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ecen314-lec9

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

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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 { φ k } : x [ n ] = summationdisplay k a k φ k [ n ] x ( t ) = summationdisplay k a k φ k ( t ) For LTI systems, knowing the response to the basic signals φ k is very useful: y [ n ] = summationdisplay k a k ˜ φ k [ n ] y ( t ) = summationdisplay k a k ˜ φ k ( t ) where ˜ φ k is the response to φ k . Previously: φ k [ n ] = δ [ n - k ] for DT systems and φ k ( t ) = δ Δ ( t - k Δ) 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 ( τ ) h ( t - τ ) Focus now: The basic signals { φ k } being the eigenfunctions of LTI systems. 1

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LTI φ k ˜ φ k = λ k φ k Eigenvalue Eigenfunction Eigenfunction in -→ Same function out with a gain By the superposition property of LTI systems: x ( t ) = summationdisplay k a k φ k ( t ) -→ y ( t ) = summationdisplay k λ k a k φ k ( t ) x [ n ] =
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ecen314-lec9 - Lecture 9 Eigenfunctions of LTI Systems Tie...

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