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| In [[mathematics]], the '''Hartley transform''' is an [[integral transform]] closely related to the [[Fourier transform]], but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the [[Fourier transform]] by [[Ralph Hartley|R. V. L. Hartley]] in 1942, and is one of many known [[List of Fourier-related transforms|Fourier-related transforms]]. 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.
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| The discrete version of the transform, the [[Discrete Hartley transform]], was introduced by [[Ronald N. Bracewell|R. N. Bracewell]] in 1983.
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| The two-dimensional Hartley transform can be computed by an analog optical process similar to an [[Fourier optics|optical Fourier transform]], with the proposed advantage that only its amplitude and sign need to be determined rather than its complex phase (Villasenor, 1994). However, optical Hartley transforms do not seem to have seen widespread use.
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| ==Definition==
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| The Hartley transform of a [[function (mathematics)|function]] ''f''(''t'') is defined by:
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| :<math> | |
| H(\omega) = \left\{\mathcal{H}f\right\}(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty
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| f(t) \, \mbox{cas}(\omega t) \mathrm{d}t,
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| </math>
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| where <math>\omega</math> can in applications be an [[angular frequency]] and
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| :<math>
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| \mbox{cas}(t) = \cos(t) + \sin(t) = \sqrt{2} \sin (t+\pi /4) = \sqrt{2} \cos (t-\pi /4)\,
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| </math>
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| 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).
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| === Inverse transform ===
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| The Hartley transform has the convenient property of being its own inverse (an [[Involution (mathematics)|involution]]):
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| :<math>f = \{\mathcal{H} \{\mathcal{H}f \}\}.</math>
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| === Conventions ===
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| 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:
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| *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—or, indeed, any pair of normalizations whose product is <math>{1}/{2\pi}</math>. (Such asymmetrical normalizations are sometimes found in both purely mathematical and engineering contexts.)
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| *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.
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| *One can use cos−sin instead of cos+sin as the kernel.
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| ==Relation to Fourier transform==
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| This transform differs from the classic Fourier transform
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| <math>F(\omega) = \mathcal{F} \{ f(t) \}(\omega)</math> in the choice of the kernel. In the Fourier transform, we have the exponential kernel:
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| <math>\exp\left({-i\omega t}\right) = \cos(\omega t) - i \sin(\omega t),</math>
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| where ''i'' is the [[imaginary number|imaginary unit]].
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| 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:
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| :<math>F(\omega) = \frac{H(\omega) + H(-\omega)}{2} - i \frac{H(\omega) - H(-\omega)}{2}.</math>
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| 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.
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| Conversely, for real-valued functions ''f''(''t''), the Hartley transform is given from the Fourier transform's real and imaginary parts:
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| :<math>\{ \mathcal{H} f \} = \Re \{ \mathcal{F}f \} - \Im \{ \mathcal{F}f \} = \Re \{ \mathcal{F}f \cdot (1+i) \}</math>
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| where <math>\Re</math> and <math>\Im</math> denote the real and imaginary parts of the complex Fourier transform.
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| == Properties ==
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| 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]]).
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| There is also an analogue of the [[convolution theorem]] for the Hartley transform. If two functions <math>x(t)</math> and <math>y(t)</math> have Hartley transforms <math>X(\omega)</math> and <math>Y(\omega)</math>, respectively, then their [[convolution]] <math>z(t) = x * y</math> has the Hartley transform:
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| :<math>Z(\omega) = \{ \mathcal{H} (x * y) \} = \sqrt{2\pi} \left( X(\omega) \left[ Y(\omega) + Y(-\omega) \right]
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| + X(-\omega) \left[ Y(\omega) - Y(-\omega) \right] \right) / 2.</math>
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| Similar to the Fourier transform, the Hartley transform of an even/odd function is even/odd, respectively.
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| === cas ===
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| The properties of the ''cas'' function follow directly from [[trigonometry]], and its definition as a phase-shifted trigonometric function <math>\mbox{cas}(t)=\sqrt{2} \sin (t+\pi /4)</math>. For example, it has an angle-addition identity of:
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| :<math>
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| 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). \,
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| </math>
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| Additionally:
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| :<math>
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| \mbox{cas} (a+b) = \cos (a) \mbox{cas} (b) + \sin (a) \mbox{cas} (-b) = \cos (b) \mbox{cas} (a) + \sin (b) \mbox{cas}(-a) \,
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| </math>
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| and its derivative is given by:
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| :<math>
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| \mbox{cas}'(a) = \frac{\mbox{d}}{\mbox{d}a} \mbox{cas} (a) = \cos (a) - \sin (a) = \mbox{cas}(-a).
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| </math>
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| == References ==
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| * {{cite journal | authorlink = Ralph Hartley | last = Hartley | first = Ralph V. L. | url = http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1694454 | 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 }}
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| * {{cite book | authorlink = Ronald N. Bracewell | last = Bracewell | first = Ronald N. | title = The Fourier Transform and Its Applications | publisher = McGraw-Hill | origyear = 1965 | edition = 2nd | year = 1978, revised 1986 | isbn = 9780070070158 }} ''(also translated into Japanese and Polish)''
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| * {{cite book | authorlink = Ronald N. Bracewell | last = Bracewell | first = Ronald N. | title = The Hartley Transform | publisher = Oxford University Press | year = 1986 | isbn = 9780195039696 }} ''(also translated into German and Russian)''
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| * {{cite journal | authorlink = Ronald N. Bracewell | last = Bracewell | 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 | year = 1994 }}
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| * {{cite journal | last = Millane | first = R. 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 | year = 1994 }}
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| * {{cite journal | last = Villasenor | first = John D. | doi = 10.1109/5.272144 | title = Optical Hartley transforms | journal = [[Proceedings of the IEEE]] | volume = 82 | issue = 3 | pages = 391–399 | year = 1994 }}
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| [[Category:Integral transforms]]
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| [[Category:Fourier analysis]]
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