Harmonic wavelet transform


In the mathematics of signal processing, the harmonic wavelet transform, introduced by David Edward Newland in 1993, is a wavelet-based linear transformation of a given function into a time-frequency representation. It combines advantages of the short-time Fourier transform and the continuous wavelet transform. It can be expressed in terms of repeated Fourier transforms, and its discrete analogue can be computed efficiently using a fast Fourier transform algorithm.

Harmonic wavelets

The transform uses a family of "harmonic" wavelets indexed by two integers j and k, given by, where
These functions are orthogonal, and their Fourier transforms are a square window function. In particular, they satisfy:
where "*" denotes complex conjugation and is Kronecker's delta.
As the order j increases, these wavelets become more localized in Fourier space and in higher frequency bands, and conversely become less localized in time. Hence, when they are used as a basis for expanding an arbitrary function, they represent behaviors of the function on different timescales.
However, it is possible to combine all of the negative orders together into a single family of "scaling" functions where
The function φ is orthogonal to itself for different k and is also orthogonal to the wavelet functions for non-negative j:
In the harmonic wavelet transform, therefore, an arbitrary real- or complex-valued function is expanded in the basis of the harmonic wavelets and their complex conjugates:
or alternatively in the basis of the wavelets for non-negative j supplemented by the scaling functions φ:
The expansion coefficients can then, in principle, be computed using the orthogonality relationships:
For a real-valued function f, and so one can cut the number of independent expansion coefficients in half.
This expansion has the property, analogous to Parseval's theorem, that:
Rather than computing the expansion coefficients directly from the orthogonality relationships, however, it is possible to do so using a sequence of Fourier transforms. This is much more efficient in the discrete analogue of this transform, where it can exploit fast Fourier transform algorithms.