Lecture Signal processing: The z – Transform - Nguyễn Công Phương

Nguyễn Công Phương

SIGNAL PROCESSING
The z – Transform


Contents
I. Introduction
II. Discrete – Time Signals and Systems
III.The z – Transform
IV. Fourier Representation of Signals
V. Transform Analysis of LTI Systems
VI. Sampling of Continuous – Time Signals
VII.The Discrete Fourier Transform
VIII.Structures for Discrete – Time Systems
IX. Design of FIR Filters
X. Design of IIR Filters
XI. Random Signal Processing
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The z – Transform
1.
2.
3.
4.
5.

The z – Transform

The Inverse z – Transform
Properties of the z – Transform
System Function of LTI Systems
LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero Locations
and Time – Domain Behavior
7. The One – Sided z – Transform
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3


The z – Transform (1)
x[n ] =

∞

∑ x[k ]δ [n − k ]

→ y[ n ] =

k =−∞

x[n ] = z , for all n
z = Re( z ) + j Im( z )
n

→ y[ n ] =

∞

∞

k =−∞

k =−∞

∑ x[k ]h[n − k ] = ∑ h[k ]x[n − k ]
∞

∑ h[k ]z

k =−∞

n−k

 ∞

=  ∑ h[k ] z − k  z n ,
 k =−∞


H (z) =

∞

∑ h[k ]z

for all n

−k

k =−∞

→ y[ n ] = H ( z ) z n ,
x[n ] = ∑ ck zkn ,
k

for all n

→ y[n] = ∑ ck H ( zk ) zkn ,

for all n

for all n

k

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4


The z – Transform (2)
X ( z) =

∞

Im( z )

z = e jω

z – plane

∑ x[n]z

−n

ω

n =−∞

0

• ROC (region of convergence):
the set of values of z for which
X(z) converges
• Zeros: the values of z for which
X(z) = 0
• Poles: the values of z for which
X(z) is infinite
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Re( z )
1

Unit circle

Im( z )

z – plane

r sin ω
0

z = re jω

r
ω Re( z )
r cos ω
5


The z – Transform (3)
Ex. 1
∞
Given x1[n ] = {1 2 3 4 5}, x2 [ n] = {1 2 3 4 5}.
↑
↑
X ( z) =
x[ n]z − n
Determine their z – transforms?
n =−∞

∑

X 1 ( z ) = x1[0]z 0 + x1[1] z −1 + x1[2] z −2 + x1[3]z −3 + x1[4]z −4
= 1 + 2 z −1 + 3z −2 + 4 z −3 + 5 z −4
ROC: entire z – plane except z = 0

X 2 ( z ) = x2 [ −2]z − ( −2 ) + x2 [ −1]z − ( −1) + x2 [0]z 0 + x2 [1]z −1 + x2 [2]z −2
= 1z 2 + 2 z + 3z 0 + 4 z −1 + 5z −2
= z 2 + 2 z + 3 + 4 z −1 + 5 z −2
ROC: entire z – plane except z = 0 & z = ∞
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The z – Transform (4)
Ex. 2

x1[n ] = δ [n], x2 [n] = δ [ n − k ], x3[n ] = δ [n + k ], k > 0

Determine their z – transforms?

X ( z) =

∞

∑ x[n]z

n =−∞

X 1 ( z ) = ... + x1[ −1] z − ( −1) + x1[0]z 0 + x1[1]z −1 + x1[2] z −2 + ...
= ... + 0 z − ( −1) + 1z 0 + 0 z −1 + 0 z −2 + ... = 1
ROC: entire z – plane

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7

−n


The z – Transform (5)
Ex. 3
1,
Find the z – transforms of the square – pulse sequence x[ n] = 
0,

X (z) =

∞

∑

n =−∞

x[n ]z

−n

0≤n≤ M
otherwise

M

= ∑1z − n
n =0

M +1
1
−
A
1 + A + A2 + A3 + ... + AN =
, if A < 1
1− A

Im

1 − z −( M +1)
→ X (z) =
1 − z −1
z

−1

Re
0

<1→ z >1

ROC: |z| > 1
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1

ROC
8

The z – Transform (6)
Ex. 4
Find the z – transforms of the sequence x[n] = anu[n]?

