信號與系統chap. 3 class notes

7/24/2019 Chap. 3 Class Notes

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Chapter 3 Fourier SeriesChapter 3 Fourier Series

Representation of Periodic SignalsRepresentation of Periodic Signals

Chapter 2 is based on representing signals asChapter 2 is based on representing signals aslinear combinations of shifted impulses.linear combinations of shifted impulses.

In this and the following two chapters, weIn this and the following two chapters, we

explore an alternative representation for signalsexplore an alternative representation for signalsand LTI systems.and LTI systems.


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3.2 THE RESPONSE OF LTI SYSTEMS TO3.2 THE RESPONSE OF LTI SYSTEMS TO

COMPLEX EXPONENTIALSCOMPLEX EXPONENTIALS

Represent signals as linear combinations of basicRepresent signals as linear combinations of basic

signals that possess the following two properties:signals that possess the following two properties:

The set of basic signals can be used toThe set of basic signals can be used toconstruct a broad and useful class of signals.construct a broad and useful class of signals.

The response of an LTI system to each signalThe response of an LTI system to each signalshould be simple enough in structure.should be simple enough in structure.

With s and z in complexWith s and z in complex


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ForFor x[nx[n] =] = zznn


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LetLet


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3.3 FOURIER SERIES REPRESENTATION OF3.3 FOURIER SERIES REPRESENTATION OF

CONTINUOUSCONTINUOUS--TIME PERIODIC SIGNALSTIME PERIODIC SIGNALS

3.3.1 Linear Combination of Harmonically3.3.1 Linear Combination of HarmonicallyRelated Complex ExponentialsRelated Complex Exponentials

Harmonically related complex exponentialHarmonically related complex exponential

Linear combinationLinear combination

is also periodic with period Tis also periodic with period T ------ F.S. representationF.S. representation


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Exp. 3Exp. 3--22 ------ x(tx(t) with fundamental freq. 2) with fundamental freq. 2,, Fig.Fig.3.43.4

------ a real periodic signala real periodic signal


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For realFor real x(tx(t)) x(tx(t) = x) = x**(t)(t)

let k =let k = --kk


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IfIf aakk in rectangular formin rectangular form

For real periodic function (3.25) = (3.31) = (3.32)For real periodic function (3.25) = (3.31) = (3.32)


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3.3.2 Determination of the Fourier Series3.3.2 Determination of the Fourier Series

Representation of a ContinuousRepresentation of a Continuous--time Periodictime Periodic

SignalSignal

DetermineDetermine aakk


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(3.38)(3.38) ------ synthesis eq.synthesis eq.

(3.39)(3.39) ------ analysis eq.analysis eq.

{{aakk}: F.S.}: F.S. coeffcoeff. or spectral. or spectral coeffcoeff. of. of x(tx(t)) aaoo: dc or constant component of: dc or constant component of x(tx(t))

------ average value ofaverage value of x(tx(t) over one) over oneperiodperiod


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Exp. 3Exp. 3--44 ------

Fig. 3.5Fig. 3.5


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3.4 CONVERGENCE OF THE FOURIER SERIES3.4 CONVERGENCE OF THE FOURIER SERIES

Every continuous periodic signal has a FourierEvery continuous periodic signal has a Fourierseries representation.series representation.

This is also true for many discontinuous signals.This is also true for many discontinuous signals.

TheThe DirichletDirichlet conditionsconditions ------ guarantees thatguarantees that x(tx(t))equalsequals its Fourier series representation, exceptits Fourier series representation, except

at isolated values of tat isolated values of t for whichfor which x(tx(t)) isisdiscontinuous.discontinuous.

At these values, the infinite series converges toAt these values, the infinite series converges to

the average of the values on either side of thethe average of the values on either side of thediscontinuity.discontinuity.


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Condition 1: Over any period,Condition 1: Over any period, x(tx(t) must be) must beabsolutelyabsolutely integrableintegrable..

Fig. 3.8(a)Fig. 3.8(a) ------ Condition 2: In any finite interval of time,Condition 2: In any finite interval of time, x(tx(t) is of) is of

bounded variation.bounded variation.

