Editing Hartley transform

{{short description|Integral transform closely related to the Fourier transform}}
In [[mathematics]], the '''Hartley transform''' ('''HT''') is an [[integral transform]] closely related to the [[Fourier transform]] (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,<ref name="Hartley_1942"/> and is one of many known [[Fourier-related transform]]s. Compared to the Fourier transform, the Hartley transform has the advantages of transforming [[real number|real]] functions to real functions (as opposed to requiring [[complex number]]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.<ref name="Bracewell_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.<ref name="Villasenor_1994"/> However, optical Hartley transforms<!-- apparently NOT abbreviated as OHT, as OHT used for something else --> do not seem to have seen widespread use.

==Definition==
The Hartley transform of a [[function (mathematics)|function]] <math>f(t)</math> is defined by:

<math display=block>
H(\omega) = \left\{\mathcal{H}f\right\}(\omega) =  \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty
f(t) \operatorname{cas}(\omega t) \, \mathrm{d}t\,,
</math>

where <math>\omega</math> can in applications be an [[angular frequency]] and

<math display=block>
\operatorname{cas}(t) = \cos(t) + \sin(t) = \sqrt{2} \sin (t+\pi /4) = \sqrt{2} \cos (t-\pi /4)\,,
</math>

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 (mathematics)|involution]]):

<math display=block>f = \{\mathcal{H} \{\mathcal{H}f \}\}\,.</math>

=== 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 <math>{1}/{\sqrt{2\pi}}</math> from the forward transform and use <math>{1}/{2\pi}</math> for the inverse&mdash;or, indeed, any pair of normalizations whose product is {{nowrap|1=<math>{1}/{2\pi}</math>.}} (Such asymmetrical normalizations are sometimes found in both purely mathematical and engineering contexts.)
*One can also use <math>2\pi\nu t</math> instead of <math>\omega t</math> (i.e., frequency instead of angular frequency), in which case the <math>{1}/{\sqrt{2\pi}}</math> coefficient is omitted entirely.
*One can use <math>\cos-\sin</math> instead of <math>\cos+\sin</math> as the kernel.

==Relation to Fourier transform==
This transform differs from the classic Fourier transform 
<math>F(\omega) = \mathcal{F} \{ f(t) \}(\omega)</math> in the choice of the kernel. In the Fourier transform, we have the exponential kernel,
{{nowrap|1=<math>\exp\left({-\mathrm{i}\omega t}\right) = \cos(\omega t) - \mathrm{i} \sin(\omega t)</math>,}}
where <math>\mathrm{i}</math> is the [[imaginary unit]].

The two transforms are closely related, however, and the Fourier transform (assuming it uses the same <math>1/\sqrt{2\pi}</math> normalization convention) can be computed from the Hartley transform via:

<math display=block>F(\omega) = \frac{H(\omega) + H(-\omega)}{2} - \mathrm{i} \frac{H(\omega) - H(-\omega)}{2}\,.</math>

That is, the real and imaginary parts of the Fourier transform are simply given by the [[even and odd functions|even and odd]] parts of the Hartley transform, respectively.

Conversely, for real-valued functions {{nowrap|1=<math>f(t)</math>,}} the Hartley transform is given from the Fourier transform's real and imaginary parts:

<math display=block>\{ \mathcal{H} f \} = \Re \{ \mathcal{F}f \} - \Im \{ \mathcal{F}f \} = \Re \{ \mathcal{F}f \cdot (1+\mathrm{i}) \}\,,</math>

where <math>\Re</math> and <math>\Im</math> denote the real and imaginary parts.

== Properties ==
The Hartley transform is a real [[linear operator]], and is [[symmetric matrix|symmetric]] (and [[Hermitian operator|Hermitian]]).  From the symmetric and self-inverse properties, it follows that the transform is a [[unitary operator]] (indeed, [[orthogonal matrix|orthogonal]]).

[[Convolution]] using Hartley transforms is<ref>{{cite book |author1=Olejniczak |editor1-last=Poularikas |title=Transforms and Applications Handbook |date=2010 |publisher=CRC Press |edition=3rd |chapter=Hartley Transform}} Equation (4.54)</ref>
<math display=block>
f(x) * g(x) = \frac{F(\omega) G(\omega) + F(-\omega) G(\omega) + F(\omega) G(-\omega) - F(-\omega) G(-\omega)}{2}
</math>
where <math>F(\omega) = \{\mathcal{H}f\}(\omega)</math> and <math>G(\omega) = \{\mathcal{H} g\}(\omega)</math>

Similar to the Fourier transform, the Hartley transform of an even/odd function is even/odd, respectively.

