Fourier Series Notes - MMU Department of Mathematics & Applied Mathematics PDF

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4 FourierSeries Contents 4.1Introduction .............. ................... ................... ...................... . 1 4.2Fourier series expansion ............ ................... ................... .............. 1 4.2.1 Properties of Periodic functions ........ ..................... .................... .. 1 4.3Fourier’s Theorem ............ ................... ................... ................... 2 4.4The Fourier Coef cients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 4.5Summary.......... .................... ...................... ................... .......5 4.6Functionsofperiod2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.7Periodicfunctionswithperiodsotherthan2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.8Even and odd functions ............ ................... ................... ............. 11 4.8.1Even function ........ ................... ................... .................... 11 4.8.2Odd function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.8.3Properties of even and odd functions ............ ..................... ............ 13 4.9Linearity property ............... ...................... ................... ............ 16 4.10Convergence of the Fourier series........ ...................... ................... .....18 4.10.1Dirichlet’s Conditions ............ .................... ..................... .... 18 4.11Half–range cosine and sine series ...... ...................... ................... ....... 21 4.11.1Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.12Integration of a Fourier series. ................... ...................... ............... 26 4.13Differentiation of a Fourier series. ................... ...................... ........... 27 4.14Orthogonal functions ............ ...................... ................... ........... 28 4.14.1De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.15Generalized Fourier series . ................... ...................... .................. 29 4.16Complex form of Fourier series..................... ...................... ............30 4.17Multiplication and Parseval’s theorem..... ...................... ................... ...33 4.17.1The multiplication theorem .................... .................... ............ 33

4–1

Chapter 4

Fourier Series

4.17.2Parseval’s theorem ........ ................... ...................... ........... 34

4.1 Introduction Therepresentationofafunctionintheformofaseriesis fairlycommonpracticeinmathematics. Probablythemostfamiliarexpansionsarepowerseriesoftheform i ({) =

" X

dq {q

q=X

inwhichtheresolvedcomponentsorbasesetcomprisethepowerfunctions 1> {> {2 > {3 > = = = > {q > = = =

Forexample, h{ = 1 + { +

" X {q {q {2 {3 + + === + + === = 2! 3! q! q! q=0

NotallfunctionscanbeexpressedasaMaclaurin’sseries,e.g. i ({) = {= Powerfunctionscomprise onlyoneexampleofabasesetfortheexpansionoffunctions. Inparticular,a Fourierseriesisan 2 expansionofaperiodicfunctioni ({) ofperiodW = inwhichthebasesetisthesetofsine $ functions,givinganexpandedrepresentationoftheform i ({) = D0 +

" X

Dq sin(q${ + !q )

q=1

AnadvantageofFourier’sseriesisthatitcanrepresentafunctioncontainingdiscontinuities. In manyengineeringproblems,however,wehavetodealwithperiodicfunctions.

4.2 Fourierseriesexpansion 4.2.1 PropertiesofPeriodicfunctions Afunctioni ({) issaidtobeperiodicifitisrepeatedatregularintervals. i ({ + W ) = i ({)

We de ne the frequency of a periodic function to be the reciprocal of its period, so that frequency=

1 1 = period W

The term circular frequency is also used in engineering, and is de ned by circularfrequency= 2 × frequency=

2 W

andismeasuredinradianspersecond. Itiscommontodroptheterm‘circular’andrefertothis simplyasthefrequencywhenthecontextisclear.

4–2

Chapter 4

Fourier Series

4.3 Fourier’sTheorem This theorem states that a periodic function that satis es certain conditions can be expressed as the sumofanumberofsinefunctionsofdifferentamplitudes,phasesandperiods. Thatis,ifi ({) isa periodicfunctionwithperiodW then i ({) = D0 + D1 sin(${ + !1 ) + D2 sin(2${ + !2 ) + = = = + Dq sin(q${ + !q ) + = = =

wheretheD0 sand!0 s areconstantsand$ = 2@W isthefrequencyofi ({).Where: D1 sin(${ + !1 ) is called the rst harmonic or the fundamental mode, and it has the same

