An integral transform which shares some features with the Fourier transform, but which (in the discrete case), multiplies the integral kernel by

(1)

instead of

(2)

The Hartley transform produces real output for a real input, and is its own inverse. It therefore can have computational advantages over the discrete Fourier transform, although analytic expressions are usually more complicated for the Hartley transform.

The discrete version of the Hartley transform can be written explicitly as

			(3)
			(4)

where denotes the Fourier transform. The Hartley transform obeys the convolution property

(5)

where

			(6)
			(7)
			(8)

Like the fast Fourier transform, there is a "fast" version of the Hartley transform. A decimation in time algorithm makes use of

			(9)
			(10)

where denotes the sequence with elements

(11)

A decimation in frequency algorithm makes use of

			(12)
			(13)

The discrete Fourier transform

(14)

can be written

			(15)
			(16)

so

(17)

Arndt, J. "The Hartley Transform (HT)." Ch. 2 in "Remarks on FFT Algorithms." http://www.jjj.de/fxt/.

Bracewell, R. N. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 1999.

Bracewell, R. N. The Hartley Transform. New York: Oxford University Press, 1986.

CITE THIS AS:

Weisstein, Eric W. "Hartley Transform." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HartleyTransform.html