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Concept# Wavelet transform

Summary

In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.
A function is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space of square integrable functions.
The Hilbert basis is constructed as the family of functions by means of dyadic translations and dilations of ,
for integers .
If under the standard inner product on ,
this family is orthonormal, it is an orthonormal system:
where is the Kronecker delta.
Completeness is satisfied if every function may be expanded in the basis as
with convergence of the series understood to be convergence in norm. Such a representation of f is known as a wavelet series. This implies that an orthonormal wavelet is self-dual.
The integral wavelet transform is the integral transform defined as
The wavelet coefficients are then given by
Here, is called the binary dilation or dyadic dilation, and is the binary or dyadic position.
The fundamental idea of wavelet transforms is that the transformation should allow only changes in time extension, but not shape. This is achieved by choosing suitable basis functions that allow for this. Changes in the time extension are expected to conform to the corresponding analysis frequency of the basis function. Based on the uncertainty principle of signal processing,
where represents time and angular frequency (, where is ordinary frequency).
The higher the required resolution in time, the lower the resolution in frequency has to be. The larger the extension of the analysis windows is chosen, the larger is the value of .
When is large,
Bad time resolution
Good frequency resolution
Low frequency, large scaling factor
When is small
Good time resolution
Bad frequency resolution
High frequency, small scaling factor
In other words, the basis function can be regarded as an impulse response of a system with which the function has been filtered.

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