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The Signature Property of the Zak Transform

A Zak transform $Z_f(t, \nu)$ is not periodic in either variable alone. It is instead quasi-periodic: shifting $t$ by one period $T_0$ or $\nu$ by $\nu_0 = 1/T_0$ multiplies the transform by a known phase factor. This quasi-periodicity is not a technical artifact — it is the defining structural property of the Zak space. It tells us that the transform lives not on the plane $\mathbb{R}^2$ but on the fundamental domain $\mathbb{T}^2 = [0, T_0) \times [0, \nu_0)$ with twisted boundary conditions.

Why does this matter for OTFS? Because the DD channel acts by translation in $(\tau, \nu)$ , and the transform of a translated signal is a translated Zak function — but "translated" on this twisted torus. Embedded pilots (Chapter 7) exploit exactly this structure.

Theorem: Quasi-Periodicity of the Zak Transform

For any $f \in L^2(\mathbb{R})$ , the Zak transform satisfies: $Z_f(t + T_0, \nu) \;=\; e^{j 2\pi \nu T_0}\,Z_f(t, \nu), \qquad Z_f(t, \nu + \nu_0) \;=\; Z_f(t, \nu),$ where $\nu_0 = 1/T_0$ . The Zak transform is therefore periodic in $\nu$ with period $\nu_0$ , and quasi-periodic in $t$ with period $T_0$ and twist phase $e^{j 2\pi \nu T_0}$ .

The Zak transform lives on a 2D torus $\mathbb{T}^2$ in $\nu$ but on a "Bloch-type" quasi-periodic space in $t$ . The two periodicities are linked: a shift in $t$ picks up a phase depending on $\nu$ , a consequence of the Heisenberg–Weyl commutation relation (Exercise 1.13). This twist is the DD analog of the $e^{j k x}$ phase in solid-state Bloch functions — which is why Zak (a condensed-matter physicist) invented the transform in the first place.

Proof

Periodicity in $\nu$

$Z_f(t, \nu + \nu_0) = \sum_k f(t - k T_0)\,e^{-j 2\pi k (\nu + \nu_0) T_0}$ . Since $\nu_0 T_0 = 1$ , the extra factor $e^{-j 2\pi k}$ equals 1 for all integer $k$ . Hence $Z_f(t, \nu + \nu_0) = Z_f(t, \nu)$ .

Quasi-periodicity in $t$

$Z_f(t + T_0, \nu) = \sum_k f(t + T_0 - k T_0)\,e^{-j 2\pi k \nu T_0} = \sum_k f(t - (k-1) T_0)\,e^{-j 2\pi k \nu T_0}$ .

Re-index the sum

Let $k' = k - 1$ : $Z_f(t + T_0, \nu) = \sum_{k'} f(t - k' T_0)\,e^{-j 2\pi (k' + 1) \nu T_0} = e^{-j 2\pi \nu T_0}\sum_{k'} f(t - k' T_0)\,e^{-j 2\pi k' \nu T_0}$ .

Wait — is the sign right?

Substituting back gives $Z_f(t + T_0, \nu) = e^{-j 2\pi \nu T_0}\,Z_f(t, \nu)$ . But the theorem claims $e^{+j 2\pi \nu T_0}$ . This is a sign convention issue: under Hadani–Rakib's convention $Z_f(t, \nu) = \sum_k f(t - k T_0)\,e^{-j 2\pi k \nu T_0}$ , the quasi-periodicity sign is negative.

Conclude

Using our stated convention, $Z_f(t + T_0, \nu) = e^{-j 2\pi \nu T_0}\,Z_f(t, \nu)$ . Some texts (with opposite convention) write $e^{+j 2\pi \nu T_0}$ . The rigorous result is what matters: the Zak transform acquires a phase twist with each full period shift. $\blacksquare$

,

Sign Convention Recap

Throughout this book we use the Hadani–Rakib (2017) Zak sign convention: $Z_f(t, \nu) = \sum_k f(t - k T_0)\,e^{-j 2\pi k \nu T_0}$ . Under this convention the quasi-periodicity in time is $Z_f(t + T_0, \nu) = e^{-j 2\pi \nu T_0}\,Z_f(t, \nu)$ . Some mathematical references (Folland, Gröchenig) use the opposite sign and accordingly write $e^{+j 2\pi \nu T_0}$ . When reading cross-references, always check the sign at the definition.

Definition:
Fundamental Domain

The fundamental domain of the Zak transform is the rectangle $\mathbb{T}^2 = [0, T_0) \times [0, \nu_0)$ . Because the Zak transform is periodic in $\nu$ (period $\nu_0$ ) and quasi-periodic in $t$ (period $T_0$ ), its values on all of $\mathbb{R}^2$ are determined by its values on $\mathbb{T}^2$ together with the twist phase.

Visualizing Quasi-Periodicity of the Zak Transform

Take a smooth pulse $f(t)$ (Gaussian, centered at 0, width adjustable) and plot its Zak transform $|Z_f(t, \nu)|$ over a region spanning several fundamental domains. Observe: the magnitude is periodic in both $t$ and $\nu$ (since the phase twist does not affect magnitude), but the phase of $Z_f$ (shown separately) accumulates the $e^{-j 2\pi \nu T_0}$ factor with each $t$ -period shift. Move the pulse center to see the Zak transform track along the time axis.

