Ferkans — Interactive Telecom Tutor

A 2D Fourier Transform on the DD Plane

The Zak transform of Chapter 2 takes a 1D time-domain signal into a 2D function on the delay-Doppler torus. What we need next is a 2D transform that relates two 2D representations — the DD plane and the TF plane. This is the symplectic Fourier transform (SFT).

"Symplectic" here refers to the sign pattern of the kernel, which ensures that the transform is its own inverse up to a sign flip, and that it respects the natural symplectic form $d\tau \wedge d\nu - df \wedge dt$ on the phase space. Practically, the SFT is simply a 2D DFT with specific sign conventions chosen so that Heisenberg-Weyl covariance is preserved.

For OTFS, the SFT is not optional — it is the operation that maps DD-domain data symbols to the TF-domain symbols handed to a standard OFDM modulator. Its inverse (the ISFFT) does the reverse at the receiver. Chapter 6 will assemble these two operations into the full transceiver.

Definition:
Continuous Symplectic Fourier Transform

For a function $F(\tau, \nu)$ on $\mathbb{R}^2$ , the symplectic Fourier transform is $(\mathcal{F}_s F)(f, t) \;=\; \iint F(\tau, \nu)\,e^{-j 2\pi (\nu t - f \tau)}\,d\tau\,d\nu.$ The inverse transform is $(\mathcal{F}_s^{-1} G)(\tau, \nu) \;=\; \iint G(f, t)\,e^{+j 2\pi (\nu t - f \tau)}\,df\,dt.$ Note the opposite signs on $f$ and $\nu$ — this is the symplectic pattern.

The SFT is almost an ordinary 2D Fourier transform, with two crucial differences: (i) the sign of the $f\tau$ term is flipped relative to the $\nu t$ term, and (ii) the output variables $(f, t)$ are the TF-plane coordinates, not generic frequency variables. These choices are what make $\mathcal{F}_s^{-1}$ equal $\mathcal{F}_s$ evaluated at $(-f, -t)$ , and what make the SFT intertwine DD translations with TF modulations — the Heisenberg-Weyl covariance we need.

,

Theorem: Symplectic Fourier Covariance Under Translation

Let $F(\tau, \nu)$ be a function on the DD plane. Translating $F$ by $(\tau_0, \nu_0)$ corresponds to modulating its SFT by phase factors: $\mathcal{F}_s\{F(\tau - \tau_0, \nu - \nu_0)\}(f, t) \;=\; e^{-j 2\pi (\nu_0 t - f \tau_0)}\,(\mathcal{F}_s F)(f, t).$ Dually, translating the TF function by $(t_0, f_0)$ corresponds to modulating the DD function.

This is the 2D analog of the standard Fourier shift theorem, specialized to the symplectic sign convention. The pair $(\mathcal{F}_s, \mathcal{F}_s^{-1})$ is designed so that translation in one domain becomes modulation in the other — exactly the covariance we need for a channel that acts by translation in the DD plane.

Proof

Apply definition

$\mathcal{F}_s\{F(\tau - \tau_0, \nu - \nu_0)\}(f, t) = \iint F(\tau - \tau_0, \nu - \nu_0)\,e^{-j 2\pi (\nu t - f \tau)}\,d\tau\,d\nu$ .

Change variables

Let $\tau' = \tau - \tau_0$ , $\nu' = \nu - \nu_0$ : $= \iint F(\tau', \nu')\,e^{-j 2\pi ((\nu' + \nu_0) t - f(\tau' + \tau_0))}\,d\tau'\,d\nu'$ .

Factor phase

$= e^{-j 2\pi (\nu_0 t - f \tau_0)}\iint F(\tau', \nu')\,e^{-j 2\pi(\nu' t - f \tau')}\,d\tau'\,d\nu' = e^{-j 2\pi (\nu_0 t - f \tau_0)}(\mathcal{F}_s F)(f, t)$ . $\blacksquare$

Definition:
Discrete Symplectic Fourier Transform (ISFFT)

For an $M \times N$ matrix $X_{DD}[\ell, k]$ on the DD grid, the inverse symplectic Fourier finite transform (ISFFT) — used at the OTFS transmitter — is $X_{TF}[n, m] \;=\; \frac{1}{\sqrt{MN}}\sum_{\ell = 0}^{M - 1}\sum_{k = 0}^{N - 1} X_{DD}[\ell, k]\,e^{j 2\pi (k n / N - m \ell / M)},$ for $n = 0, \ldots, N - 1$ and $m = 0, \ldots, M - 1$ . This produces the TF-domain grid fed to a standard OFDM modulator. The forward symplectic Fourier finite transform (SFFT), used at the receiver to return to the DD domain, is $X_{DD}[\ell, k] \;=\; \frac{1}{\sqrt{MN}}\sum_{n = 0}^{N - 1}\sum_{m = 0}^{M - 1} X_{TF}[n, m]\,e^{-j 2\pi (k n / N - m \ell / M)}.$

The ISFFT is exactly a 2D DFT/IDFT with one axis inverted. In software terms, it is np.fft.ifft(np.fft.fft(X, axis=0), axis=1) with appropriate shift conventions. We will see the code in Section 3.3.

