THE Z-TRANSFORM AND ITS APPLICATION
TO THE ANALYSIS OF LTI SYSTEMS
Contents:
2
§2.1 INTRODUCTION
3
Introduction :
 Wireless communications is fundamentally an information
transmission problem. The transmission of information
through physical media involves the transmission of
signals through systems.
 The signals transmitted and received by antennas are
waveforms that are examples of continuous-time (CT)
signals. They are CT signals because the transmission
media, the antenna/free-space combination, is a CT system.
 As such an understanding of CT signals, both in the time
and frequency domains, is required to design and analyze
a communication system.
4
Introduction :
 Most modern detection techniques sample the received
waveform and use discrete-time (DT) processing to
recover the data.
 The sampling process converts the bandlimited CT signal to
a DT signal, and the algorithm that processes the samples
of the DT signal is a DT system.
 Thus, an understanding of DT signals and systems in
both the time domain and frequency domain is required.
5
Introduction :
 There is a temptation to be familiar with either the CT world
or the DT world, but not both. This division is, in part, a
result of natural divisions in the professional world where
RF circuit designers (CT systems) and DSP algorithm
developers (DT systems) rarely have to forge a close
working relationship.
 After the decision of “where to put the A/D converter” has
been made, the two groups often work independently
from one another.
6
Introduction :
 A good system designer, however, will have equal
expertise in the time domain and frequency-domain
characteristics for both CT and DT systems. Not only
does the system designer have to know both worlds, he
must also understand the relationship between the two.
 The importance of the relationship cannot be overlooked. It
must be remembered that the samples being processed in
DT were once CT waveforms and subject to all the noise and
distortion the CT world has to offer.
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Introduction :
 This chapter assumes the student has already had a junior
level course in signals and systems and understands
frequency domain concepts for both CT signals and systems
and DT signals and systems.
 As such the basics of signals, systems, and frequency
domain concepts are reviewed only briefly. The focus of
the chapter is on the relationship between CT signals and
DT signals.
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§2.2 SIGNALS
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Signals :
10
Signals :
o Energy signals
have 𝑃 = 0 and
power signals
have infinite
energy.
o Signals that are
exactly zero
outside the range
𝑇1 ≤ 𝑡 ≤ 𝑇2 have
zero power and
are thus energy
signals as long as
𝐸 > 0.
Signals with finite nonzero energy are sometimes called Energy Signals
and signals with finite nonzero power are sometimes called Power Signals.
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Signals :
RECALL: 𝑥 𝑡 , 𝑦(𝑡) =
∞
‫׬‬−∞ 𝑥
𝑡 𝑦 ∗ (𝑡) 𝑑𝑡
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Signals :
 Autocorrelation is the correlation of a signal with a delayed copy
of itself as a function of delay.
 For non-deterministic case, it is the similarity between
observations of a random variable as a function of the time lag
between them.
 The analysis of autocorrelation is a mathematical tool for finding
repeating patterns, such as the presence of a periodic signal
obscured by noise, or identifying the missing fundamental
frequency in a signal implied by its harmonic frequencies.
 It is often used in signal processing for analyzing functions or
series of values, such as time domain signals.
13
Signals :
14
Signals :
∞
𝑥 𝑡 , 𝑦(𝑡) = න 𝑥 𝑡 𝑦 ∗ (𝑡) 𝑑𝑡
−∞
15
Signals :
 Formally, only energy signals have a Laplace or Fourier
transform.
 However, if the existence of singularity functions is
allowed, then the class of power signals that are periodic
have these transforms.
 The two singularity functions of special interest in
communication theory are the impulse function and the unitstep function.
16
Signals :
17
𝜹 𝒕 as a limit [1]:
18
Signals :
19
Signals :
20
Signals :
21
Signals :
22
Signals [2] :
If we wish to know the
exact time instant
t = nT of each sample,
we plot s(nT) as a
function of t, as
illustrated in Figure
1.5(c).
23
Signals :
24
Signals :
25
Signals :
26
Signals :
27
§2.3 SYSTEMS
28
Systems :
29
Systems :
30
Systems :
𝑥 𝑡 − 𝑡0
𝑥 𝑡 ⟼𝑦 𝑡
⟼ 𝑦 𝑡 − 𝑡0
Time Invariance
𝑥1 𝑡 ⟼ 𝑦1 𝑡 ; 𝑥2 𝑡 ⟼ 𝑦2 𝑡
𝛼𝑥1 𝑡 + 𝛽𝑥2 (𝑡) ⟼ 𝛼𝑦1 𝑡 + 𝛽𝑦2 (𝑡) Superposition
31
Systems :
32
Systems :
33
Systems :
34
Systems :
35
Systems :
𝑥 𝑛 − 𝑛0
𝑥 𝑛 ⟼𝑦 𝑛
⟼ 𝑦 𝑛 − 𝑛0
Time Invariance
𝑥1 𝑛 ⟼ 𝑦1 𝑛 ; 𝑥2 𝑛 ⟼ 𝑦2 𝑛
𝛼𝑥1 𝑛 + 𝛽𝑥2 (𝑛) ⟼ 𝛼𝑦1 𝑛 + 𝛽𝑦2 (𝑛) Superposition
36
Systems :
37
Systems :
38
Systems :
39
§2.4 FREQUENCY DOMAIN
CHARACTERIZATIONS
40
Frequency Domain Characterizations :
• Why?
