A Preliminary Discussion EE442 Analog & Digital Communication Systems Lecture...
Transcript of A Preliminary Discussion EE442 Analog & Digital Communication Systems Lecture...
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EE 442 Signal Preliminaries 1
SignalsA Preliminary Discussion
EE442 Analog & Digital Communication SystemsLecture 2
The Fourier transform of single pulse is the sinc function.
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ES 442 Signal Preliminaries 2
Communication Systems and Signals
Information converted into an electrical waveform suitable fortransmission is called a signal. Signals are time-varying.
A communication system is a collection of devices used to sendmessages or information from a source (i.e., a transmitter) to adestination (i.e., a receiver) over a communication channel (i.e., a propagation medium).
In communication systems a source generates the message or information to be communicated. The message or informationtypically falls into one of three general categories:
Voice/audioDataVideo
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ES 442 Signal Preliminaries 3
Definition and Classification of Signals – I
A signal is a function of time that represents a physical quantity.
For EE442 a signal is a waveform containing or encoding information.
A signal may be a voltage, current, electromagnetic field, or anotherphysical parameter such as air pressure in an acoustic signal.
A signal may be either deterministic or random (i.e., stochastic).Deterministic signals are by far the most common.
There are two domains in which to describe signals:(1) Time-domain waveform(2) Frequency-domain spectrum
This requires us to use Fourier analysis.
Read: Chapter 2of Agbo & Sadiku;Sections 2.1 & 2.2
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EE 442 Signal Preliminaries 4
General Classifications of Signals
Deterministic Random (Stochastic)
Periodic AperiodicQuasi-
periodicStationary
Non-stationary
Sinusoidal,Triangular,
Rectangular
Transient,Unit pulse response
ECG waveform,Temperature record
Language,Music,
etc.
Noise inElectronicCircuits
Mathematicalrepresentation
possible
Often can calculate the
waveform
Roughlyapproximate
mathematically
White GaussianNoise
Voicewaveform
Not mathematically calculable
Which of these categories don’t contain information?
http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjiwLf96o7KAhVBT2MKHS_AD6cQjRwIBw&url=http://ceng.gazi.edu.tr/dsp/periodic_signals/description.aspx&psig=AFQjCNEQf3C5H_PW_5B_3neguyvZFBr8cA&ust=1451951492289743https://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwj0_sic7I7KAhVB4WMKHbC6B9QQjRwIBw&url=https://en.wikipedia.org/wiki/Transient_response&psig=AFQjCNE08o57R7YNjC22G8M2pbnJuQsg4Q&ust=1451951796146727http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiOjOGX7Y7KAhUW2mMKHX08CqUQjRwIBw&url=http://www.swharden.com/blog/category/diy-ecg-home-made-electrocardiogram/feed/&psig=AFQjCNH17qy81UrixaykELutTcHIsXEvew&ust=1451951893470124http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwi8vOCo7o7KAhUM7GMKHVeCCFcQjRwIBw&url=http://www.school-for-champions.com/science/noise_reduction.htm&psig=AFQjCNEwM9YfDHteUAOfQ5SyBmqq0_ctww&ust=1451952337068273http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjvhc-w747KAhVV9WMKHfHbBj4QjRwIBw&url=http://archive.cnx.org/contents/fcbd1f34-bb85-442c-b25d-bd5204aea692@1/speak-and-sing-time-scaling-with-wsola&psig=AFQjCNHBW0FwyoXUps5hldB8tXyYHWTE7g&ust=1451952636484962
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ES 442 Signal Preliminaries 5
Physically Realizable Signals (Waveforms)
Meaning the signal (waveform) is measurable in a laboratory.
1. Waveform has significant non-zero values over a finitecomposite time interval.
2. Spectrum of the waveform has significant non-zero valuesover a finite frequency interval.
3. The waveform is a continuous function of time.
4. The waveform is limited to finite peak values.
5. The waveform takes on only real values (as compared tocomplex values of the format a + jb).
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ES 442 Signal Preliminaries 6
Definition and Classification of Signals – II
A signal may be either analog or digital. Sometimes the terminologyof continuous or discrete is used to distinguish between analog and digital signals.
A signal may be periodic if it has a uniform period of repetition, T, or non-periodic (i.e., aperiodic) when no uniform period of repetition exists or it does not repeat its form or shape.
Signals may be differentiated as either baseband or carrier-based.
Signals can also be distinguished by being energy signals or a powersignals. But we must be careful with this terminology. Refer to Agbo& Sadiku on page 19 of Chapter 2 for their discussion.
