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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 ...
, the Hartley transform (HT) is an
integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
closely related to the
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 ...
(FT), but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by Ralph V. L. Hartley in 1942, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
functions to real functions (as opposed to requiring
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s) and of being its own inverse. The discrete version of the transform, the discrete Hartley transform (DHT), was introduced by Ronald N. Bracewell in 1983. The two-dimensional Hartley transform can be computed by an analog optical process similar to an optical Fourier transform (OFT), with the proposed advantage that only its amplitude and sign need to be determined rather than its complex phase. However, optical Hartley transforms do not seem to have seen widespread use.


Definition

The Hartley transform of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
f(t) is defined by: H(\omega) = \left\(\omega) = \frac\int_^\infty f(t) \operatorname(\omega t) \, \mathrmt\,, where \omega can in applications be an
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
and \operatorname(t) = \cos(t) + \sin(t) = \sqrt \sin (t+\pi /4) = \sqrt \cos (t-\pi /4)\,, is the cosine-and-sine (cas) or ''Hartley'' kernel. In engineering terms, this transform takes a signal (function) from the time-domain to the Hartley spectral domain (frequency domain).


Inverse transform

The Hartley transform has the convenient property of being its own inverse (an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
): f = \\,.


Conventions

The above is in accord with Hartley's original definition, but (as with the Fourier transform) various minor details are matters of convention and can be changed without altering the essential properties: *Instead of using the same transform for forward and inverse, one can remove the / from the forward transform and use / for the inverse—or, indeed, any pair of normalizations whose product is (Such asymmetrical normalizations are sometimes found in both purely mathematical and engineering contexts.) *One can also use 2\pi\nu t instead of \omega t (i.e., frequency instead of angular frequency), in which case the / coefficient is omitted entirely. *One can use \cos-\sin instead of \cos+\sin as the kernel.


Relation to Fourier transform

This transform differs from the classic Fourier transform F(\omega) = \mathcal \(\omega) in the choice of the kernel. In the Fourier transform, we have the exponential kernel, where \mathrm is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. The two transforms are closely related, however, and the Fourier transform (assuming it uses the same 1/\sqrt normalization convention) can be computed from the Hartley transform via: F(\omega) = \frac - \mathrm \frac\,. That is, the real and imaginary parts of the Fourier transform are simply given by the even and odd parts of the Hartley transform, respectively. Conversely, for real-valued functions the Hartley transform is given from the Fourier transform's real and imaginary parts: \ = \Re \ - \Im \ = \Re \\,, where \Re and \Im denote the real and imaginary parts.


Properties

The Hartley transform is a real
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
, and is
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
(and
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
). From the symmetric and self-inverse properties, it follows that the transform is a
unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...
(indeed,
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
). There is also an analogue of the
convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...
for the Hartley transform. If two functions x(t) and y(t) have Hartley transforms X(\omega) and respectively, then their
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
z(t) = x * y has the Hartley transform: Z(\omega) = \ = \sqrt \left( X(\omega) \left Y(\omega) + Y(-\omega) \right + X(-\omega) \left Y(\omega) - Y(-\omega) \right\right) / 2\,. Similar to the Fourier transform, the Hartley transform of an even/odd function is even/odd, respectively.


cas

The properties of the ''Hartley kernel'', for which Hartley introduced the name ''cas'' for the function (from ''cosine and sine'') in 1942, follow directly from
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
, and its definition as a phase-shifted trigonometric function For example, it has an angle-addition identity of: 2 \operatorname (a+b) = \operatorname(a) \operatorname(b) + \operatorname(-a) \operatorname(b) + \operatorname(a) \operatorname(-b) - \operatorname(-a) \operatorname(-b)\,. Additionally: \operatorname (a+b) = + = \cos (b) \operatorname (a) + \sin (b) \operatorname(-a)\,, and its derivative is given by: \operatorname'(a) = \frac \operatorname (a) = \cos (a) - \sin (a) = \operatorname(-a)\,.


See also

*
cis (mathematics) is a mathematical notation defined by , where is the cosine function, is the imaginary unit and is the sine function. The notation is less commonly used in mathematics than Euler's formula, which offers an even shorter notation for but cis ...
*
Fractional Fourier transform In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the ''n''-th power, where ''n'' n ...


References

* (NB. Also translated into German and Russian.) * *


Further reading

* {{cite book , editor-first1=Kraig J. , editor-last1=Olnejniczak , editor-first2=Gerald T. , editor-last2=Heydt , chapter=Scanning the Special Section on the Hartley transform , title=Special Issue on Hartley transform , journal=
Proceedings of the IEEE The ''Proceedings of the IEEE'' is a monthly peer-reviewed scientific journal published by the Institute of Electrical and Electronics Engineers (IEEE). The journal focuses on electrical engineering and computer science. According to the ''Jou ...
, volume=82 , issue=3 , pages=372–380 , date=March 1994 , chapter-url=http://ieeexplore.ieee.org/xpl/tocresult.jsp?reload=true&isnumber=6725 , access-date=2017-10-31 , url-status=live (NB. Contains extensive bibliography.) Integral transforms Fourier analysis