In the
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
of
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, the harmonic wavelet transform, introduced by
David Edward Newland in 1993, is a
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
-based linear transformation of a given function into a
time-frequency representation. It combines advantages of the
short-time Fourier transform
The short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divid ...
and the
continuous wavelet transform
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
. It can be expressed in terms of repeated
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
s, and its discrete analogue can be computed efficiently using a
fast Fourier transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in th ...
algorithm.
Harmonic wavelets
The transform uses a family of "harmonic" wavelets indexed by two integers ''j'' (the "level" or "order") and ''k'' (the "translation"), given by
, where
:
These functions are orthogonal, and their Fourier transforms are a square
window function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the inte ...
(constant in a certain octave band and zero elsewhere). In particular, they satisfy:
:
:
where "*" denotes
complex conjugation
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
and
is
Kronecker's delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
.
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''). Hence, when they are used as a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
for expanding an arbitrary function, they represent behaviors of the function on different timescales (and at different time offsets for different ''k'').
However, it is possible to combine all of the negative orders (''j'' < 0) 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
(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''),
and
so one can cut the number of independent expansion coefficients in half.
This expansion has the property, analogous to
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates ...
, 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 (discrete ''t''), where it can exploit
fast Fourier transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in th ...
algorithms.
References
*
*
* {{cite book , editor-last1=Boashash , editor-first1=Boualem , date=2003 , title=Time Frequency Signal Analysis and Processing: A Comprehensive Reference , publisher=
Elsevier
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', th ...
, isbn=0-08-044335-4
Time–frequency analysis
Transforms
Wavelets