1) The document introduces concepts related to high frequency electronic circuits and communication systems, including dB definitions, phasors, modulation, linear modulation and transmitters.
2) It discusses phasor representation in the complex plane and how phasors can represent sinusoidal signals.
3) It covers various modulation techniques including amplitude modulation, frequency modulation, phase modulation, and linear modulation. Linear modulation uses an in-phase (I) component and quadrature (Q) component to modulate the carrier signal.
In telecommunications and signal processing, frequency modulation (FM) is the encoding of information in a carrier wave by varying the instantaneous frequency of the wave. This contrasts with amplitude modulation, in which the amplitude of the carrier wave varies, while the frequency remains constant.
In analog frequency modulation, such as FM radio broadcasting of an audio signal representing voice or music, the instantaneous frequency deviation, the difference between the frequency of the carrier and its center frequency, is proportional to the modulating signal.
In this presentation we discuss about a particular type of analog communication waves that is wideband frequency modulation. In this slide, its expression is discussed along with graphical visuals. Not forgetting its power and bandwidth as well. We also see the use of bessel function and the block diagrams that help to form this type of waves.
This presentation covers noise performance of Continuous wave modulation systems; It explains modelling of white noise , noise figure of DSB-SC, SSB, AM, FM system
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
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3. dB的定義
• , where
• Power gain
• Voltage gain
• Power (dBW)
• Power (dBm)
• Voltage (dBV)
• Voltage (dBuV)
( )dB 10 log G= ⋅ ( )aG
b
=
2
1
10 log
P
P
= ⋅
2
1
20 log
V
V
= ⋅
( )10 log
1-W
P= ⋅
( )10 log
1-mW
P= ⋅
( )20 log
1-Volt
V= ⋅
( )20 log
1- V
V
µ= ⋅
相對的相對的相對的相對的(Relative )
(比值比值比值比值, 無單位無單位無單位無單位, dB)
絕對的絕對的絕對的絕對的(Absolute )
(單位單位單位單位, dBW, dBm, dBV…)
Department of Electronic Engineering, NTUT3/40
4. In some textbooks, phasor may be
represented as
尤拉公式
• Euler’s Formula states that: cos sinjx
e x j x= +
( ) ( ) ( )
{ } { }cos Re Re
j t j j t
p p pv t V t V e V e e
ω φ φ ω
ω φ +
= ⋅ + = ⋅ = ⋅
( )cos sin
def
j
p p pV V e V V jφ
φ φ φ= ⋅ = ∠ = +• Phasor :
Don’t be confused with Vector which is commonly denoted as .A
phasor
A real signal can be represented as:
V
V
( ) ( )cospv t V tω φ= ⋅ +
Department of Electronic Engineering, NTUT4/40
5. Euler’s Trick on the Definition of e
2 3
lim 1 1
1! 2! 3!
n
x
n
x x x x
e
n→∞
= + = + + + +
…
x jx=
( ) ( )
2 3 2 4 3 5
1 1
1! 2! 3! 2! 4! 3! 5!
jx jx jxjx x x x x
e j x
= + + + + = − + − + + − + − +
… … …
• Euler played a trick : Let , where 1j = −
1
lim 1
n
n
e
n→∞
= +
6/33
2 4
cos 1
2! 4!
x x
x = − + − +…
3 5
sin
3! 5!
x x
x x= − + − +…
cos sinjx
e x j x= +
cos sinjx
e x j x−
= −
cos
2
jx jx
e e
x
−
+
=
sin
2
jx jx
e e
x
j
−
−
=
• Use and
we have
Department of Electronic Engineering, NTUT5/40
6. 座標系統
x-axis
y-axis
x-axis
y-axis
P(r,θ)
θ
r
P(x,y)
2 2
r x y= +
1
tan
y
x
θ −
=
cosx r θ=
siny r θ=
Cartesian Coordinate System Polar Coordinate System
(x,0)
(0,y)
( )cos ,0r θ
( )0, sinr θ
Projection
on x-axis
Projection
on y-axis
Department of Electronic Engineering, NTUT6/40
8. x
θ
0
π/2
π
3π/2
餘弦波形
x-axis
y-axis
θ
Go along the circle, the projection
on x-axis results in a cosine wave.
