Adjoint filter

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

${\displaystyle (h^{*})_{k}={\overline {h_{-k}}}}$

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 ${\displaystyle \ell _{2}}$ of the sequences in which the inner product is the Euclidean norm.

${\displaystyle \langle h*x,y\rangle =\langle x,h^{*}*y\rangle }$

The autocorrelation of a signal ${\displaystyle x}$ can be written as ${\displaystyle x^{*}*x}$.

Properties

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

References

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.