Data Approximation with Time-Frequency Invariant Systems

Abstract

In this paper we prove the existence of a time-frequency space that best approximates a given finite set of data. Here best approximation is in the least square sense, among all time-frequency spaces with no more than a prescribed number of generators. We provide a formula to construct the generators from the data and give the exact error of approximation. The setting is in the space of square integrable functions defined on a second countable LCA group and we use the Zak transform as the main tool.

Appendix

We give the proof of the following Lemma that has been used in Sect. 2 to prove part (2) of Theorem 2.

Lemma 1

Let $\sigma : \mathbb H_1 \longrightarrow \mathbb H_2$ be an isometric isomorphism between the Hilbert spaces $\mathbb H_1$ and $\mathbb H_2$. For a measure space (X, dμ) the map $Q_\sigma : L^2(X, \mathbb H_1) \longrightarrow L^2(X, \mathbb H_2)$ given by (Q _σf)(x) = σ(f(x)) is also an isometric isomorphism.

Proof

Let f be a measurable vector function in $L^2(X, \mathbb H_1)$, that is, for every $y\in \mathbb H_1$ the scalar function $x \longrightarrow \langle f(x) , y \rangle _{\mathbb H_1}$ is measurable. We must prove that Q _σf is also a measurable vector function in $L^2(X, \mathbb H_2)$. For $z\in \mathbb H_2$ we have

$$\displaystyle \begin{aligned}<Q_\sigma f(x),z>_{\mathbb H_2}= <\sigma(f(x)),z>_{ \mathbb H_2}=<f(x),\sigma^*(z)>_{\mathbb H_1}. \end{aligned}$$

Since σ ^∗(z) = σ ⁻¹(z) is a general element of $\mathbb H_1$, this shows that Q _σf is measurable. Moreover, for $f,g \in L^2(X, \mathbb H_1)$,

$$\displaystyle \begin{aligned} <Q_\sigma f,Q_\sigma g>_{L^2(X, \mathbb H_2)} =&\int_X<\sigma(f(x)),\sigma(g(x))>_{\mathbb H_2} d\mu(x)\\ = &\int_X<f(x), g(x)>_{\mathbb H_1} d\mu(x)=<f,g>_{L^2(X, \mathbb H_1)}\,. \end{aligned} $$

This shows that if $f\in L^2(X, \mathbb H_1)$, $Q_\sigma f \in L^2(X, \mathbb H_2)$ and that Q _σ is an isometry.

Finally, it is easy to see that $R: L^2(X, \mathbb H_2) \rightarrow L^2(X, \mathbb H_1)$ defined by Rg(x) = σ ⁻¹(g(x)) is the inverse and the adjoint of Q _σ. Therefore, Q _σ is onto. □