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}={\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

${h^{*}}^{*}=h$
$(h*g)^{*}=h^{*}*g^{*}$
$(h\leftarrow k)^{*}=h^{*}\rightarrow k$

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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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[1] 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.

[2] Koornwinder, Tom H. (1993-06-24). Wavelets: An Elementary Treatment of Theory and Applications. World Scientific. p. 70. ISBN 9789814590976.

[3] 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.