Adjoint filter

Last updated March 06, 2019

In signal processing, the adjoint filter mask $h^{*}$ of a filter mask $h$ is reversed in time and the elements are complex conjugated.^[1]^[2]^[3]

Signal processing is a subfield of mathematics, information and electrical engineering that concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements. For example, signal processing techniques are used to improve signal transmission fidelity, storage efficiency, and subjective quality, and to emphasize or detect components of interest in a measured signal.

Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. This results from systems composed solely of components classified as having a linear response. Most filters implemented in analog electronics, in digital signal processing, or in mechanical systems are classified as causal, time invariant, and linear signal processing filters.

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. For example, the complex conjugate of 3 + 4i is 3 − 4i.

(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 $\ell _{2}$ of the sequences in which the inner product is the Euclidean norm.

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and order matters. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index; it is the natural number from which the element is the image. It depends on the context or a specific convention, if the first element has index 0 or 1. When a symbol has been chosen for denoting a sequence, the nth element of the sequence is denoted by this symbol with n as subscript; for example, the nth element of the Fibonacci sequence is generally denoted F_n.

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

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

Autocorrelation, also known as serial correlation, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.

Properties

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

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References

↑ Broughton, S. Allen; Bryan, Kurt M. (2011-10-13). Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing. John Wiley & Sons. p. 141. ISBN 9781118211007.
↑ Koornwinder, Tom H. (1993-06-24). Wavelets: An Elementary Treatment of Theory and Applications. World Scientific. p. 70. ISBN 9789814590976.
↑ Andrews, Travis D.; Balan, Radu; Benedetto, John J.; Czaja, Wojciech; Okoudjou, Kasso A. (2013-01-04). Excursions in Harmonic Analysis, Volume 2: The February Fourier Talks at the Norbert Wiener Center. Springer Science & Business Media. p. 174. ISBN 9780817683795.

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[1] Broughton, S. Allen; Bryan, Kurt M. (2011-10-13). Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing. John Wiley & Sons. p. 141. ISBN 9781118211007.

[2] Koornwinder, Tom H. (1993-06-24). Wavelets: An Elementary Treatment of Theory and Applications. World Scientific. p. 70. ISBN 9789814590976.

[3] Andrews, Travis D.; Balan, Radu; Benedetto, John J.; Czaja, Wojciech; Okoudjou, Kasso A. (2013-01-04). Excursions in Harmonic Analysis, Volume 2: The February Fourier Talks at the Norbert Wiener Center. Springer Science & Business Media. p. 174. ISBN 9780817683795.