Adjoint filter

In signal processing, the adjoint filter mask $h^{\ast }$ of a filter mask $h$ is reversed in time and the elements are complex conjugated.

$(h^{\ast }{)}_k=\overline{h_{-k}}$

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.

$⟨h\ast x,y⟩=⟨x,h^{\ast }\ast y⟩$

The autocorrelation of a signal $x$ can be written as $x^{\ast }\ast x$.

Properties

• ${h^{\ast }}^{\ast }=h$
• $(h\ast g{)}^{\ast }=h^{\ast }\ast g^{\ast }$
• $(h\leftarrow k{)}^{\ast }=h^{\ast }\to k$

References

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