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.

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

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.

$⟨h*x,y⟩=⟨x,h^{*}*y⟩$

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

Properties

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