0.7 Generalizations of the basic multiresolution wavelet system  (Page 27/28)

The connections between the DTWT and DWT are:

• If the starting sequences are the scaling coefficients for the continuous multiresolution analysis at very fine scale, then the discretemultiresolution analysis generates the same coefficients as does the continuous multiresolution analysis on dyadic rationals.
• When the number of scales is large, the basis sequences of the discrete multiresolution analysis converge in shape to the basisfunctions of the continuous multiresolution analysis.

The DTWT or DMRA is often described by a matrix operator. This is especially easy if the transform is made periodic, much as the Fourierseries or DFT are. For the discrete time wavelet transform (DTWT), a matrix operator can give the relationship between a vector of inputs to give a vector of outputs. Several references on this approach are in [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] .

Continuous wavelet transforms

The natural extension of the redundant DWT in [link] is to the continuous wavelet transform (CWT), which transforms a continuous-timesignal into a wavelet transform that is a function of continuous shift or translation and a continuous scale. This transform is analogous to theFourier transform, which is redundant, and results in a transform that is easier to interpret, is shift invariant, and is valuable fortime-frequency/scale analysis. [link] , [link] , [link] , [link] , [link] , [link] , [link]

The definition of the CWT in terms of the wavelet $w\left(t\right)$ is given by

where the inverse transform is

with the normalizing constant given by

with $W\left(\text{ω}\right)$ being the Fourier transform of the wavelet $w\left(t\right)$ . In order for the wavelet to be admissible (for [link] to hold), $K<\infty$ . In most cases, this simply requires that $W\left(0\right)=0$ and that $W\left(\text{ω}\right)$ go to zero ( $W\left(\infty \right)=0$ ) fast enough that $K<\infty$ .

These admissibility conditions are satisfied by a very large set of functions and give very little insight into what basic wavelet functionsshould be used. In most cases, the wavelet $w\left(t\right)$ is chosen to give as good localization of the energy in both time and scale as possible for theclass of signals of interest. It is also important to be able to calculate samples of the CWT as efficiently as possible, usually through the DWT andMallat's filter banks or FFTs. This, and the interpretation of the CWT, is discussed in [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] .

The use of the CWT is part of a more general time-frequency analysis that may or may not use wavelets [link] , [link] , [link] , [link] , [link] .

Analogies between fourier systems and wavelet systems

In order to better understand the wavelet transforms and expansions, we will look at the various forms of Fourier transforms and expansion. If wedenote continuous time by CT, discrete time by DT, continuous frequency byCF, and discrete frequency by DF, the following table will show what the discrete Fourier transform (DFT), Fourier series (FS), discrete-timeFourier transform (DTFT), and Fourier transform take as time domain signals and produce as frequency domain transforms or series.For example, the Fourier series takes a continuous-time input signal and produces a sequence of discrete-frequency coefficients while the DTFTtakes a discrete-time sequence of numbers as an input signal and produces a transform that is a function of continuous frequency.