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.
 * $$\langle h*x, y \rangle = \langle x, h^* * y \rangle$$

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

Properties

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