elec 264 review

8/13/2019 Elec 264 Review

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Mathematical Review

sincos

:relation'

.,|z|r

z,ofphaseorangletheisand

z,ofmagnitudetheis0rwhere

-:formPolar

numbers,realareyandxand1

,:forrrectangulaor

je

sEuler

z

rez

jwhere

jyxzCartesian

j

j

+=

=

>=

=

+=


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Periodic Complex Exponential

& Sinusoidal Signal

tjte ootj o sincos

:signalssinusoidalofterms

inwrittenbecanperiod,lfundamentathe

withsignal,lexponentiacomplexThe

+=


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tjjtjj

ooo ee

Aee

AtA

+=+22

)cos(

:lexponentiacomplex

periodicofin termswritten

becanperiod,lfundamentawithsignal,sinusoidalThe

Periodic Complex Exponential& Sinusoidal Signal


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ondcycles/secorHertzinfrequencylfundamentatheisof

condradians/seinfrequencyangularlfundamentaisof2o

periodlfundamentatheis

})(

{.)sin(

})(

Re{.)cos(:followsIt

=

+=+

+=+

oT

with

tojeIMAtoAor

tojeAt

oA

Periodic Complex Exponential& Sinusoidal Signal


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increasing ,

increases the rate of oscillation

hzfo 1000=

hzfo 2000=

)cos()( += tAtx o

o

o


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The phase of the signal is the offset in the

displacement from a specified reference point at time t = 0 represents a "shift" from zero phase A change in is also referred to as a phase-shift For infinitely long sinusoids, a change in is the same as

a shift in time, such as a time-delay If x(t) is delayed (time-shifted) by 1/4 of its cycle T

whose "phase" is now -pi/2it has been shifted by pi/2

Phase shift

)cos( +tA o

Signals with different phase shift

compared to the bottom signal


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Case: x(t) is a constant

x(t) is periodic with period T

for any positive value of TThus fundamental period is

undefinedOn other hand can define

fundamental frequency to bezero, i.e., constant signal (d.c)

has zero rate of oscillation

0=o


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Play a central role in signals &

systems

Serve as building block for many

other signalsSets of harmonically related

complex exponential are periodicwith a common period To


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A necessary condition for a complex

exponentialtj oe

to be periodic with

period To is 1=oTje

,...2,1,0,2Ti.e..2ofmultipleaisthatimplieswhich

o == kkTo


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Harmonically Complex

Exponential Signals

.frequencypositive

singleaofmultiplesarethatsfrequencielfundamentawithlsexponentiaperiodic

ofsetaislexponentiacomplex

ofsetrelatedlyharmonical

.k...of

multipleintegeranbemustthen

,2

defineweIf

o

oo

o

A

ei

To

=

=


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Harmonically ComplexExponential Signals

|||k|

2

periodlfundamentaand|k|sfrequencie

lfundamentawithperiodicis)(0,k

constant.ais)(0,kFor2,...1,0,k,)(

o

o

k

T

t

tet

o

k

k

tjk

k o

=

=

==


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Harmonically Complex

Exponential Signals

.Tlengthofintervalany time

duringperiodslfundamentaitsof|k|exactly

throughgoesitaswell,asTperiodwith

periodicstillis)(harmonickthThe

o

o

tk


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General Complex Exponential

Signals

.|||C|

)(Then

.jraand|C|C

:formcartesianina

andformpolarinexpressedisCIf

numbers.complexarea''andC''bothwhere,)(

)()(

o

++==

=

+==

=

tjrttjrj

at

j

at

oo eeCee

Cetx

e

Cetx


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Signals

l.exponentiadecayingbymultipledsignalssinusoidalisx(t)0,rIf

l.exponentiagrowingbymultipledsignalssinusoidalisx(t)

0,rIf

.sinusoidalarepartsimaginary&realthe

0,rIf

)sin(||)cos(||)(

relation,sEuler'Using

=

+++== teCjteCCetx ort

o

rtat


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r>0

r


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pi=3.142;t=-10:.1:10;

f=2000;

w=2*pi*f;

r=0.1;x=zeros(size(t));

x=exp((r+w*i)*t);

theta=pi/4;

c=1*exp(i*theta);

y=c*x;

subplot(2,1,1);

plot(t,y);grid;

r=-0.1;x=zeros(size(t));

x=exp((r+w*i)*t);

theta=pi/4;

c=1*exp(i*theta);

y=c*x;

subplot(2,1,2);

plot(t,y);grid;

end;

Matlab Program for

Growing & Decaying Sinusoids


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Decaying or

Damped Sinusoids

Examples:Response of RLC circuits.

Mechanical systems having bothdamping & restoring forces e.g.

automotive suspension system.


