Newland Transform
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In the
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
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, Scalar potential, potential fields, Seismic tomograph ...
, the harmonic wavelet transform, introduced by
David Edward Newland David (; , "beloved one") was a king of ancient Israel and Judah and the third king of the United Monarchy, according to the Hebrew Bible and Old Testament. The Tel Dan stele, an Aramaic-inscribed stone erected by a king of Aram-Damas ...
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 n ...
-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 divide ...
and the
continuous wavelet transform In mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. Definition ...
. It can be expressed in terms of repeated
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
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). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
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 w(2^j t - k) \!, where :w(t) = \frac . 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. Typically, window functions are symmetric around ...
(constant in a certain octave band and zero elsewhere). In particular, they satisfy: :\int_^\infty w^*(2^j t - k) \cdot w(2^ t - k') \, dt = \frac \delta_ \delta_ :\int_^\infty w(2^j t - k) \cdot w(2^ t - k') \, dt = 0 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 numbers, then the complex conjugate of a + bi is a - ...
and \delta 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 &\te ...
. 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 \varphi(t - k) where :\varphi(t) = \frac. The function ''φ'' is orthogonal to itself for different ''k'' and is also orthogonal to the wavelet functions for non-negative ''j'': :\int_^\infty \varphi^*(t - k) \cdot \varphi(t - k') \, dt = \delta_ :\int_^\infty w^*(2^j t - k) \cdot \varphi(t - k') \, dt = 0\textj \geq 0 :\int_^\infty \varphi(t - k) \cdot \varphi(t - k') \, dt = 0 :\int_^\infty w(2^j t - k) \cdot \varphi(t - k') \, dt = 0\textj \geq 0. In the harmonic wavelet transform, therefore, an arbitrary real- or complex-valued function f(t) (in L2) is expanded in the basis of the harmonic wavelets (for all integers ''j'') and their complex conjugates: :f(t) = \sum_^\infty \sum_^\infty \left a_ w(2^j t - k) + \tilde_ w^*(2^j t - k)\right or alternatively in the basis of the wavelets for non-negative ''j'' supplemented by the scaling functions ''φ'': :f(t) = \sum_^\infty \left a_k \varphi(t - k) + \tilde_k \varphi^*(t - k) \right+ \sum_^\infty \sum_^\infty \left a_ w(2^j t - k) + \tilde_ w^*(2^j t - k)\right. The expansion coefficients can then, in principle, be computed using the orthogonality relationships: : \begin a_ & = 2^j \int_^\infty f(t) \cdot w^*(2^j t - k) \, dt \\ \tilde_ & = 2^j \int_^\infty f(t) \cdot w(2^j t - k) \, dt \\ a_k & = \int_^\infty f(t) \cdot \varphi^*(t - k) \, dt \\ \tilde_k & = \int_^\infty f(t) \cdot \varphi(t - k) \, dt. \end For a real-valued function ''f''(''t''), \tilde_ = a_^* and \tilde_k = a_k^* 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: : \begin & \sum_^\infty \sum_^\infty 2^ \left( , a_, ^2 + , \tilde_, ^2 \right) \\ & = \sum_^\infty \left( , a_k, ^2 + , \tilde_k, ^2 \right) + \sum_^\infty \sum_^\infty 2^ \left( , a_, ^2 + , \tilde_, ^2 \right) \\ & = \int_^\infty , f(x), ^2 \, dx. \end 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). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
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 (journal), Cell'', the ScienceDirect collection of electronic journals, ...
, isbn=0-08-044335-4 Time–frequency analysis Transforms Wavelets