• Which?
• Relations?
41
Frequency Domain Characterizations :
42
Frequency Domain Characterizations :
43
Frequency Domain Characterizations :
44
Frequency Domain Characterizations :
45
Frequency Domain Characterizations :
A graphical summary of the three
characterizations is illustrated in
the top portion of Figure 2.4.1.
Figure 2.4.1 The relationships
between the three domains for
describing CT LTI systems and DT
LTI systems.
The connections between CT
systems and DT systems apply
only to strictly bandlimited CT
systems sampled at or above the
minimum rate defined by the
sampling theorem.
46
Frequency Domain Characterizations :
47
Frequency Domain Characterizations :
48
Frequency Domain Characterizations :
49
Frequency Domain Characterizations :
50
Frequency Domain Characterizations :
51
Frequency Domain Characterizations :
52
Frequency Domain Characterizations :
53
Frequency Domain Characterizations :
𝑋𝑑 𝑒 𝑗𝜔 = 𝑋𝑐 𝑗Ω
𝜔
= 𝑋𝑐 𝑗
𝑇
𝜋
𝜋
for − ≤ Ω ≤
𝑇
𝑇
54
Frequency Domain Characterizations :
55
LAPLACE TRANSFORM
56
Laplace Transform :
z-plane and s-plane are related by 𝑧 = 𝑒 𝑠𝑇 .
57
Laplace Transform :
58
Laplace
Transform :
59
Laplace Transform :
60
Laplace Transform :
roots([1 2 5])
ans =
-1.0000 + 2.0000i
-1.0000 - 2.0000i
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Laplace Transform :
Note that if the coefficients 𝑎𝑛
a0 are real, then a complexvalued pole of X(s) must also
be accompanied by another
pole that is its complexconjugate.
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Laplace Transform :
63
Laplace Transform :
64
Laplace Transform :
65
Laplace Transform :
66
Laplace Transform :
67
Laplace Transform :
68
Laplace Transform :
69
Laplace Transform :
[r p k] = residue([0 1 0],[1 2 5])
r=
0.5000 + 0.2500i
0.5000 - 0.2500i
p=
-1.0000 + 2.0000i
-1.0000 - 2.0000i
k=
[]
70
Laplace Transform :
71
Laplace Transform :
72
Laplace Transform :
73
FREQUENCY NOTATION
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Frequency Notation :
75
Frequency Notation :
𝑥 𝑡 = 𝐴 cos 2𝜋𝑓0 𝑡 + 𝜃 ∀𝑡
= 𝐴 cos 𝜔0 𝑡 + 𝜃
𝑥[𝑛] = 𝑥(𝑛𝑇)
𝑥 𝑛 = 𝐴 cos 2𝜋𝑓0 𝑛𝑇 + 𝜃
𝑓0
= 𝐴 cos 2𝜋 𝑛 + 𝜃
𝑓𝑆
= 𝐴 cos 2𝜋𝐹0 𝑛 + 𝜃
= 𝐴 cos Ω0 𝑛 + 𝜃 ∀𝑛
CT Sinusoid
Sampling
DT Sinusoid
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Frequency Notation :
𝜔 : Radian / Sec.
𝜔
𝑓 = : Cycles / Sec. (Hertz)
2𝜋
Ω : Radians / Sample
Ω
𝐹 = : Cycles / Sample
2𝜋
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Frequency Notation :
−𝑓𝑠
𝑓𝑠
−
2
−2𝜋𝑓𝑠
−𝜋𝑓𝑠
0
𝑓𝑠
+
2
1
𝑓𝑠 =
𝑇𝑠
𝑓
Cycles/Sec (Hz)
0
+𝜋𝑓𝑠
2𝜋𝑓𝑠
𝜔 = 2𝜋
Rad/Sec
Cycles/sample
Rad/Samples
−1
−0.5
0
+0.5
1
𝑓
𝐹=
𝑓𝑠
−2𝜋
−𝜋
0
+𝜋
2𝜋
Ω = 2𝜋𝐹
𝑐𝑦𝑐
[
𝑠𝑒𝑐
𝑠𝑒𝑐
𝑠𝑎𝑚𝑝
Normalized Freq. Variable: 𝐹 =
 Ω = 2𝜋𝐹 =
𝑓
2𝜋
𝑓𝑠
= 𝜔𝑇
𝑓
𝑓𝑠
= 𝑓𝑇
×
=
𝑐𝑦𝑐𝑙𝑒𝑠
]
𝑠𝑒𝑐𝑜𝑛𝑑𝑠
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Frequency Notation :
𝐹 = 𝑓𝑇
Ω = 𝜔𝑇
1
𝑇=
𝑓𝑠
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CT FOURIER TRANSFORM
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DISCLAIMER
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These power point slides are NOT
SUBSTITUTE of READING TEXT
BOOK(S).
You’re ALWAYS DIRECTED to
CAREFULLY READ the relevant
book chapter and SOLVE ALL
Examples and End Problems.
82
REFERENCES :
[1] [Proakis-2014] Fundamentals of Communication Systems (2nd Ed)
[2] [Manolakis-2011] Applied Digital Signal Processing
[3]
[4]
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