Ferkans — Interactive Telecom Tutor

From TF Samples to a Continuous Waveform

The ISFFT leaves us with a TF grid $X_{TF}[n, m]$ — a set of discrete complex numbers, one per TF cell. The physical radio wants a continuous-time waveform $s(t)$ . The Heisenberg transform bridges this gap by superposing time-frequency shifted copies of a prototype pulse, one per TF cell. Under the (idealized) bi-orthogonal pulse assumption, the Heisenberg transform is exactly the OFDM modulator already used in 4G/5G.

In this section we write out the Heisenberg transform, show that it is unitary, and establish the adjoint Wigner transform (used at the receiver in Section 4). The structural point: the Heisenberg/Wigner pair is how DD-domain information travels through the physical time-frequency channel.

Definition:
Heisenberg Transform

Given a TF grid $X_{TF}[n, m]$ of size $N \times M$ , a prototype pulse $g_{tx}(t)$ , symbol duration $T_s = 1/\Delta f$ , and subcarrier spacing $\Delta f$ , the Heisenberg transform is $s(t) \;=\; \text{Heis}[X_{TF}](t) \;\triangleq\; \sum_{n = 0}^{N - 1}\sum_{m = 0}^{M - 1} X_{TF}[n, m]\,g_{tx}(t - n T_s)\,e^{j 2\pi m \Delta f(t - n T_s)}.$ The output $s(t)$ lives on the time interval $[0, NT_s) = [0, T)$ (for rectangular $g_{tx}$ ) or a slightly longer interval (for pulses with tails).

The name "Heisenberg transform" comes from the underlying non-commutative algebra of delay and Doppler operators (the Heisenberg-Weyl algebra). The transform is a sum of the projective-representation operators $\pi_{n T_s, m\Delta f}$ applied to the prototype pulse, weighted by $X_{TF}$ .

,

Theorem: The Heisenberg Transform Is Unitary (Critical Lattice)

If the prototype pulse $g_{tx}$ satisfies the Gabor-frame critical lattice condition $T_s \Delta f = 1$ with $|Z_{g_{tx}}(t, \nu)|$ bounded and bounded below on $\mathbb{T}^2$ (Chapter 2 Theorem TExistence of Gabor Frames (Critical Lattice)), then the Heisenberg transform is unitary. In particular, $\int |s(t)|^2\,dt \;=\; \sum_{n, m} |X_{TF}[n, m]|^2.$

Unitarity means no energy is gained or lost in the transform — the total power of the waveform equals the total power of the TF symbols. This is what Parseval's theorem enforces for OFDM; it holds identically for the Heisenberg transform with an appropriately chosen pulse.

Under a non-ideal pulse (not satisfying the bi-orthogonality), the transform is not strictly unitary, and small cross-symbol interference appears. For rectangular $g_{tx}$ with a CP, this mismatch is minimized (CP makes the convolution circular) but not eliminated entirely under time-varying channels.

Proof

Inner product expansion

$\|s\|^2 = \int |s|^2\,dt = \sum_{n, m, n', m'} X_{TF}[n, m]\overline{X_{TF}[n', m']}\langle g_{tx} e_{n,m}, g_{tx} e_{n', m'}\rangle$ where $e_{n, m}(t) = g_{tx}(t - n T_s)\,e^{j 2\pi m \Delta f(t - n T_s)}$ .

Gabor orthogonality

Under the bi-orthogonality assumption, $\langle e_{n, m}, e_{n', m'}\rangle = \delta_{n, n'}\delta_{m, m'}$ .

Parseval

Off-diagonal terms vanish, leaving $\|s\|^2 = \sum_{n, m} |X_{TF}[n, m]|^2$ . Unitarity follows. $\blacksquare$

Definition:
Wigner Transform (Adjoint of Heisenberg)

At the receiver, the Wigner transform recovers the TF grid from a received waveform $r(t)$ : $Y_{TF}[n, m] \;=\; \text{Wig}[r][n, m] \;\triangleq\; \int r(t)\,\overline{g_{rx}(t - n T_s)}\,e^{-j 2\pi m \Delta f(t - n T_s)}\,dt,$ where $g_{rx}(t)$ is the receive prototype pulse. When $g_{rx} = g_{tx}$ (matched filtering), the Wigner transform is the adjoint of the Heisenberg transform — composing them returns the identity, up to the noise from the channel.

Theorem: Heisenberg-Wigner Roundtrip

Under the critical-lattice bi-orthogonal pulse assumption with $g_{rx} = g_{tx}$ (matched filter), for a clean (noise-free, LTI-channel-free) reception: $\text{Wig} \circ \text{Heis} \;=\; \text{Identity} \quad \text{on } \mathbb{C}^{N \times M}.$

Start with a TF grid, go to time-domain waveform, come back to TF grid with the adjoint — perfect recovery. This is the baseline for the full OTFS chain: any imperfection comes from (i) the actual (time-varying) channel, (ii) noise, (iii) pulse-shape mismatch. Sections 3 and 4 quantify these imperfections.

Proof

Compose

$\text{Wig}[\text{Heis}[X_{TF}]][n, m] = \int \text{Heis}[X_{TF}](t)\,\overline{g(t - n T_s)}\,e^{-j 2\pi m \Delta f(t - n T_s)}\,dt$ .

Substitute

Expand the Heisenberg sum: $= \sum_{n', m'} X_{TF}[n', m']\int g(t - n' T_s)\overline{g(t - n T_s)}\,e^{j 2\pi[(m' - m)\Delta f(t - n' T_s) - m\Delta f(n' - n) T_s/(\cdot)]}\,dt$ .

