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In this chapter, we develop the properties of wavelet systems in terms of the underlying filter banks associated with them. This is an expansion andelaboration of the material in Chapter: Filter Banks and the Discrete Wavelet Transform , where many of the conditions and properties developed from a signal expansion point of view in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients are now derived from the associated filter bank. The Mallat algorithm uses a special structure of filters anddownsamplers/upsamplers to calculate and invert the discrete wavelet transform. Such filter structures have been studied for over three decades in digitalsignal processing in the context of the filter bank and transmultiplexer problems [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] . Filter bank theory, besides providing efficient computational schemes for wavelet analysis, also gives valuable insights into theconstruction of wavelet bases. Indeed, some of the finer aspects of wavelet theory emanates from filter bank theory.
A filter bank is a structure that decomposes a signal into a collection of subsignals. Depending on the application, these subsignals helpemphasize specific aspects of the original signal or may be easier to work with than the original signal. We have linear or non-linear filter banks depending on whether or not the subsignals depend linearly on the original signal. Filter banks were originally studied in thecontext of signal compression where the subsignals were used to “represent” the original signal. The subsignals (called subbandsignals) are downsampled so that the data rates are the same in the subbands as in the original signal—though this is not essential. Keypoints to remember are that the subsignals convey salient features of the original signal and are sufficient to reconstruct the original signal.
[link] shows a linear filter bank that is used in signal compression (subband coding). The analysis filters $\left\{{h}_{i}\right\}$ are used to filter the input signal $x\left(n\right)$ . The filtered signals are downsampled to give the subband signals. Reconstruction of the original signal is achieved by upsampling, filtering and adding upthe subband signals as shown in the right-hand part of [link] . The desire for perfect reconstruction (i.e., $y\left(n\right)=x\left(n\right)$ ) imposes a set of bilinear constraints (since all operations in [link] are linear) on the analysis and synthesis filters. This also constrains the downsampling factor, $M$ , to be at most the number of subband signals, say $L$ . Filter bank design involves choosing filters $\left\{{h}_{i}\right\}$ and $\left\{{g}_{i}\right\}$ that satisfy perfect reconstruction and simultaneously give informative and useful subbandsignals. In subband speech coding, for example, a natural choice of desired frequency responses—motivated by the nonuniform sensitivity ofthe human ear to various frequency bands—for the analysis and synthesis filters is shown in [link] .
In summary, the filter bank problem involves the design of the filters ${h}_{i}\left(n\right)$ and ${g}_{i}\left(n\right)$ , with the following goals:
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