ecen314-lec9

ecen314-lec9 - Lecture 9: Eigenfunctions of LTI Systems Tie...

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Unformatted text preview: Lecture 9: Eigenfunctions of LTI Systems Tie Liu October 3, 2011 1 Signal decompositions using eigenfunctions of LTI systems Signal decompositions using a set of basic signals { k } : x [ n ] = summationdisplay k a k k [ n ] x ( t ) = summationdisplay k a k k ( t ) For LTI systems, knowing the response to the basic signals k is very useful: y [ n ] = summationdisplay k a k k [ n ] y ( t ) = summationdisplay k a k k ( t ) where k is the response to k . Previously: k [ n ] = [ n- k ] for DT systems and k ( t ) = ( t- k ) ( 0 eventually) for CT systems, leading to the well-known convolution formulas for LTI systems: y [ n ] = summationdisplay k =- x [ k ] h [ n- k ] y ( t ) = integraldisplay - x ( ) h ( t- ) d Focus now: The basic signals { k } being the eigenfunctions of LTI systems. 1 LTI k k = k k Eigenvalue Eigenfunction Eigenfunction in- Same function out with a gain By the superposition property of LTI systems:...
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ecen314-lec9 - Lecture 9: Eigenfunctions of LTI Systems Tie...

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