulse?
2D circular convolution and its FFT-based computation(Review Telecom Ch. 4)
Self-check: Can you express $(f \star\star g)[\ell, k]$ on an $M \times N$ grid and compute it via 2D FFT?
Bello's four system functions(Review OTFS Ch. 1 (s02))
Self-check: Can you name all four functions and state the Fourier relations between them?

Notation for This Chapter

Symbols introduced in this chapter. See also the NGlobal Notation Table master table.

Symbol	Meaning	Introduced
$x_{DD}(\tau, \nu), y_{DD}(\tau, \nu)$	Transmit and receive signals represented in the continuous DD domain	s01
$\star\star$	Two-dimensional convolution in delay and Doppler	s01
$X_{DD}[\ell, k], Y_{DD}[\ell, k]$	Transmit and receive symbols on the discrete DD grid	s02
$\ell_i, k_i$	Discrete delay and Doppler indices of the $i$ -th path: $\ell_i = \lfloor \tau_i W \rfloor$ , $k_i = \lfloor \nu_i T \rfloor$	s02
$l_{\max}, k_{\max}$	Maximum delay index $l_{\max} = \lceil \tau_{\max} W\rceil$ and maximum Doppler index $k_{\max} = \lceil f_D T\rceil$	s02
$\mathbf{H}_{DD}$	The $MN \times MN$ DD channel matrix acting on vectorized data	s03
$\pi_{\tau_i, \nu_i}$	The delay- $\tau_i$ Doppler- $\nu_i$ shift operator on a function of two variables	s01

← Ch 3 Continuous-Time DD Input-Output Relation