# 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 a_{0} with a low-pass filter
h_{1}[n] = h[-n] and a high-pass filter
g_{1}[n] = g[-n] and then subsamples the
output:

a_{1} [n] = a_{0} *
h_{1} [2n]

and

d_{1} [n] = a_{0}* g_{1} [2n]
.
A reconstructed signal a_{2} is obtained
by filtering the zero expanded signals with a dual low-pass filter
h_{2} and a dual high-pass filter g_{2}. If z(x)
denotes the signal obtained from x by inserting a zero between every
sample, this can be written as:

a_{2} [n] = z(a_{1})
* h_{2} [n] + z(d_{1}) * g_{2}
[n] .
The following
figure illustrates the decomposition and reconstruction
process.

The filter bank is said to be a *perfect
reconstruction filter bank* when a_{2} = a_{0} .
If, additionally, h = h_{2} and g = g_{2}, 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 g_{2} filters can easily
be derived from the h and h_{2} filters, and the filter
synthesis is equivalent to solving

where h and h_{2} are trigonometric
polynomials.

From
filters to wavelets