Adjoint filter explained

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} }\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 ^{*} → k

Notes and References

1. Book: Broughton, S. Allen. Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing. Bryan. Kurt M.. 2011-10-13. John Wiley & Sons. 9781118211007. 141. en.
2. Book: Koornwinder, Tom H.. Wavelets: An Elementary Treatment of Theory and Applications. 1993-06-24. World Scientific. 9789814590976. 70. en.
3. Book: Andrews, Travis D.. Excursions in Harmonic Analysis, Volume 2: The February Fourier Talks at the Norbert Wiener Center. Balan. Radu. Benedetto. John J.. Czaja. Wojciech. Okoudjou. Kasso A.. 2013-01-04. Springer Science & Business Media. 9780817683795. 174. en.