Adjoint filter

In signal processing, the adjoint filter mask [math]\displaystyle{ h^* }[/math] of a filter mask [math]\displaystyle{ h }[/math] is reversed in time and the elements are complex conjugated.

[math]\displaystyle{ (h^*)_k = \overline{h_{-k}} }[/math]

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 [math]\displaystyle{ \ell_2 }[/math] of the sequences in which the inner product is the Euclidean norm.

[math]\displaystyle{ \langle h*x, y \rangle = \langle x, h^* * y \rangle }[/math]

The autocorrelation of a signal [math]\displaystyle{ x }[/math] can be written as [math]\displaystyle{ x^* * x }[/math].

Properties

• [math]\displaystyle{ {h^*}^* = h }[/math]
• [math]\displaystyle{ (h*g)^* = h^* * g^* }[/math]
• [math]\displaystyle{ (h\leftarrow k)^* = h^* \rightarrow k }[/math]