Fig. 3.8 (b)Fig. 3.8 (b) ------

Condition 3Condition 3 ------ In any finite interval of time,In any finite interval of time,there are only a finite number of discontinuities.there are only a finite number of discontinuities.

Fig. 3.8 (c)Fig. 3.8 (c) ------


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Gibbs phenomenonGibbs phenomenon

------

Fig. 3.9Fig. 3.9

1.1. xxNN(t(t) has the average value of the discontinuity) has the average value of the discontinuity

2. for any other value of t = t2. for any other value of t = t11,,

3. constant 9% overshot3. constant 9% overshot

3 5 PROPERTIES OF CONTINUOUS3 5 PROPERTIES OF CONTINUOUS TIMETIME


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3.5 PROPERTIES OF CONTINUOUS3.5 PROPERTIES OF CONTINUOUS--TIMETIME

FOURIER SERIESFOURIER SERIES

3.5.1 Linearity3.5.1 Linearity


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3.5.2 Time Shifting3.5.2 Time Shifting

yy((tt) =) = xx((ttttoo))

letlet == ttttoo

i.e.i.e.

When a periodic signal is shifted in time, theWhen a periodic signal is shifted in time, the

magnitudes of its Fourier series coefficientsmagnitudes of its Fourier series coefficientsremain unaltered, i.e., |remain unaltered, i.e., |aakk| = || = |bbkk|.|.

3 5 3 Time Reversal3 5 3 Time Reversal


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3.5.3 Time Reversal3.5.3 Time Reversal

yy((tt) =) = xx((--tt))

letlet kk == --mm

Time reversal applied to a continuousTime reversal applied to a continuous--time signaltime signal

results in a time reversal of the correspondingresults in a time reversal of the correspondingsequence of F.S. coefficients.sequence of F.S. coefficients.

IfIf x(tx(t) is even) is even x(x(--tt) =) = x(tx(t)) aa--kk

== aakk

aakk

is evenis even

x(tx(t) is odd) is odd x(x(--tt) =) = --x(tx(t) a) a--kk == --aakk aakk is oddis odd


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3.5.4 Time Scaling3.5.4 Time Scaling

IfIf x(tx(t) is periodic with period T, then) is periodic with period T, then x(x(tt), where), where is a positive real number, is periodic withis a positive real number, is periodic with

period T/period T/..

The Fourier coefficients have not changed, theThe Fourier coefficients have not changed, the

Fourier series representation has changed.Fourier series representation has changed.

Because the fundamental frequency hasBecause the fundamental frequency has

changed.changed.


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3.5.5 Multiplication3.5.5 Multiplication

x(tx(t) and) and y(ty(t) are both periodic with period T.) are both periodic with period T.

discretediscrete--time convolutiontime convolution


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3.5.6 Conjugation and Conjugate Symmetry3.5.6 Conjugation and Conjugate Symmetry

Take * on both side and let k =Take * on both side and let k = --mm

x(tx(t) real) real x(tx(t) = x*(t)) = x*(t) aa--kk == aakk**

x(tx(t) real and even) real and even aa--kk == aakk** aakk == aakk**

aakk = a= a--kk aakk = a= a--kkaakk real and evenreal and even

x(tx(t) real and odd) real and odd aakk

pure imaginary and oddpure imaginary and odd


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3.5.73.5.7 ParsevalParsevalss Relation for ContinuousRelation for Continuous--TimeTime

Periodic SignalsPeriodic Signals

|a|akk||22 : average power in the: average power in the kkthth harmonicharmonic

component ofcomponent of x(tx(t))

The total average power in a periodic signalThe total average power in a periodic signalequals the sum of the average powers in all ofequals the sum of the average powers in all of

its harmonic components.its harmonic components. Tab. 3.1Tab. 3.1 ------ summary of propertiessummary of properties

average poweraverage power


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Exp. 3.6Exp. 3.6 ------ fromfrom Exp. 3Exp. 3--55, T = 4, T, T = 4, T11 = 1= 1

Fig. 3.10Fig. 3.10 ------

x(tx(t)) aakk

x(tx(t1)1)

--1/21/2

g(tg(t))


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Exp. 3.9Exp. 3.9 ------ x(tx(t) = ?) = ?

