0.7 Generalizations of the basic multiresolution wavelet system  (Page 21/28)

All of these methods are very signal and problem dependent and, in some cases, can give much better results than the standard M-band or waveletpacket based methods.

Local trigonometric bases

In the material up to this point, all of the expansion systems have required the translation and scaling properties of [link] and the satisfactionof the multiresolution analysis assumption of [link] . From this we have beenable to generate orthogonal basis systems with the basis functions having compact support and, through generalization to M-band wavelets and waveletpackets, we have been able to allow a rather general tiling of the time-frequency or time-scale plane with flexible frequency resolution.

By giving up the multiresolution analysis (MRA) requirement, we will be able to create another basis system with a time-frequency tiling somewhat the dualof the wavelet or wavelet packet system. Much as we saw the multiresolution system dividing the frequency bands in a logarithmic spacing for the $M=2$ systems and a linear spacing for the higher $M$ case, and a rather general form for the wavelet packets, we will now develop the local cosine and local sine basis systems for a more flexible time segmenting of the time-frequency plane. Rather than modifying the MRA systems by creatingthe time-varying wavelet systems, we will abandon the MRA and build a basis directly.

What we are looking for is an expansion of a signal or function in the form

where the functions ${\chi }_{j,k}\left(t\right)$ are of the form (for example)

Here $w_k\left(t\right)$ is a window function giving localization to the basis function and $\alpha$ , $\beta$ and $\gamma$ are constants the choice of which wewill get to shortly. $k$ is a time index while $n$ is a frequency index. By requiring orthogonality of these basis functions, thecoefficients (the transform) are found by an inner product

We will now examine how this can be achieved and what the properties of the expansion are.

Fundamentally, the wavelet packet system decomposes $L^2\left(\text{ℝ}\right)$ into a direct sum of orthogonal spaces, each typically covering a certain frequencyband and spanned by the translates of a particular element of the wavelet packet system. With wavelet packets time-frequency tilingwith flexible frequency resolution is possible. However, the temporal resolution is determined by the frequency band associated witha particular element in the packet.

Local trigonometric bases [link] , [link] are duals of wavelet packets in thesense that these bases give flexible temporal resolution. In this case, $L^2\left(\text{ℝ}\right)$ is decomposed into a direct sum of spaces each typically covering a particular time interval. The basis functionsare all modulates of a fixed window function.

One could argue that an obvious approach is to partition the time axis into disjointbins and use a Fourier series expansion in each temporal bin. However, since the basis functions are “rectangular-windowed” exponentialsthey are discontinuous at the bin boundaries and hence undesirable in the analysis of smooth signals. If one replaces the rectangularwindow with a “smooth” window, then, since products of smooth functions are smooth, one can generate smooth windowed exponential basisfunctions. For example, if the time axis is split uniformly, one is looking at basis functions of the form $\left\{w,(t-k),e^{\iota 2\pi nt}\right\},k,n\in \mathbf{Z}$ for some smooth window function $w\left(t\right)$ . Unfortunately, orthonormality disallows the function $w\left(t\right)$ from being well-concentrated in time or in frequency - which is undesirable for time frequency analysis. More precisely,the Balian-Low theorem (see p.108 in [link] ) states that the Heisenberg product of $g$ (the product of the time-spread and frequency-spread which is lower bounded by the Heisenberg uncertainty principle) is infinite.However, it turns out that windowed trigonometric bases (that use cosines and sinesbut not exponentials) can be orthonormal, and the window can have a finite Heisenberg product [link] . That is the reason why we are lookingfor local trigonometric bases of the form given in [link] .