0.2 A multiresolution formulation of wavelet systems  (Page 6/8)

For the general wavelet expansion of [link] or [link] , Parseval's theorem is

with the energy in the expansion domain partitioned in time by $k$ and in scale by $j$ . Indeed, it is this partitioning of the time-scale parameter plane that describes the DWT. If the expansion system is a tight frame,there is a constant multiplier in [link] caused by the redundancy.

Daubechies [link] , [link] showed that it is possible for the scaling function and the wavelets to have compact support (i.e., be nonzero onlyover a finite region) and to be orthonormal. This makes possible the time localization that we desire. We now have a framework for describingsignals that has features of short-time Fourier analysis and of Gabor-based analysis but using a new variable, scale. For the short-timeFourier transform, orthogonality and good time-frequency resolution are incompatible according to the Balian-Low-Coifman-Semmes theorem [link] , [link] . More precisely, if the short-time Fourier transform is orthogonal, either the time or the frequency resolution is poor and thetrade-off is inflexible. This is not the case for the wavelet transform. Also, note that there is avariety of scaling functions and wavelets that can be obtained by choosing different coefficients $h\left(n\right)$ in [link] .

Donoho [link] has noted that wavelets are an unconditional basis for a very wide class of signals. This means wavelet expansions of signalshave coefficients that drop off rapidly and therefore the signal can be efficiently represented by a small number of them.

We have first developed the basic ideas of the discrete wavelet system using a scaling multiplier of 2 in the defining [link] . This is called a two-band wavelet system because of the two channels or bands in the related filter banks discussed in Chapter: Filter Banks and the Discrete Wavelet Transform and Chapter: Filter Banks and Transmultiplexers . It is also possible to define a more general discrete waveletsystem using $\phi \left(t\right)={\sum }_{n}h\left(n\right)\sqrt{M}\phi (Mt-n)$ where $M$ is an integer [link] . This is discussed in Section: Multiplicity-M (M-Band) Scaling Functions and Wavelets . The details of numerically calculating the DWT are discussed in Chapter: Calculation of the Discrete Wavelet Transform where special forms for periodic signals are used.

Display of the discrete wavelet transform and the wavelet expansion

It is important to have an informative way of displaying or visualizing the wavelet expansion and transform. This is complicated in that the DWTis a real-valued function of two integer indices and, therefore, needs a two-dimensional display or plot. This problem is somewhat analogous toplotting the Fourier transform, which is a complex-valued function.

There seem to be five displays that show the various characteristics of the DWT well:

1. The most basic time-domain description of a signal is the signal itself (or, for most cases, samples of the signal) but it gives no frequency orscale information. A very interesting property of the DWT (and one different from the Fourier series) is for a high starting scale $j_0$ in [link] , samples of the signal are the DWT at that scale. This is an extreme case, but it shows the flexibility of the DWT and willbe explained later.
2. The most basic wavelet-domain description is a three-dimensional plot of the expansion coefficients or DWT values $c\left(k\right)$ and $d_j\left(k\right)$ over the $j,k$ plane. This is difficult to do on a two-dimensional page or display screen, but we show a form of that in [link] and [link] .
3. A very informative picture of the effects of scale can be shown by generating time functions $f_j\left(t\right)$ at each scale by summing [link] over $k$ so that
   $$f\left(t\right)=f_{j_0}+\sum _j{f}_j\left(t\right)$$
   where
   $$f_{j_0}=\sum _{k}c\left(k\right)\phi (t-k)$$
   $$f_j\left(t\right)=\sum _k{d}_j\left(k\right)2^{j/2}\psi (2^{j}t-k).$$
   This illustrates the components of the signal at each scale and is shown in  [link] and [link] .
4. Another illustration that shows the time localization of the wavelet expansion is obtained by generating time functions $f_k\left(t\right)$ at each translation by summing [link] over $k$ so that
   $$f\left(t\right)=\sum _k{f}_k\left(t\right)$$
   where
   $$f_k\left(t\right)=c\left(k\right)\varphi (t-k)+\sum _j{d}_j\left(k\right)2^{j/2}\psi (2^{j}t-k).$$
   This illustrates the components of the signal at each integer translation.
5. There is another rather different display based on a partitioning of the time-scale plane as if the time translation index and scale index werecontinuous variables. This display is called “tiling the time-frequency plane." Because it is a different type of display and is developed andillustrated in Chapter: Calculation of the Discrete Wavelet Transform , it will not be illustrated here.