Hartley transform explained

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.

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

f(t) is defined by:

$$H(\backslash omega)\; =\; \backslash left\backslash (\backslash omega)\; =\; \backslash frac\backslash int\_^\backslash inftyf(t)\; \backslash operatorname(\backslash omega\; t)\; \backslash ,\; \backslash mathrmt\backslash ,,$$

where

\omega can in applications be an angular frequency and

$$\backslash operatorname(t)\; =\; \backslash cos(t)\; +\; \backslash sin(t)\; =\; \backslash sqrt\; \backslash sin\; (t+\backslash pi\; /4)\; =\; \backslash sqrt\; \backslash cos\; (t-\backslash pi\; /4)\backslash ,,$$

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):

$$f\; =\; \backslash \backslash ,.$$

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

{ 1 }/{\sqrt{ 2 \pi}} from the forward transform and use { 1 }/{ 2 \pi} 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 { 1 }/{\sqrt{ 2 \pi}} 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) = l{F} \{ f(t) \}(\omega) 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

1 /\sqrt{ 2 \pi} normalization convention) can be computed from the Hartley transform via:

$$F(\backslash omega)\; =\; \backslash frac\; -\; \backslash mathrm\; \backslash frac\backslash ,.$$

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:

$$\backslash \; =\; \backslash Re\; \backslash \; -\; \backslash Im\; \backslash \; =\; \backslash Re\; \backslash \backslash ,,$$

where

\Re and \Im denote the real and imaginary parts.

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).

Convolution using Hartley transforms is^[1] $$f(x)\; *\; g(x)\; =\; \backslash frac$$where

F(\omega) = \{ l{H}f\}(\omega) and G(\omega) = \{ l{H} g\}(\omega)

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, and its definition as a phase-shifted trigonometric function For example, it has an angle-addition identity of:

$$2\; \backslash operatorname\; (a+b)\; =\; \backslash operatorname(a)\; \backslash operatorname(b)\; +\; \backslash operatorname(-a)\; \backslash operatorname(b)\; +\; \backslash operatorname(a)\; \backslash operatorname(-b)\; -\; \backslash operatorname(-a)\; \backslash operatorname(-b)\backslash ,.$$

Additionally:

$$\backslash operatorname\; (a+b)\; =\; +\; =\; \backslash cos\; (b)\; \backslash operatorname\; (a)\; +\; \backslash sin\; (b)\; \backslash operatorname(-a)\backslash ,,$$

and its derivative is given by:

$$\backslash operatorname\text{'}(a)\; =\; \backslash frac\; \backslash operatorname\; (a)\; =\; \backslash cos\; (a)\; -\; \backslash sin\; (a)\; =\; \backslash operatorname(-a)\backslash ,.$$

See also

• cis (mathematics)
• Fractional Fourier transform

References

• Book: Ronald N. Bracewell . Bracewell . Ronald N. . The Hartley Transform . . Stanford, California, USA . New York, NY, USA . Oxford Engineering Science Series . 19 . 1986 . 1 . 0-19-503969-6. (NB. Also translated into German and Russian.)
• Ronald N. Bracewell . Bracewell . Ronald N. . 10.1109/5.272142 . Aspects of the Hartley transform . . 82 . 3 . 381–387 . 1994.
• Millane . Rick P. . Rick Millane . 10.1109/5.272146 . Analytic properties of the Hartley transform . . 82 . 3 . 413–428 . 1994.

Further reading

• Book: Kraig J. . Olnejniczak . Gerald T. . Heydt . Scanning the Special Section on the Hartley transform . Special Issue on Hartley transform . . 82 . 3 . 372–380 . March 1994 . https://ieeexplore.ieee.org/xpl/tocresult.jsp?reload=true&isnumber=6725 . 2017-10-31 . (NB. Contains extensive bibliography.)

Notes and References

1. Book: Olejniczak . Poularikas . Transforms and Applications Handbook . 2010 . CRC Press . 3rd . Hartley Transform. Equation (4.54)