ENSC380Lecture 28

Objectives:

• z-Transform• Properties of z-transform

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Atousa Hajshirmohammadi, SFU

z-Transform

• z-Transform is defined for DT signals, just as Laplace transform is defined forCT signals.

• The relationship of the “z-Transform” to DTFT is the same as that of Laplacetransform to CTFT.

• Recall DTFT:

X(F ) =∞∑

n=−∞

x[n]e−j2πFn or X(jΩ) =∞∑

n=−∞

x[n]e−jΩn

• Now replace Ω with a general complex variable S = Σ + jΩ:

X(S) =∞∑

n=−∞

x[n]e−(Σ+jΩ)n

• For convenience in notations e(Σ+jΩ) is replaced with z, thus z-Transform isdefined as:

X(z) =∞∑

n=−∞

x[n]z−n

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Examples

• Find the Laplace transform of x(t) = eαtu(t) for α > 0.

• Find the z-transform of x[n] = αnu[n], for α > 0.

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Region of Convergence

• As seen in the previous example, for z-transform the region of convergence(ROC) is defined based on the magnitude of z, i.e., |z|. Thus the ROC coversrings or circles in the complex plane.

• The values of z for which the transform is equal to “zero” or “infinity” are calledthe “zeros” and “poles”, respectively.

• Find the ROC and poles and zeros of the z-transform of x[n] = α−nu[−n], forα > 0.

• Find the z-transform (ROC and poles and zeros) of x[n] = 2nu[n] + 3nu[−n].

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Transfer Function

• Similar to other transforms (FT or Laplace transform) z-Transform of a DTsystem is the ratio between the z-transforms of its output and input signals:

• In other words the multiplication-convolution property applies to z-Transform aswell:

y[n] = x[n] ∗ h[n] then Y (z) = X(z).H(z)

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Atousa Hajshirmohammadi, SFU

Unilateral z-Transform

• Analogous to unilateral Laplace transform, the unilateral z-transform is definedas:

X(z) =∞∑

n=0

x[n]z−n

• For causal signals (x[n] = 0 for n < 0), this is the same as the bilateral ztransform.

• From now on will refer to the unilateral z-transform, simply as the z-transform.

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Properties

Here we list the properties of z-transform and using these properties and AppendixG, will try to solve some related examples.

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PropertiesIf two causal DT signals form these transform pairs,

then the following properties hold for the z transform.

Linearity

Time Shifting

Delay:

Advance:

g n[ ] Z← → ⎯ G z( ) and h n[ ] Z← → ⎯ H z( )

α g n[ ]+ β h n[ ] Z← → ⎯ α G z( )+ β H z( )

g n − n0[ ] Z← → ⎯ z−n0 G z( ) , n0 ≥ 0

g n + n0[ ] Z← → ⎯ zn0 G z( )− g m[ ]z− m

m=0

n0 −1

∑⎛ ⎝ ⎜

⎞ ⎠ ⎟ , n0 > 0

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Atousa Hajshirmohammadi, SFU

PropertiesChange of Scale

Initial Value Theorem

z-Domain Differentiation

Convolution in Discrete Time

α n g n[ ] Z← → ⎯ G

z

α⎛ ⎝

⎞ ⎠

g 0[ ]= limz→∞

G z( )

−ng n[ ] Z← → ⎯ z

ddz

G z( )

g n[ ]∗h n[ ] Z← → ⎯ H z( )G z( )

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Atousa Hajshirmohammadi, SFU

PropertiesDifferencing

Accumulation

Final Value Theorem

(if the limit exists)

g n[ ]− g n −1[ ] Z← → ⎯ 1− z−1( )G z( )

g m[ ]

m=0

n

∑ Z← → ⎯ z

z −1G z( )=

11− z−1 G z( )

limn→∞

g n[ ]= limz→1

z −1( )G z( )

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Atousa Hajshirmohammadi, SFU

Example 1

A DT system has the following z-transform. Find the difference equation that definesthis system.

H(z) =Y (z)

X(z)=

z − 1/2

z2 − z + 2/9

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Example 2

First find the z-transform of x[n] = e−n/40u[n]. Then use the “change of scale”property to find the z-transform of:

x1[n] = e−n/40 sin(2πn

8)u[n]

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