ENSC380 Lecture 28 Objectives: z-TransformUnilateral z-Transform • Analogous to unilateral Laplace...
Transcript of ENSC380 Lecture 28 Objectives: z-TransformUnilateral z-Transform • Analogous to unilateral Laplace...
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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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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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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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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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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