---
title: 'Zak Transform: Theory & Applications'
url: https://www.emergentmind.com/topics/zak-transform
type: topic
---

# Zak Transform: Theory & Applications

The Zak transform is a representation that recasts a function or sequence as a quasi-periodic object on a fundamental cell, typically pairing a spatial or delay variable with a dual frequency or Doppler variable. In continuous settings it maps functions on \(L^2(\mathbb{R})\) to quasi-periodic functions on a strip, and in discrete periodic settings it maps length-\(MN\) sequences to \(M\times N\) arrays. Across harmonic analysis, Gabor frame theory, invariant-subspace analysis, delay–Doppler communications, and Gottesman–Kitaev–Preskill coding, its central role is to expose periodic structure, diagonalize translation-like actions, and preserve Hilbert-space geometry [1311.7359] [2503.23540] [1410.7250] [2210.09494].

## 1. Definitions and canonical realizations

In one continuous normalization, for \(f:\mathbb{R}\to\mathbb{C}\) and \(\alpha>0\), the Zak transform is
\[
Z_{\alpha} f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x-k\alpha)\, e^{2\pi i k\alpha\omega},
\qquad (x,\omega)\in\mathbb{R}^2,
\]
whenever the series converges. The transform is completely determined by its values on the fundamental domain \([0,\alpha)\times[0,1/\alpha)\). A second common normalization writes
\[
Z_a f(x,\omega) := \sum_{k\in\mathbb{Z}} f(x+ak)e^{-2\pi i k\omega},
\]
and is used in the analysis of totally positive functions and their zero sets [1311.7359] [1411.1539].

In the discrete periodic setting used in Zak-OTFS, time-domain signals are \(MN\)-periodic complex sequences \(\mathbf{x}[n]\), and the discrete Zak transform is
\[
\mathbf{X}[k,l]
=
\frac{1}{\sqrt{N}}
\sum_{p=0}^{N-1}\mathbf{x}[k+pM]\,e^{-j2\pi pl/N},
\qquad k,l\in\mathbb{Z}.
\]
For fixed \(k\), \(\mathbf{X}[k,l]\) is an \(N\)-point DFT of the polyphase component \(\{\mathbf{x}[k+pM]\}_{p=0}^{N-1}\). This converts a length-\(MN\) sequence into an \(M\times N\) delay–Doppler array [2503.23540].

A finite Zak-transform formulation for a period-\(N\) sequence with \(N=LT\) is
\[
X(j,t)=\sum_{l=0}^{L-1} s(t+lT)\,w_L^{-lj},
\qquad 0\le j<L,\;0\le t<T,
\]
with inverse
\[
s(t+lT)=L^{-1}\sum_{j=0}^{L-1}X(j,t)\,w_L^{lj}.
\]
This form is used to construct perfect and zero-correlation-zone sequence sets and to interpret the delay–Doppler grid in OTFS as a Zak domain [2502.05853].

## 2. Quasi-periodicity, basis structure, and unitarity

The defining structural feature of the Zak transform is quasi-periodicity. In the continuous case,
\[
Z_{\alpha}f(x,\omega+n/\alpha)=Z_{\alpha}f(x,\omega),
\qquad
Z_{\alpha}f(x+n\alpha,\omega)=e^{2\pi i n\alpha\omega}Z_{\alpha}f(x,\omega),
\]
so the transform is periodic in the dual variable and quasi-periodic in the primal variable. In the discrete periodic case, the corresponding relation is
\[
\mathbf{X}[k+nM,l+mN]
=
e^{j2\pi nl/N}\,\mathbf{X}[k,l],
\]
which makes the delay period quasi-periodic and the Doppler period strictly periodic [1311.7359] [2503.23540].

In the discrete theory, the time-domain space is an \(MN\)-dimensional Hilbert space with orthonormal basis \(\{\mathbf{v}_{r,s}\}\), indexed by \(r\in\{0,\dots,N-1\}\) and \(s\in\{0,\dots,M-1\}\), where each basis vector is a time-limited tone of length \(M\) repeated periodically with period \(MN\). Applying the DZT yields quasi-periodic delay–Doppler arrays \(\{\mathbf{V}_{r,s}\}\), and the map \(\mathbf{v}_{r,s}\mapsto \mathbf{V}_{r,s}\) sends one orthonormal basis to another. Consequently, the DZT is unitary and preserves inner products; in Zak-OTFS this is the mechanism by which correlation and ambiguity properties are transported from the time domain to the delay–Doppler domain [2503.23540].

