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Lecture 2 - Discrete Time System Input x[n x y[n = f...

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Discrete Time System y [ n ] = f { x [ n k ] = n-k ], y [ n-k ] } x [ n ] g214 y [ n ] x n n y Input x [ n ] Output y [ n ] IF x 1 n n y 1 Linear System Homogeneity Property x 1 [ n ] g214 y 1 [ n ] THEN a x 1 [ n ] g214 ____ Input x [ n ] Output y [ n ] ax 1 n n ay 1 Linear System x 1 n n y 1 x 2 n n y 2 IF x [ n ] g214 y [ n ] Additivity Property x [ n ] = x 1 + x 2 n n y [ n ] = y 1 + y 2 1 1 x 2 [ n ] g214 y 2 [ n ] THEN x 1 [n] +x 2 [n] g214 ________ Input x [ n ] Output y [ n ]
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Linear System IF Additivity + Homogeneity g214 Linearity x 1 [ n ] g214 y 1 [ n ] x 2 [ n ] g214 y 2 [ n ] THEN ax 1 [n] +bx 2 [n] g214 _____________ Input x [ n ] Output y [ n ] Linear System Checking Difference Equation For Linearity: Nonlinear if Diff Equ. contains either: A constant term y [n] = Products of inputs or output terms y [n] = x 1 n n y 1 Time-Invariant System IF x 1 [ n ] g214 y 1 [ n ] Shift-Invariance Property O n n x 2 [ n ] = x 1 [ n-3 ] y 2 [ n ] = y 1 [ n-3 ] x 2 [ n ] = x 1 [ n-k ] x 2 [ n ] g214 y 2 [ n ] Input x [ n ] Output y [ n ] THEN y 2 [ n ] = _________ x 2 [ n ] is x 1 [ n ] shifted by k y 2 [ n ] is y 1 [ n ] __________
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Time-Invariant System Shift-Invariance Property So, shift-invariance (a time-invariant system) implies:
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