ecen314-lec6

# ecen314-lec6 - Lecture 6 Properties of DT Convolution Tie...

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Unformatted text preview: Lecture 6: Properties of DT Convolution Tie Liu September 13, 2011 Property 1 (The sifting property): x [ n ] ∗ δ [ n − n ] = x [ n − n ]. In particular, x [ n ] ∗ δ [ n ] = x [ n ]. Proof. By definition, x [ n ] ∗ δ [ n − n ] = ∞ summationdisplay k = −∞ x [ k ] δ [ n − k − n ] Note that δ [ n − k − n ] = braceleftbigg 1 , k = n − n , k negationslash = n − n We conclude that x [ n ] ∗ δ [ n − n ] = x [ n − n ]. Interpretation: x [ n ] y [ n ] = x [ n- n ] δ [ n- n ] Delay by n Property 2 (The commutative property): y [ n ] = x [ n ] ∗ h [ n ] = h [ n ] ∗ x [ n ] Proof. By definition, x [ n ] ∗ h [ n ] = ∞ summationdisplay k = −∞ x [ k ] h [ n − k ] and h [ n ] ∗ x [ n ] = ∞ summationdisplay k = −∞ h [ k ] x [ n − k ] Let k ′ = n − k . We have k = n − k ′ and hence x [ n ] ∗ h [ n ] = ∞ summationdisplay k ′ = −∞ x [ n − k ′ ] h [ k ′ ] Note that k are k ′ are simply running variables. We conclude that x [ n ] ∗ h [ n ] = h [ n ] ∗ x [ n ]....
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ecen314-lec6 - Lecture 6 Properties of DT Convolution Tie...

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