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.¹²³

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

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 ${\displaystyle x^{*}*x}$.

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