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The connections between the DTWT and DWT are:
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] .
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 $W\left(\text{\omega}\right)$ 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 $W\left(\text{\omega}\right)$ 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] .
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.
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