Some Preliminary Discussion EE442 Analog & Digital Communication Systems Lecture 2 · 2018. 1....

EE 442 Signal Preliminaries 1

SignalsSome Preliminary Discussion

EE442 Analog & Digital Communication SystemsLecture 2

The Fourier transform of single pulse is the sinc function.

ES 442 Signal Preliminaries 2

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 describing 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 deterministic or random (i.e., stochastic).

There are two domains in which to describe a signal:(1) Time-domain waveform(2) Frequency-domain spectrum

This brings us to 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


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 are also 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 manysinusoidal waveforms.

Example:RectangularWaveform

Synthesizedfrom the Sum of

SinusoidalSignals

with many more harmonics added of decreasing amplitude.

fundamental f

third harmonic 3f

fifth harmonic 5f


Sinusoidal Signal (Line Spectra) of Periodic Waveform

fundamental f

3rd harmonic

5th harmonic . . . .

Frequency f

f(1/T)

3f 5f 7f 9f

T

A

Trignometic Fourier series

Refer to Section 2.5of Agbo & Sadiku;

Pages 26 to 33


Sinusoidal Signals Build Up a Periodic Waveform


Sinusoidal Signals Generating a Periodic Square Wave

https://en.wikipedia.org/wiki/Fourier_series

t

https://en.wikipedia.org/wiki/Fourier_series


Rotating phasor Generating a Cosine Waveform

Re

Im

A

A-A

tim

eRotatingPhasor:

CCW rotation is positive angle

or positive frequency

Complex Plane

Time evolvingprojection ontohorizontal axisyields cosine

waveform


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


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

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

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?”


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

via 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


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) Parameter is time-varying(3) Parameter takes on any value

within a defined range (said to becontinuous values)

(1) Represents a sequence of numbers or “states”

(2) Numbers change in discrete time (said to be time-varying)

(3) Numbers are restricted a finite setof 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

18EE 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 do you classify these two waveforms?

(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

can’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 . . .

Some Preliminary Discussion EE442 Analog & Digital Communication Systems Lecture 2 · 2018. 1....

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