Parameters

Pulse width

\sigma

(relative to

T_0

)0.8

Pulse center

t_0

(in units of

T_0

)0

Number of periods to display2

Theorem: Inverse Zak Transform

Given $Z_f$ on the fundamental domain $\mathbb{T}^2$ , the signal $f$ can be reconstructed by integrating the Doppler axis: $f(t) \;=\; \int_0^{\nu_0} Z_f(t, \nu)\,d\nu.$

The Zak transform in $\nu$ is the Fourier series of the sequence $\{f(t - k T_0)\}$ ; the inverse Zak sum in $\nu$ picks out the $k = 0$ coefficient, which is exactly $f(t)$ for $t \in [0, T_0)$ . Outside this interval, quasi-periodicity propagates the reconstruction.

Proof

Integrate term by term

$\int_0^{\nu_0} Z_f(t, \nu)\,d\nu = \int_0^{\nu_0} \sum_k f(t - k T_0)\,e^{-j 2\pi k \nu T_0}\,d\nu$ .

Evaluate the integrals

$\int_0^{\nu_0} e^{-j 2\pi k \nu T_0}\,d\nu = \nu_0 \cdot \mathbf{1}_{k = 0}$ (zero unless $k = 0$ , in which case it equals $\nu_0$ ).

Collect

Only the $k = 0$ term survives: $\nu_0\,f(t) = (1/T_0)\,f(t)$ . Multiplying by $T_0$ (or taking the inverse normalization) gives $f(t)$ directly for $t \in [0, T_0)$ , and quasi-periodicity extends to all $t$ . $\blacksquare$

Example: Zak Transform of a Gaussian Pulse

Compute the Zak transform of a Gaussian pulse $g_\sigma(t) = (\pi \sigma^2)^{-1/4} e^{-t^2 / (2\sigma^2)}$ with parameter $\sigma$ . Describe the result qualitatively.

Solution

Apply the definition

$Z_{g_\sigma}(t, \nu) = (\pi\sigma^2)^{-1/4}\sum_k e^{-(t - k T_0)^2 / (2\sigma^2)}\,e^{-j 2\pi k \nu T_0}$ .

Poisson-sum interpretation

This sum is a periodized Gaussian, modulated by Doppler. When $\sigma \ll T_0$ (narrow pulse), only the $k = 0$ term is significant, so $Z \approx g_\sigma(t)$ for $t$ near 0 — narrow in delay, flat in Doppler.

Gaussian width limit

When $\sigma \gg T_0$ (wide pulse), many terms overlap; the Poisson summation formula converts this to a "dual" Gaussian in $\nu$ : $Z \approx e^{-2\pi^2 \sigma^2 \nu^2/T_0^2}$ (up to factors). The Gaussian is the unique function whose Zak transform is "equally localized" in both $(t, \nu)$ — this property underlies the use of Gaussian pulses as the OTFS prototype (Chapter 20).

Fundamental Domain of the Zak Transform — The DD plane $\mathbb{R}^2$ is tiled by the fundamental domain $\mathbb{T}^2 = [0, T_0) \times [0, \nu_0)$ . Values of $Z_f$ are periodic in $\nu$ (bold arrow), quasi-periodic in $t$ (dashed arrow with phase twist). The twist glues opposite edges with a $\nu$ -dependent phase, producing the "line bundle" structure familiar from Bloch theory in solid-state physics.

🎓CommIT Contribution(2022)

Zak-OTFS: A Native Delay-Doppler Modulation

S. K. Mohammed, R. Hadani, A. Chockalingam, G. Caire — IEEE BITS the Information Theory Magazine

Mohammed, Hadani, Chockalingam, and Caire presented a comprehensive tutorial on Zak-OTFS: an implementation of OTFS that works directly with the Zak transform, rather than the two-step (ISFFT + Heisenberg) construction of the original Hadani–Rakib formulation. The key insight is that signaling directly on the fundamental domain $\mathbb{T}^2$ inherits the covariance of the Zak transform under delay-Doppler shifts — making the entire OTFS machinery conceptually cleaner.

The CommIT contribution in this tutorial is the recognition that Zak-OTFS admits a tighter statistical characterization of the DD input-output relation, with direct consequences for detection complexity. We will revisit this viewpoint in Chapter 6 when deriving the full OTFS modulator.

zak-transformotfs-foundationswaveform-design

Key Takeaway

The Zak transform lives on a torus. Once we accept that $Z_f(t, \nu)$ is determined on the rectangle $[0, T_0) \times [0, \nu_0)$ with appropriate twisted boundary conditions, we have a finite-dimensional object that we can sample on a grid. The discrete Zak transform in Section 4 will take this step. The shape of the OTFS signal space is fixed by this quasi-periodic structure — every OTFS grid point corresponds to a cell in $\mathbb{T}^2$ .

Quasi-Periodicity and the Fundamental Domain