Ferkans — Interactive Telecom Tutor

Reversing the Transmitter

The OTFS receiver is the adjoint of the transmitter: Wigner transform (matched filter + sampling) → SFFT → DD-grid estimate. Under ideal bi-orthogonal pulses and no noise/channel, the chain is exactly invertible (Theorem THeisenberg-Wigner Roundtrip from §2 plus Theorem TSFFT and ISFFT Are Inverses from Ch 3). Under the actual channel — the 2D DD convolution of Chapter 4 — the receiver output is a convolved DD grid, and the detector (Chapter 8) removes the convolution to recover the transmitted data.

OTFS Receiver (Full Chain)

Complexity:

O(MN \log(MN))

Input: Received waveform

r(t)

(after RF down-conversion), receive pulse

g_{rx}(t)

, CP length

L_{CP}

Output: DD-domain estimate

\hat{X}_{DD}

1. Stage 1 — CP removal: For each OFDM symbol

n = 0, \ldots, N-1

,

discard the first

L_{CP}

samples of the

n

-th symbol interval.

2. Stage 2 — Wigner transform: Compute

Y_{TF}[n, m] = \int r(t)\,\overline{g_{rx}(t - n T_s)}\,e^{-j 2\pi m \Delta f(t - n T_s)}\,dt

,

implemented as a length-

M

DFT per OFDM symbol.

3. Stage 3 — SFFT: Apply the forward symplectic FFT:

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

.

4. Stage 4 — Detection: Pass

\hat{X}_{DD}

to the DD-domain detector

(MMSE, MP, etc.) of Chapter 8 to recover the data.

5. Return the decoded QAM symbols.

Theorem: End-to-End DD Input-Output

Assume bi-orthogonal prototype pulses with $g_{rx} = g_{tx}$ , cyclic prefix $L_{CP} \geq l_{\max}$ , and an integer-Doppler $P$ -path channel. Then the DD-domain receiver output satisfies $\hat{X}_{DD}[\ell, k] \;=\; \sum_{i=1}^{P} h_i\,e^{j 2\pi \alpha_i(\ell, k)}\,X_{DD}[(\ell - \ell_i)\bmod M,\,(k - k_i)\bmod N] \;+\; W_{DD}[\ell, k],$ where $W_{DD}$ is i.i.d. $\mathcal{CN}(0, \sigma^2)$ — the same 2D circular convolution with sparse kernel we derived from first principles in Chapter 4.

Everything comes together: the transmitter (ISFFT + Heisenberg) puts the DD-grid data onto a TF-grid carrier waveform; the channel (modeled as a 2D DD convolution via Chapter 4) shifts and scales; the receiver (Wigner + SFFT) reverses the TF-grid carrier and returns to DD. The net operation, from DD input to DD output, is exactly the 2D circular convolution predicted by Chapter 4. Nothing is lost; nothing is added except the expected AWGN.

Proof

Concatenate transforms

Transmitter: $s(t) = \text{Heis}[\text{ISFFT}[X_{DD}]](t)$ . Channel: $r(t) = (h \star\star)(s)(t) + w(t)$ — the 2D DD convolution of Chapter 4. Receiver: $\hat{X}_{DD} = \text{SFFT}[\text{Wig}[r]]$ .

Use Wig-Heis invertibility

Under bi-orthogonal pulses, $\text{Wig} \circ \text{Heis} = \text{Id}$ (Theorem THeisenberg-Wigner Roundtrip). Therefore on the noise-free, LTI path, $\text{Wig}[h \star\star (\text{Heis}[X_{TF}])] = h \star\star_{TF} X_{TF}$ , the TF-domain convolution corresponding to the DD-domain shift.

Translate to DD

Applying the SFFT to both sides commutes with the DD convolution (by the SFT covariance property, Chapter 3). The net result is $\hat{X}_{DD} = h \star\star_{DD} X_{DD}$ , matching Theorem TDiscrete DD Input-Output Relation (Integer Doppler).

Add noise

The AWGN $w(t)$ passes through the Wigner + SFFT (both unitary) to yield i.i.d. $\mathcal{CN}(0, \sigma^2)$ on the DD grid. $\blacksquare$

Key Takeaway

The full OTFS chain realizes the DD-domain 2D convolution. From QAM bits entering the DD-grid mapper at the transmitter, through the Heisenberg modulator and physical channel, back through the Wigner demodulator and SFFT at the receiver, the end-to-end operation is exactly the 2D circular convolution $\hat{X}_{DD} = h \star\star X_{DD} + \mathbf{w}$ of Chapter 4. The detector now has a simple target: invert this convolution. This is the point of the whole transceiver construction.

