Electronic Design
Analog signals
Minh Thuy LE
School of Electrical and Electronic Engineering (SEEE),
Hanoi University of Science and Technology (HUST), Vietnam
thuy.leminh@hust.edu.vn
19-Feb-24
Electronic Design – Le Minh Thuy
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Contents
1. Signals definition and representation
2. Frequency Spectrum of Signals
3. Analog and Digital Signals
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Electronic Design – Le Minh Thuy
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Signal Definition & Representation
• signal – contains information
• e.g. voice of radio announcer reading the news
• process – an operation which allows an observer to
understand this information from a signal
• generally done electrically
• transducer – device which converts signal from nonelectrical to electrical form
• e.g. microphone (sound to electrical)
19-Feb-24
Electronic Design – Le Minh Thuy
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Signal Definition & Representation
• Signal representation
o
Thevenin form – represent the signals via voltage
source 𝑣𝑆 (𝑡) and series resistance 𝑅𝑆
▪ Preferable 𝑅𝑆 is small
o
Norton form – represent the signals via current
source 𝑖𝑆 (𝑡) and parallel resistance 𝑅𝑆
▪ Preferable 𝑅𝑆 is high
Figure 1: Two alternative representations of a signal source: (a) the
Thévenin form; (b) the Norton form.
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Electronic Design – Le Minh Thuy
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Spectrum of Signals
• spectrum – defines the a time-domain signal in
terms of the strength of harmonic components
• decomposition – of a periodic function into the
(possibly infinite) sum of simpler oscillating
functions
⇒ Fourier series representation of a periodic signal
𝑓 𝑥 :
∞
𝑎0
𝑓 𝑥 =
+ ෍ [𝑎𝑘 cos 𝑘𝑥 + 𝑏𝑘 sin(𝑘𝑥)]
2
𝑘=1
(1)
1 𝜋
1 𝜋
𝑎𝑘 = න 𝑓 𝑥 cos 𝑘𝑥 𝑑𝑥, 𝑘 ≥ 0 , 𝑏𝑘 = න 𝑓 𝑥 sin 𝑘𝑥 𝑑𝑥, 𝑘 ≥ 1
𝜋 −𝜋
𝜋 −𝜋
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Electronic Design – Le Minh Thuy
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Spectrum of Signals
• Example 1:
The square wave signal below can be represented as:
∞
𝑎0
(2)
𝑓 𝑥 =
+ ෍ [𝑎𝑘 cos 𝑘𝑥 + 𝑏𝑘 sin(𝑘𝑥)]
2
𝑘=1
4𝑉𝑎
1
1
=
[sin 𝜔0 𝑡 + sin 3𝜔0 𝑡 + sin 5𝜔0 𝑡 + ⋯ ]
𝜋
3
5
Figure 2:
Periodic
square
wave
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Electronic Design – Le Minh Thuy
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Spectrum of Signals
• Example 1:
Therefore, we have the frequency spectrum (also called
as line spectrum)
Figure 3: The frequency spectrum (also known as the line spectrum) of
the periodic square wave of Fig. 2
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Electronic Design – Le Minh Thuy
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Spectrum of Signals
• Root Mean Square of a Fourier series:
Root mean square magnitude over a period of time 𝑇 of
a function 𝑓(𝑡) is given by:
𝑡0 +𝑇
1
2
1
𝐹𝑟𝑚𝑠 =
න
𝑓 𝑡 2 𝑑𝑡
𝑇 𝑡0
Therefore, the RMS of a Fourier series is:
∞
𝑎02 1
𝐹𝑟𝑚𝑠 =
+ ෍ 𝑎𝑘2 + 𝑏𝑘2
4 2
(3)
1
2
(4)
𝑘=1
Note: Fourier Transform can be applied to a non-periodic
function of time, but it will yield a continuous frequency
spectrum.
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Electronic Design – Le Minh Thuy
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Analog and Digital Signals
• analog signal – is
continuous with respect to
both value and time
• discrete-time signal – is
continuous with respect to
value but sampled at
discrete points in time
• digital signal – is quantized
(applied to values) as well
as sampled at discrete
points in time
a. Analog signal
b. Discrete-time signal
c. Digital signal
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Electronic Design – Le Minh Thuy
Figure 4: Different types
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of signals
Analog and Digital Signals
sampling
• Question: Are digital and
binary synonymous?
quantization
Figure 5: Relations between three types of signal
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Electronic Design – Le Minh Thuy
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Exercise
• Exercise 1: prove equation (2)
• Exercise 2: calculate the RMS of (2)
• Exercise 3: find the Fourier series of these two
signal (Amplitude of the signal = 1), calculate
the RMS
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Electronic Design – Le Minh Thuy
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