View source for Linear shift-invariant systems



Electrical networks or optical imaging systems transform their input (e.g. voltages or light intensities) as a function of time and/or of space. In general such one- or more-dimensional transformations ''S'' map some input functions ''f''(''x'',''y'',''z'';''t'') into some output functions ''g''(''x'',''y'',''z'';''t''):

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[[File:hepa_img616.gif|319x96px]]

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The transformation is called a linear system ''L'', if the following equation holds for all functions ''f''<sub>1</sub>, ''f''<sub>2</sub> and any ''a'' and ''b'':

[[File:hepa_img617.gif|389x16px]]

i.e. an arbitrary function that can be expressed as a sum of several elementary excitations will be transformed by a linear system as the superposition of the output of these excitations. In general:

[[File:hepa_img618.gif|446x16px]]

''L'' is called ''shift-invariant'' if and only if a shift (translation) of the input causes the same shift of the output:

[[File:hepa_img619.gif|470x16px]]

Electrical networks or optical systems are usually treated as time- and space-invariant, respectively.

To simplify the notation and to derive the computational aspects, we choose a one-dimensional discrete system. With

[[File:hepa_img620.gif|346x39px]]

we can write the identity:

[[File:hepa_img621.gif|333x34px]]

Application of the linear operator ''L'' produces:

[[File:hepa_img622.gif|429x34px]]

which is the superposition sum of the shift-varying ''impulse response'' ''h''(''k'';''i''). If ''L'' is shift-invariant, i.e. ''h''(''k''-''i'') = ''L''[''d''(''k''-''i'')], the equation can be written in form of a [[Convolution#37|convolution]]

[[File:hepa_img623.gif|331x34px]]

or abbreviated:

[[File:hepa_img624.gif|288x16px]]

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[[File:hepa_img625.gif|406x355px]]

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The impulse response ''h'' is called the ''point spread function'' in the two-dimensional case.

If ''F'',''G'' and ''H'' are the Fourier transforms of ''f'',''g'' and ''h'', respectively,

[[File:hepa_img626.gif|291x15px]]

with the frequency response or transfer function ''H'' of the linear shift-invariant system ''L''. For more [[File:hepa_img1.gif|14x8px]] e.g. [[References 1#kunt80|Kunt80]] details and more references or [[References 1#good68|Goodman68]].

[[Category:W.Krisher and R.Bock]]
[[Category:Data analysis]]
[[Category:Statistics]]

{{Sourceattribution|Linear shift-invariant systems|1}}