Exercises

$Z_1(t, \nu) = \sum_k 1 \cdot e^{-j 2\pi k \nu T_0} = \sum_k e^{-j 2\pi k \nu T_0}$.

$\sum_k e^{-j 2\pi k \nu T_0} = (1/T_0) \sum_m \delta(\nu - m/T_0) = \nu_0\,\sum_m \delta(\nu - m \nu_0)$. The Zak transform is a Dirac comb in Doppler, independent of $t$.

A constant in time has no delay structure but has all its energy at Doppler zero — and, by periodicity, at every integer multiple of $\nu_0$. This confirms that the Zak transform of a pure baseband signal is a delta at the origin.

$Z_{\alpha f + \beta g}(t, \nu) = \sum_k (\alpha f + \beta g)(t - k T_0)\,e^{-j 2\pi k \nu T_0} = \alpha \sum_k f(t - k T_0)\,e^{-j 2\pi k \nu T_0} + \beta \sum_k g(t - k T_0)\,e^{-j 2\pi k \nu T_0}$.

The two sums are $Z_f$ and $Z_g$, respectively. Hence $Z_{\alpha f + \beta g} = \alpha Z_f + \beta Z_g$. $\blacksquare$

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

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

$Z_\phi(t, \nu) = \sum_k e^{j 2\pi \nu_1 (t - k T_0)}\,e^{-j 2\pi k \nu T_0} = e^{j 2\pi \nu_1 t}\sum_k e^{-j 2\pi k (\nu - \nu_1) T_0}$.

The sum is $\nu_0\,\sum_m \delta(\nu - \nu_1 - m \nu_0)$.

$Z_\phi(t, \nu) = \nu_0\,e^{j 2\pi \nu_1 t}\,\sum_m \delta(\nu - \nu_1 - m \nu_0)$. The tone at Doppler $\nu_1$ has a Zak transform localized at the Doppler line $\nu = \nu_1$ (and its periodic shifts), with a phase that varies linearly with $t$. Geometrically, a tone lives on a horizontal line in the DD plane.

$\int_{\mathbb{T}^2} Z_f(t, \nu)\,\overline{Z_g(t, \nu)}\,dt\,d\nu = \int_0^{T_0}\int_0^{\nu_0}\sum_{k, l} f(t - k T_0)\,\overline{g(t - l T_0)}\,e^{-j 2\pi (k - l)\nu T_0}\,dt\,d\nu$.

$\int_0^{\nu_0} e^{-j 2\pi (k - l) \nu T_0}\,d\nu = \nu_0\,\mathbf{1}_{k = l}$. The sum collapses to $k = l$: $= \nu_0 \int_0^{T_0}\sum_k f(t - k T_0)\,\overline{g(t - k T_0)}\,dt$.

$\int_0^{T_0} f(t - k T_0)\,\overline{g(t - k T_0)}\,dt$ shifts as $k$ varies across $\mathbb{Z}$. Summing over $k$ covers the whole real line: $= \nu_0 \int_{\mathbb{R}} f(t)\,\overline{g(t)}\,dt = (1/T_0)\,\langle f, g\rangle$. $\blacksquare$

$\cos(2\pi k_0 n / (MN)) = \frac{1}{2}[e^{j 2\pi k_0 n / (MN)} + e^{-j 2\pi k_0 n / (MN)}]$.

DZT of $e^{j 2\pi k_0 n / (MN)}$: $\sqrt{N}\,e^{j 2\pi k_0 n / (MN)}\,\mathbf{1}_{k = m_0}$ (by the tone-DZT example). DZT of $e^{-j 2\pi k_0 n / (MN)}$: $\sqrt{N}\,e^{-j 2\pi k_0 n / (MN)}\,\mathbf{1}_{k = -m_0 \bmod N}$.

$Z_x[n, k] = \frac{\sqrt{N}}{2}\left[e^{j 2\pi k_0 n / (MN)}\,\mathbf{1}_{k = m_0} + e^{-j 2\pi k_0 n / (MN)}\,\mathbf{1}_{k = -m_0 \bmod N}\right]$. A real cosine tone produces DZT impulses at two symmetric Doppler bins, consistent with the real-signal Doppler spectrum symmetry.

