0.3 Filter banks and the discrete wavelet transform  (Page 4/6)

Our reason for discussing filtering and up-sampling here is that is exactly what the synthesis operation [link] does. This equation is evaluated by up-sampling the $j$ scale coefficient sequence $c_j\left(k\right)$ , which means double its length by inserting zeros between each term, then convolving it with the scaling coefficients $h\left(n\right)$ . The same is done to the $j$ level wavelet coefficient sequence and the results are added to give the $j+1$ level scaling function coefficients. This structure is illustrated in [link] where $g_0\left(n\right)=h\left(n\right)$ and $g_1\left(n\right)=h_1\left(n\right)$ . This combining process can be continued to any levelby combining the appropriate scale wavelet coefficients. The resulting two-scale tree is shown in [link] .

Input coefficients

One might wonder how the input set of scaling coefficients $c_{j+1}$ are obtained from the signal to use in the systems of [link] and [link] . For high enough scale, the scaling functions act as “delta functions" with the inner product to calculatethe high scale coefficients as simply a sampling of $f\left(t\right)$ [link] , [link] . If the samples of $f\left(t\right)$ are above the Nyquist rate, they are good approximations to the scaling coefficients at that scale,meaning no wavelet coefficients are necessary at that scale. This approximation is particularly good if moments of the scaling function arezero or small. These ideas are further explained in Section: Approximation of Scaling Coefficients by Samples of the Signal and Chapter: Calculation of the Discrete Wavelet Transform .

An alternative approach is to “prefilter" the signal samples to make them a better approximation to the expansion coefficients. This isdiscussed in [link] .

This set of analysis and synthesis operations is known as Mallat's algorithm [link] , [link] . The analysis filter bank efficiently calculates the DWT using banks of digital filters and down-samplers, andthe synthesis filter bank calculates the inverse DWT to reconstruct the signal from the transform. Although presented here as a method ofcalculating the DWT, the filter bank description also gives insight into the transform itself and suggests modifications and generalizations thatwould be difficult to see directly from the wavelet expansion point of view. Filter banks will be used more extensively in the remainder of thisbook. A more general development of filter banks is presented in Section: Multiplicity-M (M-Band) Scaling Functions and Wavelets .

Although a pure wavelet expansion is possible as indicated in [link] and [link] , properties of the wavelet are best developed and understood through the scaling function. This is certainly true if thescaling function has compact support because then the wavelet is composed of a finite sum of scaling functions given in [link] .

In a practical situation where the wavelet expansion or transform is being used as a computational tool in signal processing or numerical analysis,the expansion can be made finite. If the basis functions have finite support, only a finite number of additions over $k$ are necessary. If the scaling function is included as indicated in [link] or [link] , the lower limit on the summation over $j$ is finite. If the signal is essentially bandlimited, there is a scale above which there is little orno energy and the upper limit can be made finite. That is described in Chapter: Calculation of the Discrete Wavelet Transform .