Quadrature Mirror Filter - Further Description

Further Description

The earliest wavelets were based on expanding a function in terms of rectangular steps, the Haar wavelets. This is usually a poor approximation, whereas Daubechies wavelets are among the simplest but most important families of wavelets. A linear filter that is zero for “smooth” signals, given a record of points is defined as:

It is desirable to have it vanish for a constant, so taking the order for example:

And to have it vanish for a linear ramp so that:

A linear filter will vanish for any, and this is all that can be done with a fourth order wavelet. Six terms will be needed to vanish a quadratic curve and so on given the other constraints to be included. Next an accompanying filter may be defined as:

This filter responds in an exactly opposite manner, being large for smooth signals and small for non-smooth signals. A linear filter is just a convolution of the signal with the filter’s coefficients, so the series of the coefficients is the signal that the filter responds to maximally. Thus, the output of the second filter vanishes when the coefficients of the first one are input into it. The aim is to have:

Where the associated time series flips the order of the coefficients because the linear filter is a convolution, and so both have the same index in this sum. A pair of filters with this property are defined as quadrature mirror filters. Even if the two resulting bands have been subsampled by a factor of 2, the relationship between the filters means that approximately perfect reconstruction is possible. That is, the two bands can then be upsampled, filtered again with the same filters and added together, to reproduce the original signal exactly (but with a small delay). (In practical implementations, numeric precision issues in floating-point arithmetic may affect the perfection of the reconstruction.)