Adjoint filter

In signal processing, the adjoint filter mask [math]\displaystyle{ h^* }[/math] of a filter mask [math]\displaystyle{ h }[/math] is reversed in time and the elements are complex conjugated.[1][2][3]

[math]\displaystyle{ (h^*)_k = \overline{h_{-k}} }[/math]

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 [math]\displaystyle{ \ell_2 }[/math] of the sequences in which the inner product is the Euclidean norm.

[math]\displaystyle{ \langle h*x, y \rangle = \langle x, h^* * y \rangle }[/math]

The autocorrelation of a signal [math]\displaystyle{ x }[/math] can be written as [math]\displaystyle{ x^* * x }[/math].

## Properties

* [math]\displaystyle{ {h^*}^* = h }[/math]
* [math]\displaystyle{ (h*g)^* = h^* * g^* }[/math]
* [math]\displaystyle{ (h\leftarrow k)^* = h^* \rightarrow k }[/math]

## 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.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Adjoint filter. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Adjoint_filter&oldid=54971"