Harmonic wavelet transform
Encyclopedia
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 transform
s, and its discrete analogue can be computed efficiently using a fast Fourier transform
algorithm.
, where
These functions are orthogonal, and their Fourier transforms are a square window function
(constant in a certain octave band and zero elsewhere). 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 (frequency) and in higher frequency bands, and conversely become less localized in time (t). Hence, when they are used as a basis
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 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
, 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
algorithms.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
of signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, the harmonic wavelet transform, introduced by David Edward Newland in 1993, is a wavelet
Wavelet
A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have...
-based linear transformation of a given function into a time-frequency representation
Time-frequency representation
A time–frequency representation is a view of a signal represented over both time and frequency. Time–frequency analysis means analysis into the time–frequency domain provided by a TFR...
. It combines advantages of the short-time Fourier transform
Short-time Fourier transform
The short-time Fourier transform , or alternatively short-term Fourier transform, 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....
and the continuous wavelet transform
Continuous wavelet transform
A continuous wavelet transform is used to divide a continuous-time function into wavelets. Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization...
. It can be expressed in terms of repeated Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
s, and its discrete analogue can be computed efficiently using a fast Fourier transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...
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, whereThese functions are orthogonal, and their Fourier transforms are a square window function
Window function
In signal processing, a window function is a mathematical function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation...
(constant in a certain octave band and zero elsewhere). 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 (frequency) and in higher frequency bands, and conversely become less localized in time (t). Hence, when they are used as a basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
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
whereThe function φ is orthogonal to itself for different k and is also orthogonal to the wavelet functions for non-negative j:
Harmonic wavelet transform
In the harmonic wavelet transform, therefore, an arbitrary real- or complex-valued function (in L2Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
) 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
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...
, 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
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...
algorithms.