ECTE301 Notes Week1

7/28/2019 ECTE301 Notes Week1

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ECTE301

Digital Signal Processing


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Why this subject -- Examples

Telephone

Phone Phonevoice voicevoltage/current


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Why this subject-- Examples

CD Music Recording and playing

Mic Quantization

Music

Sampling

Speaker D/A

CD

Decoding

Encoding

Music


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Why this subject?

ECTE301 Digital signal processing willprovide you with the fundamental

knowledge about signals and systems,

and basic skills of DSP system design


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Why this subject?

Signals: what, why

Digital signals: why, how

systems

Digital signal processing: what, why,

how


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What are Signals

Signals are something that carry information;

Signals appear in physical forms: observable

measurable

Usually varies with time --- functions of time

Examples: voice, measurement of

metrological quantities, etc


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How do Signals look like

Speech signal


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How do Signals look like

noise


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Signal Classification

Continuous in time, continuous in

amplitude analogy signals

Discrete in time, continuous in

amplitude discrete signals

Discrete in time, discrete in amplitude

digital signals


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Why Signal Processing

Signal processing refers to the work of

manipulating signals so that information

carried can be expressed, transmitted,

restored etc in a more efficient and

reliable way

Efficiency: Lest resource usage

Reliability: Lest error


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Why Signal Processing?

The purposes of SP can be: Enhance the signals (noise reduction, interference

elimination etc)

Present signal in an efficient way (source coding)

Extract feature of signals

.


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What is Signal Processing?

System

Input Output


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Why DSP?

Signals are expressed as digital numbers Signal processing can be realized by

mathematical operations using computers

Digital signals can be the results of sampling

continuous-time signals (such as voice

signals) or discrete in nature (such as text in

emails)


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What is DSP?

DSP

DSP chips or

General purpose

computer

Input Output

Input and output are discrete sequences.


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Systems

Something that can manipulate,change,record, or transmit signals.

We will try to describe the system by

mathematical tools We will also try to describe the tasks of

DSP by mathematical operations


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What DSP

Signal analysis: spectral analysis, discreteorthogonal transforms etc

Behavior of DSP systems: system impulse

response, system functions, and input-output

relationship;


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What are covered in the

subject? General concepts

The simplest signals sinusoidal signals

Spectrum representation of signals

Sampling of analog signals Filtering of digital signals FIR filters

Analysis of FIR filters

Z-Transform

IIR filters

Spectral analysis of signals


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Digital Signal Processing

Week 1

Sinusoidal Signals


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Sinusoidal Signals

The reasons for looking at sinusoidal signalsare:

The most widely used;

The simplest signal that could be used to carry

information; Other signals can be expressed by combinations

of sinusoidal signals


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Outline

What are sinusoidal signals Description of sinusoidal signals

Amplitude, Frequency and phase

Direct expression Complex exponential signals complex

sinusoidal signals

Phasor operations

Relationship between real sinusoidsand complex exponentials


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Sinusoidal signals

where


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Sinusoidal signals

)2cos()cos()( 00 tfAtAtx

A -- Amplitude

-- radian frequency

-- cyclic frequency

-- initial phase

-- instantaneous phase

0

0f

t0


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Sinusoidal signals

Period vs frequency

)cos()( 0 tAtx

)(

)cos(

))(cos()(

000

000

tx

TtA

TtATtx

kT 200

0

0

2

kT

0

0

2

TThe smallest


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Phase shift vs Time shift

)cos(

))(cos()(

00

0

tA

tAtx

0

The time shift

The phase shift

Exercise 2.2, 2.3, 2.4, 2.5


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Complex Exponential Signals

A complex exponential signal

)sin()cos(

)(

00

)( 0

tjAtA

Aetxtj

Sinusoidal in nature

Two components with 90 degrees ofphase difference


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Complex Exponential Signals Complex

Numbers

Complex exponential signals can be

manipulated by the operations of

complex numbers

Use polar form of expressions phasors

Multiplications of two complex numbers

Additions of two complex numbers


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Numbers

Product of two complex numbers Phasor

rotation

)(

21213

2211

21

21 ,

i

jj

errzzz

erzerz


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Numbers

Addition of two complex numbers

phasor addition

)sinsin(coscos

,

22112211

213

2211

21

rrjrr

zzzerzerz

jj


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Numbers

Addition of complex exponentials ofsame frequency is still an complexexponentials with the same frequency

