TOPICS
Search

Inverse Hilbert Transform

The inverse Hilbert transform is the inverse of the Hilbert transform. The Hilbert transform and inverse Hilbert transform are given by the integral transform pair

g(y)=H[f(x)]
(1)
=1/piPVint_(-infty)^infty(f(x)dx)/(x-y)
(2)
f(x)=H^(-1)[g(y)]
(3)
=-1/piPVint_(-infty)^infty(g(y)dy)/(y-x)
(4)

(Bracewell 1999), where the Cauchy principal value is taken in each of the integrals.

The opposite convention, with denominator y-x in the forward transform, is also common (NIST DLMF, eqn. 1.14.41; King 2009, vol. 2, p. 6). The Wolfram Language functions HilbertTransform[f, x, y] and InverseHilbertTransform[g, y, x] use this opposite sign convention, so their results are the negatives of the corresponding transforms shown above (though the normalization by 1/pi is the same).

See also

Cauchy Principal Value, Fourier Transform, Hilbert Transform, Integral Transform

Explore with Wolfram|Alpha

References

Bracewell, R. "The Hilbert Transform." The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 267-272, 1999.King, F. W. Hilbert Transforms, Vol. 2. Cambridge, England: Cambridge University Press, p. 6, 2009.NIST Digital Library of Mathematical Functions. "Hilbert Transform." §1.14(v). https://dlmf.nist.gov/1.14.v.

Cite this as:

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

Subject classifications