In previous chapters we encountered various signal models, continuous and discrete, for signals of both finite and infinite duration. We then developed notions of Fourier analysis appropriate to several of these models. Wavelet theory is similar in that it can be approached from both the continuous point of view and the discrete point of view, for signals of either finite or infinite length. As with Fourier analysis, wavelet analysis in the continuous setting involves more technical details and does not allow one to get to the computational applications very quickly. In this chapter we will thus focus on discrete signals only. We’ll examine some continuous models in the next chapter.
However, it is easier to first carry out the analysis for signals that are bi-infinite in extent, for example, signals in L2(Z) or L∞(Z) rather than RN. In specific computational examples where we have a finite length signal x we will embed x in L2(Z) by zero extension or some other technique, and then apply techniques for bi-infinite signals.
Wavelet analysis is in part motivated by the need to analyze signals whose frequency content varies over the duration of the signal. This is one application where Fourier methods do not perform well, for the basic Fourier waveforms are global in nature. As a result standard Fourier techniques do not really recognize or exploit the fact that the frequency content of a signal or image can vary considerably from one point to ...