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Impulse Function

time

volt


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Delayed Impulse


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Summation Operation of x[n]


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Discrete-Time Complex

Exponential Sequence

sequence.time-discrete

with thedealingwhenperferredisformformer

the,previouslydescribedhavewesignallexponentia

time-continuousthesimilar toisformhisAlthough t

.where,][-:formfollowing

in thesequencetheexpresscanely weAlternativ

numbers.complexgeneralinareandCwhere

,][

eCenx

Cnx

n

n

==

=


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Real Exponential Signalnnxge 1.1*2][.. =1where*][ >= nCnx


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Real Exponential Signal10where*][


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Real Exponential Signal01-where*][


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Real Exponential Signal1where*][


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Real Exponential Signal

Real-valued discrete exponentials

are used to describe:1) Population growth as function of

generation2) Total return on investment as a

function of day, month or

quarter


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Sinusoidal Signals

njjnjj

o

oo

nj

o

o

nj

o

n

oo

o

o

ee

A

ee

A

nA

njne

nAnx

enx

Cenx

+=+

+=

+=

=

===

22)cos(

sincos-:relationsEuler'From

radians.ofunitshaveand

boththeness,dimensionlasnTaking

).cos(][

:signalsinusoidaltorelatedcloselyissignalThis

.][

number.imaginaryanpurelybej&1Clet,][


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Discrete-time SinusoidalSignals

)12/2cos(][ nnx =

)31/8cos(][ nnx =

)6/cos(][ nnx =


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Discrete-time Sinusoidal

Signals

These discrete-time signals possessed:

1) Infinite total energy2) Finite average power


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Signals

lly.exponentiagrowingsinusoidal1,|a|

lly,exponentiadecayingsinusoidal1,|a|

.sinusoidalareparts&,1||

)sin(||||)cos(||||][

.e||,e|C|C

:formpolarinandCWriting

signals.sinusoidaland

lsexponentiarealofin termsdinterpretebecan

lexponentiacomplextime-discretegeneralThe

ojj

>

1

+


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Discrete-time Sinusoidal

Signals

12

1

2

,12/2becauseperiodic

)12/2cos(][

o

o

=

=

=

nnx

31

4

2,31/8becauseperiodic

)31/8cos(][o

o ==

=

nnx

numberrational2

,6/1becauseperiodicnot)6/cos(][

o

o

==

nnx


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Fundamental Period & Frequency of

discrete-time complex exponential

)2

m(N

-:aswrittenisperiodlfundamentaThe

,N

2isfrequencylfundamentaIts

N,periodlfundamentawithperiodicisex[n]If

o

jo

=

=

=

m

o

n


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nt oo jj eandesignaltheofComparison

.ofvalues

distinctforsignalsDistinct

ee

o

jj oo

nt

.ofchoiceanyforPeriodic o

ofrequencylFundamenta

o

undefined

2

:0

:0

periodlFundamenta

o

o

=

2ofmultiplesbyseparated

offor valuessignalsIdentical o

mand0Nintegerssomefor

,

2

ifonlyPeriodic o

>

= N

m

m

frequencylFundamenta 0

)

2

(:0

:0

periodlFundamenta

o

o

om

undefined

=

Harmonically related


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periodic exponential sequence

Considering periodic exponentialswith common period N samples:

1,...0,kfor,][ )/2( == nNjkk en

This set of signals possessfrequencies which are multiples of

N/2



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In continuous-time case

][e][

2/N)njk(2

)/2)((

neen

k

nj

nNNkj

Nk

==

=

+

+

tTjk

e)/2(

are all distinct signals for ,....2,1,0 =k



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Therefore , there are only N distinct

periodic exponentials in the discrete

harmonic sequences.

)][][(e.g.above.theof

onetoidenticalis][otherAny

][.......

,][,][,1][

0N

k

/)1(2

1

/42

/21

nn

n

en

enenn

NnNj

N

NnjNnjo

=

=

===


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o Mathematical review

o Transformation of the independent variable

o LTI Signals & Systems

o Convolutiono Frequency representation of signals

o Fourier Series

o Fourier Transform

ELEC264: Signals And Systems

Review

Dr. Aishy AmerConcordia University

Electrical and Computer Engineering

Figures and examples in these course slides are taken from the following sources:

A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997

M.J. Roberts, Signals and Systems, McGraw Hill, 2004

J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

http://metalab.uniten.edu.my/~zainul/images/Signals&Systems/

T f i f


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Transformation ofIndependent Variable

Aircraft control system:

Input correspond to pilot action theseaction are transformed by electrical &mechanical system of the aircraft tochanges to aircraft trust or positioncontrol surfaces such as the rudder& ailerons

finally these changes affect thedynamics & kinematics such as the

aircraft velocity and heading

M difi i f i d d


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Modification of independentvariable (time axes)

Introducing several basic properties of signals& systems through elementarytransformations.