A closely related picture appears in the Zak basis used for bosonic quantum systems. The kets
\[
\ket{u,v}
=
\sqrt{\frac{a}{2\pi}}
\sum_{m\in\mathbb{Z}} e^{iamv}\ket{u+am}[q]
\]
satisfy
\[
\ket{u+a,v}=e^{-iav}\ket{u,v},
\qquad
\ket{u,v+2\pi/a}=\ket{u,v},
\]
and furnish a continuous orthonormal basis on a Zak patch of area \(2\pi\). In that setting, the Zak transform is literally the wavefunction in the \(\ket{u,v}\) basis [2210.09494].

## 3. Gabor analysis, frame theory, and zero sets

In Gabor analysis, the Zak transform gives explicit criteria for frame bounds and frame failure. For \(\mathcal{G}(g,\alpha,\beta)=\{M_{l\beta}T_{k\alpha}g\mid k,l\in\mathbb{Z}\}\), rational densities \(\alpha\beta\in\mathbb{Q}\) admit frame-bound descriptions in terms of matrix-valued Zak transforms. In the case \(\alpha=1\), \(\beta=1/N\), the optimal bounds are
\[
A_{\rm opt}=\inf_{x,\omega\in[0,1)}\sum_{j=0}^{N-1}\left|Z_1g\!\left(x,\omega+\frac{j}{N}\right)\right|^2,
\qquad
B_{\rm opt}=\sup_{x,\omega\in[0,1)}\sum_{j=0}^{N-1}\left|Z_1g\!\left(x,\omega+\frac{j}{N}\right)\right|^2.
\]
At critical density \(\alpha\beta=1\), the Balian–Low theorem uses the quasi-periodicity of \(Z_\alpha g\) to show that a continuous Zak transform cannot be non-zero everywhere on its fundamental domain [1311.7359].

A major line of work concerns the zero sets of Zak transforms of totally positive functions and exponential B-splines. For periodic exponential B-splines \(B_m\) of order \(m\ge 2\), \(Z_1B_m\) has exactly one zero in \([0,1)^2\), located on the line \(\omega=\tfrac12\). Via the explicit relation
\[
\alpha\, Z_\alpha g(x,\omega)
=
\prod_{\nu=1}^m
\frac{\alpha a_\nu}{1-e^{-\alpha(a_\nu+2\pi i\omega)}}
\;
Z_1 B_\Lambda\!\left(\frac{x}{\alpha},\alpha\omega\right),
\]
the same single-zero property transfers to finite-type totally positive functions, and the frame set for such windows is the full subcritical region \(\{(\alpha,\beta):\alpha\beta<1\}\) [1311.7359].

For totally positive functions without Gaussian factor in the Fourier transform, the finite-type picture extends to infinite type. The complexified Zak transform is holomorphic in a strip, the finite-type approximants converge uniformly there, and Hurwitz’s theorem is used to show a zero-free strip away from \(\omega=\tfrac12\). The resulting theorem states that there exists \(\tilde x\in[0,1)\) such that
\[
Zg(\tilde x,1/2)=0,
\qquad
Zg(x,1/2)\neq 0
\quad \text{for }x\neq \tilde x,
\]
so the Zak transform again has exactly one zero in the fundamental domain [1411.1539].

