- Definition Inverse transform Conventions
- Relation to Fourier transform
- Properties cas
- See also
- References
- Further reading
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, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming real functions to real functions (as opposed to requiring complex numbers) 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.[2]
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.[3] However, optical Hartley transforms do not seem to have seen widespread use.
Definition
The Hartley transform of a function f(t) is defined by:
where 
can in applications be an angular frequency and
is the cosine-and-sine 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):
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
instead of 
(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
in the choice of the kernel. In the Fourier transform, we have the exponential kernel:
where i is the imaginary unit.
The two transforms are closely related, however, and the Fourier transform (assuming it uses the same 
normalization convention) can be computed from the Hartley transform via:
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 f(t), the Hartley transform is given from the Fourier transform's real and imaginary parts:
where 
and 
denote the real and imaginary parts of the complex Fourier transform.
Properties
The Hartley transform is a real linear operator, and is symmetric (and Hermitian). From the symmetric and self-inverse properties, it follows that the transform is a unitary operator (indeed, orthogonal).
There is also an analogue of the convolution theorem for the Hartley transform. If two functions 
and 
have Hartley transforms 
and 
, respectively, then their convolution 
has the Hartley transform{{citation needed|date=January 2018}}:
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 function (from cosine and sine) in 1942,[5] follow directly from trigonometry, and its definition as a phase-shifted trigonometric function 
. For example, it has an angle-addition identity of:
Additionally:
and its derivative is given by:
See also
- cis (mathematics)
- Fractional Fourier transform
References
2. ^1 {{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}}
3. ^1 {{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.)
}}
- {{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. |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 |journal=Proceedings of the IEEE |volume=82 |issue=3 |pages=372–380 |date=March 1994 |url=http://ieeexplore.ieee.org/xpl/tocresult.jsp?reload=true&isnumber=6725 |access-date=2017-10-31 |dead-url=no}} (NB. Contains extensive bibliography.)