Apply bi-orthogonality

The cross-term integrates to $\delta_{n, n'}\delta_{m, m'}$ under bi-orthogonal pulse assumption: only the $(n', m') = (n, m)$ term survives.

Collect

$\text{Wig}[\text{Heis}[X_{TF}]][n, m] = X_{TF}[n, m]$ . Roundtrip is identity. $\blacksquare$

Heisenberg-Wigner Pair as the TF Modulator-Demodulator — The Heisenberg transform (top) maps the TF grid to a continuous waveform by superposing time-frequency shifted prototype pulses. The Wigner transform (bottom) does the reverse via matched filtering and FFT. Together they form the TF-domain modulator-demodulator of OTFS — structurally identical to the OFDM (IFFT+CP / CP-removal+FFT) pair but with the explicit time-frequency-shift interpretation of the underlying operators.

Why This Is Just OFDM With Better Names

Under the rectangular-pulse, critical-lattice assumption, the Heisenberg transform reduces exactly to an OFDM modulator:

Per-OFDM-symbol inverse DFT: the inner sum over $m$ .
Per-frame serialization: the outer sum over $n$ .

The only "new" thing the Heisenberg transform brings is the operator interpretation: time-shift $\mathcal{T}_{nT_s}$ and frequency-shift $\mathcal{F}_{m\Delta f}$ as independent operators that collectively generate the TF lattice. This operator view makes the generalization to non-rectangular pulses (and non-integer lattice densities) conceptually clean.

For practical OTFS deployment, the take-away is: the modulator is OFDM. Only the precoder (ISFFT) is different.

Heisenberg Transform of a Simple TF Grid

Enter a simple TF grid pattern (single symbol, row, column, random) and observe the resulting time-domain OTFS waveform $s(t)$ . Watch how the waveform structure reflects the TF data: a single TF cell produces a windowed sinusoid; a TF row produces a multitone burst; random data produces a noise-like Gaussian waveform.

Parameters

Subcarriers

M

8

OFDM symbols

N

4

TF pattern

Display

Example: Waveform of a Single TF Cell

Let $X_{TF}[n, m] = \mathbf{1}_{(n, m) = (0, 3)}$ — single TF-grid cell at the first OFDM symbol, fourth subcarrier. Compute $s(t) = \text{Heis}[X_{TF}](t)$ with rectangular $g_{tx}$ of width $T_s$ .

Solution

Substitute

$s(t) = g_{tx}(t)\,e^{j 2\pi \cdot 3 \cdot \Delta f \cdot t}$ for $t \in [0, T_s)$ , zero elsewhere.

Interpret

A single complex sinusoid at frequency $3\Delta f$ , time- gated to the first OFDM symbol. Exactly what one expects from OFDM.

Compare with OTFS full frame

If instead the DD grid had a single entry $X_{DD}[\ell_0, k_0] = 1$ , the ISFFT would produce $X_{TF}[n, m] = e^{j 2\pi(k_0 n/N - m \ell_0/M)}/\sqrt{MN}$ — a plane wave across the whole TF grid. The Heisenberg transform would then give a time-domain waveform that is a sum of $MN$ sinusoids, spread uniformly over the entire frame. OTFS's spreading is visible in both DD and time domains.

🎓CommIT Contribution(2022)

Zak-OTFS: Direct Modulation on the DD Plane

S. K. Mohammed, R. Hadani, A. Chockalingam, G. Caire — IEEE BITS the Information Theory Magazine

Mohammed, Hadani, Chockalingam, and Caire propose Zak-OTFS, an alternative construction that bypasses the two-stage ISFFT + OFDM approach. In Zak-OTFS, the transmit waveform is constructed directly as an inverse Zak transform of the DD grid data: $s(t) \;=\; Z^{-1}[X_{DD}](t) \;=\; \sqrt{T_0}\int_0^{\nu_0} X_{DD}(t, \nu)\,d\nu$ (for the continuous version; the discrete version uses a length- $N$ inverse DFT on blocks of length $M$ ).

Under idealized bi-orthogonal pulses, Zak-OTFS produces the same time-domain waveform as Hadani-Rakib OTFS. The advantages of Zak-OTFS are conceptual clarity (one transform instead of two) and a cleaner treatment of pulse-shape effects in the DD domain. The CommIT contribution in this formulation is the rigorous demonstration of equivalence under critical-lattice conditions, and the identification of a clean way to handle fractional Doppler via Zak-domain windowing (see Chapter 10).

For the remainder of this book we use the Hadani-Rakib two-step form (widely adopted in the literature) but note the Zak-OTFS alternative where relevant.

zak-otfsmodulationcaire-otfs

The Heisenberg Transform: TF Grid to Time

From TF Samples to a Continuous Waveform

Definition: Heisenberg Transform

Theorem: The Heisenberg Transform Is Unitary (Critical Lattice)

Inner product expansion

Gabor orthogonality

Parseval

Definition: Wigner Transform (Adjoint of Heisenberg)

Theorem: Heisenberg-Wigner Roundtrip

Compose

Substitute

Apply bi-orthogonality

Collect

Heisenberg-Wigner Pair as the TF Modulator-Demodulator

Why This Is Just OFDM With Better Names

Heisenberg Transform of a Simple TF Grid

Parameters

Example: Waveform of a Single TF Cell

Substitute

Interpret

Compare with OTFS full frame

Zak-OTFS: Direct Modulation on the DD Plane

Definition:
Heisenberg Transform

Definition:
Wigner Transform (Adjoint of Heisenberg)