Sinusoidal waves relate to a Circle
very closely.
Complete going along the circle to
finish a cycle, and the angle θ
rotates with 2π rads and you are
back to the original starting-point
and. Complete another cycle again,
sinusoidal waveform in one period
repeats again. Keep going along the
circle, the waveform will
periodically appear.
Department of Electronic Engineering, NTUT8/40
9. 複數平面(I)
It seems to be the same thing with x-y plan, right?
• Carl Friedrich Gauss (1777-1855) defined the complex plan.
He defined the unit length on Im-axis is equal to “j”.
A complex Z = x + jy can be denoted as (x, yj) on the complex plan.
(sometimes, ‘j’may be written as ‘i’which represent imaginary)
Re-axis
Im-axis
Re-axis
Im-axis
P(r,θ)
θ
r
P(x,yj)
2 2
r x y= +
1
tan
y
x
θ −
=
cosx r θ=
siny r θ=
(x,0j)
(0,yj)
( )cos ,0r θ
( )0, sinr θ
( )1j = −
Department of Electronic Engineering, NTUT9/40
10. 複數平面(II)
Re-axis
Im-axis
1
Every time you multiply something by j, that thing will rotate 90 degrees.
1j = − 2
1j = − 3
1j = − − 4
1j =
1*j=j
j
j*j=-1
-1
-j
-1*j=-j -j*j=1
(0.5,0.2j)
(-0.2, 0.5j)
(-0.5, -0.2j)
(0.2, -0.5j)
• Multiplying j by j and so on:
Department of Electronic Engineering, NTUT10/40
11. 正弦波
Re-axis
Im-axis
P(x,y)
x
y
r
θ θθ
y = rsinθ
θ
0 π/2 π 3π/2 2π
To see the cosine waveform, the same operation can be applied to trace out
the projection on Re-axis.
Department of Electronic Engineering, NTUT11/40
12. 相量表示法 (I) – 以sine為基底
( ) ( ) { } { }sin Im Imj j t j j
sv t A t Ae e Ae eφ ω φ θ
ω φ= + = =
Re-axis
Im-axis
P(A,ϕ)
y = Asinϕ
θ
0 π/2 π 3π/2 2π
ϕ
tθ ω=
Given the phasor denoted as a point on the complex-plan, you should know it
represents a sinusoidal signal. Keep this in mind, it is very important!
time-domain waveform
Department of Electronic Engineering, NTUT12/40
13. 相量表示法 (II) – 以cosine為基底
( ) ( ) { } { }cos Re Rej j t j j
sv t A t Ae e Ae eφ ω φ θ
ω φ= + = =
Re-axis
Im-axis
P(A, ϕ)
y = Acos ϕ
θ
0 π/2 π 3π/2 2π
ϕ
tθ ω=
time-domain waveform
Department of Electronic Engineering, NTUT13/40
14. 相量表示法 (III)
( ) ( ) { }1
1 1 1 1sin Im j j t
v t A t Ae eφ ω
ω φ= + =
Re-axis
Im-axis
P(A1, ϕ1)
ϕ1
P(A2, ϕ2)
P(A3, ϕ3)
θ
0 π/2 π 3π/2 2π
tθ ω=
A1sin ϕ1
( ) ( ) { }2
2 2 2 2sin Im j j t
v t A t A e eφ ω
ω φ= + =
( ) ( ) { }3
3 3 3 3sin Im j j t
v t A t A e eφ ω
ω φ= + =
A2sin ϕ2
A3sin ϕ3
Department of Electronic Engineering, NTUT14/40
15. 到處都是相量
• Circuit Analysis, Microelectronics:
Phasor is often constant.
• Field and Wave Electromagnetics, Microwave Engineering:
Phasor varies with the propagation distance.
• Communication System:
Phasor varies with time (complex envelope, envelope, or
equivalent lowpass signal of the bandpass signal).
( ) ( )5cos 1000 30sv t t= + 5 30sV = ∠
( ) ( ) ( ) ( ) ( )
{ }, cos cos Re
j x t j x t
v x t A x t B x t Ae Be
β ω β ω
β ω β ω − − +
= − + + = +
( ) j x j x
V x Ae Beβ β−
= +
( ){ }Re j t
V x e ω
=
Department of Electronic Engineering, NTUT15/40
16. 調變(調制)
• Why modulation?
