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

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