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

$\phi(t - k T_0) = \phi(t)$ for all $k \in \mathbb{Z}$ (since $\phi$ has period $T_0$). Therefore $Z_{\phi f}(t, \nu) = \phi(t)\sum_k f(t - k T_0)\,e^{-j 2\pi k \nu T_0} = \phi(t)\,Z_f(t, \nu)$. $\blacksquare$

Exercise 5 with $f = g$ gives $\int_{\mathbb{T}^2} |Z_f|^2\,dt\,d\nu = (1/T_0)\,\|f\|^2$, or $T_0 \int_{\mathbb{T}^2} |Z_f|^2\,dt\,d\nu = \|f\|^2$. Thus $Z$ is isometric onto its image in $L^2(\mathbb{T}^2, T_0\,dt\,d\nu)$.

Every quasi-periodic $F(t, \nu)$ satisfying the required boundary conditions can be written as $Z_f$ for some $f \in L^2(\mathbb{R})$ — this follows from the inverse formula.

From Theorem TInverse Zak Transform: $f(t) = \int_0^{\nu_0} Z_f(t, \nu)\,d\nu$ for $t \in [0, T_0)$ and extended by decomposition to all $t \in \mathbb{R}$. The inverse is a single integral, confirming the linear bijection. $\blacksquare$

$Z_{g_\sigma}(0, \nu) = (\pi\sigma^2)^{-1/4}\sum_k e^{-(k T_0)^2 / (2\sigma^2)}\,e^{-j 2\pi k \nu T_0}$.

With $q = e^{-T_0^2 / (2\sigma^2)}$ (real and in $(0, 1)$) and $z = -\pi \nu T_0$: $\sum_k q^{k^2} e^{2jkz} = \theta_3(z | q) = \theta_3(-\pi\nu T_0 \,|\, e^{-T_0^2/(2\sigma^2)})$.

Applying Poisson summation to the Gaussian yields the dual form $Z_{g_\sigma}(0, \nu) = (\pi\sigma^2)^{-1/4}(\sigma \sqrt{2\pi}/T_0)\sum_m e^{-2\pi^2 \sigma^2 (\nu - m \nu_0)^2}$. In the narrow-Gaussian regime ($\sigma \ll T_0$), the first form is useful (one term dominates); in the wide-Gaussian regime ($\sigma \gg T_0$), the dual form is useful (the Gaussian in $\nu$ dominates).

$y[n] = x[(n - 1) \bmod MN]$. For $n = 0$, $y[0] = x[MN - 1]$, which is the last entry of the last block.

For $n \neq 0$, the shift stays within a block: $Z_y[n, k] = Z_x[n - 1, k]$. For $n = 0$, the shift crosses the block boundary, picking up a phase: $Z_y[0, k] = e^{-j 2\pi k / N}\,Z_x[M - 1, k]$. Combining: $Z_y[n, k] = Z_x[(n-1) \bmod M, k]\,(\text{phase twist if wrapping})$.

The phase twist is the discrete analog of the continuous quasi-periodicity $e^{-j 2\pi \nu T_0}$. Circular shift in time becomes twisted-circular shift in delay — the same structure observed in the continuous case. $\blacksquare$

Let $n_0 = n_0' + m_0 M$ with $n_0' \in \{0, \ldots, M-1\}$, and $k_0 = k_0'\,M + k_0''$ with $k_0'' \in \{0, \ldots, M-1\}$.

$e^{j 2\pi k_0 n / (MN)} = e^{j 2\pi k_0'' n / (MN)} \cdot e^{j 2\pi k_0' n / N}$. The first factor is a within-period phase; the second factor is a constant-across-$n'$ phase that effectively shifts the Doppler index.

Computing $Z_y[n, k]$ using these substitutions, the Doppler-shift component yields a shift in the $k$-index: $k \mapsto k - k_0'$ mod $N$. The time-shift component yields the usual delay shift $n \mapsto n - n_0'$ mod $M$.

