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

Because the basis functions of all four Fourier transforms are periodic, the transform of a periodic signal (CT or DT) is a function of discretefrequency. In other words, it is a sequence of series expansion coefficients. If the signal is infinitely long and not periodic, thetransform is a function of continuous frequency and the inverse is an integral, not a sum.

A bit of thought and, perhaps, referring to appropriate materials on signal processing and Fourier methods will make this clear and show why so manyproperties of Fourier analysis are created by the periodic basis functions.

Also recall that in most cases, it is the Fourier transform, discrete-time Fourier transform, or Fourier series that is needed but it is the DFT thatcan be calculated by a digital computer and that is probably using the FFT algorithm. If the coefficients of a Fourier series drop off fast enoughor, even better, are zero after some harmonic, the DFT of samples of the signal will give the Fourier series coefficients. If a discrete-timesignal has a finite nonzero duration, the DFT of its values will be samples of its DTFT. From this, one sees the relation of samples of asignal to the signal and the relation of the various Fourier transforms.

Now, what is the case for the various wavelet transforms? Well, it is both similar and different. The table that relates the continuous anddiscrete variables is given by where DW indicates discrete values for scale and translation given by $j$ and $k$ , with CW denoting continuous values for scale and translation.

We have spent most this book developing the DWT, which is a series expansion of a continuous time signal. Because the waveletbasis functions are concentrated in time and not periodic, both the DTWT and DWT will represent infinitely long signals. In most practical cases,they are made periodic to facilitate efficient computation. Chapter: Calculation of the Discrete Wavelet Transform gives the details of how the transform is made periodic. The discrete-time, continuous wavelet transform (DTCWT) is seldom used andnot discussed here.

The naming of the various transforms has not been consistent in the literature and this is complicated by the wavelet transforms having twotransform variables, scale and translation. If we could rename all the transforms, it would be more consistent to useFourier series (FS) or wavelet series (WS) for a series expansion that produced discrete expansion coefficients, Fourier transforms (FT) orwavelet transforms (WT) for integral expansions that produce functions of continuous frequency or scale or translation variable together with DT(discrete time) or CT (continuous time) to describe the input signal. However, in common usage, only the DTFT follows this format!

Recall that the difference between the DWT and DTWT is that the input to the DWT is a sequence of expansion coefficients or a sequence of innerproducts while the input to the DTWT is the signal itself, probably samples of a continuous-time signal. The Mallat algorithm or filter bankstructure is exactly the same. The approximation is made better by zero moments of the scaling function (see Section: Approximation of Scaling Coefficients by Samples of the Signal ) or by some sort of prefiltering ofthe samples to make them closer to the inner products [link] .

As mentioned before, both the DWT and DTWT can be formulated as nonperiodic, on-going transforms for an exact expansion of infinite durationsignals or they may be made periodic to handle finite-length or periodic signals. If they are made periodic (as in Chapter: Calculation of the Discrete Wavelet Transform ), then there is an aliasing that takes place in the transform. Indeed, the aliasinghas a different period at the different scales which may make interpretation difficult. This does not harm the inverse transform whichuses the wavelet information to “unalias" the scaling function coefficients. Most (but not all) DWT, DTWT, and matrix operatorsuse a periodized form [link] .