Z transform pairs-2 - ECE306 z-TRANSFORM PROPERTIES The...

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z -TRANSFORM PROPERTIES The index-domain signal is x [ n ] for -∞ < n < ; and the z -transform is: X ( z ) = X n =-∞ x [ n ] z - n ⇐⇒ x [ n ] = 1 2 π j I X ( z ) z n dz z The ROC is the set of complex numbers z where the z -transform sum converges. Signal: x [ n ] - ∞ < n < z -Transform: X ( z ) Region of Convergence x [ n ], x 1 [ n ] and x 2 [ n ] X ( z ) , X 1 ( z ) and X 2 ( z ) R x , R 1 and R 2 ax 1 [ n ] + bx 2 [ n ] aX 1 ( z ) + bX 2 ( z ) contains R 1 R 2 x [ n - n ] z - n X ( z ) R x except for the possible addition or deletion of z = 0 or z = ∞ z n x [ n ] X ( z / z ) | z | R x n x [ n ] - z dX ( z ) dz R x except for the possible addition or deletion of z = 0 or z = ∞ x * [ n ] X * ( z * ) R x < e { x [ n ] } 1 2 [ X ( z ) + X * ( z * ) ] contains R x = m { x [ n ] } 1 2 j [ X ( z ) - X * ( z * ) ] contains R x x [ - n ] X ( 1 / z ) 1 / R x = { z : z - 1 R x } x 1 [ n ] * x 2 [ n ] X 1 ( z ) · X 2 ( z ) contains R 1 R 2 x 1 [ n ] · x 2 [ n ] 1 2 π j I X 1 (v) X 2 ( z /v) d v v contains R 1 R 2 Parseval’s Theorem: X n =-∞ x 1 [ n ] x * 2 [ n ] = 1 2 π j I X 1 (v) X * 2 ( 1 /v * ) d v v Initial Value Theorem: x [ n ] = 0 , for n < 0 =⇒ lim z →∞ X ( z ) = x [0]
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z -TRANSFORM PAIRS The index-domain signal is x [ n ] for -∞ < n < ; and the z -transform is: X ( z ) = X n =-∞ x [ n ] z - n ⇐⇒ x [ n ] = 1 2 π j I X ( z ) z n dz z The ROC is the set of complex numbers z where the z -transform sum converges. Signal: x [ n ] - ∞ < n < z -Transform: X ( z ) Region of Convergence δ [ n ] 1 All z δ [ n - n ] z - n | z | > 0, if n > 0 | z | < , if n < 0 u [ n ] 1 1 - z - 1 | z
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