6. Time-Frequency Algorithms Using Malvar—Wilson Wavelets
Society for Industrial and Applied Mathematics eBooks · 2001
6.1 Introduction This chapter continues the time-frequency analysis of Chapter 5. We will introduce algorithms that allow us to decompose a given signal s into a linear combination of time-frequency atoms. The time-frequency atoms that we use are denoted by fR and are “coded” by Heisenberg rectangles R with sides parallel to the axes and with area 1 or 2π, depending on the normalization. If R = [a, b] × [α, β], we require that the function fR be essentially supported on the interval [a, b] and that its Fourier transform f^R be essentially supported on [α, β] and the opposite frequencies [−β, −α]. We also want the algorithmic structure of fR to be simple and explicit to facilitate numerical processing in real time. The decomposition s (t) = ∑ j=0 ∞ αj ƒ Rj (t) 6.1 cannot be unique, and we take advantage of this flexibility by looking for optimal decompositions, which for our purposes means that they contain the fewest possible terms. The point of view of Ville (and of numerous other signal-processing experts) is that it is first necessary to understand the physics of the process and that “the algorithms will follow.” A careful reading of Ville's fundamental paper [254] suggests the following algorithm for finding the optimal decomposition (6.1): (1) Compute the Wigner—Ville transform W(t, ξ) of f; (2) define the domains Ωj of the time-frequency plane by 2−j−1 ≤ |W(t, ξ)| < 2−j, j ≥ 0; and (3) optimally cover Ωj with Heisenberg boxes Rj,k. One should then use these boxes to write the optimal decomposition (6.1). This program appears unrealistic, and one of the main objections is this: The domains Ωj may have complicated structures, and thus the Heisenberg boxes may provide poor coverings for the Ωj. This situation can be improved if the set of horizontal and vertical Heisenberg boxes is enlarged to include oblique boxes, but this means that we will need other time-frequency atoms. These new atoms will be introduced in section 6.11. For the time being we will be less ambitious and stay with horizontal and vertical Heisenberg boxes. The time-frequency atoms that we use are completely explicit. They are either Malvar—Wilson wavelets or wavelet packets, and we will immediately write down the atomic decompositions of the type (6.1).