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Three Views of the Same Object

The Zak transform can feel abstract because it is defined by a formal infinite sum. It turns out that the Zak transform is intimately connected to two more familiar time-frequency objects: the short-time Fourier transform (STFT) and the Gabor expansion. Understanding these connections grounds the Zak transform in classical signal processing and, more importantly, tells us how to approximate it numerically.

Definition:
Short-Time Fourier Transform

For a signal $f(t)$ and a window function $g(t)$ , the short-time Fourier transform (STFT) is $V_g f(t, \nu) \;=\; \int_{-\infty}^{\infty} f(s)\,\overline{g(s - t)}\,e^{-j 2\pi \nu s}\,ds.$ The STFT measures the "frequency content of $f$ near time $t$ " through the window $g$ .

When the window $g$ is a unit-height rectangle of width $T_0$ , the STFT becomes closely related to the Zak transform — the integral collapses to a sum over a single period, and the full Zak transform emerges by considering all periods.

Theorem: Zak–STFT Relation

Let $g(t) = \mathbf{1}_{[0, T_0)}(t)$ be the rectangular window. Then $Z_f(t, \nu) \;=\; e^{j 2\pi \nu t}\,V_g f(t, \nu) \quad \text{for } t \in [0, T_0).$ That is, the Zak transform on the fundamental domain equals the STFT with a rectangular window, up to a phase factor.

The STFT takes a short-time snapshot of $f$ through a window; the Zak transform does the same, but with the window being the indicator of one fundamental period $T_0$ . The two objects agree inside the fundamental rectangle and are extended differently: the STFT is a continuous function of all $(t, \nu) \in \mathbb{R}^2$ , while the Zak transform is quasi-periodic and determined by its values on $\mathbb{T}^2$ .

Proof

Restrict the STFT to one period

For $t \in [0, T_0)$ and $g = \mathbf{1}_{[0, T_0)}$ : $V_g f(t, \nu) = \int_{t}^{t + T_0} f(s)\,e^{-j 2\pi \nu s}\,ds$ .

Change variables

Let $s = t + u$ with $u \in [0, T_0)$ : $V_g f(t, \nu) = e^{-j 2\pi \nu t}\int_0^{T_0} f(t + u)\,e^{-j 2\pi \nu u}\,du$ .

Connect to Zak

The Fourier coefficients of the sampled sequence $\{f(t - k T_0)\}$ — which is what $Z_f(t, \nu)$ computes — match this inner integral after a sign change in the shift variable. Rearranging yields $Z_f(t, \nu) = e^{j 2\pi \nu t}\,V_g f(t, \nu)$ on the fundamental domain. $\blacksquare$

Gabor Frames and Why They Matter for OTFS

The Gabor system is a lattice of time-frequency shifted copies of a prototype pulse $g$ : $\{g_{m, n}(t) = g(t - m T_0)\,e^{j 2\pi n \nu_0 t}\}_{m, n \in \mathbb{Z}}$ . A signal can be expanded in this system: $f(t) \;=\; \sum_{m, n} c_{m, n}\,g_{m, n}(t),$ with coefficients $c_{m, n}$ determined by inner products with a dual window. This is exactly the OTFS modulator structure: the transmit signal is a sum of delay-Doppler shifted copies of a prototype pulse, with coefficients equal to the data symbols placed on the DD grid.

The central question — when is this expansion unique? — is answered by the Zak transform. For the critical lattice $T_0 \cdot \nu_0 = 1$ , the dual window is a 1:1 inverse of the prototype in Zak coordinates: $\tilde{g}(t) = Z^{-1}[1 / Z_g(t, \nu)]$ . Good prototypes have $Z_g$ bounded away from zero; "bad" prototypes (like narrow Gaussians with $\sigma \ll T_0$ ) have zeros in $Z_g$ , making the Gabor expansion ill-conditioned.

Theorem: Existence of Gabor Frames (Critical Lattice)

Consider the Gabor lattice $\{g(t - m T_0)\,e^{j 2\pi n \nu_0 t}\}_{m, n \in \mathbb{Z}}$ with $T_0 \nu_0 = 1$ (critical density). The system is a Riesz basis of $L^2(\mathbb{R})$ if and only if $0 < \underset{(t, \nu) \in \mathbb{T}^2}{\operatorname{ess\,inf}} |Z_g(t, \nu)| \;\leq\; \underset{(t, \nu) \in \mathbb{T}^2}{\operatorname{ess\,sup}} |Z_g(t, \nu)| < \infty.$ The dual window is $\tilde{g} = Z^{-1}[1/Z_g]$ .

The Zak transform of the prototype pulse $g$ directly determines whether the Gabor system is well-conditioned. Zeros in $Z_g$ correspond to "holes" in the tiling — the Gabor system fails to span certain directions of the signal space. For OTFS design, this is a concrete pulse-shaping criterion: choose $g$ so that $|Z_g|$ is bounded away from zero on $\mathbb{T}^2$ .

Proof

Setup

Compute the frame operator $S_g = \sum_{m, n}\langle \cdot, g_{m, n}\rangle\,g_{m, n}$ . In the Zak representation, $S_g$ acts as multiplication by $|Z_g(t, \nu)|^2$ .

Condition number

$S_g$ is invertible iff the multiplier $|Z_g|^2$ is bounded and bounded below. Equivalently: $\operatorname{ess\,inf} |Z_g|^2 > 0$ .

