Chapter Summary

Key Points

1.
The Zak transform is a 1D-to-2D map with a twist. It takes a time-domain signal $f(t)$ and produces a function $Z_f(t, \nu)$ on the delay-Doppler plane, defined by $Z_f(t, \nu) = \sum_k f(t - k T_0)\,e^{-j 2\pi k \nu T_0}$ . Although the output has two variables, it is determined by its values on the fundamental rectangle $\mathbb{T}^2 = [0, T_0) \times [0, \nu_0)$ with quasi-periodic boundary conditions.
2.
Covariance under delay-Doppler shifts is the defining property. A signal $f$ that has been delayed by $\tau_0$ and Doppler-shifted by $\nu_0$ produces a Zak transform that is simply a translate of $Z_f$ by $(\tau_0, \nu_0)$ , up to a phase factor. This is the structural reason the Zak transform is the correct signal space for a DD channel: the channel acts by translation, and the transform intertwines this action with translation on the torus.
3.
Quasi-periodicity, not periodicity. $Z_f(t + T_0, \nu) = e^{-j 2\pi \nu T_0}\,Z_f(t, \nu)$ and $Z_f(t, \nu + \nu_0) = Z_f(t, \nu)$ . The phase twist $e^{-j 2\pi \nu T_0}$ is the DD-plane reflection of the Heisenberg–Weyl commutation $\mathcal{T}\mathcal{E} = e^{-j\theta}\mathcal{E}\mathcal{T}$ . Every OTFS pilot design (Chapter 7) must respect this twist.
4.
The Zak transform unifies STFT and Gabor analysis. On the fundamental domain $\mathbb{T}^2$ , the Zak transform equals the STFT with a rectangular window of width $T_0$ . The Gabor expansion at the critical density $T_0 \nu_0 = 1$ is invertible if and only if the Zak transform of the prototype pulse is bounded away from zero. This gives a concrete pulse-shaping criterion for OTFS (treated fully in Chapter 20).
5.
The discrete Zak transform is a column-wise FFT in disguise. For a time-domain vector $\mathbf{x} \in \mathbb{C}^{MN}$ , the discrete Zak transform $Z_x[n, k]$ is computed by reshaping $\mathbf{x}$ into an $N \times M$ matrix and taking a column-wise $N$ -point DFT. This costs $O(MN \log N)$ operations — no worse than OFDM — and is unitary, so the inverse exists as a standard iDFT.

Looking Ahead

With the Zak transform in hand, we can move a time-domain signal to the DD plane. But the OTFS transmitter actually lives in the time-frequency plane: data symbols placed on the DD grid are first transformed to the TF grid by an inverse symplectic Fourier transform (ISFFT), then converted to a time-domain waveform by standard OFDM modulation. Chapter 3 develops the symplectic Fourier transform — the 2D DFT on the DD plane that bridges DD and TF. Chapter 4 then uses both transforms to derive the 2D convolution input-output relation promised at the end of Chapter 1. The pieces come together in Chapter 6, where the full OTFS transceiver chain is built.

The Discrete Zak Transform Exercises