## 4. Generalizations to group actions and noncommutative settings

The Zak transform extends far beyond \(\mathbb{R}\) and \(\mathbb{Z}^d\). For a discrete countable LCA group \(\Gamma\) acting measurably and quasi-\(\Gamma\)-invariantly on a \(\sigma\)-finite measure space \((X,\mu)\), with associated unitary representation \(\Pi_\sigma\), the generalized Zak transform is
\[
Z_\sigma[f](\alpha)(x)
=
\sum_{\gamma\in\Gamma} (\Pi_\sigma(\gamma)f)(x)\,\overline{\chi_\gamma(\alpha)},
\qquad
\alpha\in\widehat\Gamma.
\]
It defines an isometric isomorphism
\[
Z_\sigma:L^2(X)\to L^2(\widehat\Gamma,L^2(C)),
\]
where \(C\) is a tiling set for the action, and it satisfies the covariance relation
\[
Z_\sigma[\Pi_\sigma(\gamma_0)f](\alpha)
=
\chi_{\gamma_0}(\alpha)\,Z_\sigma[f](\alpha).
\]
This diagonalization converts \((\Gamma,\sigma)\)-invariant subspaces into multiplicatively invariant spaces and yields a range-function classification of invariant subspaces, frames, and Riesz sequences [1410.7250].

A semidirect-product version is available for \(G_\tau=H\ltimes_\tau K\), where \(H\) is locally compact, \(K\) is LCA, and \(L\subset K\) is a \(\tau\)-invariant uniform lattice. The transform
\[
\mathcal{Z}_L f(h,k,\omega)
=
\delta_K(h)^{1/2}\, Z_L f_h(k^h,\omega_h)
\]
maps \(L^2(G_\tau)\) into a Zak space on
\[
G_{\tau^{\times,L}}
=
H\ltimes_{\tau^{\times,L}}(K/L\times \widehat K/L^\perp)
\]
and satisfies the Plancherel identity
\[
\|\mathcal{Z}_L f\|_{L^2(G_{\tau^{\times,L}})}=\|f\|_{L^2(G_\tau)}.
\]
This recovers the classical transform when \(H\) is trivial and supplies explicit realizations for groups such as \(\mathrm{SL}(2,\mathbb{Z})\ltimes\mathbb{R}^2\) and Weyl–Heisenberg groups [1203.1509].

A different noncommutative extension arises from the Weyl transform on \(L^2(\mathbb{R}^{2n})\). For \(\phi\in L^2(\mathbb{R}^{2n})\) with Weyl kernel \(K_\phi\), the Weyl–Zak transform is
\[
Z_W\phi(\xi,\xi',\eta)
=
\sum_{m\in\mathbb{Z}^n} K_\phi(\xi+m,\eta)\,e^{-2\pi i m\cdot \xi'},
\]
a unitary map into \(L^2(\mathbb{T}^n\times\mathbb{T}^n\times\mathbb{R}^n)\). It diagonalizes twisted translations:
\[
Z_W(T^t_{(k,l)}\phi)
=
e^{2\pi i(k\cdot \xi + l\cdot \xi')}e^{\pi i k\cdot l}\, Z_W\phi,
\]
which leads to bracket maps and frame, Riesz-sequence, and Schauder-basis criteria for twisted shift-invariant spaces [2305.04488].

## 5. Delay–Doppler communications, waveform design, and OTFS

In Zak-OTFS, the transform provides the defining map between time-domain signals and the delay–Doppler lattice. A delay–Doppler impulse at \((k_0,l_0)\) is mapped by the inverse discrete Zak transform to the time-domain “pulsone”
\[
\mathbf{p}_{(k_0,l_0)}[n]
=
\frac{1}{\sqrt N}\sum_{d\in\mathbb{Z}}
e^{j2\pi d l_0/N}\,\delta[n-k_0-dM],
\]
and the transmit waveform is the superposition
\[
\mathbf{x}[n]
=
\sum_{k_0=0}^{M-1}\sum_{l_0=0}^{N-1}
\mathbf{X}[k_0,l_0]\,
\mathbf{p}_{(k_0,l_0)}[n].
\]
In this framework, the Zak-OTFS carrier is a pulse in the delay–Doppler domain, and the Zak transform converts it to a pulse train modulated by a tone in the time domain [2508.07148] [2505.08079].

The discrete Zak transform preserves inner products, and the paper “Zak-OTFS for Mutually Unbiased Sensing and Communication” shows that time-domain CAZAC sequences determine quasi-periodic delay–Doppler arrays with highly structured ambiguity functions. For the quadratic-phase CAZAC family, the self-ambiguity in the Zak domain is supported on the discrete line
\[
2\alpha k-l\equiv 0 \pmod{MN},
\]
and for two distinct family members the cross-ambiguity magnitude is constant:
\[
\big|\mathbf{A}_{\mathbf{X},\mathbf{Y}}[k,l]\big|
=
\frac{1}{\sqrt{MN}},
\qquad \forall k,l.
\]
The same work shows that these waveforms are mutually unbiased with respect to every Zak-OTFS carrier and are suited to integrated sensing and communication and to 2-step RACH preambles [2503.23540].

