Hartley Transform

The Hartley Transform is an integral transform which shares some features with the Fourier transform, but which, in the most common convention, multiplies the integral kernel by

cas(2pinut)=cos(2pinut)+sin(2pinut)

(1)

instead of by e^(-2piift), giving the transform pair

H(f) = [画像:int_(-infty)^inftyV(t)cas(2pift)dt]

(2)

V(t) = [画像:int_(-infty)^inftyH(f)cas(2pift)df]

(3)

(Bracewell 1986, p. 10, Bracewell 1999, p. 179).

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.

In the discrete case, the kernel is multiplied by

[画像: cos((2pikn)/N)+sin((2pikn)/N) ]

(4)

instead of

[画像: e^(-2piikn/N)=cos((2pikn)/N)-isin((2pikn)/N). ]

(5)

The discrete version of the Hartley transform--using an alternate convention with the plus sign replaced by a minus sine can be written explicitly as

H[a] = [画像:1/(sqrt(N))sum_(n=0)^(N-1)a_n[cos((2pikn)/N)-sin((2pikn)/N)]]

(6)

= RF[a]-IF[a],

(7)

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

H[a*b]_k=1/2(A_kB_k-A^__kB^__k+A_kB^__k+A^__kB_k),

(8)

where

a^__0 = a_0

(9)

a^__(n/2) = a_(n/2)

(10)

a^__k = a_(n-k).

(11)

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

H_n^(left)[a] = H_(n/2)[a^(even)]+XH_(n/2)[a^(odd)]

(12)

H_n^(right)[a] = H_(n/2)[a^(even)]-XH_(n/2)[a^(odd)],

(13)

where X denotes the sequence with elements

[画像: a_ncos((pin)/N)-a^__nsin((pin)/N). ]

(14)

A decimation in frequency algorithm makes use of

H_n^(even)[a] = H_(n/2)[a^(left)+a^(right)]

(15)

H_n^(odd)[a] = H_(n/2)[X(a^(left)-a^(right))].

(16)

The discrete Fourier transform

[画像: A_k=F[a]=sum_(n=0)^(N-1)e^(-2piikn/N)a_n ]

(17)

can be written

[画像:[A_k; A_(-k)]] = [画像:sum_(n=0)^(N-1)[e^(-2piikn/N) 0; 0 e^(2piikn/N)]_()_(F)[a_n; a_n]]

(18)

= [画像:sum_(n=0)^(N-1)1/2[1-i 1+i; 1+i 1-i]_()_(T^(-1))[cos((2pikn)/N) sin((2pikn)/N); -sin((2pikn)/N) cos((2pikn)/N)]_()_(H)1/2[1+i 1-i; 1-i 1+i]_()_(T)[a_n; a_n],]

(19)

F=T^(-1)HT.

(20)

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References

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.

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Hartley Transform

Cite this as:

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

Hartley Transform

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