Hartley transformation

The Hartley transformation , abbreviated HT , is in functional analysis - a branch of mathematics  - a linear integral transformation with reference to the Fourier transformation and, like this, a frequency transformation . In contrast to the complex Fourier transform, the Hartley transform is a real transform. It is named after Ralph Hartley , who introduced it in 1942.

The Hartley transformation also exists in discrete form, the discrete Hartley transformation, abbreviated DHT, which is used in digital signal processing and image processing . This form was published by RNBracewell in 1994.

definition

The Hartley transform of a function f ( t ) is defined as:

${\displaystyle H(\omega )={\mathcal {H}}(f)(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)\,{\mbox{cas}}(\omega t)\mathrm {d} t}$

with the angular frequency ω and the abbreviation:

${\displaystyle {\mbox{cas}}(t)=\cos(t)+\sin(t)={\sqrt {2}}\sin(t+\pi /4)={\sqrt {2}}\cos(t-\pi /4)}$

which is referred to as the "Hartley core".

In the literature there are also different definitions regarding the factor , which normalize this factor to 1 and the factor occurs in the inverse Hartley transformation . ${\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}$ ${\displaystyle {\tfrac {1}{2\pi }}}$

Inverse transformation

The Hartley transformation is inverse to itself as defined above, making it an involutive transformation:

${\displaystyle f={\mathcal {H}}({\mathcal {H}}(f))}$

Relation to the Fourier transform

The Fourier transform

${\displaystyle F(\omega )={\mathcal {F}}(f)(\omega )}$

differs due to its complex core:

${\displaystyle \exp \left({-\mathrm {i} \omega t}\right)=\cos(\omega t)-\mathrm {i} \sin(\omega t)}$

with the imaginary unit from the purely real core of the Hartley transformation. With the appropriate choice of normalization factors, the Fourier transformation can be calculated directly from the Hartley transformation: ${\displaystyle i}$ ${\displaystyle \operatorname {cas} (\omega t)}$

${\displaystyle F(\omega )={\color {darkred}{\sqrt {2\pi }}}\left({\frac {H(\omega )+H(-\omega )}{2}}-\mathrm {i} {\frac {H(\omega )-H(-\omega )}{2}}\right)}$

The red correction factor disappears when using the alternative definition mentioned above without ${\displaystyle {\sqrt {2\pi }}}$ ${\displaystyle {\frac {1}{\sqrt {2\pi }}}}$

The real or imaginary part of the Fourier transformation is formed by the even and odd parts of the Hartley transformation.

Hartley Core Relationships

For the "Hartley kernel" , the following relationships can be derived from the trigonometric functions: ${\displaystyle {\mbox{cas}}(t)}$

The addition theorem:

${\displaystyle 2{\mbox{cas}}(a+b)={\mbox{cas}}(a){\mbox{cas}}(b)+{\mbox{cas}}(-a){\mbox{cas}}(b)+{\mbox{cas}}(a){\mbox{cas}}(-b)-{\mbox{cas}}(-a){\mbox{cas}}(-b)\,}$

and

${\displaystyle {\mbox{cas}}(a+b)=\cos(a){\mbox{cas}}(b)+\sin(a){\mbox{cas}}(-b)=\cos(b){\mbox{cas}}(a)+\sin(b){\mbox{cas}}(-a)\,}$

The derivation is given as:

${\displaystyle {\frac {{\mbox{d cas}}(a)}{{\mbox{d }}a}}=\cos(a)-\sin(a)={\mbox{cas}}(-a)}$

literature

• Bernd Jähne : Digital image processing . 6th edition. Springer, 2005, ISBN 3-540-24999-0 .
• Ronald Newbold Bracewell: The Hartley Transform . 1st edition. Oxford University Press, 1986, ISBN 0-19-503969-6 .

Individual evidence

1. ↑ Ralph Hartley: A more symmetrical Fourier analysis applied to transmission problems . In: Institute of Radio Engineers (Ed.): Proceedings of the IRE . tape 30 , no. 3 , March 1942, ISSN  0096-8390 , p. 144-150 ( IEEE Xplore Digital Library [accessed August 25, 2010]).
2. ^ RN Bracewell: Aspects of the Hartley transform . In: Proceedings of the IRE . No. 82 (3) , 1994, doi : 10.1109 / 5.272142 .