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 can be expressed in terms of repeated Fourier transforms, and its discrete analogue can be computed efficiently using a fast Fourier transform algorithm.
The transform uses a family of "harmonic" wavelets indexed by two integers j (the "level" or "order") and k (the "translation"), given by
As the order j increases, these wavelets become more localized in Fourier space (frequency) and in higher frequency bands, and conversely become less localized in time (t).
However, it is possible to combine all of the negative orders (j < 0) together into a single family of "scaling" functions
(in L2) is expanded in the basis of the harmonic wavelets (for all integers j) 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(t),
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.