1.1. x(tx(t) is a real signal.) is a real signal.2.2. x(tx(t) is periodic with period T = 4, and it has) is periodic with period T = 4, and it hasF.S. coefficientsF.S. coefficients aakk..

3.3. aakk = 0 for k > 1.= 0 for k > 1.4. The signal with Fourier coefficients4. The signal with Fourier coefficients bbkk = e= e

--jjk/2k/2aa--kkis odd.is odd.

5.5.

from (3)from (3) aakk = a= a--11,, aaoo, a, a11from (2), T = 4from (2), T = 4

from (1)from (1) aa11 = a= a--11**

1/2)(4

2

41 = dttx


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from (4),from (4),

x(x(--tt)) aa--kksignal shift by 1 to the right (i.e., replace by tsignal shift by 1 to the right (i.e., replace by t --1)1)

aakkeeikik/2/2

x(x(--(t(t1)) =1)) = x(x(--tt + 1)+ 1) bbkk = e= e--jkjk/2/2aa--kk

x(tx(t) real) real x(x(--tt + 1) is real+ 1) is real

bboo = 0, b= 0, b--11 == -- bb11

must be oddmust be odd

F (5)F (5)


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From (5),From (5),average power foraverage power for x(tx(t) = that of) = that of x(x(--tt +1)+1)

bb11

== --bb--11

|b|b11| =| = , and b, and b

11has to be purelyhas to be purely

imaginaryimaginary bb11 = j/2 or= j/2 orj/2j/2

bboo

= 0= 0 aaoo

= 0= 0

aa11 = e= e--jj/2/2bb11 == --jbjb--11 = jb= jb11

== --1/21/2 oror 1/21/2

x(tx(t) =) = --coscos ((t/2) ort/2) or cos(cos(t/2)t/2)


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3.6 FOURIER SERIES REPRESENTATION OF3.6 FOURIER SERIES REPRESENTATION OF

DISCRETEDISCRETE--TIME PERIODIC SIGNALSTIME PERIODIC SIGNALS

A finite series, no issues of convergenceA finite series, no issues of convergence

3.6.1 Linear Combinations of Harmonically3.6.1 Linear Combinations of Harmonically

Related Complex ExponentialsRelated Complex Exponentials

Harmonically related but only N distinct signalsHarmonically related but only N distinct signalsin the set, i.e.,in the set, i.e.,

3.6.2 Determination of the Fourier Series3.6.2 Determination of the Fourier Series


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3.6.2 Determination of the Fourier Series

Representation of a Periodic SignalRepresentation of a Periodic Signal

the sum over one period of the values of athe sum over one period of the values of aperiodic complex exponential is zero, unless thatperiodic complex exponential is zero, unless thatcomplex exponential is a constant.complex exponential is a constant.

= 0, unless k= 0, unless kr = 0 or anr = 0 or an

integer multiple of Ninteger multiple of N


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Take k = 0,Take k = 0,, N, N11

Take k = 1,Take k = 1,, N, N

ffoo[n[n] =] = ffNN[n[n]] aaoo == aaNN

aakk repeat periodically with period Nrepeat periodically with period N

aakk is a sequence defined for all values of k, butis a sequence defined for all values of k, butwhere only N successive elements in thewhere only N successive elements in thesequence will be used in the Fourier seriessequence will be used in the Fourier series

representation.representation.