X (z) =

∞

∑

n =−∞

∞

x[n ]z − n = ∑ a z

1 + A + A2 + A3 + ... =

n =0

n −n

∞

= ∑ (az −1 )n
n=0

1

, if A < 1
1− A

→ X (z) =

az −1 < 1 → z > a
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1
z
=
1 − az −1 z − a
Zero: z = 0
Pole: z = a
ROC: |z| > a
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The z – Transform (7)
Ex. 4
Find the z – transforms of the sequence x[n] = anu[n]? X ( z ) =

1
z
=
1 − az −1 z − a

Zero: z = 0; pole: z = a; ROC: |z| > a
1

0< a <1

a >1

a =1

1

…
…

…

…

0

…

n

n

0

Im

a

n

0

Im

1 Re

Im

1 Re

1

0

0

ROC

…

1

ROC

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a Re

0

ROC

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The z – Transform (8)
Ex. 5
n≥0

0,
Find the z – transforms of the sequence x[ n] = −a u[ −n − 1] =  n
 −a ,
n

X (z) =

∞

∑

n =−∞

−1

−1

x[n ]z − n = − ∑ a n z − n = − ∑ (az −1 )n
n =−∞

n<0

n =−∞

0

…

n
…

= −a −1 z(1 + a −1 z + a −2 z 2 + ...)

1
1 + A + A + A + ... =
, if A < 1
1− A
2

3

1
z
Zero: z = 0
→ X ( z ) = −a z
=
1 − a −1 z z − a Pole: z = a
a −1 z < 1 → z < a ROC: |z| < a
z
x[n] = a n u[n ] → X ( z ) =

Zero: z = 0; pole: z = a; ROC: |z| > a
z −a
−1

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The z – Transform (9)
Ex. 6
 a n ,
Find the z – transforms of the sequence x[ n] =  n
 −b ,

X (z) =

∞

∑

n =−∞

x[n ]z

−n

−1

n<0

∞

= − ∑ b z + ∑ an z−n
n =−∞

n −n

−1

n =0

z
If b z < 1 → z < b → − ∑ b z =
z −b
n =−∞
∞
z
−1
n −n
If az < 1 → z > a → ∑ a z =
z−a
n =0
−1

n≥0

n −n

z

z
→ X ( z) =
+
z−b z−a
Zero: z = 0
Pole: z = a, b
ROC: a < |z| < b

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The z – Transform (10)
a

…

0

…

0

…

n

…
…

n

0

n

…
b

Im

a

Im

Re

Re

0
ROC

ROC: |z| > a

Im

0

0

b

ROC

ROC: |z| < a
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a

Re
b

ROC

ROC: a < |z| < b
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The z – Transform (11)
Ex. 7
Find the z – transforms of the sequence x[ n] = r n (cos ω0n)u[n ], r > 0, 0 ≤ ω0 ≤ 2π

X (z) =

∞

∑

x[n ]z

−n

n =−∞

∞

= ∑ r n (cos ω0n ) z − n
n =0

1 jθ 1 − jθ
e = cos θ + j sin θ → cos θ = e + e
2
2
jθ

1 ∞
1 ∞
jω0 −1 n
→ X ( z) = ∑ ( re z ) + ∑ ( re − jω0 z −1 )n
2 n =0
2 n =0

e jθ = cos θ + j sin θ = cos2 θ + sin 2 θ = 1
re jω0 z −1 < 1 & re − jω0 z −1 < 1 → rz −1 < 1 → z > r
→ X (z) =

1
1
1

1
+
, ROC : z > r
jω0 −1
− jω0 −1
2 1 − re z
2 1 − re z
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The z – Transform (12)
Ex. 7
Find the z – transforms of the sequence x[ n] = r n (cos ω0n)u[n ], r > 0, 0 ≤ ω0 ≤ 2π

1
1
1
1
+
2 1 − re jω0 z −1 2 1 − re − jω0 z −1
(1 − re − jω0 z −1 ) + (1 − re jω0 z −1 )
2 − rz −1 ( e − jω0 + e jω0 )
=
=
jω0 −1
− jω0 −1
2(1 − re z )(1 − re z ) 2[1 − 2(r cos ω0 ) z −1 + r 2 z −2 ]

X (z) =

e jθ = cos θ + j sin θ → e jθ + e − jθ = 2 cosθ
Im

p1
r Re

0 z1 z2

1 − r(cos ω0 ) z −1
→ X (z) =
1 − 2(r cos ω0 ) z −1 + r 2 z −2
z( z − r cos ω0 )
=
( z − re jω0 )( z − re − jω0 )
Zero: z1 = 0; z2 = rcosω0

p2
ROC

Pole : p1 = re jω0 ; p2 = re − jω0
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ROC: |z| > r

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The z – Transform (13)

• ROC
– There is no pole inside a ROC
– The ROC is a connected region
– For finite duration sequences, the ROC is the
entire z – plane, sometimes except for z = 0 and
z=∞

• The z – transform
– We need both X(z) and its ROC
– X(z) is not defined outside the ROC
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16


The z – Transform (14)

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17


The z – Transform
1.
2.
3.
4.
5.