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ES 442 Signal Preliminaries 7
Energy Signal versus Power Signal
2( )gE g t dt
−
=
Given signal g(t) (can either be current or voltage)
Energy Signal
2
2
21lim ( )
T
T
gT
P g t dtT→ −
=
Power Signal
0 < Eg < ∞
→ Deterministic & non-periodic
signals
0 < Pg < ∞
→ Periodic & random signals
We shall call this a“Finite Energy Signal”
We shall call this a“Finite Power Signal”
Refer to page 19 of Agbo & Sadiku
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EE 442 Signal Preliminaries 8
Why So Much Emphasis on Sinusoidal Signals?All practical waveforms can be analyzed and constructed from manyharmonically-related sinusoidal waveforms.
Example:Rectangularwaveform
synthesizedfrom the sum of
sinusoidalsignals
with many more harmonics added of decreasing amplitude.
fundamental f
third harmonic 3f
fifth harmonic 5f
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ES 442 Signal Preliminaries 9
Fourier Series Expressing a Periodic Square Waveform
( )0 0 01
( ) cos ( ) sin ( )n nn
f t a a n b n
=
= + +
DC AC
0
2; T
T
= is the period
0 0 0
0 0 0
1 2 2( ) , ( ) cos( ) , ( )sin( )
T T T
n na f t dt a f t n t dt b f t n t dtT T T
= = =
Trigonometric format:
We compute the coefficients using
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ES 442 Signal Preliminaries 10
https://slideplayer.com/slide/1663100/
https://slideplayer.com/slide/1663100/
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ES 442 Signal Preliminaries 11
Signal Spectra of Periodic Square Waveform
fundamental f
3rd harmonic
5th harmonic . . . .
Frequency f
f
(1/T)3f 5f 7f 9f
T
A
Trigonometric Fourier series
Refer to Section 2.5of Agbo & Sadiku;
Pages 26 to 33
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ES 442 Signal Preliminaries 12
Signal Spectra of Periodic Square Waveform
https://gifer.com/en/CUAS
https://gifer.com/en/CUAS
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ES 442 Signal Preliminaries 13
Sinusoidal Signals Constructing a Periodic Waveform
Spectrum Analyzer Display
Oscilloscope Display
Two Viewpoints: Time Domain and Frequency Domain
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ES 442 Signal Preliminaries 14
Sinusoidal Signals Generating a Periodic Square Wave
https://en.wikipedia.org/wiki/Fourier_series
t =
https://www.youtube.com/watch?v=k8FXF1KjzY0
Recommended:
https://en.wikipedia.org/wiki/Fourier_serieshttps://www.youtube.com/watch?v=k8FXF1KjzY0
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ES 442 Signal Preliminaries 15
Review of Phasors
Phasors are used only to represent sinusoidal waveshapes.
Definition: A complex number c is a phasor if it represents a sinusoidalWaveform; for example
where the phasor is
is a rotating phasor and is distinguished from phasor c.
Note: Magnitude |c | is usually a peak value, but sometimes an RMS value,where RMS stands for “root-mean square.”
( ) 00( ) cos Rej t
g t c t c c e = + =
j cc c e=
0j t j cc e +
c
cStatic phasor
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EE 442 Signal Preliminaries 16
Rotating Phasor Generating a Cosine Waveform
Re
Im
A
A-A
tim
eRotatingPhasor:
Note: CCW rotation is a positive angle
or positive frequency
Complex Plane
Time evolvingprojection ontohorizontal axisyields cosine
waveform
RotatingPhasor
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EE 442 Signal Preliminaries 17
Certainly You Remember Euler’s Identity
cos( ) sin( )
Let 2 , then
e cos( ) sin( )
cos( ) sin( )
cos( ) sin( )
jx
j t
j t
e x j x
x ft t
t j t
e t j t
t j t
−
=
= =
= +
= − + −
= −
Because cosine is an even function and sine is an odd function.
( )
( )
1cos( )
21
sin( )2
j t j t
j t j t
t e e
t e ej
−
−
= +
= −
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ES 442 Signal Preliminaries 18
https://te.m.wikipedia.org/wiki/దస్్తరం:Simple_harmonic_motion_animation_2.gif
Sine and Cosine Waves are in Quadrature
https://te.m.wikipedia.org/wiki/దస్త్రం:Simple_harmonic_motion_animation_2.gif
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EE 442 Signal Preliminaries 19
Conjugate Phasor Representation of Sines & Cosines
2 2
cos(2 )2
j ft j ft
fte e
−
+=
2 2
sin(2 )2
j ft j ft
ftj
e e
−−
=
ImIm
ReRe
2j fte 2j fte
2j fte −
2j fte −−CCW CCW
CW
CW
Positive frequency (CCW)Negative frequency (CW)
Complex Plane
Rotating Phasors Counter rotatingvectors (or phasors)
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EE 442 Signal Preliminaries 20
Forming a Cosine Signal With Conjugate Phasors
2 21 12 2
cos(2 )j ft j ft
ft e e −= +Im
Re
2j fte
2j fte −Projection onto real-axis:
Time t evolution
Amplitude
0
Counter rotatingvectors (phasors)Euler’s formula
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EE 442 Signal Preliminaries 21
Forming a Sine Signal With Conjugate Phasors
Im
Re
2j fte 2j fte −−
2 21 12 2
sin(2 )j ft j ft
j jft e e −= −
Time tevolution
Amplitude
0
Projection onto imaginary-axis:
Counter rotatingvectors (phasors)
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EE 442 Signal Preliminaries 22
How Do We Explain Negative Frequencies?
