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Copyright Monash University 2009

Signal

ProcessingFirst

Lecture1

Sinusoids


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READING ASSIGNMENTS ThisLecture:

Chapter2,

pp.

917

AppendixA:

Complex

Numbers

AppendixB:MATLAB

Chapter1:Introduction

2


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CONVERGING FIELDS

3

EECmpE

Math

Applications

Physics

Computer

Science


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COURSE OBJECTIVE Studentswillbeableto:

Understandmathematical descriptions

of

signalprocessingalgorithms andexpress

those

algorithms

as

computer

implementations (MATLAB)

Whatareyourobjectives?

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WHY USE DSP ? Mathematicalabstractions leadto

generalizationand

discovery

of

new

processingtechniques

Computerimplementationsareflexible

Applicationsprovideaphysical context

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Fourier Everywhere Telecommunications

Sound&Music

CDROM,Digital

Video

FourierOptics

XrayCrystallography

ProteinStructure

&

DNA

ComputerizedTomography

MagneticResonanceImaging:MRI

Radioastronomy MechanicalEngineering Ref:Prestini,TheEvolutionofAppliedHarmonicAnalysis

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LECTURE OBJECTIVES Writegeneralformulaforasinusoidal

waveform,or

signal

Fromtheformula,plotthesinusoidversus

time

Whatsasignal?

Itsafunction of

time,

x(t)

or

s(t)

inthemathematicalsense

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TUNING FORK EXAMPLE CDROMdemo

Ais

at

440

Hertz

(Hz)

WaveformisaSINUSOIDALSIGNAL

Computerplot

looks

like

asine

wave

Thisshouldbethemathematicalformula:

8

))440(2cos( tA


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TUNING FORK A-440 Waveform

9

ms3.2

85.515.8

T

Hz435

3.2/1000

/1

Tf

Time (sec)


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SPEECH EXAMPLE Morecomplicatedsignal(BAT.WAV)

Waveformx(t) is

NOT

aSinusoid

Theorywilltellus

x(t) is

approximately

asum

of

sinusoids

FOURIERANALYSIS

Breakx(t) intoitssinusoidalcomponents

Calledthe

FREQUENCY

SPECTRUM

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Speech Signal: BAT

Nearly

Periodic in

Vowel

RegionPeriodis(Approximately)T=0.0065sec

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DIGITIZE the WAVEFORM x[n] isaSAMPLEDSINUSOID

Alist

of

numbers

stored

in

memory

Sampleat11,025samplespersecond

Called

the

SAMPLING

RATE

of

the

A/DTimebetweensamplesis

1/11025=90.7microsec

Outputvia

D/A

hardware

(at

Fsamp)

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STORING DIGITAL SOUND x[n] isaSAMPLEDSINUSOID

Alist

of

numbers

stored

in

memory

CDrateis44,100samplespersecond

16bit

samples

Stereouses2channels

Numberofbytesfor1minuteis

2X(16/8)X60X44100=10.584Mbytes

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SINES and COSINES AlwaysusetheCOSINEFORM

Sineisaspecialcase:

14

))440(2cos( tA

)cos()sin(2

tt


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SINUSOIDAL SIGNAL

FREQUENCY

Radians/sec

Hertz(cycles/sec)

PERIOD (in

sec)

AMPLITUDE

Magnitude

PHASE

15

A tcos( )

( )2 f

Tf

1 2


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EXAMPLE of SINUSOID GiventheFormula

Makeaplot

16

)2.13.0cos(5 t


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PLOT COSINE SIGNAL

FormuladefinesA,,and

17

5 0 3 12cos( . . ) t

A 5

0.3

1.2


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PLOTTING COSINE SIGNAL from

the FORMULA

Determine period:

Determineapeak locationbysolving

Zero crossingisT/4beforeorafter

Positive&NegativepeaksspacedbyT/218

)2.13.0cos(5 t

3/203.0/2/2 T

0)2.13.0(0)( tt


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PLOT the SINUSOID

UseT=20/3andthepeaklocationatt=4

19

)2.13.0cos(5 t

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TRIG FUNCTIONS CircularFunctions

CommonValues

sin(k)=0

cos(0)=1

cos(2k)=1andcos((2k+1) )=1

cos((k+0.5)

)=0

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