=== cas ===
<!-- Section header used in redirects -->
The properties of the ''Hartley kernel'', for which Hartley introduced the name ''cas'' for the function (from ''cosine and sine'') in 1942,<ref name="Hartley_1942"/><ref name="Bracewell_1999"/> follow directly from [[trigonometry]], and its definition as a phase-shifted trigonometric function {{nowrap|1=<math>\operatorname{cas}(t)=\sqrt{2} \sin (t+\pi /4)=\sin(t)+\cos(t)</math>.}} For example, it has an angle-addition identity of:

<math display=block>
2 \operatorname{cas} (a+b) = \operatorname{cas}(a) \operatorname{cas}(b) + \operatorname{cas}(-a) \operatorname{cas}(b) + \operatorname{cas}(a) \operatorname{cas}(-b) - \operatorname{cas}(-a) \operatorname{cas}(-b)\,.
</math>

Additionally:

<math display=block> 
\operatorname{cas} (a+b) = {\cos (a) \operatorname{cas} (b)} + {\sin (a) \operatorname{cas} (-b)} = \cos (b) \operatorname{cas} (a) + \sin (b) \operatorname{cas}(-a)\,,
</math>

and its derivative is given by:

<math display=block>
\operatorname{cas}'(a) = \frac{d}{da} \operatorname{cas} (a) = \cos (a) - \sin (a) = \operatorname{cas}(-a)\,.
</math>

== See also ==
* [[cis (mathematics)]]
*[[Fractional Fourier transform]]

== References ==
{{reflist|refs=
<ref name="Hartley_1942">{{cite journal |author-link=Ralph Hartley |author-last=Hartley |author-first=Ralph V. L. |url=https://www.researchgate.net/publication/3468825 |title=A More Symmetrical Fourier Analysis Applied to Transmission Problems |journal=[[Proceedings of the IRE]] |volume=30 |issue=3 |pages=144–150 |date=March 1942 |doi=10.1109/JRPROC.1942.234333 |s2cid=51644127 |archive-date=2019-04-05 |access-date=2017-10-31 |archive-url=https://web.archive.org/web/20190405073552/https://www.researchgate.net/publication/3468825_A_More_Symmetrical_Fourier_Analysis_Applied_to_Transmission_Problems |url-status=live }}</ref>
<ref name="Bracewell_1983">{{cite journal |author-link=Ronald N. Bracewell |author-last=Bracewell |author-first=Ronald N. |title=Discrete Hartley transform |journal=[[Journal of the Optical Society of America]] |volume=73 |issue=12 |pages=1832&ndash;1835 |date=1983|doi=10.1364/JOSA.73.001832 |bibcode=1983JOSA...73.1832B |s2cid=120611904 }}</ref>
<ref name="Villasenor_1994">{{cite journal |author-last=Villasenor |author-first=John D. |doi=10.1109/5.272144 |title=Optical Hartley transforms |journal=[[Proceedings of the IEEE]] |volume=82 |issue=3 |pages=391–399 |date=1994}}</ref>
<ref name="Bracewell_1999">{{cite book |author-link=Ronald N. Bracewell |author-last=Bracewell |author-first=Ronald N. |title=The Fourier Transform and Its Applications |publisher=[[McGraw-Hill]] |edition=3 |orig-year=1985, 1978, 1965 |date=June 1999 |isbn=978-0-07303938-1}} (NB. Second edition also translated into Japanese and Polish.<!-- Unknown for third edition -->)</ref>
}}
* {{cite book |author-link=Ronald N. Bracewell |author-last=Bracewell |author-first=Ronald N. |title=The Hartley Transform |publisher=[[Oxford University Press, Inc.]] |location=Stanford, California, USA |publication-place=New York, NY, USA |series=Oxford Engineering Science Series |volume=19 |date=1986 |edition=1 |isbn=0-19-503969-6}} (NB. Also translated into German and Russian.)
* {{cite journal |author-link=Ronald N. Bracewell |author-last=Bracewell |author-first=Ronald N. |doi=10.1109/5.272142 |title=Aspects of the Hartley transform |journal=[[Proceedings of the IEEE]] |volume=82 |issue=3 |pages=381–387 |date=1994}}
* {{cite journal |author-last=Millane |author-first=Rick P. |authorlink1=Rick Millane |doi=10.1109/5.272146 |title=Analytic properties of the Hartley transform |journal=[[Proceedings of the IEEE]] |volume=82 |issue=3 |pages=413–428 |date=1994}}

== 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 |publisher=[[Proceedings of the IEEE]] |volume=82 |issue=3 |pages=372–380<!-- 447? --> |date=March 1994 |chapter-url=https://ieeexplore.ieee.org/xpl/tocresult.jsp?reload=true&isnumber=6725 |access-date=2017-10-31 }} (NB. Contains extensive bibliography.)

[[Category:Integral transforms]]
[[Category:Fourier analysis]]