$Z_y[n, k] = e^{j \text{phase}}\,Z_x[(n - n_0') \bmod M, (k - k_0') \bmod N]$ with phase $e^{j 2\pi k_0'' (n - n_0') / (MN)}$ and additional quasi-periodicity twists when the modular shifts wrap. $\blacksquare$

$Z_f(t, -\nu) = \sum_k f(t - k T_0)\,e^{j 2\pi k \nu T_0}$ (using $-\nu$). Since $f$ is real, $f(t - k T_0)$ is real, so we can take the conjugate of each term freely.

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

The two expressions are identical: $Z_f(t, -\nu) = \overline{Z_f(t, \nu)}$. This is the discrete-time analog of the real-signal Fourier-transform conjugate symmetry, lifted to the DD plane. $\blacksquare$

In Zak coordinates, $Z_{g_{m,n}}(t, \nu) = e^{j 2\pi n \nu_0 t}\,Z_g(t - m T_0, \nu - n \nu_0)$. The inner product is $\langle g_{m,n}, g\rangle = T_0 \int_{\mathbb{T}^2} Z_{g_{m,n}}(t, \nu)\,\overline{Z_g(t, \nu)}\,dt\,d\nu$.

Using quasi-periodicity, $Z_{g_{m,n}} = e^{j\phi_{m,n}(t,\nu)}\,Z_g$ for a specific phase. The inner product becomes $\int_{\mathbb{T}^2} e^{j\phi_{m,n}(t,\nu)}\,|Z_g(t, \nu)|^2\,dt\,d\nu$.

For this to equal $\delta_{m,0}\delta_{n,0}$, we need $|Z_g(t, \nu)|^2$ to Fourier-transform (in $(m, n)$) to a delta at the origin — i.e., $|Z_g(t, \nu)|^2 =$ constant $= 1$ (after normalization). Hence $|Z_g| = 1$ everywhere on $\mathbb{T}^2$.

The rectangular pulse $g(t) = \mathbf{1}_{[0, T_0)}(t)/\sqrt{T_0}$ has $|Z_g| = 1/\sqrt{T_0}$ flat on $\mathbb{T}^2$ — bi-orthogonal after normalization. No smooth pulse achieves this, but pulses close to rectangular can get arbitrarily close.

$T_0 \nu_0 = 1$: the Gabor lattice has density 1 per unit DD area. The prototype pulse occupies one cell per symbol.

$T_0 \nu_0 > 1$: more cells than symbols. The Gabor frame is redundant; the dual window is non-unique. Reconstruction is more robust (over-specified least squares) but wastes potential capacity.

$T_0 \nu_0 < 1$: fewer cells than symbols. The Gabor system cannot span $L^2(\mathbb{R})$; reconstruction is impossible. This is the Balian–Low theorem for the Gabor setting.

OTFS operates at exactly critical density — one data symbol per DD cell. This balances capacity (no wasted cells) against well-conditionedness (no missing information). The sensitivity to pulse shape at critical density is what motivates the pulse-design discussion of Chapter 20.

The fractional Fourier transform $\mathcal{F}^\alpha$ rotates the DD plane by angle $\alpha$. At $\alpha = \pi/2$ it is the ordinary Fourier transform, exchanging delay and Doppler axes.

A linear canonical transform with symplectic matrix $\begin{pmatrix}a & b\\ c & d\end{pmatrix}$ (with $ad - bc = 1$) acts on the DD plane by $(\tau, \nu) \mapsto (a\tau + b\nu, c\tau + d\nu)$. The Zak transform intertwines this with the corresponding unitary on $L^2(\mathbb{R})$.

A DD-plane shear ($b \neq 0$) corresponds to a chirp modulation in time. A DD-plane rotation corresponds to fractional Fourier transform — this is the basis of chirp-OTFS and affine-OTFS variants, which reshape the DD signal space for different channel types. For a reference treatment see Mohammed et al. (2024) "Linear Canonical Transform OTFS."

Whether there exists a universally optimal LCT for a given channel statistics is an open problem. See Chapter 22.