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

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