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8/17/2019 9 Lec9 DSP Z Transform
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Lecture 9
DIGITAL SIGNAL
PROCESSING
(DSP)
1
Z-TRANSFORM
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Z-transform
Transform techniques are an important tool
in the analysis of signals and Linear time-
invariant (LTI) systems.
The z-transform plays the same role in the
analysis of discrete-time signals and LTI
systems as
The Laplace transform does in the analysis
of continuous-time signals and LTI systems.
2
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Z-transform3
The z-transform of a sequence x[n•] is
In both cases z is a continuous complex
variable.
n
n z n x z X ][][
Z
X[z]x[n]
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Example: Determine the z-transforms of the
following finite-duration signal.
x1[n]=(1, 2, 5, 7, 0, 1)
Z-transform4
5321
1
543210
1
5
01
1
7521][
107521][
][][
][][
z z z z z X
z z z z z z z X
z n x z X
z n x z X
n
n
n
n
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Z-transform
Example: Determine the z-transforms of the
following finite-duration signal
5
312
1
321012
1
3
21
1
1752][
107521][
][][
][][
z z z z z X
z z z z z z z X
z n x z X
z n x z X
n
n
n
n
x[n]= [1, 2, 5, 7, 0, 1]
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Z-transform & Fourier transform6
In general
We may obtain the Fourier transform from
the z-transform by making the substitution
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Z-transform & Fourier transform7
For r=1 this becomes the Fourier transform of
x[n•]
The Fourier transform therefore corresponds to
the z-transform evaluated on the unit circle:
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Region of Convergence (ROC)
The Fourier transform does not converge for
all sequences.
Similarly, the z-transform does not converge
for all sequences or for all values of z.
The set of values of z for which the z-transform
converges is called
the region of convergence (ROC).
8
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Region of Convergence (ROC)
The z-transform therefore exists (or converges)
if
This leads to the condition for the existence of
the z-transform.
9
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Region of Convergence (ROC)
If the ROC includes the unit circle
Then the Fourier transform will converge.
10
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Region of Convergence (ROC)
Most useful z-transforms can be expressed in the form
where P(z) and Q(z) are polynomials in z.
The values of z for which P(z) = 0 are called the
zeros of X(z).
The values of z for which Q(z) = 0 are called thepoles of X(z).
11
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Right-sided exponential sequence12
Thissequence is
right-sided
because it
is nonzero
only for
n ≥ 0
0 < a < 1 x[n] = anu[n]
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Right-sided exponential sequence13
ROC
ROC
0 < a < 1 x[n] = anu[n]
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Right-sided exponential sequence14
ROC
0 < a < 1 x[n] = anu[n]
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Right-sided exponential sequence15
ROCoutside
a circle of
radius (a)
ROC
0 < a < 1 x[n] = anu[n]
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Right-sided exponential sequence16
The Fourier transform of x[n]•exists
ROC includes the unit circle
x[n] = anu[n]
ROC
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Right-sided exponential sequence
The Fourier transform of x[n]•doesn’t exist ROC doesn’t include the unit circle
x[n] is exponentially growing, and the sum
therefore does not converge.
17
If
x[n] = anu[n]
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Left-sided exponential sequence18
This sequence is left-sided because it is
nonzero only for n ≤ -1
0 < a < 1
x[n] = -anu[-n-1]
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Left-sided exponential sequence19
x[n] = -anu[-n-1]
ROC
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Left-sided exponential sequence20
ROC
inside
a circle of
radius (a)
x[n] = -anu[-n-1]
ROC
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Left-sided exponential sequence21
x[n] = -anu[-n-1]
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Region of Convergence (ROC)
left-sided sequence & right-sided sequence
23
Left-sided sequence
x[n] = -anu[-n-1]
Right-sided sequence
x[n] = anu[n]
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Sum of two Exponentials24
x[n] = (1/2)nu[n]+(-1/3)nu[n]
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Sum of two Exponentials25
x[n] = (1/2)nu[n]+(-1/3)nu[n]
ROC
ROC
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Sum of two Exponentials26
x[n] = (1/2)nu[n]+(-1/3)nu[n]
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Sum of two Exponentials27
× PolesO Zeros
ROC
│Z│> 1/2
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Region of Convergence (ROC)28
Left-sided sequence
ROC │Z││a│
x[n] = anu[n]
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Z-transform Pairs29
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Z-transform Pairs30
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Z-transform Pairs31