Communication
Bandwidth
Antenna Size
Security, avoid Interferes, etc.
Voice
Electric signal
Audio
Equipment
Audio
Equipment
Modulator Demodulator
Electric signal
Voice
Department of Electronic Engineering, NTUT16/40
17. 振幅調變(Amplitude Modulation)
( ) ( ) cos2m BB cs t s t A f tπ= ⋅
Baseband real signal
Voice
Electric signal
Audio
Equipment
Audio
Equipment
Modulator Demodulator
Electric signal
Voice
( )BBs t
cos2 cA f tπ
Carrier (or local)
High-frequency sinusoid
Amplitude-modulated signal
(AM signal)
Department of Electronic Engineering, NTUT17/40
18. 頻率調變(Frequency Modulation)
( ) ( ){ }cos 2m c f BBs t A f K s t tπ = + ⋅
Voice
Electric signal
Audio
Equipment
Audio
Equipment
Modulator Demodulator
Electric signal
Voice
Baseband real signal
( )BBs t
cos2 cA f tπ
Carrier (or local)
High-frequency sinusoid
Frequency-modulated signal
(FM signal)
Department of Electronic Engineering, NTUT18/40
19. 相位調變(Phase Modulation)
Voice
Electric signal
Audio
Equipment
Audio
Equipment
Modulator Demodulator
Electric signal
Voice
( ) ( )cos 2m c p BBs t A f t K s tπ = +
( )cos 2 c BBA f t tπ φ= +
Baseband real signal
( )BBs t
cos2 cA f tπ
Carrier (or local)
High-frequency sinusoid
Phase-modulated signal
(PM signal)
Department of Electronic Engineering, NTUT19/40
20. 線性調變(Linear Modulation)
( ) ( ) ( )cos 2m BB c BBs t A t f t tπ φ= ⋅ +
Voice
Electric signal
Audio
Equipment
Audio
Equipment
Modulator Demodulator
Electric signal
Voice
Baseband real signal
( )BBs t
cos2 cA f tπ
Carrier (or local)
High-frequency sinusoid
Linear-modulated signal
( )BBs t ( ) ( ), ?BB BBA t tφ
Department of Electronic Engineering, NTUT20/40
21. 線性調變之數學推導
• Consider a modulated signal
( ) ( ) ( ) ( ) ( )
{ }2
cos 2 Re c BBj f t t
m BB c BB BBs t A t f t t A t e
π φ
π φ
+
= ⋅ + = ⋅
( ) ( )
( ) ( ) ( ){ }2 2
Re Re cos sinBB c cj t j f t j f t
BB BB BB BBA t e e A t t j t eφ π π
φ φ = ⋅ = ⋅ +
( ) ( ) ( )
( ) ( ) ( )cos sinBBj t
l BB BB BB BBs t A t e A t t j t
φ
φ φ= ⋅ = ⋅ +
( ) ( ) ( ) ( ) ( ) ( )cos sinBB BB BB BBA t t jA t t I t jQ tφ φ= ⋅ + ⋅ = +
( ) ( ) ( ) ( ){ }Re cos2 sin2m c cs t I t jQ t f t j f tπ π= + ⋅ +
( ) ( )cos2 sin 2c cI t f t Q t f tπ π= −
Time-varying phasor (information in both amplitude and phase)
( )BBs t : real
( )ls t : complex
Modulated signal is the linear combination of I(t), Q(t), and the carrier. Thus the linear modulator
is also called “I/Q Modulator,” and it is an universal modulator.
Department of Electronic Engineering, NTUT21/40
22. 線性調變器
• The modulator accomplishes the mathematical operation.
