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Lecture12_ExampleZTransforms

# Lecture12_ExampleZTransforms - Properties of the...

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& \$ % Properties of the Z -Transform Linearity: a 1 x 1 [ n ] + a 2 x 2 [ n ] ←→ a 1 X 1 ( z ) + a 2 X 2 ( z ) , RoC = R x 1 R x 2 Time Shifting Property: x [ n - n 0 ] ←→ z - n 0 X ( z ) , RoC = R x ( except possible addition/deletion of z = 0 or z = ) Exponential Weighting: z n 0 x [ n ] ←→ X ( z - 1 0 z ) , RoC = | z 0 | R x The poles of the Z -transform are scaled by | z 0 |

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& \$ % Linear Weighting nx ( n ) ←→ - z dX ( z ) dz , RoC = R x ( except possible addition/deletion of z = 0 or z = ) Time Reversal x [ - n ] ←→ X ( z - 1 ) , RoC = 1 R x Convolution x [ n ] * y [ n ] ←→ X ( z ) Y ( z ) , RoC = R x R y Multiplication x [ n ] w [ n ] ←→ 1 2 πj Z X ( v ) w ( z v ) v - 1 dv
& \$ % Inverse Z -Transform Examples Using long division: Causal sequence 1 1 - az - 1 , RoC = | z | > | a | = 1 + az - 1 + az - 2 + az - 3 + · · · IZT (1 + az - 1 + a 2 z - 2 + a 3 z - 3 + · · · ) = a n u [ n ] Using long division: Noncausal sequence 1 1 - az - 1 , RoC = | z | < | a | Here the IZT is computed as follows: IZT ( 1 1 - az - 1 ) = IZT ( z - a + z ) This results in: IZT ( - a - 1 z + a - 2 z 2 + a - 3 z 3 + · · · ) = - a n u [ - n - 1]

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& \$ % Inverse Z -transform - using Power series expansion X ( z ) = log
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