What is a metaplectic chirp?

A metaplectic chirp is a chirp signal — a complex exponential with quadratic (or higher‑order polynomial) phase — viewed through the lens of the metaplectic representation, which is the unitary group action that implements linear symplectic transformations on phase space (time–frequency space) [1][2].

Plain-signal-processing picture

A classical chirp is a signal of the form

$$c(t) = e^{i\pi \alpha t^2}$$

(or a Gaussian-windowed version, called a chirplet) — a sinusoid whose instantaneous frequency sweeps linearly in time [3]. In discrete form on the cyclic group $\mathbb{Z}_n$, one takes second‑degree characters $\psi^{[c]}(m) = e^{i\pi c m^2 / n}$, and "chirp multiplication" is the operator $R_c v(m) = \psi^{[c]}(m),v(m)$ [2].

Metaplectic picture

The metaplectic group $\mathrm{Mp}(2d,\mathbb{R})$ is the double cover of the symplectic group $\mathrm{Sp}(2d,\mathbb{R})$, and its unitary representation on $L^2(\mathbb{R}^d)$ — the Segal–Shale–Weil (oscillator) representation — intertwines the Schrödinger representation of the Heisenberg group with linear symplectic changes of the time–frequency plane [1][2]. Inside this representation, the generators are:

• the Fourier transform (which rotates the time–frequency plane by 90°),
• translations / modulations (which shift in time or frequency),
• chirp multiplications $R_c$ (which shear the time–frequency plane — mapping the momentum axis to a tilted line of slope $c$) [2].

A metaplectic chirp is therefore one of these chirp-multiplication operators (or the function $\psi^{[c]}$ itself) interpreted as the metaplectic operator corresponding to a symplectic shear

$$A = \begin{pmatrix} 1 & 0 \\ c & 1 \end{pmatrix} \in \mathrm{Sp}(2,\mathbb{R}).$$

Together with the Fourier transform — which corresponds to the rotation

$$J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$

— and translations, metaplectic chirps generate the full metaplectic representation [2].

Why they matter

Metaplectic chirps are the building blocks of the free metaplectic transform and its generalizations of the linear canonical transform, which appear in time–frequency analysis of non-stationary signals, quantum optics, and radar processing [4][5]. They also underlie metaplectic Gabor frames, where chirp-modulated Gaussians replace plain modulated Gaussians to better track signals whose time–frequency content lies along tilted or curved ridges rather than horizontal lines [6]. In the discrete Heisenberg setting on $\mathbb{Z}_n$, every metaplectic operator factors into a product of a Fourier transform and a finite number of chirp multiplications — a discrete analog of the Iwasawa decomposition of $\mathrm{Sp}(2,\mathbb{R})$ [2].

So: a metaplectic chirp is a quadratic-phase exponential regarded as the unitary realization, inside the metaplectic representation, of a symplectic shear of the time–frequency plane.