Review of Linear System Theory

The following is a (very) brief review of linear system theory and Fourier analysis. I work primarily with discrete signals, but each result developed in this review has a parallel in terms of continuous signals and systems. I assume the reader is familiar with linear algebra (as reviewed in my handout Geometric Review of Linear Algebra), and least squares estimation (as reviewed in my handout Least Squares Optimization). 1 Linear shift-invariant systems A system is linear if it obeys the principle of superposition: the response to a weighted sum of any two inputs is the (same) weighted sum of the responses to each individual input. A system is shift-invariant (also called translation-invariant for spatial signals, or time-invariant for temporal signals) if the response to any input shifted by any amount ∆ is equal to the re- sponse to the original input shifted by amount ∆. These two properties are completely independent: a system can have one of them, both or neither [think of an example of each of the 4 possibilities]. [via]

http://www.cns.nyu.edu/~eero/NOTES/linearSyst...

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