Exercises

$X_{TF}[n, m] = (1/2)[X_{DD}[0, 0]\,e^{j\pi(0 - 0)} + X_{DD}[1, 1]\,e^{j\pi(n - m)}] = (1/2)[1 + e^{j\pi(n - m)}]$.

$X_{TF}[0, 0] = (1/2)(1 + 1) = 1$. $X_{TF}[0, 1] = (1/2)(1 - 1) = 0$. $X_{TF}[1, 0] = (1/2)(1 - 1) = 0$. $X_{TF}[1, 1] = (1/2)(1 + 1) = 1$. So the ISFFT of this diagonal DD pattern is a diagonal TF pattern.

$s(t) = (1/\sqrt{T_s})\sum_{m=0}^{3} e^{j 2\pi m \Delta f t}$ for $t \in [0, T_s)$.

$\sum_{m=0}^{3} e^{j 2\pi m \Delta f t}$ is the Dirichlet kernel of order 4 at $\Delta f t$. It peaks at $t = 0, T_s$ (periodic).

A 4-subcarrier OFDM symbol with all unit data is a time-compressed pulse at the start of each symbol interval — the classical "OFDM high PAPR symbol." OTFS has the same behavior.

$X_{TF}[n, m] = (1/\sqrt{MN})\sum_{\ell, k} 1 \cdot e^{j 2\pi(k n/N - m\ell/M)}$ $= (1/\sqrt{MN})\,\sum_\ell e^{-j 2\pi m\ell/M}\,\sum_k e^{j 2\pi k n/N}$.

$\sum_\ell e^{-j 2\pi m\ell/M} = M\,\mathbf{1}_{m = 0}$. $\sum_k e^{j 2\pi k n/N} = N\,\mathbf{1}_{n = 0}$.

$X_{TF}[n, m] = \sqrt{MN}\,\mathbf{1}_{n = 0}\mathbf{1}_{m = 0}$. A single non-zero TF cell — the DC component on the first OFDM symbol.

Through the Heisenberg transform, this produces a DC tone for one symbol duration. The channel shifts this DC content by the path Dopplers. The receiver recovers a single-tone structure, SFFT of which is the all-ones DD grid again. A perfectly symmetric example.

$X_{DD} \to \text{ISFFT} \to X_{TF} \to \text{Heis} \to s \to \text{Wig} \to \hat{Y}_{TF} \to \text{SFFT} \to \hat{X}_{DD}$.

By Theorem THeisenberg-Wigner Roundtrip, $\text{Wig} \circ \text{Heis} = \text{Id}$. So $\hat{Y}_{TF} = X_{TF}$.

$\text{SFFT}(X_{TF}) = \text{SFFT}(\text{ISFFT}(X_{DD})) = X_{DD}$ by Theorem TSFFT and ISFFT Are Inverses.

$\hat{X}_{DD} = X_{DD}$. Under ideal pulses, no channel, and no noise, the OTFS transceiver is exact. $\blacksquare$

For $t \in [n T_s, (n+1) T_s)$: $s(t) = (1/\sqrt{T_s})\sum_m X_{TF}[n, m]\,e^{j 2\pi m \Delta f(t - n T_s)}$.

With $\Delta t = T_s/M$ (Nyquist): $s[n M + \ell] = (1/\sqrt{M T_s})\sum_m X_{TF}[n, m]\,e^{j 2\pi m \ell/M}$. This is exactly the $M$-point inverse DFT of the $n$-th column of $X_{TF}$ (up to normalization).

The CP is added by copying the last $L_{CP}$ samples of this IDFT output to the front — the standard OFDM practice.

Heisenberg transform with rectangular pulse = OFDM IFFT + CP. This is the content of "OTFS is precoded OFDM."

OTFS waveform: $s(t) = (1/\sqrt{MN})\sum_{\ell, k} X_{DD}[\ell, k]\,\phi_{\ell, k}(t)$. With i.i.d. QPSK $X_{DD}$, each $s(t)$ sample is a sum of $MN$ i.i.d. phase-shifted terms.

By CLT, $s(t)$ is approximately $\mathcal{CN}(0, 1/MN \cdot MN) = \mathcal{CN}(0, 1)$. Absolute value squared is Exponential(1).

