Fourier (1) - Petra Christian University

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Z Transform (2)
Hany Ferdinando
Dept. of Electrical Eng.
Petra Christian University
Overview
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Unilateral Z transform
Z transform in LTI system
Convolution and deconvolution
Frequency response analysis
Applications
Z Transform (1) - Hany Ferdinando
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Unilateral Z Transform
The general formula of z transform is
X(z) 

 x(n)z
n
n  
This is bilateral z transform. Consider that the
range of n is from –∞ to ∞. For Unilateral z
transform, the formula becomes

X(z)   x(n)zn
n 0
Z Transform (1) - Hany Ferdinando
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Unilateral Z Transform
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All properties of bilateral z transform
can be used in unilateral z transform,
except the shifting property
For this, one can derived it from the
formula
This property is important in solving
difference equation
Z Transform (1) - Hany Ferdinando
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Z Transform in LTI System
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The analysis of discrete-time LTI system
cannot be separated from z transform.
If X(z) is input, H(z) is impulse response of a
system and Y(z) is output of that system,
then Y(z) = H(z)X(z) (see convolution
property)
H(z) is referred to as the transfer function of
the system
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Z Transform in LTI System
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The stability and causality can be
associated with constraints on the pole-zero
pattern and RoC of the H(z)
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If the system is causal, then the RoC of H(z) will
be outside the outermost pole
If the system is stable, then the RoC of H(z)
must include the unit circle
If the system is stable and causal, then both
consequences above are fulfilled
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Convolution and Deconvolution
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y = h * u in the time domain becomes
Y = HU in the z-domain
Therefore, we can write it as
y  Z Hz Uz 
1
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Hz is h in the z-domain and Uz is u in the
z-domain
Z-1[ ] is inverse Z transform
Z Transform (1) - Hany Ferdinando
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Convolution and Deconvolution
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h = 2k, k ≥ 0 and u = 2-k, k ≥ 0.
Convolve h and u
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Find H(z) and U(z), don’t forget the RoC
Multiply H(z) and U(z)
Combine the RoCs
Find the inverse of their multiplication
result
Z Transform (1) - Hany Ferdinando
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Convolution and Deconvolution
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h = {1,2,3} and y = {1,1,2,-1.3}. Find u
if y = h*u
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Find H(z) and Y(z)  it’s easy
Find U(z) from Y(z)/H(z)
Then take inverse Z transform from U(z)
to get u
Z Transform (1) - Hany Ferdinando
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Frequency Response
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It is used to evaluate the digital filter
The procedures:
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Substitute z with ejq
Separate real and imaginary part
Calculate the magnitude and the phase
angle
Draw both results (for test, it is not
necessary)
Z Transform (1) - Hany Ferdinando
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Application
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To solve linear difference equation
To characterize the transfer function of
discrete-time LTI system
To design digital filter (it is in DSP
course)
Z Transform (1) - Hany Ferdinando
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Next…
We have finished to discuss the z transform.
No other way to understand the z transform
well unless you exercise yourself.
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Signals and Linear System by Robert A. Gabel,
chapter 6, p 349-363
Signals and Systems by Alan V. Oppenheim,
chapter 9, p 573-603
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