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]
\( (h^*)_k = \overline{h_{-k}} \)
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 \( \ell _{2} \) of the sequences in which the inner product is the Euclidean norm.
\( \langle h*x, y \rangle = \langle x, h^* * y \rangle \)
The autocorrelation of a signal x can be written as \( x^* * x. \)
Properties
\( {h^*}^* = h \)
\( (h*g)^* = h^* * g^* \)
\( (h\leftarrow k)^* = h^* \rightarrow k \)
References
Broughton, S. Allen; Bryan, Kurt M. (2011-10-13). Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing. John Wiley & Sons. p. 141. ISBN 9781118211007.
Koornwinder, Tom H. (1993-06-24). Wavelets: An Elementary Treatment of Theory and Applications. World Scientific. p. 70. ISBN 9789814590976.
Andrews, Travis D.; Balan, Radu; Benedetto, John J.; Czaja, Wojciech; Okoudjou, Kasso A. (2013-01-04). Excursions in Harmonic Analysis, Volume 2: The February Fourier Talks at the Norbert Wiener Center. Springer Science & Business Media. p. 174. ISBN 9780817683795.
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