( ) ( ) ( ) ( ) ( ){ }Re cos sin cos2 sin 2m BB BB BB c cs t A t t j t f t j f tφ φ π π= ⋅ + +
( ) ( ) ( ) ( )cos cos2 sin sin 2BB BB c BB BB cA t t f t A t t f tφ π φ π= −
( ) ( )cos2 sin 2c cI t f t Q t f tπ π= −
( )I t
cos ctω
sin ctω−
( )Q t
( )ms t
( )I t
cos ctω
sin ctω
( )Q t
( )ms t
+
− 90
( )I t
cos ctω
( )Q t
( )ms t
Department of Electronic Engineering, NTUT
I component Q component
I channel Q channel
22/40
23. 線性發射機架構
• Linear Transmitter
90
( )I t
cos ctω
( )Q t
( )ms t
Power Amplifier
(PA)
Antenna
Baseband
Processor
90
cos ctω
( )ms t
Power Amplifier
(PA)
Antenna
Matching /
BPF
Matching
( )I t
( )Q t
Baseband
Processor
Department of Electronic Engineering, NTUT23/40
24. 線性解調變
( ) ( ) ( ) ( ) ( )cos 2 cos2 sin2m BB c BB c cs t A t f t t I t f t Q t f tπ φ π π= ⋅ + = −
( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1
cos2 cos 2 sin2 cos2 cos4 1 sin4 sin0
2 2
m c c c c c cs t f t I t f t Q t f t f t I t f t Q t f tπ π π π π π= − ⋅ = ⋅ + − ⋅ +
( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1
sin2 cos2 sin2 sin 2 sin4 sin0 1 cos4
2 2
m c c c c c cs t f t I t f t f t Q t f t I t f t Q t f tπ π π π π π− = − + = − ⋅ + + ⋅ −
( ) ( ) ( )cos4 sin 4
2 2 2
c c
I t I t Q t
f t f tπ π
= + −
( ) ( ) ( )sin4 cos4
2 2 2
c c
Q t I t Q t
f t f tπ π
= − +
?
Receiver
( )ms t ( )BBs t
Received modulated signal:
Multiplied by “cosine”:
Multiplied by “−−−− sine”:
High-frequency components
(should be filtered out)
High-frequency components
(should be filtered out)
Department of Electronic Engineering, NTUT24/40
25. 線性解調器
( )I t
cos ctω
sin ctω−
( )Q t
( )ms t
LPF
LPF
( )I t
( )Q t
( )ms t
LPF
LPF
90
cos ctω
( ) ( ) ( )
( ) ( )BBj t
l BBs t A t e I t jQ t
φ
= ⋅ = +
( ) ( ) ( )2 2
BBA t I t Q t= +
( )
( )
( )
1
tanBB
Q t
t
I t
φ −
=
Baseband
Processing
Original Information (or data)
( )I t
( )Q t
Department of Electronic Engineering, NTUT25/40
26. 線性接收機架構
• Linear Receiver (direct conversion)
90
( )I t
cos ctω
( )Q t
( )ms t
Low Noise Amplifier
(LNA)
Baseband
Processor
LPF
LPF
Matching /
BPF
90
( )I t
cos ctω
( )Q t
( )ms t
Low Noise Amplifier
(LNA)
Baseband
Processor
LPF
LPF
Matching
Department of Electronic Engineering, NTUT26/40
27. 調變訊號的頻譜
• Fourier Series Representations
• Non-periodic Waveform and Fourier Transform
• Spectrum of a Real Signal
• AM, PM, and Linear Modulated Signal
• Concept of Complex Envelope
Department of Electronic Engineering, NTUT27/40
28. 傅立葉級數
• There are three forms to represent the Fourier Series of a
periodic signal :
Sine-cosine form
Amplitude-phase form
Complex exponential form
( ) ( )0 1 1
1
cos sinn n
n
x t A A n t B n tω ω
∞
=
= + +∑
( ) ( )0 1
1
cosn n
n
x t C C n tω φ
∞
=
= + +∑
( ) 1jn t
n
n
x t X e ω
∞
=−∞
= ∑
( )x t
Department of Electronic Engineering, NTUT
t
x(t)
t
t
t
( )X jω
ω
1f 13 f 15 f
.etc
T1
1 1C φ∠
2 2C φ∠
3 3C φ∠
28/40
29. Sine-Cosine Form
( )0 0
area under curve in one cycle
period T
1 T
A x t dt
T
= =∫
( ) 10
2
cos , for 1 but not for 0
T
nA x t n tdt n n
T
ω= ≥ =∫
( ) 10
2
sin , for 1
T
nB x t n tdt n
T
ω= ≥∫
is the DC term
(average value over one cycle)
• Other than DC, there are two components appearing at a given
harmonic frequency in the most general case: a cosine term
with an amplitude An and a sine term with an amplitude Bn.