“The existence of the spectrum at negative frequencies is somewhat disturbing to some people because, by definition, the frequency (number
of repetitions per second) is a positive quantity.
How do we interpret a negative frequency – f0?”
https://www.researchgate.net/post/Can_anyone_explain_the_concept_of_negative_frequency
https://www.researchgate.net/post/Can_anyone_explain_the_concept_of_negative_frequency
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ES 442 Signal Preliminaries 23
How Do We Explain Negative Frequencies?
“The existence of the spectrum at negative frequencies is somewhat disturbing to some people because, by definition, the frequency (number
of repetitions per second) is a positive quantity.
How do we interpret a negative frequency – f0?”
Negative frequencies are a mathematical construct to analyze
real signals using a complex number framework. It requires the
use of double-sided spectra. A complex number can be made
real by adding its conjugate to it (e.g., (a + jb) + (a - jb) = 2a. A
real sinusoid can be represented using complex exponentials by
using the sum of e(jωt) and its complex conjugate e(-jωt). This is
where the negative frequency idea comes from.
https://www.researchgate.net/post/Can_anyone_explain_the_concept_of_negative_frequency
Answer:
https://www.researchgate.net/post/Can_anyone_explain_the_concept_of_negative_frequency
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EE 442 Signal Preliminaries 24
Analog Signals and Digital Signals
All signals are analog signals – the differentiator is what they represent!
Analog Signals Digital Signals
(1) A parameter of the signal represents a physical parameter
(2) Physical parameter is time-varying(3) Parameter takes on any value
within a defined range (said to becontinuous valued)
(1) Represents a sequence of numbers or “states”
(2) Numbers change in discrete time (said to be time-varying)
(3) Numbers are restricted to a finiteset of discrete values
Waveforms:
Analog signal
Digital signal
Waveformsas commonly
drawn in textbooks
https://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwj18bqyo5jKAhVK0mMKHTRECEMQjRwIBw&url=https://www.engr.colostate.edu/~dga/mech307/lectures.html&psig=AFQjCNHlbVfs1lHvsVBJbIQH_m-sgL1noA&ust=1452275838527585
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25EE 442 Signal Preliminaries
Analog & Digital Signals: Continuous versus Discrete Valued
Analog & continuous
Analog & discrete
t
tn
Digital & continuous
Digital & discrete
t
tn
0 1 0 0 1 1 0 1 1 1 0 0 1 1 0 1 1 0
0 1 0 0 1 1 0 1 1 1 0 0 1 1 0 1 1 0
String of values
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EE 442 Signal Preliminaries 26
Quiz: How would you classify waveform A and waveform B?
(1) Continuous, (2) Discrete, (3) Analog, (4) Digital
Waveform A (gray) Waveform B (red)
time
Am
plit
ud
e
https://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjUkZ_e5o7KAhXHKGMKHapGAakQjRwIBw&url=https://en.wikibooks.org/wiki/Control_Systems/Sampled_Data_Systems&psig=AFQjCNHXcJiNn1LSl0Zk44BQQwvJPzu-4Q&ust=1451950313313808
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EE 442 Signal Preliminaries 27
Example B: Bit Sequence of 10001010111 . . .
Amplitude
Time
+½A
-½A
1 0 0 0 1 0 1 0 1 1 1
Amplitude
Time
+½A
-½A
High state Represents
a “1”
Low state Represents
a “0”
1 0 1 0 1 0 1 0 1 0 Square
Waveformshown
This is a periodic waveform.
NO communication.Why?
“Information” isBeing transmitted.
Why?
Example A: Bit Sequence of 10101010101 . . .
A pure sinusoidal waveformor a square waveform
doesn’t transmit information.
Time variation alone is not sufficient to communicate information
Not a periodic waveform.
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EE 442 Signal Preliminaries 28
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EE 442 Signal Preliminaries 29
Bandwidth Definitions
The bandwidth of a signal provides a measure of the extent of spectral contentof the signal for positive frequencies. What does significant mean?
1. 3-dB Bandwidth – The separation (along positive frequency axis) between The points where the amplitude drops to of its peak value (½ power points). 1 2
2. Null-to-null Bandwidth – For example, for the sinc function the bandwidthwould be the frequency width from -1/T to 1/T (null-to-null points).
3. Root-mean-square (RMS) Bandwidth – Defined as
−
−
=
22
2
( )
( )RMS
f G fBW
G f
And there are numerous other bandwidth definitions . . .