The z-Transform and Its Properties
The z-Transform and Its Properties
The z-Transform and Its Properties
Reference:
Sections 3.1 and 3.2 of
Professor Deepa Kundur
John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:
Principles, Algorithms, and Applications, 4th edition, 2007.
University of Toronto
Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
The z-Transform and Its Properties
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3.1 The z-Transform
The z-Transform and Its Properties
The Direct z-Transform
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Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
3.1 The z-Transform
Region of Convergence
Direct z-Transform:
X (z) =
∞
X
x(n)z −n
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the region of convergence (ROC) of X (z) is the set of all values
of z for which X (z) attains a finite value
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The z-Transform is, therefore, uniquely characterized by:
n=−∞
I
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Notation:
X (z)
≡
1. expression for X (z)
2. ROC of X (z)
Z{x(n)}
Z
x(n) ←→ X (z)
Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
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Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
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The z-Transform and Its Properties
3.1 The z-Transform
The z-Transform and Its Properties
ROC Families: Finite Duration Signals
Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
The z-Transform and Its Properties
ROC Families: Infinite Duration Signals
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3.2 Properties of the z-Transform
Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
The z-Transform and Its Properties
z-Transform Properties
Property
Notation:
3.1 The z-Transform
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3.2 Properties of the z-Transform
Linearity Property
Linearity:
Time shifting:
Time Domain
x(n)
x1 (n)
x2 (n)
a1 x1 (n) + a2 x2 (n)
x(n − k)
z-Domain
X (z)
X1 (z)
X2 (z)
a1 X1 (z) + a2 X2 (z)
z −k X (z)
z-Scaling:
Time reversal
Conjugation:
z-Differentiation:
Convolution:
an x(n)
x(−n)
x ∗ (n)
n x(n)
x1 (n) ∗ x2 (n)
X (a−1 z)
X (z −1 )
X ∗ (z ∗ )
dX (z)
−z dz
X1 (z)X2 (z)
ROC
ROC: r2 < |z| < r1
ROC1
ROC2
At least ROC1 ∩ ROC2
At least ROC, except
z = 0 (if k > 0)
and z = ∞ (if k < 0)
|a|r2 < |z| < |a|r1
1
< |z| < r1
r1
2
ROC
r2 < |z| < r1
At least ROC1 ∩ ROC2
Z
x1 (n) ←→ X1 (z), ROC1
Z
x2 (n) ←→ X2 (z), ROC2
Z
a1 x1 (n) + a2 x2 (n) ←→ a1 X1 (z) + a2 X2 (z),
At least ROC1 ∩ ROC2
among others . . .
Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
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Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
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The z-Transform and Its Properties
3.2 Properties of the z-Transform
The z-Transform and Its Properties
Linearity Property
3.2 Properties of the z-Transform
Time Shifting Property
Let x(n) = a1 x1 (n) + a2 x2 (n)
∞
X
X (z) =
x(n)z −n
Z
x(n) ←→ X (z),
ROC
n=−∞
∞
X
(a1 x1 (n) + a2 x2 (n))z −n
=
n=−∞
∞
X
= a1
Z
∞
X
x1 (n)z −n + a2
n=−∞
x(n − k) ←→ z −k X (z),
x2 (n)z −n
n=−∞
At least ROC
except z = 0 (k > 0) or
z = ∞ (k < 0)
= a1 X1 (z) + a2 X2 (z)
ROC: At least ROC1 ∩ ROC2
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The z-Transform and Its Properties
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Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
3.2 Properties of the z-Transform
The z-Transform and Its Properties
Time Shifting Property
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3.2 Properties of the z-Transform
Time Shifting Property: ROC
Let y (n) = x(n − k).