Reconstruction formula

The dual window satisfies $S_g \tilde{g} = g$ , so in Zak coordinates $|Z_g|^2 \cdot Z_{\tilde{g}} = Z_g$ , giving $Z_{\tilde{g}} = \overline{Z_g}^{-1}$ . Applying the inverse Zak transform yields the dual window. $\blacksquare$

,

Zak Transform of a Chirp

A chirp $\phi(t) = e^{j \pi \alpha t^2}$ with sweep rate $\alpha$ has a Zak transform that is localized along a diagonal of the fundamental domain — the DD-plane signature of a linear frequency sweep. As $\alpha$ varies, the diagonal tilts. Notice that at critical sweep rates (where $\alpha T_0^2$ is an integer), the Zak transform "folds" onto itself due to the Heisenberg–Weyl group's metaplectic action.

Parameters

Chirp rate

\alpha\, T_0^2

1

Period

T_0

(

\mu

s)4

Duration (in

T_0

units)4

Gabor-Friendly vs Gabor-Hostile Pulses

Plot $|Z_g(t, \nu)|$ for three prototype pulses: (1) a rectangle of width $T_0$ (flat Zak, ideal Gabor), (2) a Gaussian with $\sigma = 0.3\,T_0$ (narrow Gaussian — zeros in Zak, breaks Gabor expansion at critical lattice), and (3) a raised-cosine with roll-off 0.5 (smooth, no zeros). Observe how the lowest value of $|Z_g|$ over the fundamental domain predicts the condition number of the Gabor expansion.

Parameters

Prototype pulse

Raised-cosine roll-off

\alpha

0.25

Gaussian

\sigma/T_0

0.4

Historical Note: Joshua Zak and the 1967 Paper

1967

Joshua Zak, then at the Technion in Haifa, published "Finite Translations in Solid-State Physics" in the October 1967 issue of Physical Review Letters. His motivation had nothing to do with signal processing — he was studying the spectrum of electrons in a crystal under an applied magnetic field (the integer quantum Hall effect, decades before its discovery). To handle the two conflicting periodicities (crystal lattice in space, magnetic phase in momentum), he introduced what is now called the Zak transform.

Two decades later, Augustus Janssen (Philips Research Labs, Eindhoven) recognized that Zak's construction solved a central problem in Gabor analysis: characterizing when the Gabor system at critical density is a Riesz basis. Janssen's 1988 paper "The Zak Transform: a Signal Transform for Sampled Time-Continuous Signals" brought Zak's idea into signal processing.

A further three decades later, Hadani and Rakib realized that the Zak transform was exactly the tool needed to build a waveform on the delay-Doppler plane. OTFS was born. Solid-state physics, harmonic analysis, wireless communications — the same mathematical object serves all three.

The Gabor Expansion Is the OTFS Modulator

Looking ahead: in Chapter 6 we will write the OTFS transmit waveform as $x(t) \;=\; \sum_{\ell = 0}^{M-1}\sum_{k=0}^{N-1} X[\ell, k]\,g(t - \ell\,\Delta\tau)\,e^{j 2\pi k\,\Delta\nu\, t},$ where $g(t)$ is a prototype pulse and $X[\ell, k]$ are the data symbols on the DD grid. This is exactly a Gabor expansion. The theorems of this section tell us when this expansion is invertible (reconstruction at the receiver) and how to choose the pulse $g$ (bounded $|Z_g|$ from above and below). Chapter 20 revisits pulse shaping with this in mind.

Common Mistake: The "Ideal OTFS Pulse" Is a Fiction

Mistake:

Early OTFS papers (Hadani 2017, Raviteja 2018) assume a prototype pulse $g$ that is simultaneously rectangular in time and rectangular in frequency — i.e., a "bi-orthogonal" pulse with flat Zak transform on both sides. This is not achievable by any finite-energy signal.

Correction:

No $L^2$ function can be compactly supported in both time and frequency (this is the uncertainty principle). The "ideal pulse" is a pedagogical device. Practical OTFS uses a raised-cosine or truncated Gaussian pulse, and the non-ideality introduces second-order correction terms to the DD input-output relation. These corrections are captured by off-diagonal entries in the DD channel matrix (Chapter 4) and are small for typical system parameters — but they are never zero. A clean treatment is Mohammed–Hadani–Chockalingam–Caire (2022).

Gabor frame

A Gabor frame is a collection of time-frequency shifted copies of a prototype pulse $g$ , $\{g(t - m T_0)\,e^{j 2\pi n \nu_0 t}\}_{m, n \in \mathbb{Z}}$ , that spans $L^2(\mathbb{R})$ . When $T_0 \nu_0 = 1$ (critical density) and the Zak transform $Z_g$ is bounded away from zero, the Gabor system is a Riesz basis. The OTFS modulator is a Gabor expansion at critical density.

Related: Zak Transform, Otfs Modulation

Relation to STFT and Gabor Frames

Three Views of the Same Object

Definition: Short-Time Fourier Transform

Theorem: Zak–STFT Relation

Restrict the STFT to one period

Change variables

Connect to Zak

Gabor Frames and Why They Matter for OTFS

Theorem: Existence of Gabor Frames (Critical Lattice)

Setup

Condition number

Reconstruction formula

Zak Transform of a Chirp

Parameters

Gabor-Friendly vs Gabor-Hostile Pulses

Parameters

Historical Note: Joshua Zak and the 1967 Paper

The Gabor Expansion Is the OTFS Modulator

Common Mistake: The "Ideal OTFS Pulse" Is a Fiction

Gabor frame

Definition:
Short-Time Fourier Transform