Title: The z-Transform
1The z-Transform
2Content
- Introduction
- z-Transform
- Zeros and Poles
- Region of Convergence
- Important z-Transform Pairs
- Inverse z-Transform
- z-Transform Theorems and Properties
- System Function
3The z-Transform
4Why z-Transform?
- A generalization of Fourier transform
- Why generalize it?
- FT does not converge on all sequence
- Notation good for analysis
- Bring the power of complex variable theory deal
with the discrete-time signals and systems
5The z-Transform
6Definition
- The z-transform of sequence x(n) is defined by
Fourier Transform
7z-Plane
Fourier Transform is to evaluate z-transform on a
unit circle.
8z-Plane
9Periodic Property of FT
Can you say why Fourier Transform is a periodic
function with period 2??
10The z-Transform
11Definition
- Give a sequence, the set of values of z for which
the z-transform converges, i.e., X(z)lt?, is
called the region of convergence.
ROC is centered on origin and consists of a set
of rings.
12Example Region of Convergence
ROC is an annual ring centered on the origin.
13Stable Systems
- A stable system requires that its Fourier
transform is uniformly convergent.
- Fact Fourier transform is to evaluate
z-transform on a unit circle. - A stable system requires the ROC of z-transform
to include the unit circle.
14Example A right sided Sequence
15Example A right sided Sequence
For convergence of X(z), we require that
16Example A right sided Sequence ROC for
x(n)anu(n)
Which one is stable?
17Example A left sided Sequence
18Example A left sided Sequence
For convergence of X(z), we require that
19Example A left sided Sequence ROC for
x(n)?anu(? n?1)
Which one is stable?
20The z-Transform
21Represent z-transform as a Rational Function
where P(z) and Q(z) are polynomials in z.
Zeros The values of zs such that X(z) 0
Poles The values of zs such that X(z) ?
22Example A right sided Sequence
ROC is bounded by the pole and is the exterior of
a circle.
23Example A left sided Sequence
ROC is bounded by the pole and is the interior of
a circle.
24Example Sum of Two Right Sided Sequences
ROC is bounded by poles and is the exterior of a
circle.
ROC does not include any pole.
25Example A Two Sided Sequence
ROC is bounded by poles and is a ring.
ROC does not include any pole.
26Example A Finite Sequence
N-1 zeros
ROC 0 lt z lt ?
ROC does not include any pole.
N-1 poles
Always Stable
27Properties of ROC
- A ring or disk in the z-plane centered at the
origin. - The Fourier Transform of x(n) is converge
absolutely iff the ROC includes the unit circle. - The ROC cannot include any poles
- Finite Duration Sequences The ROC is the entire
z-plane except possibly z0 or z?. - Right sided sequences The ROC extends outward
from the outermost finite pole in X(z) to z?. - Left sided sequences The ROC extends inward from
the innermost nonzero pole in X(z) to z0.
28More on Rational z-Transform
Consider the rational z-transform with the pole
pattern
Find the possible ROCs
29More on Rational z-Transform
Consider the rational z-transform with the pole
pattern
Case 1 A right sided Sequence.
30More on Rational z-Transform
Consider the rational z-transform with the pole
pattern
Case 2 A left sided Sequence.
31More on Rational z-Transform
Consider the rational z-transform with the pole
pattern
Case 3 A two sided Sequence.
32More on Rational z-Transform
Consider the rational z-transform with the pole
pattern
Case 4 Another two sided Sequence.
33The z-Transform
- Important
- z-Transform Pairs
34Z-Transform Pairs
35Z-Transform Pairs
36The z-Transform
37The z-Transform
- z-Transform Theorems and Properties
38Linearity
Overlay of the above two ROCs
39Shift
40Multiplication by an Exponential Sequence
41Differentiation of X(z)
42Conjugation
43Reversal
44Real and Imaginary Parts
45Initial Value Theorem
46Convolution of Sequences
47Convolution of Sequences
48The z-Transform
49Shift-Invariant System
y(n)x(n)h(n)
x(n)
H(z)
X(z)
Y(z)X(z)H(z)
50Shift-Invariant System
X(z)
Y(z)
51Nth-Order Difference Equation
52Representation in Factored Form
Contributes poles at 0 and zeros at cr
Contributes zeros at 0 and poles at dr
53Stable and Causal Systems
Causal Systems ROC extends outward from the
outermost pole.
54Stable and Causal Systems
Stable Systems ROC includes the unit circle.
55Example
- Consider the causal system characterized by
56Determination of Frequency Response from
pole-zero pattern
- A LTI system is completely characterized by its
pole-zero pattern.
Example
57Determination of Frequency Response from
pole-zero pattern
- A LTI system is completely characterized by its
pole-zero pattern.
H(ej?)?
?H(ej?)?
58Determination of Frequency Response from
pole-zero pattern
- A LTI system is completely characterized by its
pole-zero pattern.
Example
?H(ej?) ?1?(?2 ?3 )
59Example