Z
Y (z) =
=
x(n) = δ(n) ←→
∞
X
n=−∞
∞
X
y (n)z −n
I
Example: For k = −1
y (n) = x(n − (−1)) = x(n + 1) = δ(n + 1)
(x(n − k))z
−n
n=−∞
y (n) = δ(n + 1)
Let m = n − k
∞
∞
X
X
−(m+k)
x(m)z
=
x(m)z −m z −k
=
"
I
∞
X
Z
←→
Y (z) = z, ROC: entire z-plane
except z = ∞
Example: For k = 1
y (n) = x(n − (1)) = x(n − 1) = δ(n − 1)
m=−∞
m=−∞
= z −k
X (z) = 1, ROC: entire z-plane
#
x(m)z −m
= z −k X (z)
Z
y (n) = δ(n − 1) ←→
Y (z) = z −1 , ROC: entire z-plane
m=−∞
Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
except z = 0
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The z-Transform and Its Properties
3.2 Properties of the z-Transform
The z-Transform and Its Properties
Scaling in the z-Domain
3.2 Properties of the z-Transform
Scaling in the z-Domain
Let y (n) = an x(n)
Y (z) =
Z
x(n) ←→ X (z),
ROC: r1 < |z| < r2
=
Z
an x(n) ←→ X (a−1 z),
ROC: |a|r1 < |z| < |a|r2
=
∞
X
y (n)z −n
n=−∞
∞
X
n=−∞
∞
X
an x(n)z −n
x(n)(a−1 z)−n
n=−∞
−1
= X (a z)
ROC: r1 < |a−1 z| =
Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
The z-Transform and Its Properties
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3.2 Properties of the z-Transform
< r2 ≡ |a|r1 < |z| < |a|r2
Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
The z-Transform and Its Properties
z-Transform Properties
Property
Notation:
|z|
|a|
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3.2 Properties of the z-Transform
Convolution Property
Linearity:
Time shifting:
Time Domain
x(n)
x1 (n)
x2 (n)
a1 x1 (n) + a2 x2 (n)
x(n − k)
z-Domain
X (z)
X1 (z)
X1 (z)
a1 X1 (z) + a2 X2 (z)
z −k X (z)
z-Scaling:
Time reversal
Conjugation:
z-Differentiation:
Convolution:
an x(n)
x(−n)
x ∗ (n)
n x(n)
x1 (n) ∗ x2 (n)
X (a−1 z)
X (z −1 )
X ∗ (z ∗ )
dX (z)
−z dz
X1 (z)X2 (z)
ROC
ROC: r2 < |z| < r1
ROC1
ROC2
At least ROC1 ∩ ROC2
ROC, except
z = 0 (if k > 0)
and z = ∞ (if k < 0)
|a|r2 < |z| < |a|r1
1
< |z| < r1
r1
2
ROC
r2 < |z| < r1
At least ROC1 ∩ ROC2
x(n) = x1 (n) ∗ x2 (n) ⇐⇒ X (z) = X1 (z) · X2 (z)
among others . . .
Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
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Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
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The z-Transform and Its Properties
3.2 Properties of the z-Transform
The z-Transform and Its Properties
Convolution using the z-Transform
Common Transform Pairs
Basic Steps:
1. Compute z-Transform of each of the signals to convolve (time
domain → z-domain):
X1 (z) = Z{x1 (n)}
X2 (z) = Z{x2 (n)}
2. Multiply the two z-Transforms (in z-domain):
X (z) = X1 (z)X2 (z)
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1
2
3
4
5
6
7
cos(ω0 n)u(n)
8
sin(ω0 n)u(n)
9
an
10
an sin(ω0 n)u(n)
cos(ω0 n)u(n)
z-Transform, X (z)
1
1
1−z −1
1
1−az −1
az −1
(1−az −1 )2
1
1−az −1
az −1
(1−az −1 )2
1−z −1 cos ω0
1−2z −1 cos ω0 +z −2
z −1 sin ω0
1−2z −1 cos ω0 +z −2
1−az −1 cos ω0
1−2az −1 cos ω0 +a2 z −2
1−az −1 sin ω0
1−2az −1 cos ω0 +a2 z −2
Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
Signal, x(n)
δ(n)
u(n)
an u(n)
nan u(n)
−an u(−n − 1)
−nan u(−n − 1)
7
cos(ω0 n)u(n)
sin(ω0 n)u(n)
9
an
cos(ω0 n)u(n)
10
an
sin(ω0 n)u(n)
z-Transform, X (z)
1
1
1−z −1
1
1−az −1
az −1
(1−az −1 )2
1
1−az −1
az −1
(1−az −1 )2
1−z −1 cos ω0
1−2z −1 cos ω0 +z −2
z −1 sin ω0
1−2z −1 cos ω0 +z −2
1−az −1 cos ω0
1−2az −1 cos ω0 +a2 z −2
1−az −1 sin ω0
1−2az −1 cos ω0 +a2 z −2
ROC
All z
|z| > 1
|z| > |a|
|z| > |a|
|z| < |a|
|z| < |a|
|z| > 1
|z| > 1
|z| > |a|
|z| > |a|
Professor Deepa Kundur (University of Toronto) The z-Transform and Its Properties
3.2 Properties of the z-Transform
The z-Transform and Its Properties
Common Transform Pairs
Signal, x(n)
δ(n)
u(n)
an u(n)
nan u(n)
−an u(−n − 1)
−nan u(−n − 1)
1
2
3
4
5
6
8
3. Find the inverse z-Transformof the product (z-domain → time
domain):
x(n) = Z −1 {X (z)}
The z-Transform and Its Properties
3.2 Properties of the z-Transform
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3.2 Properties of the z-Transform
Common Transform Pairs
ROC
All z
|z| > 1
|z| > |a|
|z| > |a|
|z| < |a|
|z| < |a|
|z| > 1
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z-Transform expressions that are a fraction of polynomials in
z −1 (or z) are called rational.
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z-Transforms that are rational represent an important class of
signals and systems.
|z| > 1
|z| > |a|
|z| > |a|
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