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.{{Cite book|url=https://books.google.com/books?id=PLcC2gmtv3kC|title=Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing|last=Broughton|first=S. Allen|last2=Bryan|first2=Kurt M.|date=2011-10-13|publisher=John Wiley & Sons|isbn=9781118211007|pages=141|language=en}}{{Cite book|url=https://books.google.com/books?id=VQa3CgAAQBAJ|title=Wavelets: An Elementary Treatment of Theory and Applications|last=Koornwinder|first=Tom H.|date=1993-06-24|publisher=World Scientific|isbn=9789814590976|pages=70|language=en}}{{Cite book|url=https://books.google.com/books?id=bcJJAAAAQBAJ|title=Excursions in Harmonic Analysis, Volume 2: The February Fourier Talks at the Norbert Wiener Center|last=Andrews|first=Travis D.|last2=Balan|first2=Radu|last3=Benedetto|first3=John J.|last4=Czaja|first4=Wojciech|last5=Okoudjou|first5=Kasso A.|date=2013-01-04|publisher=Springer Science & Business Media|isbn=9780817683795|pages=174|language=en}}

: $(h^*)_k = \overline{h_{-k}}$

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

Category:Digital signal processing