Peak over one frame of length $NT_s \cdot M$ samples: $\max |s|^2$ is the max of $MN$ Exp(1) variables. $\mathbb{E}\text{PAPR} \approx \log(MN) = \log(4096) \approx 8.3$ (≈9 dB).

Same calculation. Both OTFS and OFDM with i.i.d. QAM have PAPR $\sim \log(MN)$ dB. No advantage for either. Practical OFDM uses PAPR reduction techniques (clipping, tone reservation); OTFS can use the same techniques.

$s(t) = \sum_{n=0}^{N-1}\sum_{m=0}^{M-1} X_{TF}[n, m]\,g_{tx}(t - n T_s)\,e^{j 2\pi m\Delta f(t - n T_s)}$. The outer sum over $n$ decouples: each $n$ is a separate OFDM symbol supported on $[n T_s, (n+1)T_s)$.

Each OFDM symbol is an $M$-point IFFT: $O(M \log M)$ per symbol.

$N$ such IFFTs: $O(NM \log M)$. Add $O(L_{CP} N)$ for CP prepending. Total: $O(MN \log M)$. With $M, N \sim 2^{10}$, this is $\sim 10^7$ ops per frame — comfortable realtime.

From Theorem: $|\text{cross-talk}| \lesssim \max |Z_g\,\bar{Z_{g^\perp}} - 1|$. For raised-cosine with roll-off $\alpha$, the Zak-magnitude variation is roughly $\alpha/2 = 0.125$.

Cross-talk $\sim 0.125$, i.e., $\sim -18$ dB relative to the signal. Moderate. Acceptable for QPSK/16-QAM; marginal for 256-QAM.

$\alpha = 0.1$ (very narrow roll-off) gives cross-talk $\sim -26$ dB — adequate for most QAM orders. The trade-off is spectral containment: smaller $\alpha$ means wider side-lobes in the frequency domain. Typical deployment chooses $\alpha = 0.1 - 0.25$.

$\sum_{\ell, k} \mathbb{E}|X_{DD}|^2 = MN\,P_{\text{sym}}$.

Unitarity: $\sum_{n, m}\mathbb{E}|X_{TF}|^2 = \sum_{\ell, k}\mathbb{E}|X_{DD}|^2 = MN\,P_{\text{sym}}$.

Unitarity of Heisenberg: $\int \mathbb{E}|s(t)|^2\,dt = \sum_{n, m}\mathbb{E}|X_{TF}|^2 = MN\,P_{\text{sym}}$. Frame duration: $NT_s$. Average power: $P_{\text{frame}} = MN\,P_{\text{sym}}/(NT_s) = M\,P_{\text{sym}}/T_s$. This is the same as OFDM with $M$ subcarriers — power is preserved by the precoder. $\blacksquare$

$\hat{Y}_{TF}[n, m] = H(m\Delta f)\,X_{TF}[n, m]$ — diagonal in both indices, independent of $n$.

$\hat{X}_{DD}[\ell, k] = \text{SFFT}(H \cdot X_{TF})[\ell, k] = \sum_{n, m} H(m\Delta f)\,X_{TF}[n, m]\,e^{-j 2\pi(k n/N - m\ell/M)}/\sqrt{MN}$.

The sum separates into a TF-frequency-dependent factor times the $X_{TF}$ contributions. Expanding fully, the DD matrix is a convolution matrix: $\hat{X}_{DD}[\ell, k] = \sum_{\ell'}\tilde{H}[\ell - \ell']\,X_{DD}[\ell', k]$ with $\tilde{H}$ the inverse DFT of $H(m\Delta f)$ — the delay-axis taps of the LTI channel.

LTI in TF = delay-only convolution in DD (no Doppler coupling). This is the $P$-path channel of Chapter 4 restricted to $(k_i = 0)$ for all $i$.

function OTFS_receive(r, channel_estimate, noise_var):
    # Stage 1: CP removal, segment into N OFDM symbols
    symbols = segment(r, N, T_s, L_CP)

    # Stage 2: per-symbol FFT (Wigner)
    Y_TF = [fft(s, M) for s in symbols]

    # Stage 3: SFFT
    Y_DD = sfft(Y_TF, M, N)  # 2D FFT, unitary

    # Stage 4: MMSE detection in DD domain
    # H_DD diagonalized: apply diag MMSE
    H_freq = sfft_eigenvalues(channel_estimate, M, N)
    X_DD_est = (conj(H_freq) * Y_DD) / (abs(H_freq)**2 + noise_var)
    X_DD_hat = isfft(X_DD_est, M, N)  # return to DD

    return demap_qam(X_DD_hat)

Stage 1: $O(MN + N L_{CP})$. Stage 2 (N FFTs): $O(NM\log M)$. Stage 3 (SFFT): $O(MN\log(MN))$. Stage 4 (MMSE): $O(MN\log(MN))$ for the 2D FFT; $O(MN)$ for the element-wise diag MMSE. Stage 5 (QAM demap): $O(MN)$. Total: $O(MN\log(MN))$.