Examples of elementary transformation:- time shift, x(t-t0), x[n-n0]

time reversal, x(-t), x[-n].

time scaling, x(0.5t), x[2n].

and combinations of these. x(at+b), x[an-b],where a & b are signed constants*.

Shifti i ht l i


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Shifting right or laggingsignal x(t)

X(t)

tX(t-t0)

0

0 t

t0

is a positive value

Shifti l ft l di i l


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Shifting left or leading signalx(t)

X(t)

tX(t+t1)

0

0 t

-t1

Folded or Flipped x(t)


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Folded or Flipped x(t)=x(-t), time reversal

Signal Flip about y axes


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Signal Flip about y- axes

x[-n], time reversal


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Examples

x[n]

x[n-5]


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Examples

x[n]

x[n+5]


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Example 1.1x(t)

x(t+1), x(t) shifted left by 1sect

t

1

1

0

0

12

1 2-1

Tables of x(t) & x(t+1) & x(


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Tables of x(t) & x(t+1) & x(-

t+1)

t x(t) x(t+1) x(-t+1)-2 0 0 0

-1 0 1 0

0 1 1 11 1 0 1

2 0 0 0

3 0 0 0


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Example 1.1x(t+1) is x(t) shifted left by 1

x(-t+1) is x(t+1) flipped about t=0t

t

1

1

0

0

12

1 2-1

-1


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Example 1.1 Alternative 1x(t-1) is x(t) shifted right by 1se

x(-t+1)=x(-1(t-1))

Flip about axis t=1

t

t

1

1

0

0

12

1 2-1


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Example 1.1, Method 2x(-t), flip about axis t=0

x(-t+1), shift right (because -t) by 1t

t

1

1

0

0

1 2

1 2-1

-1


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Example 1.1x(t)

x(t+1), x(t) shifted left by 1sect

t

1

1

0

0

12

1 2-1


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o Fourier Series

o Fourier Transform


Review








Topic 1:


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Topic 1:

LTI Signals & Systems

Topic 1:


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Topic 1:


Topic 1:


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op c


Topic 1:


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p



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o Fourier Series

o Fourier Transform


Review









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Topic 2: Convolution


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Types of Systems

Time-invariant, linear, causal, memory, stable,invertible Need to be able to prove whether a system is linear

and time-invariant

Convolution Sum Know how to compute impulse response h[n]

Convolution Integral Graphical Convolution

Convolution Properties Stability and Impulse Response


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Convolution:

1. Write down equations for both signals2. Reflect and shift one of the two signals (preferably the finite

duration signal)3. Label the edge(s) of the reflected/shifted signal with

t and/or t-a appropriately4. Shift signal to the left until there is no overlap between the two

signals5. Identify number of different cases in which there is overlap

between the two signals

6. For each of the above cases, evaluate the convolution integralwith the limits of integration determined by the region ofoverlap expressed as a function of t ( see step 3)


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o

Convolutiono Frequency representation of signals

o Fourier Series

o Fourier Transform


Review









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Waveform Representation

Waveform representation

Plot of the signal value vs. time

Sound amplitude, temperaturereading, stock price, ..

Mathematical representation: x(t)

x: variable value

T: independent variable


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Sample Speech Waveform


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Sample Music Waveform


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Sinusoidal Signals

Sinusoidal signals: important because they can be used to synthesize any signal An arbitrary signal can be expressed as a sum of many sinusoidal signals with

different frequencies, amplitudes and phases

Phase shift: how much the max. of the sinusoidal signal is shifted away from t=0 Music notes are essentially sinusoids at different frequencies

Complex Exponential


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Signals

)sin(2)cos(2

:ForumlaEuler

)()(:Note:ConjugateComplex

phase)(initialshiftphasetheis

)sin()cos(||)(

:SignallExponentiaComplex

sincos:tionrepresentaPolar

taniszofPhase

||iszofMagnitude

:tionrepresentaCartesian

:NumberComplex

***

00

)(

1

22

0

jeeee

realarezzandzzjbaz

tzjtzeztx

zjzezz

a

bz

baz

jbaz

tj

j

==+

+=

+++==

+==

==

+=

+=

+

o

o

o

o

What is frequency of an


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arbitrary signal?

