• Source: Adjoint filter

In signal processing, the adjoint filter mask




h

∗




{\displaystyle h^{*}}

of a filter mask



h


{\displaystyle h}

is reversed in time and the elements are complex conjugated.




(

h

∗



)

k


=



h

−
k


¯




{\displaystyle (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




ℓ

2




{\displaystyle \ell _{2}}

of the sequences in which the inner product is the Euclidean norm.




⟨
h
∗
x
,
y
⟩
=
⟨
x
,

h

∗


∗
y
⟩


{\displaystyle \langle h*x,y\rangle =\langle x,h^{*}*y\rangle }


The autocorrelation of a signal



x


{\displaystyle x}

can be written as




x

∗


∗
x


{\displaystyle x^{*}*x}

.




Properties









h

∗




∗


=
h


{\displaystyle {h^{*}}^{*}=h}





(
h
∗
g

)

∗


=

h

∗


∗

g

∗




{\displaystyle (h*g)^{*}=h^{*}*g^{*}}





(
h
←
k

)

∗


=

h

∗


→
k


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





References

Kata Kunci Pencarian:

• Kategori gelanggang
• Aljabar Heyting
• Adjoint filter
• Kalman filter
• Fast wavelet transform
• Radon transform
• Limit (category theory)
• Polyphase matrix
• Discrete Laplace operator
• Glossary of order theory
• Ridge regression
• Complete lattice