# Perfect Reconstruction and Conjugate Mirror Filter Banks

A perfect reconstruction filter bank decomposes a signal by filtering and subsampling.

It reconstructs it by inserting zeroes, filtering and summation.

## Definition

A (discrete) two-channel multirate filter bank convolves a signal a0 with a low-pass filter h1[n] = h[-n] and a high-pass filter g1[n] = g[-n] and then subsamples the output:

a1 [n] = a0 * h1 [2n]
and
d1 [n] = a0* g1 [2n] .

A reconstructed signal a2 is obtained by filtering the zero expanded signals with a dual low-pass filter h2 and a dual high-pass filter g2. If z(x) denotes the signal obtained from x by inserting a zero between every sample, this can be written as:

a2 [n] = z(a1) * h2 [n] + z(d1) * g2 [n] .

The following figure illustrates the decomposition and reconstruction process. The filter bank is said to be a perfect reconstruction filter bank when a2 = a0 . If, additionally, h = h2 and g = g2, the filters are called conjugate mirror filters.

## Caracterization

Perfect reconstruction filter banks are caracterized in a theorem by Vetterli. When the filters have a finite impulse response, the g and g2 filters can easily be derived from the h and h2 filters, and the filter synthesis is equivalent to solving where h and h2 are trigonometric polynomials.

From filters to wavelets