Ferkans — Interactive Telecom Tutor

OTFS = QAM Symbols × 2D SFT

With all three pieces — the DD grid, the ISFFT, and the standard OFDM Heisenberg transform — we can now state what OTFS is, operationally:

OTFS maps QAM symbols placed on the delay-Doppler grid to a time-domain waveform via the composition: ISFFT → OFDM modulator.

This two-step construction is due to Hadani–Rakib (2017) and is the universally-adopted OTFS modulator. An alternative, mathematically cleaner construction — Zak-OTFS — works directly with the discrete Zak transform and is treated in Chapter 6. Both constructions produce the same time-domain waveform in the idealized limit of bi-orthogonal pulse shapes.

This section examines the ISFFT-plus-OFDM construction from the DD-grid perspective and sets up the notation used in subsequent chapters.

Definition:
OTFS Modulator (Hadani–Rakib Construction)

The OTFS transmit signal is generated by three operations:

Place QAM symbols $X_{DD}[\ell, k]$ on the $M \times N$ DD grid.
Apply ISFFT to obtain $X_{TF}[n, m] = \text{ISFFT}(X_{DD})[n, m]$ .
Apply OFDM modulation (Heisenberg transform) to $X_{TF}$ : the time-domain waveform is $x(t) \;=\; \sum_{n = 0}^{N-1}\sum_{m = 0}^{M-1} X_{TF}[n, m]\,g_{tx}(t - n T_s)\,e^{j 2\pi m \Delta f(t - n T_s)},$ where $g_{tx}(t)$ is the pulse shape (typically a rectangular window of width $T_s$ plus cyclic prefix). The overall mapping DD grid → time-domain waveform is a unitary linear map from $\mathbb{C}^{MN}$ to a finite-dimensional subspace of $L^2([0, T + T_{CP}))$ .

Theorem: OTFS is a Gabor Expansion with DD Coefficients

Applying the definition of the ISFFT, the OTFS transmit signal can be written directly as a delay-Doppler Gabor expansion: $x(t) \;=\; \frac{1}{\sqrt{MN}}\sum_{\ell = 0}^{M-1}\sum_{k = 0}^{N-1} X_{DD}[\ell, k]\,\phi_{\ell, k}(t),$ where $\phi_{\ell, k}(t)$ is a prototype pulse delayed by $\ell \Delta\tau$ and Doppler-shifted by $k \Delta\nu$ : $\phi_{\ell, k}(t) \;=\; \sum_{n = 0}^{N-1}\sum_{m = 0}^{M-1} e^{j 2\pi (k n / N - m \ell / M)}\,g_{tx}(t - n T_s)\,e^{j 2\pi m \Delta f(t - n T_s)}.$

The OTFS transmit signal is a sum of delay-Doppler shifted copies of a prototype pulse, with coefficients equal to the data symbols on the DD grid. This is exactly the Gabor expansion structure discussed in RThe Gabor Expansion Is the OTFS Modulator.

The functions $\phi_{\ell, k}$ are designed to occupy (approximately) the $(\ell, k)$ -th cell of the DD grid. Their mutual orthogonality (up to the ideal bi-orthogonal pulse limit) is what makes OTFS invertible at the receiver.

Proof

Substitute ISFFT into OFDM modulator

$x(t) = \sum_{n, m} \frac{1}{\sqrt{MN}}\sum_{\ell, k} X_{DD}[\ell, k]\,e^{j 2\pi(k n / N - m \ell / M)}\,g_{tx}(t - n T_s)\,e^{j 2\pi m \Delta f(t - n T_s)}$ .

Rearrange sums

Swap the order of summation (finite sums — no convergence issue): $x(t) = \frac{1}{\sqrt{MN}}\sum_{\ell, k} X_{DD}[\ell, k]\,\phi_{\ell, k}(t)$ where $\phi_{\ell, k}$ is as stated.

Interpretation

Each $\phi_{\ell, k}$ is a "template" signal for the DD cell $(\ell, k)$ . The OTFS transmit signal is a linear combination of these templates with coefficients $X_{DD}[\ell, k]$ — exactly the Gabor expansion structure predicted by Chapter 2. $\blacksquare$

Key Takeaway

OTFS is the Gabor expansion on the critical DD lattice. Each data symbol $X_{DD}[\ell, k]$ contributes a delay-Doppler shifted copy of a prototype pulse to the transmit waveform. The symbolic expansion $x(t) = \sum_{\ell, k} X_{DD}[\ell, k]\,\phi_{\ell, k}(t)$ is the direct implementation of Chapter 2's Gabor-lattice theory. The ISFFT-plus-OFDM formulation is the FFT-friendly way to compute this expansion, but the underlying structure is the Gabor lattice on $[0, T] \times [0, W]$ at critical density.

