: In audio/voice codecs, a quadrature mirror filter pair is often used to implement a filter bank that splits an input signal into two bands.
The resulting high-pass and low-pass signals are often reduced by a factor of 2, giving a critically sampled two-channel representation of the original signal.
The analysis filters are often related by the following formula in addition to quadrate mirror property: where
In other words, the power sum of the high-pass and low-pass filters is equal to 1.
A linear filter that is zero for “smooth” signals, given a record of
is defined as It is desirable to have it vanish for a constant, so taking the order
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.
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.
[1] 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.
(In practical implementations, numeric precision issues in floating-point arithmetic may affect the perfection of the reconstruction.)