will also be in the ROC X z N 2 X n x n z n left sided sequence Particular

Will also be in the roc x z n 2 x n x n z n left

This preview shows page 138 - 148 out of 312 pages.

will also be in the ROC. X ( z ) = N 2 X n = -∞ x ( n ) z - n left - sided sequence Particular cases: if N 2 > 0 then the ROC does not include z = 0 if N 2 0 then the ROC includes z = 0 TU Darmstadt Elektrotechnik und Informationstechnik 13
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CoE 4TL4, Hamilton 200 Property 6: If x ( n ) is a two-sided sequence, and if the circle | z | = r 0 is in the ROC, then the ROC will be a ring in the z -plane that includes the circle | z | = r 0 . X ( z ) = X n = -∞ x ( n ) z - n two - sided sequence Any two-sided sequence can be represented as a direct sum of a right-sided and left-sided sequences = the ROC of this composite signal will be the intersection of the ROC’s of the components. TU Darmstadt Elektrotechnik und Informationstechnik 13
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CoE 4TL4, Hamilton 200 To property 1: a) the ROC must be a connected region, b) the ROC cannot be nonsymmetric Re Im Im Re a) b) TU Darmstadt Elektrotechnik und Informationstechnik 13
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CoE 4TL4, Hamilton 200 To property 6: intersection of the ROC’s of right-sided and left-sided sequences Re Im Im Re Im Re U = TU Darmstadt Elektrotechnik und Informationstechnik 14
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CoE 4TL4, Hamilton 200 To properties 2, 4, 5, 6: pole-zero pattern and three possible ROC’s that correspond to X ( z ) = h (1 - 1 3 z - 1 )(1 - 1 . 3 z - 1 ) i - 1 Im Re 1 Im Re Im Re 1 1 a) right-sided b) left-sided c) two-sided TU Darmstadt Elektrotechnik und Informationstechnik 14
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CoE 4TL4, Hamilton 200 4.3 The Inverse z -Transform We obtained that X ( z ) fl fl fl z = re = F{ x ( n ) r - n } Applying the inverse DTFT, we get x ( n ) = r n F - 1 { X ( re ) } = r n 1 2 π Z π - π X ( re ) e jωn = 1 2 π Z π - π X ( re | {z } z )( re | {z } z ) n = 1 2 πj I X ( z ) z n - 1 dz ←- dz = jre TU Darmstadt Elektrotechnik und Informationstechnik 14
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CoE 4TL4, Hamilton 200 Remarks on inverse z -transform: H · · · dz denotes integration around a closed circular contour centered at the origin and having the radius r the value of r must be chosen so that the contour of integration | z | = r belongs to the ROC contour integration in the complex plane may be a complicated task simpler alternative procedures exist for obtaining the sequence from its z -transform TU Darmstadt Elektrotechnik und Informationstechnik 14
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CoE 4TL4, Hamilton 200 4.4 Alternative Methods for Inverse z -Transform Inspection method: consists simply of becoming familiar with (or recognizing “by inspection” ) certain transform pairs. Examples: a n u ( n ) 1 1 - az - 1 , | z | > | a | δ ( n - m ) z - m , z 6 = 0 if m > 0 , z 6 = if m < 0 sin( ωn ) u ( n ) [sin ω ] z - 1 1 - [2 cos ω ] z - 1 + z - 2 , | z | > 1 TU Darmstadt Elektrotechnik und Informationstechnik 14
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CoE 4TL4, Hamilton 200 Extended inspection method: consists of expressing a complicated X ( z ) as a sum of simpler terms and then applying to each term the inspection method . Example: X ( z ) = z 2 (1 - 0 . 5 z - 1 )(1 + z - 1 )(1 - z - 1 ) = z 2 - 0 . 5 z - 1 + 0 . 5 z - 1 = x ( n ) = δ ( n + 2) - 0 . 5 δ ( n + 1) - δ ( n ) + 0 . 5 δ ( n - 1) TU Darmstadt Elektrotechnik und Informationstechnik 14
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CoE 4TL4, Hamilton 200 4.5 Properties of z -Transform Linearity: If X ( z ) = Z{ x ( n ) } and Y ( z ) = Z{ y ( n ) } then a X ( z ) + b Y ( z ) = Z{ a x ( n ) + b y ( n ) } Also if x ( n ) = Z - 1 { X ( z ) } and y ( n ) = Z - 1 { Y ( z ) } then a x ( n ) + b y ( n ) = Z - 1 { a X ( z ) + b Y ( z ) } with ROC = ROC x ROC y TU Darmstadt
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