Prototype Pulses $\phi_{\ell, k}(t)$ in OTFS

For various $(\ell_0, k_0)$ DD grid positions, plot the prototype pulse $\phi_{\ell_0, k_0}(t)$ . Observe that it is localized in time near $t = \ell_0 \Delta\tau \pmod{T_s}$ and oscillates at the Doppler rate $k_0 \Delta\nu$ . The collection $\{\phi_{\ell, k}\}$ tiles the time-frequency plane at critical density.

Parameters

Delay index

\ell_0

4

Doppler index

k_0

2

Delay bins

M

16

Doppler bins

N

16

OTFS Transmitter Chain: QAM Grid to Waveform

A 16×4 DD grid populated with random QPSK symbols. We animate: (1) the DD grid display, (2) the ISFFT producing a TF-grid image, (3) the OFDM Heisenberg transform producing the time-domain waveform. Each step is labelled with the corresponding math expression. The final waveform is the OTFS transmit signal, ready for the channel.

Example: A 2×2 OTFS End-to-End Example

Build the OTFS transmit signal for $M = N = 2$ and BPSK data $X_{DD}[0, 0] = +1, X_{DD}[0, 1] = -1, X_{DD}[1, 0] = +1, X_{DD}[1, 1] = +1$ . Assume a rectangular pulse $g_{tx}(t) = (1/\sqrt{T_s})\,\mathbf{1}_{[0, T_s)}(t)$ . Compute: (a) $X_{TF}$ , (b) $x(t)$ .

Solution

ISFFT

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

OFDM modulation

For $t \in [0, T_s)$ (first OFDM symbol): $x(t) = X_{TF}[0, 0]\,g_{tx}(t) + X_{TF}[0, 1]\,g_{tx}(t)\,e^{j 2\pi \Delta f t}$ $\;\;= g_{tx}(t)(1 - e^{j 2\pi \Delta f t})$ . For $t \in [T_s, 2 T_s)$ (second OFDM symbol): $x(t) = X_{TF}[1, 0]\,g_{tx}(t - T_s) + X_{TF}[1, 1]\,g_{tx}(t - T_s)\,e^{j 2\pi \Delta f(t - T_s)}$ $\;\;= g_{tx}(t - T_s)(1 + e^{j 2\pi \Delta f(t - T_s)})$ .

Interpret

The resulting waveform is a piecewise OFDM signal. Notice how the destructive interference in the first symbol ( $1 - e^{j\omega t}$ has a zero at $\omega t = 0$ ) and constructive interference in the second symbol are functions of the DD symbol placement — this is how OTFS spreads information across the waveform.

🎓CommIT Contribution(2024)

A Three-Part Tutorial on OTFS

W. Yuan, R. Schober, G. Caire — IEEE Communications Magazine

The Yuan–Schober–Caire tutorial (three parts, 2023–2024) is the single best entry point to the mathematical structure of OTFS for readers who already know OFDM. Part I (OFDM ↔ OTFS relationship), Part II (waveform design and receiver), and Part III (ISAC applications) together give a complete picture.

The CommIT group's involvement in this tutorial and in the broader standardization discussions around OTFS is what anchors the book's cell-free (Ch. 17) and LEO-satellite (Ch. 18) CommIT contributions in the next parts of the library. Part III in particular is the conceptual bridge between the DD-domain waveform machinery developed here and the ISAC-specific analysis in Chapter 12.

otfs-tutorialwaveform-designcaire

Common Mistake: The Ideal Bi-Orthogonal Pulse Assumption

Mistake:

Many derivations assume the prototype pulse $g_{tx}$ is bi-orthogonal — i.e., $\langle g_{tx}(\cdot - n T_s)\,e^{j 2\pi m \Delta f\cdot}, g_{rx}(\cdot)\rangle = \delta_{n, 0}\delta_{m, 0}$ for a matched receiver pulse $g_{rx}$ . This is not achievable exactly for any finite-energy pulse (Balian-Low theorem).

Correction:

With the rectangular pulse $g_{tx} = g_{rx} = (1/\sqrt{T_s})\mathbf{1}_{[0, T_s)}$ and a cyclic prefix of length $\geq \tau_{\max}$ , the bi-orthogonality condition is satisfied exactly for the discrete-time OFDM grid — but only approximately for the Doppler direction. The approximate nature shows up as off-diagonal entries in the DD channel matrix (Chapter 4), typically small for $f_D T_s \ll 1$ . For $f_D T_s$ not small (LEO, HST at mmWave), pulse shaping matters more — see Chapter 20.

OTFS as a 2D SFT of QAM Symbols