The z – Transform

The Inverse z – Transform
Properties of the z – Transform
System Function of LTI Systems
LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero Locations
and Time – Domain Behavior
7. The One – Sided z – Transform
sites.google.com/site/ncpdhbkhn

18


The Inverse z – Transform (1)

x[n] =

1

∫
2π j

n −1

C

X ( z ) z dz

b0 + b1 z −1 + b2 z −2 ... + bN −1 z − ( N −1)
X ( z) =

−1
−2
−N
1 + a1 z + a2 z ... + a N z

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The Inverse z – Transform (2)
Ex. 1
1 + z −1
X (z) =
(1 − z −1 )(1 − 0.2 z −1 )

1 + z −1
K1
K2
=
+
(1 − z −1 )(1 − 0.2 z −1 ) 1 − z −1 1 − 0.2 z −1

→ 1 + z −1 = K1 (1 − 0.2 z −1 ) + K 2 (1 − z −1 )
z = 1 → 1 + 1 = K1(1 − 0.2 × 1) + K 2 (1 − 1)

→ K1 = 2.5

z = 0.2 → 1 + 5 = K1 (1 − 0.2 × 5) + K 2 (1 − 5)

→ K 2 = −1.5

→ X ( z) =
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2.5
1.5
−
1 − z −1 1 − 0.2 z −1
20


The Inverse z – Transform (3)
Ex. 1
1 + z −1
2.5
1. 5
X (z) =
=
−
(1 − z −1 )(1 − 0.2 z −1 )
1 − z −1 1 − 0.2 z −1
 2.5
1
→ 2.5u[ n]
−1

a u[n] →
,
ROC

:
z
>
a

1 − az −1
→ 1 − z
 −1.5 → −1.5(0.2) n u[n ]
If z > 1
1 − 0.2 z −1
n

→ x[ n] = 2.5u[ n] − 1.5(0.2) n u[n ]

 2.5
1
→ −2.5u[ − n − 1]
−1

− a u[ − n − 1] →
,
ROC
:
z
<
a

1 − az −1
→ 1 − z
 −1.5 → 1.5(0.2)n u[ − n − 1]

If z < 0.2
1 − 0.2 z −1
n

→ x[ n] = −2.5u[ − n − 1] + 1.5(0.2) n u[ − n − 1]

 2.5
1 − z −1 → −2.5u[ − n − 1]
If 0.2 < z < 1 → 
 −1.5 → −1.5(0.2) n u[n ]
1 − 0.2 z −1
→ x[ n] = −2.5u[ − n − 1] − 1.5(0.2) n u[n ]
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21


The Inverse z – Transform (4)
Ex. 2
1 + z −1
X (z) =
1 − z −1 + 2.5 z −2
1 − z −1 + 2.5z −2 = 0 → p1,2 = 0.5 ± j1.5 = 1.58e ± j1.25

1 + z −1
K1
K2
=
+
1 − z −1 + 2.5 z −2 1 − p1z −1 1 − p2 z −1

→ 1 + z −1 = K1(1 − p2 z −1 ) + K 2 (1 − p1z −1 )
z = p1 → 1 + 1 / p1 = K1 (1 − p2 / p1 ) + K 2 (1 − 1) → K1 = 0.5 − j 0.67 = 0.83e − j 0.93
z = p2 → 1 + 1 / p2 = K1 (1 − 1) + K 2 (1 − p1 / p2 ) → K 2 = 0.5 + j 0.67 = 0.83e j 0.93

→ x[n] = 0.83e − j 0.93 (1.58e j1.25 )n u[n] + 0.83e j 0.93 (1.58e− j1.25 )n u[n]
= 0.83(1.58) n (e j (1.25n−0.93) + e − j (1.25n −0.93) )
e j (1.25n −0.93) + e− j (1.25n−0.93) = 2 cos(1.25n − 0.93)

→ x[ n] = 1.67(1.58)n cos(1.25n − 0.93)u[ n] = 1.67(1.58) n cos(1.25n − 53.13o )u[n ]
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22


The z – Transform
1.
2.
3.
4.
5.

The z – Transform
The Inverse z – Transform
Properties of the z – Transform
System Function of LTI Systems
LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero Locations
and Time – Domain Behavior

7. The One – Sided z – Transform
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23


Properties of the z – Transform

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24


The z – Transform
1.
2.
3.
4.
5.

The z – Transform
The Inverse z – Transform
Properties of the z – Transform
System Function of LTI Systems
LTI Systems Characterized by Linear
Constant – Coefficient Difference Equations
6. Connections between Pole – Zero Locations
and Time – Domain Behavior
sites.google.com/site/ncpdhbkhn

25