Modulation space

Modulation spaces are a family of Banach spaces defined by the behavior of the short-time Fourier transform with respect to a test function from the Schwartz space. They were originally proposed by Hans Georg Feichtinger and are recognized to be the right kind of function spaces for time-frequency analysis. Feichtinger's algebra, while originally introduced as a new Segal algebra, is identical to a certain modulation space and has become a widely used space of test functions for time-frequency analysis.

Modulation spaces are defined as follows. For $$ 1\leq p,q \leq \infty $$, a non-negative function $$ m(x,\omega) $$ on $$\mathbb{R}^{2d}$$ and a test function $$ g \in \mathcal{S}(\mathbb{R}^d) $$, the modulation space $$ M^{p,q}_m(\mathbb{R}^d) $$ is defined by


 * $$ M^{p,q}_m(\mathbb{R}^d) = \left\{ f\in \mathcal{S}'(\mathbb{R}^d)\ :\ \left(\int_{\mathbb{R}^d}\left(\int_{\mathbb{R}^d} |V_gf(x,\omega)|^p m(x,\omega)^p dx\right)^{q/p} d\omega\right)^{1/q} < \infty\right\}.$$

In the above equation, $$ V_gf $$ denotes the short-time Fourier transform of $$ f $$ with respect to $$ g $$ evaluated at $$ (x,\omega) $$, namely


 * $$V_gf(x,\omega)=\int_{\mathbb{R}^d}f(t)\overline{g(t-x)}e^{-2\pi it\cdot \omega}dt=\mathcal{F}^{-1}_{\xi}(\overline{\hat{g}(\xi)}\hat{f}(\xi+\omega))(x).$$

In other words, $$ f\in M^{p,q}_m(\mathbb{R}^d) $$ is equivalent to $$ V_gf\in L^{p,q}_m(\mathbb{R}^{2d}) $$. The space $$ M^{p,q}_m(\mathbb{R}^d) $$ is the same, independent of the test function $$ g \in \mathcal{S}(\mathbb{R}^d) $$ chosen. The canonical choice is a Gaussian.

We also have a Besov-type definition of modulation spaces as follows.


 * $$ M^s_{p,q}(\mathbb{R}^d) = \left\{ f\in \mathcal{S}'(\mathbb{R}^d)\ :\ \left(\sum_{k\in\mathbb{Z}^d} \langle k \rangle^{sq} \|\psi_k(D)f\|_p^q\right)^{1/q} < \infty\right\}, \langle x\rangle:=|x|+1$$,

where $$\{\psi_k\}$$ is a suitable unity partition. If $$m(x,\omega)=\langle \omega\rangle^s$$, then $$M^s_{p,q}=M^{p,q}_m$$.

Feichtinger's algebra
For $$ p=q=1 $$ and $$ m(x,\omega) = 1 $$, the modulation space $$ M^{1,1}_m(\mathbb{R}^d) = M^1(\mathbb{R}^d)$$ is known by the name Feichtinger's algebra and often denoted by $$ S_0 $$ for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators. $$ M^1(\mathbb{R}^d)$$ is a Banach space embedded in $$ L^1(\mathbb{R}^d) \cap C_0(\mathbb{R}^d) $$, and is invariant under the Fourier transform. It is for these and more properties that $$ M^1(\mathbb{R}^d)$$ is a natural choice of test function space for time-frequency analysis. Fourier transform $$\mathcal{F}$$ is an automorphism on $$M^{1,1}$$.