(A complete cycle can also be noted
from )~
2 2
T T−
Department of Electronic Engineering, NTUT29/40
30. Amplitude-Phase Form
( ) ( )0 1
1
cosn n
n
x t C C n tω φ
∞
=
= + +∑
( ) ( )0 1
1
sinn n
n
x t C C n tω θ
∞
=
= + +∑
2 2
n n nC A B= +
• The sum of two or more sinusoids of a given frequency is
equivalent to a single sinusoid at the same frequency.
• The amplitude-phase form of the Fourier series can be
expressed as either
or
0 0C A= is the DC term
is the net amplitude of a given component at frequency nf1,
since sine and cosine phasor forms are always
perpendicular to each other.
where
Department of Electronic Engineering, NTUT30/40
31. Complex Exponential Form (I)
1
1 1cos sinjn t
e n t j n tω
ω ω= +
1
1 1cos sinjn t
e n t j n tω
ω ω−
= −
1 1
1cos
2
jn t jn t
e e
n t
ω ω
ω
−
+
=
1 1
1sin
2
jn t jn t
e e
n t
j
ω ω
ω
−
−
=
cos sinjx
e x j x= +
cos sinjx
e x j x−
= −
cos
2
jx jx
e e
x
−
+
=
sin
2
jx jx
e e
x
j
−
−
=
Recall that
• Euler’s formula
1
nω is called the positive frequency, and 1
nω− the negative frequency
From Euler’s formula, we know that both positive-frequency and negative-
frequency terms are required to completely describe the sine or cosine
function with complex exponential form.
Here
1jn t
e ω
1jn t
e ω−
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32. Complex Exponential Form (II)
1 1jk t jk t
k kX e X eω ω−
−+ ( )where kkX X− =
( ) 1jn t
n
n
x t X e ω
∞
=−∞
= ∑
( ) 1
0
1 T
jn t
nX x t e dt
T
ω−
= ∫
• The general form of the complex exponential form of the
Fourier series can be expressed as
where Xn is a complex value
• At a given real frequency kf1, (k>0), that spectral representation
consists of
The first term is thought of as the “positive frequency” contribution, whereas the second is the
corresponding “negative frequency” contribution. Although either one of the two terms is a
complex quantity, they add together in such a manner as to create a real function, and this
is why both terms are required to make the mathematical form complete.
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33. 當週期趨近無限大
T 2T 3T 4T 5T
( )x t
f
nX
T 2T
T
T
f
nX
f
nX
f
nX
Single pulse T → ∞
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34. 傅立葉轉換
( ) ( )X f F x t= F ( ) ( )1
x t F X f−
= F
( ) ( ) j t
X f x t e dtω
∞
−
−∞
= ∫
( ) ( ) j t
x t X f e dfω
∞
−∞
= ∫
• Fourier transformation and its inverse operation :
• The actual mathematical processes involved in these operations
are as follows:
2 fω π=
• The Fourier transform is, in general, a complex function
and has both a magnitude and an angle:
( )X f
( ) ( ) ( )
( ) ( )j f
X f X f e X f fφ
φ= = ∠
( )X f
f
For the nonperiodic signal, its spectrum is continuous, and, in
general, it consists of components at all frequencies in the
range over which the spectrum is present.
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35. 調變譜 (I)
• From Euler’s Formula :
• AM signal (DSB-SC)
cos
2
jx jx
e e
x
−
+
=
A “real signal” is composed of positive and negative frequency components.
