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 with respect to 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

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