Filter Banks
Filter banks allow signals to be decomposed into subbands. In this way, parallel powerful processing can be easily applied. Also, the decomposition paves the way for signal compression procedures. Due to these reasons, the interest on filter banks has sig
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Filter Banks
1.1 Introduction Filter banks allow signals to be decomposed into subbands. In this way, parallel powerful processing can be easily applied. Also, the decomposition paves the way for signal compression procedures. Due to these reasons, the interest on filter banks has significantly grown along years, so today there is large body of theory on this matter. The chapter starts with an introduction to new concepts and to architectural elements associated to filter banks. Most of the chapter focuses on FIR filters, so after the introductory section, the next one treats in detail the aspects of FIR filters that are relevant for their use in filter banks. The fourth section attacks the main issue, which is perfect reconstruction. The question is that the original signal should be recovered after decomposition into subbands. A series of mathematical conditions for this to happen are discovered, and then a classification of filter banks is derived. This is done in the most simple context: 2-channel filter banks. The chapter continues with the introduction of filter bank structures and design approaches, and then with extensions to M-channel filter banks and to multidimensional signals (for instance, images). An important point is that wavelets, which will be the theme of the next chapter, are strongly related to filter banks, at least for real-time implementation. Concerning notation, the filter coefficients would be h(0) . . . h(N – 1), so it has N terms (filter length), or they would be h(–L) . . . h(L) with L = (N – 1)/2. In some cases we use 2k + 1 instead of (N − 1) to highlight that N is even. The frequency responses of the filters are between 0 and π. The basic reference literature for this chapter is [24, 27, 44, 45, 49]. Other more specific references will be given in appropriate places.
© Springer Science+Business Media Singapore 2017 J.M. Giron-Sierra, Digital Signal Processing with Matlab Examples, Volume 2, Signals and Communication Technology, DOI 10.1007/978-981-10-2537-2_1
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1 Filter Banks
1.2 Filter Banks and Multirate Systems Nowadays a lot of people use compressed information. This is the case of MP3, MP4, etc., for audio and video. It can be recognized as a problem of efficient use of a limited resource: the storage capacity, or perhaps the bandwidth of Internet. In real life signal processing applications, like in mobile phones, generic or specific microprocessors are employed. These units do have processing rate limits, and this should be taken into account. Thus there is a problem of efficiency. And there are several alternatives to deal with it. Processing work could be distributed. Resources could be shared. Tasks could be tailored. Distribution could be done by decomposition in the frequency domain, by using sets of filters. It also can be done with decomposition in the time domain, for instance in multiplexing style. Tailoring could be done by adapting the sampling rate to the signal frequencies, thus leading to multirate systems. Figure 1.1 presents a basic example of single
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