( ) ( )cos2m cs t A t f tπ=
Two-sided amplitude frequency spectrum
( ) ( )2 1000 2 10001
50cos 2 1000
2
j t j t
t e eπ π
π × − ×
× = +
2525
0 Hz 1 kHz1 kHz−
f
One-sided amplitude frequency spectrum
50
0 Hz 1 kHz
( )50cos 2 1000tπ ×
f
t( ) ( )BBs t A t=
f
f
cf0 Hzcf−
0 Hz
USBLSB
USBLSBLSBUSB
cos2 cf tπ
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“real signal”
35/40
36. Phase
Modulator
調變譜 (II)
t( )BBs t
f
0 Hz
USBLSB
cos2 cf tπ
( ) ( )2 2
2 2
c cj t j tj f t j f tA A
e e e e
φ φπ π− −
= +
( ) ( )( )cos 2m cs t A f t tπ φ= +
( )
{ } ( )
{ }2 2
Re Rec c
j f t t j t j f t
A e A e e
π φ φ π+
= ⋅ = ⋅
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“real signal”
f
cf0 Hzcf−
USBLSBLSBUSB
“complex”“complex” “real”
• PM signal
Complex conjugate
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37. 調變譜 (III)
I/Q
Modulator
t( )BBs t
f
0 Hz
USBLSB
cos2 cf tπ
( ) ( ) ( ) ( )2 2
2 2
c cj t j tj f t j f tA t A t
e e e eφ φπ π− −
= +
( ) ( ) ( )( )cos 2m cs t A t f t tπ φ= +
( ) ( )
{ }2
Re cj t j f t
A t e eφ π
= ⋅
“real signal”
• I/Q modulated signal
( )I t
( )Q t
f
cf0 Hzcf−
USBLSBLSBUSB
“complex”“complex” “real”
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Complex conjugate
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38. 複數波包的概念 (I)
• Bandpass real signal :
( ) ( ) ( )( )
( ) ( ) ( ) ( )2 2
cos 2
2 2
c cj t j tj f t j f t
m c
A t A t
s t A t f t t e e e e
φ φπ π
π φ − −
= + = +
( ) ( )
( ) ( )2 21 1
2 2
c cj t j tj f t j f t
A t e e A t e eφ φπ π− −
= +
( )ls t ( )ls t∗
( )lS f∗
( )lS f
Complex timed value
Spectrum
( ) ( )
( ) ( )2 21 1
2 2
c cj t j tj f t j f t
A t e e A t e eφ φπ π− −
= +
( ) 2 cj f t
ls t e π
⋅ ( ) 2 cj f t
ls t e π−∗
⋅
( )l cS f f∗
− −( )l cS f f−
Complex timed value
Spectrum
( ) ( ) ( )
1
2
m l c l cS f S f f S f f∗
= − + − −
f
cf0 Hzcf−
USBLSBLSBUSB
( )
1
2
l cS f f−( )
1
2
l cS f f∗
− −
Spectrum of the bandpass signal
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39. 複數波包的概念 (II)
• Equivalent low-pass signal (complex envelope):
f
0 Hz
( )lS f
cfcf−
( ) 21
2
cj f t
ls t e π
⋅( ) 21
2
cj f t
ls t e π−∗
⋅
( ) ( ) ( )
( ) ( )j t
ls t A t e I t jQ t
φ
= = +
( ) ( ) ( )
1
2
m l c l cS f S f f S f f∗
= − + − −
f
cf0 Hzcf−
USBLSBLSBUSB
( ) ( )
1
2
I t jQ t+
Spectrum of the bandpass signal
( ) ( )
1
2
I t jQ t−
( )ms t
( ) ( ) ( )
( ) ( )BBj t
ls t A t e I t jQ t
φ
= = +
complex envelope
( ) ( ) ( ) ( ) ( ) 2
cos 2 Re cj t j f t
m cs t A t f t t A t e eφ π
π φ = ⋅ + = ⋅
( ) ( ){ }2
Re cj f t
I t jQ t e π
= +
complex envelope
carriercarrier2 cj f t
e π
carrier
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40. 本章總結
• In this chapter, the phasor was introduced to manifest itself in
the mathematical operation for communication engineering.
• A modulated signal is a linear combination of I(t), Q(t), and
the carrier. This mathematical combination can be realized
with a practical circuitry, say, “modulator.”
• The demodulation is the decomposition of the modulated
signal, which is the reverse process to recover the baseband
signal I(t) and Q(t).
• The modulated signal can be viewed as a complex envelope
carried by a sinusoidal carrier. With this equivalent lowpass
form to represent a bandpass system, the mathematical
analysis can be easily simplified.
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