)(1

11

)(

000

00)(

tjtjjtj

N

k

jk

N

k

tjj

k

N

k

tj

k

AeeAeeeA

eeAeAtx

k

kk


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Relationship between real sinusoids and

complex exponentials

A real sinusoid can be considered as the real

part of a complex exponential

tjjtjeAeAetA 00 ReRe)cos(

)(

0

is called the phasor of the sinusoid, which

contains the amplitude and initial phase of the

signal

jAe


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Addition of real sinusoids

Combination of sinusoids with the same

frequency is still a sinusoid with the same

frequency

N

k

kk tAtx1

0 )cos()(


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Addition of real sinusoids Direct computation

)cos(

sincos

sinsincoscos

sinsincoscos

)cos()(

0

00

0

1

0

1

1

00

1

0

tA

tDtC

tAtA

ttA

tAtx

N

k

kk

N

k

kk

N

k

kkk

N

k

kk

22 DCA C

D1tan

N

k

kk

N

k

kk

AD

AC

1

1

sin

cos


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Addition of real sinusoids phasor

addition

Direct computation is very complex

)cos(

Re

Re

ReRe

)cos()(

0

1

11

)(

1

0

0

0

00

tA

eAe

eeA

eeAeA

tAtx

tjj

tjN

k

j

k

N

k

tjj

k

N

k

tj

k

N

k

kk

k

kk

N

k

j

k

j keAAe1

Addition of phasors


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Addition of real sinusoids phasor

addition

Phasor addition rule

Obtain the phasor representation of each of

the individual signals

Add the phasors together, convert the result

into polar form

Multiply by to get

Exercise 2.9

jAe)( 0 tjAetje 0


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Product of real sinusoids

Multiplication of sinusoids results in sinusoids

with difference frequencies: Principle of

modulation

))()cos(())()cos((21

)cos()cos()()(

)cos()(

)cos()(

2121212121

22112121

2222

1111

ttAA

ttAAtxtx

tAtx

tAtx


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Spectrum Representation

Spectrum is a graphical representation of thefrequency content of a signal.


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Signals


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Spectrum of a sinusoid Spectrum of a sum of sinusoids

Spectrum of a multiplication of sinusoids

Beat notes Amplitude modulation


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Spectrum of a sinusoid

We use the frequencies and complexmagnitudes of complex exponentials to denote

the spectrum of a signal, that is, for

tjtj eAAetx 00 )()(

We have

0

jAeA


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Spectrum of a sinusoid

Hence for a real sinusoid, the spectrum shouldbe

tjtj

tjtj

eAeA

eeA

tAtx

00

00

*

)()(

02

)cos()(

0

jeA

2

0

jeA

2


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Spectrum of a sum of sinusoids

N

k

tjj

ktjj

k

N

k

tjtjk

N

k

kkk

k

k

k

k

kkkk

eeA

eeA

ee

A

tAtx

1

1

)()(

1

22

2

)cos()(


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Spectrum of a sum of sinusoids

1

1

2

1 jeA

12

2

22 jeA

2

2

22 jeA

1

2

1 jeA


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Spectrum of Product of sinusoids

)()(cos)()(cos2

1)cos()cos()()(

)cos()(

)cos()(

2121212121

22112121

2222

1111

ttAA

ttAAtxtx

tAtx

tAtx


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Spectrum of Product of sinusoids

)()()()(

)()()()(21

21

)()(22222

)()(11111

22212221

22212221

2222

1111

4

)()(2

)cos()(

2)cos()(

tjtj

tjtj

tjtj

tjtj

ee

eeAA

txtx

eeA

tAtx

eeAtAtx


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Spectrum of product of sinusoids

21

21 )( 21 )( 21

)(21 21

4

jeAA

)(21 21

4

jeAA

)(21 21

4

jeAA

)(21 21

4

jeAA

21


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Examples

sc

cs

ff

tftfAtx

),2cos()2cos()(

Beat notes

Amplitude modulation


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Examples Beat Note


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Examples Beat note


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Examples AM signals


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Examples AM signal


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Tutorial Questions

Problems:2.3, 2.5, 2.7, 2.10, 2.16, 2.18, 3.2, 3.3, 3.5

ECTE301 Notes Week1

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