Sinusoidal signals have a distinct (unique) frequency

An arbitrary signal x(t) does not have a unique frequency x(t) can be decomposed into many sinusoidal signals

with different frequencies, each with different magnitudeand phase

Spectrumof x(t): the plot of the magnitudes and phasesof different frequency components

Fourier analysis: find spectrum for signals

Bandwidthof x(t): the spread of the frequencycomponents with significant energy existing in a signal


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Frequency content in signals


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Frequency content in signals A constant : only zero frequency component (DC

component)

A sinusoid : Contain only a single frequency component Periodic signals : Contain the fundamental frequency and

harmonics : Line spectrum Slowly varying : contain low frequency only

Fast varying : contain very high frequency Sharp transition : contain from low to high frequency Music: :

contain both slowly varying and fast varyingcomponents, wide bandwidth

: increasing , increases

h f ill i

)cos( +tA o o


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the rate of oscillation

hzfo 1000=

hzfo 2000=

o

S l


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Spectrum analyzer

x1(t) & x2(t) look similar; A spectrum analyzer reveals the difference:x2(t) contains a sinusoid that causes the two large spikes

Fourier Analysis: Types of signals


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Fourier Analysis: Types of signals


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Fourier Analysis: Types of

i l


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signals

All continuous signals are

CT but not all CT signalsare continuous

Fo rier Anal sis


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Fourier Analysis

Discrete inTime

Periodic inFrequency

Continuousin Time

Aperiodic in

Frequency

Aperiodic in Time

Continuous in Frequency

Periodic in TimeDiscrete in Frequency

=

=

=

k

tjkk

T

tjk

k

eatx

dtetxT

a

0

0

)(

P-CTDT:SeriesFourierInverseCT

)(1

DTP-CT:SeriesFourierCT

T

0

T

=

+

=

+

=

2

2

2

)(2

1][

DTPCT:TransformFourierDTInverse

][)(

PCTDT:TransformFourierDT

deeXnx

enxeX

njj

n

njj

=

=

=

=

1

0

NN

1

0

NN

0

0

][1

][

P-DTP-DTSeriesFourierDTInverse

][][

P-DTP-DTSeriesFourierDT

N

k

knj

N

n

knj

ekXN

nx

enxkX

=

=

dejXtx

dtetxjX

tj

tj

)(2

1)(

CTCT:TransformFourierCTInverse

)()(

CTCT:TransformFourierCT

Relations Among Fourier

Methods


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Methods



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o Convolution

o Frequency representation of signals

o Fourier Series

o Fourier Transform

Review








Topic 3: Fourier Series


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A Fourier Series/transform is unique,

i.e., no two same signals in time givethe same function in frequency



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Topic 3: Fourier Series Fourier Series:

Exponential Fourier series

Trigonometric Fourier Series

=

=

==

k

tkfj

T

tkfj

ekXtx

TfdtetxT

kX

0

0

2

0

2

)()(

)1(/1)(1)(

=

=

++=

=

==

==

1

0

1

00

0

)2sin()2cos()(

)0(

)(thatnote)}(Im{2

)2()(thatnote)}(Re{2

k

k

k

k

ksk

kck

tkfbtkfaatx

Xa

bkXkXb

akXkXa



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Topic 3: Fourier Series Fourier Series

Symmetries of signal (even and odd) Even signal: X(k) is purely real

Odd signal: X(k) is purely imaginary

Properties of Fourier series Shifting property

)()( 002

0 kXettx tkfj


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Topic 3:

Fourier Series


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Fourier Series


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Topic 4: Fourier Transform


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Topic 4: Fourier Transform Fourier transform for aperiodic signals

Understand the meaning of the inverse Fouriertransform

Sketch the spectrum

Determine the bandwidth of the signal from itsspectrum

Know how to interpret a spectrogram plot



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Fourier Series versus

Fourier Transform


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Fourier Transform Fourier Series (FS): a representation of periodic

signals as a linear combination of complex

exponentials Fourier Transform (FT): apply to signals that are not

periodic

Aperiodic signals can be viewed as a periodic signal

with an infinite period The CT Fourier Series is a good analysis tool for

systems with periodic excitation but the CT FourierSeries cannot represent an aperiodic signal for all time

The CT Fourier transform can represent an aperiodicsignal for all time

DTFT: Summary


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DT Fourier Transform represents a discrete

time aperiodic signal as a sum of infinitelymany complex exponentials, with the

frequency varying continuously in (-, )

DTFT is periodic only need to determine it for


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DT-FT Summary: a quiz


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A discrete-time LTI system has impulse response

Find the output y[n] due to input

(Suggestion: work with and using the convolution property)

Solution This can be solved using convolution of h[n] and x[n]. However, the point was to use the convolution in time

multiplication in frequency property.

Therefore, It can be readily shown that

Therefore,

][2

1][ nunh

n

=

][7

1][ nunx

n

=

)()()(][*][][ jjj eXeHeYnxnhny ==

( ) 1,1

1)(][][

elec 264 review

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