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This chapter gives a brief discussion of several areas of application. It is intended to show what areas and what tools are being developed and togive some references to books, articles, and conference papers where the topics can be further pursued. In other words, it is a sort of annotatedbibliography that does not pretend to be complete. Indeed, it is impossible to be complete or up-to-date in such a rapidly developing newarea and in an introductory book.

In this chapter, we briefly consider the application of wavelet systems from two perspectives. First, we look at wavelets as a tool for denoisingand compressing a wide variety of signals. Second, we very briefly list several problems where the application of these tools shows promise or hasalready achieved significant success. References will be given to guide the reader to the details of these applications, which are beyond thescope of this book.

Wavelet-based signal processing

To accomplish frequency domain signal processing, one can take the Fourier transform (or Fourier series or DFT) of a signal, multiply some of theFourier coefficients by zero (or some other constant), then take the inverse Fourier transform. It is possible to completely remove certaincomponents of a signal while leaving others completely unchanged. The same can be done by using wavelet transforms to achieve wavelet-based,wavelet domain signal processing, or filtering. Indeed, it is sometimes possible to remove or separate parts of a signal that overlap in both timeand frequency using wavelets, something impossible to do with conventional Fourier-based techniques.

Transform-Based Signal Processor
Transform-Based Signal Processor

The classical paradigm for transform-based signal processing is illustrated in [link] where the center “box" could be either a linear or nonlinear operation. The “dynamics" of the processing are allcontained in the transform and inverse transform operation, which are linear. The transform-domain processing operation has no dynamics; it isan algebraic operation. By dynamics, we mean that a process depends on the present and past, and by algebraic, we mean it depends only on thepresent. An FIR (finite impulse response) filter such as is part of a filter bank is dynamic. Each output depends on the current and a finitenumber of past inputs (see [link] ). The process of operating point-wise on the DWT of a signal is static or algebraic. It does notdepend on the past (or future) values, only the present. This structure,which separates the linear, dynamic parts from the nonlinear static parts of the processing, allows practical and theoretical results that areimpossible or very difficult using a completely general nonlinear dynamic system.

Linear wavelet-based signal processing consists of the processor block in [link] multiplying the DWT of the signal by some set of constants (perhaps by zero). If undesired signals or noise can beseparated from the desired signal in the wavelet transform domain, they can be removed by multiplying their coefficients by zero. This allows amore powerful and flexible processing or filtering than can be achieved using Fourier transforms. The result of this total process is a linear,time-varying processing that is far more versatile than linear, time-invariant processing. The next section gives an example of using theconcentrating properties of the DWT to allow a faster calculation of the FFT.

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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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