# Dyadic Wavelet Transform

Dyadic wavelet transforms are **scale samples** of wavelet
transforms following a geometric sequence
of ratio 2. Time is not sampled.

This transform uses dyadic wavelets.

It is implemented by perfect reconstruction filter banks.

## Definition

The dyadic wavelet transform of f is defined
by

It defines a stable complete representation if its
Heisenberg boxes cover all of the frequency axis, that is, if there
exist A et B such that

The family of dyadic wavelets is a frame of
L^{2}(R).

## Wavelet synthesis

To build dyadic wavelets, it is sufficient to
satisfy the previous condition. To do so, it is possible to proceed
as for the construction of orthogonal and biorthogonal wavelet
bases, using conjugate mirror or perfect reconstruction filter
banks.

The wavelets satisfy then scaling equations and
the fast
dyadic wavelet transform is implemented
using filter banks.

## Implementation

The fast dyadic wavelet
transform uses the same filters as for the
computation of the fast wavelet transform of a discrete signal,
except that no subsampling is performed.

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