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Theorem: SFFT and ISFFT Are Inverses

The ISFFT and SFFT are mutual inverses: $\text{SFFT} \circ \text{ISFFT} = \text{identity}$ on $\mathbb{C}^{M \times N}$ , and similarly $\text{ISFFT} \circ \text{SFFT} = \text{identity}$ . Both transforms are unitary with respect to the Frobenius inner product.

The ISFFT and SFFT are 2D DFTs with opposite signs. Their composition is the 2D identity, just as $\text{DFT}^{-1} \cdot \text{DFT} = I$ in 1D.

Proof

Compose

$\text{SFFT}(\text{ISFFT}(X_{DD}))[\ell', k'] = \frac{1}{MN}\sum_{n, m}\sum_{\ell, k} X_{DD}[\ell, k]\,e^{j 2\pi((k - k')n/N + (m \ell' - m \ell)/M)}$ .

Sum over $n$ and $m$

$\sum_n e^{j 2\pi (k - k') n/N} = N\,\mathbf{1}_{k = k'}$ and $\sum_m e^{j 2\pi m(\ell' - \ell)/M} = M\,\mathbf{1}_{\ell = \ell'}$ .

Collect

The compound sum collapses to $X_{DD}[\ell', k']$ , i.e., the identity. Unitarity follows from the $1/\sqrt{MN}$ normalization. $\blacksquare$

Symplectic Fourier Transform of a Simple DD-Grid Pattern

Place a single symbol on the DD grid at $(\ell_0, k_0)$ and observe its image on the TF grid after the ISFFT: it is a 2D complex exponential with frequency $\ell_0$ in one axis and $k_0$ in the other. The SFT "spreads" a point in DD into a complex plane wave in TF. Conversely, a point on the TF grid transforms to a plane wave on the DD grid — two dual pictures.

Parameters

Delay index

\ell_0

4

Doppler index

k_0

6

Delay bins

M

16

Doppler bins

N

16

Display

Example: ISFFT of a Single DD-Grid Symbol

Compute the ISFFT of a DD-grid matrix with a single nonzero entry: $X_{DD}[\ell, k] = \mathbf{1}_{(\ell, k) = (\ell_0, k_0)}$ .

Solution

Apply the ISFFT

$X_{TF}[n, m] = \frac{1}{\sqrt{MN}}\,e^{j 2\pi (k_0 n / N - m \ell_0 / M)}$ .

Interpret

The single DD impulse becomes a 2D complex exponential on the TF grid with angular frequencies $(\ell_0 / M, k_0 / N)$ — a tilted plane wave. This is the "spread" characteristic of the DD domain: every DD-grid symbol is spread across the entire TF grid, which is why OTFS is robust to localized TF-grid erasures (a single subcarrier hit by interference does not take out any particular DD symbol).

Implication for diversity

The spreading property is the root of OTFS's diversity gain. A DD symbol is transmitted on all TF cells, so it passes through every channel realization — the effective diversity order can be as high as the number of resolvable paths $P$ , as we will prove in Chapter 9.

From DD Grid to TF Plane Waves

A single symbol is placed on the DD grid at position

(\ell_0, k_0)

. The ISFFT spreads this symbol into a 2D complex plane wave across the TF grid, with horizontal frequency

k_0 / N

and vertical frequency

\ell_0 / M

. Placing two symbols shows linearity: the result is the sum of two plane waves. With

MN

symbols placed at random, the TF grid looks like interference of

MN

plane waves — this is the OTFS transmit signal.

Key Takeaway

A DD impulse is a TF plane wave — and vice versa. The symplectic Fourier transform relates "point-like" signals in one domain to "spread" signals in the other. A data symbol placed on a single DD grid point contributes a complex exponential to every TF grid cell. This total spreading across the TF grid is what gives OTFS its diversity: the information in each data symbol is physically transmitted through every channel realization on the TF grid, so as long as enough of the TF grid arrives with usable SNR, the data symbol can be recovered.

Why Not Just a 2D DFT?

One might ask: why define a "symplectic" Fourier transform when an ordinary 2D DFT would produce the same spreading? The answer is covariance. The SFT is designed so that a DD-plane translation corresponds exactly to a TF-plane modulation with the right sign pattern to match the Heisenberg-Weyl group action. An ordinary 2D DFT does not do this — signs would be wrong, and the DD channel's action on the transmitted TF signal would not reduce to a 2D DD-plane convolution. In Chapter 4 we will see this covariance used explicitly when deriving the DD input-output relation.

🔧Engineering Note

ISFFT Implementation in a Standard OFDM Modem

The ISFFT can be computed as a 2D FFT with one axis inverted, at a cost of $O(MN(\log M + \log N))$ . For 5G-NR-compatible numerologies with $M = 1200$ and $N = 14$ , this is $\sim 3 \times 10^5$ operations — trivial on modern silicon.

The practical consequence: OTFS can be deployed as a software precoder on top of existing OFDM hardware. The receiver reverses the operation with an SFFT, also a 2D FFT. No new RF chains, no new ADC/DAC, no new power amplifier design. This is the engineering reason OTFS has a realistic path to 6G deployment.

Practical Constraints

•
2D FFT complexity: $O(MN \log(MN))$ — negligible compared to OFDM IFFT
•
Re-uses existing OFDM silicon with firmware/software update only
•
Zero-change-to-hardware deployment is feasible for standalone OTFS

The Symplectic Fourier Transform