Transmitter: data → DFT (1D, length $M$) → OFDM mod. Receiver: OFDM demod → IDFT (1D) → detector.

Transmitter: data → ISFFT (2D symplectic FFT) → OFDM mod. Receiver: OFDM demod → SFFT (2D) → detector.

Both are "precoded OFDM": a linear transform of the data before OFDM, reversed at the receiver. The key difference is the dimensionality and sign structure of the precoder:

• DFT-s-OFDM: 1D DFT. Reduces PAPR; Doppler-agnostic.
• OTFS: 2D symplectic DFT. The symplectic sign pattern is what gives the DD-geometric interpretation; 2D is what enables the DD-grid packaging.

ISFFT is symplectic (mixed signs on the two axes); ordinary 2D DFT is not. The symplectic structure is what makes the DD-plane translations (channel action) correspond to TF modulations with the right signs. Without it, the 2D precoding does not give a DD-convolution channel. This is why DFT-s-OFDM does not solve mobility: it's not 2D, and it's not symplectic.

$s(t) = \sum_{n, m} X_{TF}[n, m]\,g_{tx}(t - n T_s)\,e^{j 2\pi m\Delta f(t - n T_s)}$.

For i.i.d. zero-mean QAM $X_{TF}$ with unit power: $\mathbb{E}[s(t)\,s^*(t - \tau)] = \sum_m |G(f - m\Delta f)|^2$ after Fourier transform (roughly).

$S(f) = \sum_{m = 0}^{M-1} |G_{tx}(f - m\Delta f)|^2$. The sum of $M$ shifted pulse spectra gives the full-bandwidth spectrum.

Determined by $|G_{tx}|^2$'s side lobes. Rectangular pulse: sinc-squared side lobes, slow $1/f^2$ roll-off. RRC pulse: very tight spectral containment, fast $f^{-4}$ roll-off. Regulators require specific OOB emission masks; OTFS inherits the same pulse-shape considerations as OFDM.

Data $X_{DD}$ → ISFFT → $X_{TF}$ → OFDM mod → $s(t)$. This is OTFS.

Data $X_{DD}$ → OFDM mod (?!) → waveform. But OFDM mod expects a TF-grid input, not a DD grid; the signature would be wrong.

The orders are not equivalent. The ISFFT converts a DD grid to a TF grid; only then can OFDM consume it. Applying OFDM directly to DD-grid data would treat the DD indices as if they were TF indices — a category error.

OTFS's ISFFT is the semantic bridge: DD-grid data → TF-grid signal. Without it, the DD structure is lost. The order "precode-first, modulate-second" is essential.

Time signal: $s[n_T] = (1/\sqrt{N})\sum_k Z^{-1}[X_{DD}][n, k]\,e^{j 2\pi k n_T/N}$ where $n = n_T \bmod M$ (within-block) and $n_T = \lfloor n_T/M\rfloor$ (block index). Equivalently, time signal $s[\cdot]$ is the inverse discrete Zak transform.

Time signal: ISFFT of $X_{DD}$ then IDFT over $m$ axis per block. Compose: time sample $s[n_T] = (1/\sqrt{MN})\sum_{\ell, k} X_{DD}[\ell, k]\,e^{j 2\pi \text{(composite phase)}}$.

Expanding both and comparing exponents (critical lattice, rectangular pulse) reveals the same time-sample formula. Numerical verification: the two produce bit-identical output under these assumptions.

Under non-rectangular pulses or non-critical lattices, the two formulations diverge. Zak-OTFS keeps the DD-plane interpretation cleanly; Hadani-Rakib may acquire residual cross-talk. For research purposes, Zak-OTFS is the preferred formulation (see Mohammed et al., 2022).