Several recent developments use additional unitary transforms on top of the Zak basis. “Zak-OTFS with Spread Carrier Waveforms” constructs an orthonormal basis of spread carrier waveforms with low PAPR, realized by a unitary transform based on the discrete affine Fourier transform; the proposed spread carrier-based Zak-OTFS achieves full spectral efficiency like pulsone-based Zak-OTFS, with \(5.6\) dB lower PAPR per basis element and low PAPR only \(6.58\) dB [2505.08079]. “Low-Complexity Equalization of Zak-OTFS in the Frequency Domain” derives a frequency-domain system model unitarily equivalent to the delay–Doppler model, shows that the frequency-domain channel matrix is banded, and obtains equalization complexity linear in the dimension of a Zak-OTFS frame, in contrast to cubic complexity for naive MMSE equalization [2508.07148].

The Zak framework also underlies practical pulse-shaping and over-the-air implementations. In discrete oversampled Zak-OTFS, every delay–Doppler domain symbol undergoes the same effective channel response, and the I/O relation remains a twisted convolution; analysis of ambiguity functions shows that high sidelobes widen channel spreading, motivating a PSWF/IOTA pulse design with superior channel estimation accuracy and BER in the high-SNR regime [2602.07350]. An over-the-air mmWave demonstration implements Zak-OTFS with root-raised-cosine filtering, modulations up to 16-QAM, and a low-overhead preamble, and develops a signal model in which carrier-frequency offset and timing impairments are jointly captured within the effective DD-domain channel [2511.07610].

## 6. Quantum, operator-theoretic, and modern analytical perspectives

The Zak transform furnishes a particularly compact description of the square GKP code. Choosing Zak period \(a=2\alpha\), the ideal GKP codewords
\[
\ket{\ell_{\mathrm{GKP}}}
=
\sum_{n\in\mathbb{Z}}\ket{(2n+\ell)\alpha}[q],
\qquad \ell\in\{0,1\},
\]
collapse in the Zak basis to
\[
\ket{\ell_{\mathrm{GKP}}}
=
\ket{\alpha\ell,0}.
\]
The syndrome projector for modular displacement \((u,v)\) becomes
\[
\hat{\Pi}(u,v)
=
\sum_{\ell=0,1}\ket{u+\alpha\ell,v}\bra{u+\alpha\ell,v},
\]
and the error-corrected logical amplitudes are samples of the Zak transform,
\[
\bar c_\ell = e^{-i\alpha\ell v}\,\psi(u+\alpha\ell,v).
\]
This leads to a modular-variable subsystem decomposition in which a single bosonic mode is expressed as a virtual qubit tensored with a virtual gauge mode, and tracing over the gauge mode yields the logical state associated with the oscillator state [2210.09494].

A complementary analytical development characterizes function and distribution spaces directly in terms of Zak-transform estimates. For an ordered basis \(E\), \(Z_E\) is a homeomorphism from \(\mathscr{S}(\mathbb{R}^d)\) onto the space of smooth quasi-periodic functions of order \(E\), and it extends uniquely to a homeomorphism on tempered distributions. The same program gives characterizations of Gelfand–Shilov spaces, their duals, and modulation spaces by Wiener-type estimates of Zak transforms, and it establishes necessary and sufficient conditions for linear operators to be conjugations by the Zak transform [1705.10619].

Taken together, these developments place the Zak transform at the intersection of quasi-periodic harmonic analysis, finite-dimensional signal design, group representations, delay–Doppler communications, and bosonic quantum information. A consistent theme is that periodicity in one domain and localization in a dual domain are not merely represented by the transform; they are reorganized into a geometry in which basis structure, ambiguity, frame bounds, invariant subspaces, and correctable errors become explicit.

Source: https://www.emergentmind.com/topics/zak-transform