Adjoint filter

In signal processing, the adjoint filter mask h* of a filter mask h is reversed in time and the elements are complex conjugated.[1][2][3]

(h*)k=h−k‾

Its name is derived from the fact that the convolution with the adjoint filter is the adjoint operator of the original filter, with respect to the Hilbert space ℓ2 of the sequences in which the inner product is the Euclidean norm.

⟨h*x,y⟩=⟨x,h**y⟩

The autocorrelation of a signal x can be written as x**x.

## Properties

* h**=h
* (h*g)*=h**g*
* (h<-k)*=h*->k

## References

1. ↑ Broughton, S. Allen; Bryan, Kurt M. (2011-10-13) (in en). Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing. John Wiley & Sons. pp. 141. ISBN 9781118211007. https://books.google.com/books?id=PLcC2gmtv3kC.
2. ↑ Koornwinder, Tom H. (1993-06-24) (in en). Wavelets: An Elementary Treatment of Theory and Applications. World Scientific. pp. 70. ISBN 9789814590976. https://books.google.com/books?id=VQa3CgAAQBAJ.
3. ↑ Andrews, Travis D.; Balan, Radu; Benedetto, John J.; Czaja, Wojciech; Okoudjou, Kasso A. (2013-01-04) (in en). Excursions in Harmonic Analysis, Volume 2: The February Fourier Talks at the Norbert Wiener Center. Springer Science & Business Media. pp. 174. ISBN 9780817683795. https://books.google.com/books?id=bcJJAAAAQBAJ.

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