Ferkans — Interactive Telecom Tutor

The Three-Stage Transmitter

We have all the pieces; now we assemble them. The OTFS transmitter produced by Hadani and Rakib in 2017 has three stages:

Place QAM data symbols on the discrete DD grid $X_{DD}[\ell, k]$ .
Precode with the ISFFT to obtain a TF-grid signal $X_{TF}[n, m]$ .
Modulate with a standard OFDM-like Heisenberg transform to produce the time-domain waveform $s(t)$ .

The elegance of this design is that stage 3 is identical to an OFDM transmitter. Only stage 2 — the ISFFT precoder — distinguishes OTFS from OFDM. Concretely, if you have 5G OFDM hardware, you can deploy OTFS by adding a software ISFFT block before the existing OFDM modulator. This is the concrete engineering argument for OTFS deployability.

We outline the full chain here and elaborate each stage in Sections 2-3 (transmitter) and Section 4 (receiver).

OTFS Transmitter (Hadani-Rakib Three-Stage Form)

Complexity:

O(MN \log(MN))

Input: QAM data symbols

\{X_{DD}[\ell, k]\}_{\ell=0,\ldots,M-1;\,k=0,\ldots,N-1}

, prototype pulse

g_{tx}(t)

, CP length

L_{CP}

Output: Time-domain OTFS signal

s(t)

1. Stage 1 — Symbol placement: Form the

M \times N

matrix

X_{DD}

from the data stream. Pilot and guard entries (Chapter 7) replace some data cells.

2. Stage 2 — ISFFT: Apply the inverse symplectic FFT:

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

.

3. Stage 3 — Heisenberg transform: Apply OFDM-like modulation with cyclic prefix:

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)}

.

4. Return

s(t)

, ready for RF transmission.

In software, the ISFFT is a 2D FFT with one axis flipped — $O(MN\log(MN))$ . The Heisenberg transform is $N$ length- $M$ IFFTs, also $O(MN \log M)$ . Both are well within the capability of 5G NR silicon.

,

Definition:
OTFS Transmit Signal

The OTFS transmit signal is $s(t) \;=\; \frac{1}{\sqrt{MN}}\sum_{\ell, k} X_{DD}[\ell, k]\,\phi_{\ell, k}(t),$ where the basis waveforms are $\phi_{\ell, k}(t) \;=\; \sum_{n = 0}^{N-1}\sum_{m = 0}^{M-1} e^{j 2\pi(k n/N - m \ell/M)}\,g_{tx}(t - n T_s)\,e^{j 2\pi m \Delta f(t - n T_s)}.$ Each basis $\phi_{\ell, k}$ is a delay- $\ell$ Doppler- $k$ shifted copy of a prototype "cell pulse," with the DD grid acting as the coefficient lattice.

,

Key Takeaway

OTFS is a Gabor expansion on the critical DD lattice. Each data symbol $X_{DD}[\ell, k]$ contributes a delay-Doppler-shifted copy $\phi_{\ell, k}(t)$ of a prototype pulse to the transmit waveform. The ISFFT-plus-OFDM two-step is just an FFT-accelerated way to compute this Gabor expansion. The mathematical essence is: "send one delay-Doppler-shifted pulse per DD cell, superposed." This is the direct realization of the Chapter 2 Gabor analysis.

OTFS Transmitter Block Diagram — Three-stage OTFS transmitter: QAM symbols enter the DD-grid mapper, pass through the ISFFT precoder, then through the OFDM-like Heisenberg modulator (IFFT + CP + serial-to-parallel + DAC). The output is a time-domain waveform ready for RF conversion. The ISFFT block is the only addition beyond standard OFDM hardware — it can be a software precoder on top of a 5G OFDM transmitter.

OTFS Transmitter Step-by-Step

Watch the signal transform at each stage: (1) DD grid with QPSK symbols, (2) TF grid after ISFFT, (3) time-domain waveform after Heisenberg modulation. Adjust grid size and see how the stage transforms change. This is the operational sequence for a full OTFS frame generation.

Parameters

Delay bins

M

8

Doppler bins

N

8

DD input pattern

Random seed3

Example: Smallest Complete OTFS Frame: $M = N = 2$

Build an OTFS signal for $M = N = 2$ with QPSK data $X_{DD} = \begin{pmatrix}1+j & -1-j\\ 1-j & -1+j\end{pmatrix}$ and a rectangular prototype pulse. Compute $X_{TF}$ and the resulting time-domain waveform.

Solution

ISFFT to $X_{TF}$

$X_{TF}[n, m] = (1/2)\sum_{\ell, k} X_{DD}[\ell, k]\,e^{j\pi(k n - m \ell)}$ . Expanding: $X_{TF}[0, 0] = (1/2)[(1+j) + (-1-j) + (1-j) + (-1+j)] = 0$ . Similarly: $X_{TF}[0, 1] = (1/2)[(1+j) - (-1-j) + (1-j) - (-1+j)] = 2$ . $X_{TF}[1, 0] = (1/2)[(1+j) + (-1-j) - (1-j) - (-1+j)] = 0$ . $X_{TF}[1, 1] = (1/2)[(1+j) - (-1-j) - (1-j) + (-1+j)] = 0$ . So $X_{TF}[0, 1] = 2$ and all others zero — all the data has been concentrated at one TF cell by the ISFFT. This is a particular feature of this symmetric input.

Heisenberg transform

Only $X_{TF}[0, 1] = 2$ is non-zero. $s(t) = 2\,g_{tx}(t)\,e^{j 2\pi \Delta f t}$ for $t \in [0, T_s)$ . A single subcarrier tone on the first OFDM symbol, zero on the second.

Interpret

This particular $X_{DD}$ produces a "spread" TF image that collapses to a single TF cell. The ISFFT exchanges the roles of DD and TF: the original $X_{DD}$ had four non-zero cells; $X_{TF}$ has one. Different inputs produce spread images in the opposite direction — illustrating the spreading duality.

🚨Critical Engineering Note

Deploying OTFS on Existing 5G NR Hardware

A key practical argument for OTFS is deployability: OTFS can be added to existing 5G NR infrastructure as a software-only precoder. Concretely:

Transmitter: an ISFFT block is added after the UE data mapper, before the OFDM modulator. This is a 2D FFT — a software operation.
Receiver: a SFFT block is added after the OFDM demodulator. Also software.
RF front-end: unchanged. Same ADC/DAC, same PA, same filters.
Channel estimation pilots: replaced with embedded pilot scheme (Chapter 7). A firmware update to the baseband.

No change to hardware. No change to cell-search, timing, or frequency synchronization. The key realization is that OTFS's waveform is identical to OFDM's — only the data mapping changes. This is the engineering argument behind CommIT's cell-free OTFS deployment studies: existing O-RAN radio units work as-is.

Practical Constraints

•
ISFFT/SFFT are 2D FFTs — $O(MN \log(MN))$ additional operations per frame
•
Firmware update to existing 5G modems feasible; no silicon changes required
•
OTFS can coexist with OFDM on the same TDD/FDD air interface by frame type

📋 Ref: O-RAN.WG4.CUS.0-R004-v11.00

The OTFS Transmitter: Three Stages