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, and scientific measurements. Signal processing techniq ...

, the adjoint filter mask

h^*

of a

filter mask A respirator is a device designed to protect the wearer from inhaling hazardous atmospheres including fumes, vapours, gases and particulate matter such as dusts and airborne pathogens such as viruses. There are two main categories of respir ...

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, then) the complex conjugate of a + bi is equal to 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 a Euclidean vector 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 ...

of the original filter, with respect to the

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

\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 calle ...

s in which the inner product is the

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

. :

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

The

autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...

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