The Inverse z-TransformIn science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it's the exact opposite. Paul DiracContent and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000 Prentice Hall Inc.

*The Inverse Z-TransformFormal inverse z-transform is based on a Cauchy integralLess formal ways sufficient most of the timeInspection methodPartial fraction expansionPower series expansionInspection MethodMake use of known z-transform pairs such as

Example: The inverse z-transform of

*Inverse Z-Transform by Partial Fraction ExpansionAssume that a given z-transform can be expressed as

Apply partial fractional expansion

First term exist only if M>NBr is obtained by long divisionSecond term represents all first order polesThird term represents an order s pole There will be a similar term for every high-order pole Each term can be inverse transformed by inspection

*Partial Fractional Expression

Coefficients are given as

Easier to understand with examples

*Example: 2nd Order Z-TransformOrder of nominator is smaller than denominator (in terms of z-1)No higher order pole

*Example ContinuedROC extends to infinity Indicates right sided sequence

*Example #2Long division to obtain Bo

*Example #2 ContinuedROC extends to infinityIndicates right-sides sequence

*Inverse Z-Transform by Power Series ExpansionThe z-transform is power series

In expanded form

Z-transforms of this form can generally be inversed easilyEspecially useful for finite-length seriesExample

*Z-Transform Properties: LinearityNotation

Linearity

Note that the ROC of combined sequence may be larger than either ROCThis would happen if some pole/zero cancellation occursExample:

Both sequences are right-sidedBoth sequences have a pole z=aBoth have a ROC defined as |z|>|a|In the combined sequence the pole at z=a cancels with a zero at z=aThe combined ROC is the entire z plane except z=0We did make use of this property already, where?

*Z-Transform Properties: Time ShiftingHere no is an integerIf positive the sequence is shifted rightIf negative the sequence is shifted leftThe ROC can change the new term mayAdd or remove poles at z=0 or z=Example

*Z-Transform Properties: Multiplication by ExponentialROC is scaled by |zo|All pole/zero locations are scaledIf zo is a positive real number: z-plane shrinks or expandsIf zo is a complex number with unit magnitude it rotatesExample: We know the z-transform pair

Lets find the z-transform of

*Z-Transform Properties: DifferentiationExample: We want the inverse z-transform of

Lets differentiate to obtain rational expression

Making use of z-transform properties and ROC

*Z-Transform Properties: Conjugation

Example

*Z-Transform Properties: Time ReversalROC is invertedExample:

Time reversed version of

*Z-Transform Properties: ConvolutionConvolution in time domain is multiplication in z-domainExample:Lets calculate the convolution of

Multiplications of z-transforms is

ROC: if |a|1 if |a|>1 ROC is |z|>|a|Partial fractional expansion of Y(z)