The Complete OTFS Chain: Transmitter, Channel, Receiver — Top row (transmitter): QAM → DD grid → ISFFT → TF grid → Heisenberg (IFFT + CP + DAC) → antenna. Middle (channel): time-varying multipath, which in the DD domain is a 2D sparse convolution. Bottom row (receiver): antenna → ADC → CP removal → Wigner (FFT) → TF grid → SFFT → DD grid → detector. The ISFFT and SFFT are the only OTFS-specific blocks; the rest is identical to OFDM.

Signal Flow Through the Full OTFS Transceiver

End-to-end animation of a random OTFS frame from QAM bits through the full transceiver chain. Watch the data transform at each stage: (1) DD grid with QPSK, (2) TF grid after ISFFT (plane-wave mixture), (3) time-domain waveform, (4) received waveform after the channel, (5) TF grid after Wigner, (6) DD grid after SFFT (now the original plus a 2D shifted copy per path). The channel's action is a 2D convolution — visibly, the output DD grid is a shifted version of the input.

End-to-End OTFS Chain Simulation

Generate a random OTFS frame, pass it through a user-defined $P$ -tap channel with AWGN, and display the received DD grid. Compare with the transmitted DD grid: the receiver output is the transmit grid convolved with the channel, shifted by each path's $(\ell_i, k_i)$ . Adjust SNR to see the noise floor relative to the signal peaks.

Parameters

Delay bins

M

16

Doppler bins

N

8

Number of paths

P

3

SNR (dB)15

l_{\max}

2

Seed4

Why OTFS Noise Is Clean

The AWGN in the DD domain, $W_{DD}[\ell, k]$ , is i.i.d. $\mathcal{CN}(0, \sigma^2)$ — exactly the same distribution as the time-domain AWGN before any processing. This is a direct consequence of the Wigner-SFFT pair being unitary: unitary transforms preserve i.i.d. Gaussian vectors exactly.

This is a quiet but important fact. In OFDM, the noise after the receiver FFT is also i.i.d. Gaussian — that is why OFDM's per-subcarrier Gaussian channel analysis works. OTFS inherits this clean noise structure. The detector (Chapter 8) gets a noise model it can reason about cleanly.

Example: Worked Example: Two-Path Channel, 2×2 OTFS Frame

A 2×2 OTFS frame transmits $X_{DD}[0, 0] = 1$ , $X_{DD}[1, 1] = 1$ (all others zero). The channel has two paths: $(h_1, \ell_1, k_1) = (1, 0, 0)$ (LOS) and $(h_2, \ell_2, k_2) = (0.5, 1, 0)$ (one-delay-shift echo). Ignore noise. Compute the received DD grid $\hat{X}_{DD}$ .

Solution

Path 1 (LOS)

Shifts by $(0, 0)$ , scales by $h_1 = 1$ : preserves $X_{DD}$ as-is. Contribution: $\hat{X}_{DD,1}[0, 0] = 1, \hat{X}_{DD,1}[1, 1] = 1$ .

Path 2 (echo)

Shifts by $(1, 0)$ , scales by $h_2 = 0.5$ : $\hat{X}_{DD,2}[(\ell - 1)\bmod 2, k] = 0.5\,X_{DD}[\ell, k]$ ... With the modular shift: $\hat{X}_{DD,2}[1, 0] = 0.5$ (from $X[0, 0]$ ), $\hat{X}_{DD,2}[0, 1] = 0.5$ (from $X[1, 1]$ ).

Sum

$\hat{X}_{DD} = \hat{X}_{DD,1} + \hat{X}_{DD,2}$ : $\hat{X}_{DD}[0, 0] = 1, \hat{X}_{DD}[0, 1] = 0.5,$ $\hat{X}_{DD}[1, 0] = 0.5, \hat{X}_{DD}[1, 1] = 1$ .

Detection

The detector knows the channel; it reverses the shift and sum to recover the original sparse $(0, 0)$ and $(1, 1)$ values. For this simple case zero-forcing works perfectly (the channel matrix is invertible).

Why This Matters: From Transceiver to Channel Estimation

We have built the OTFS transceiver assuming the channel is known. In practice, the receiver must estimate the $(h_i, \ell_i, k_i)$ triples from pilot signals. Chapter 7 develops the embedded pilot scheme (CommIT cell-free OTFS contribution): a single pilot in the DD grid surrounded by a guard region, sufficient to recover all $P$ path parameters with minimal overhead. The end-to-end chain defined here is the framework into which the estimation step plugs.

The OTFS Receiver