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Exp. 3.10Exp. 3.10 ------ x[nx[n] =] = sinsinoonn

Fig. 3.13Fig. 3.13 ------ N = 5N = 5


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Exp. 3.12Exp. 3.12 ------ Fig. 3.16Fig. 3.16,, x[nx[n] = 1, for] = 1, for NN11


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Exp. 3.12Exp. 3.12 Fig. 3.16Fig. 3.16,, x[nx[n] 1, for] 1, for NN11 nn NN11

let m = n + Nlet m = n + N11

Geometric seriesGeometric series

Fig 3 17Fig 3 17 ------ aakk for 2Nfor 2N11 + 1 = 5 N+ 1 = 5 N11 = 2 N = 10= 2 N = 10


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Fig. 3.17Fig. 3.17 --- aakk for 2Nfor 2N11 + 1 = 5, N+ 1 5, N11 = 2, N = 10, 2, N 10,20, 4020, 40

Convergence ?Convergence ? ------ Fig. 3.18Fig. 3.18

N = 9, 2NN = 9, 2N11 + 1 = 5+ 1 = 5

for M = 4for M = 4 x[nx[n] =] = x[nx[n]] There are no convergence issues and no GibbThere are no convergence issues and no Gibbss

phenomenonphenomenon

3.7 PROPERTIES OF DISCRETE3.7 PROPERTIES OF DISCRETE--TIME FOURIERTIME FOURIER


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SERIESSERIES

3.7.1 Multiplication3.7.1 Multiplication

Tab. 3.2Tab. 3.2

both periodic with period Nboth periodic with period N

periodic convolutionperiodic convolution

3.7.2 First Difference3.7.2 First Difference


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------ Time shifting and linearityTime shifting and linearity

3.7.33.7.3

Parseval'sParseval's

Relation for DiscreteRelation for Discrete

--TimeTime

Periodic SignalsPeriodic Signals

the average power in a periodic signal equalsthe average power in a periodic signal equals

the sum of the average powers in all of itsthe sum of the average powers in all of itsharmonic components.harmonic components.

3.8 FOURIER SERIES AND LTI SYSTEMS3.8 FOURIER SERIES AND LTI SYSTEMS


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3.8 FOURIER SERIES AND LTI SYSTEMS

ContContttx(tx(t) =) = eestst y(ty(t) =) = H(s)eH(s)estst

focus onfocus on Re{sRe{s} = 0, i.e., s =} = 0, i.e., s =jj

DiscDisctt

x[nx[n] =] = zznn y[ny[n] =] = H(z)zH(z)znn

focus on |z| = 1, i.e., z =focus on |z| = 1, i.e., z = eejj

FrequencyFrequencyresponseresponse

Take contTake cont t as an e amplet as an example


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Take contTake contt as an examplet as an example

The effect of the LTI system is to modifyThe effect of the LTI system is to modify

individually each of the Fourier coefficients of theindividually each of the Fourier coefficients of theinput through multiplication by the value of theinput through multiplication by the value of thefrequency response at the correspondingfrequency response at the corresponding

frequency.frequency.

F.S. coefficient for outputF.S. coefficient for output


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In discIn disc -- tt

1/(1+jk1/(1+jkoo))

3.9 FILTERING3.9 FILTERING

3 9 1 Frequency3 9 1 Frequency Shaping FiltersShaping Filters


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3.9.1 Frequency3.9.1 Frequency--Shaping FiltersShaping Filters

Fig. 3.22Fig. 3.22 ------ dB (decibel) = 20dB (decibel) = 20 log|H(jlog|H(j)|)| Fig. 3.23Fig. 3.23 ------ y(ty(t) =) = dx(t)/dtdx(t)/dt

x(tx(t) =) = eejjtt y(ty(t) =) =jjeejjtt

used to enhance rapid variations or transitions ofused to enhance rapid variations or transitions ofsignals.signals.

Fig. 3.24Fig. 3.24 ------ enhance edge in picture processingenhance edge in picture processing x(tx(t11, t, t22)) ------ brightness of the imagebrightness of the image


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3.9.2 Frequency3.9.2 Frequency--Selective FiltersSelective Filters


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Select some bands and reject othersSelect some bands and reject others LowpassLowpass filterfilter

HighpassHighpass filterfilter

BandpassBandpass filterfilter

Cutoff frequenciesCutoff frequencies

PassbandPassband,, stopbandstopband Fig. 3.26Fig. 3.26 ------ idealideal lowpasslowpass

Fig. 3.27Fig. 3.27 ------ idealideal highpasshighpass

idealideal bandpassbandpass

End of Chap. 2End of Chap. 2


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信號與系統chap. 3 class notes

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