signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...

, the adjoint filter mask

h^*

of a filter mask

h

is reversed in time and the elements are

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...

d. :

(h^*)_k = \overline

Its name is derived from the fact that the convolution with the adjoint filter is the

adjoint operator In mathematics, specifically in operator theory, each linear operator A on an inner product space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where \l ...

of the original filter, with respect to the

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

\ell_2

of the

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

s in which the inner product is the

Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...

. :

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

The

autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...

of a signal

x

can be written as

x^* * x

Properties

^* = h

(h*g)^* = h^* * g^*

(h\leftarrow k)^* = h^* \rightarrow k

References

Digital signal processing {{signal-processing-stub