a preliminary discussion ee442 analog & digital communication systems lecture...

EE 442 Signal Preliminaries 1

SignalsA Preliminary Discussion

EE442 Analog & Digital Communication SystemsLecture 2

The Fourier transform of single pulse is the sinc function.

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


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


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


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).


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.


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


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


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


https://slideplayer.com/slide/1663100/

https://slideplayer.com/slide/1663100/


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


Signal Spectra of Periodic Square Waveform

https://gifer.com/en/CUAS

https://gifer.com/en/CUAS


Sinusoidal Signals Constructing a Periodic Waveform

Spectrum Analyzer Display

Oscilloscope Display

Two Viewpoints: Time Domain and Frequency Domain


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


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


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


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

−

−

= +

= −


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


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)


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


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)


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



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.


Answer:



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

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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


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


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.


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 . . .

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