frequency$ astheparentfunctioni ({). Dq sin(q${ + !q ) iscalledthenth harmonic,andithasfrequencyq$,whichisq timesthatof

thefundamental. Dq denotestheamplitudeofthenth harmonicand!q isitsphaseangle,measuringthelagorlead

ofthenth harmonicwithreferencetoapure sinewaveofthesamefrequency. Since Dq sin(q${ + !q ) = (Dq cos !q ) sin q${ + (Dq sin !q ) cos q${ = eq sin q${ + dq cos q${

where eq = Dq cos !q > dq = Dq sin !q " X d0 theexpansionmaybewrittenasi ({) = + (dq cos q${ + eq sin q${) whered0 = 2D0 . 2 q=1 TheexpansioniscalledtheFourierseriesexpansionofthefunctioni ({),andthedv andev are

calledtheFouriercoef cients.

4.4 The Fourier Coef cients Thefollowingintegrals,inwhichW = 2@$ areveryimportant. ; A ? 0 q= 6 0 R f+W 1. f cos q${ g{ = A = W q=0 2.

3.

4.

5.

R f+W f

R f+W f

R f+W f

R f+W f

sin q${ g{ = 0

;q ; A ?0 sin p${ sin q${ g{ = A = 1W 2 cos p${ cos q${ g{ =

; A ? 0

A = 1W 2

cos p${ sin q${ g{ = 0

p 6= q p = q 6= 0 p 6= q p = q 6= 0 ; p andq

4–3

Chapter 4

Fourier Series

The following de nite integrals does pop up a lot when we use Fourier series. The numbers p andq occurringintheseformulaeareintegersotherthanzero. Asanexercise,PROVEthem. R ;q 1. 3 sin q{g{ = 0

2.

3.

4. 5.

R

3

R

0

R

0

sin q{g{ = 0

sin q{g{ =

; A ? 0

A = 2 q

cos q{g{ = 0

R

3

;q

;q even ;q odd ;q

cos2 q{g{ = 

q 6= 0

R

sin q{ cos p{g{ = 0 ; q andp ; A ? 0 q 6= p R 7. 3 cos q{ cos p{g{ = A =  q=p

6.

8.

3

R

3 sin q{ sin p{g{ =

; A ?0

A =

q 6= p q=p

Exercise 1 Evaluatethefollowingintegrals: R 1. 0 { sin q{g{ 2. 3. 4.

R

0

R

0

R3

0

ANSWERS ¸ 2 2 > q odd;  > q even q q ¸  1  2 > q odd;0> q even q ¸  4 > q odd;0> q even q ¸  54 54  > q odd; 2 2 > q even q  q2  2 

{ cos q{g{ { sin q{g{  {2 cos

R 2

³ q{ ´ 3



sin q{g{

g{

Theresultsconstitutetheorthogonalityrelationsforsineandcosinefunctions,andshowthattheset offunctions {1> cos ${> cos 2${> = = = > cos q${> sin ${> sin 2${> = = = > sin q${}

isanorthogonalsetoffunctionsontheintervalf  {  f + W . Thechoiceoff isarbitraryinthese results,itonlybeingnecessarytointegrateoveraperiodofdurationW .

4–4

Chapter 4

Fourier Series

Integratingtheserieswithrespectto{ overtheperiod{ = f to{ = f + W ,,wehave ¶ Z f+W Z f+W " X 1 Z f+W µ Z f+W i ({)g{ = cos q${g{ + e sin q${g{ g{ + q d0 dq f f f f 2 q=1 " X 1 [dq (0) + eq (0)] = d0 (W ) + 2 q=1 =

1 W d0 2

Thus

d0

andwecanseethattheconstantterm

Z 1 1 g+W i ({)g{ d0 = 2 W g 2 intheFourierseriesexpansionrepresentsthemeanvalueof

thefunctioni ({) overoneperiod. Hence 2 W

d0 =

Z

f+W

i ({)g{

f

Toobtainthisresult,wehaveassumedthatterm-by-termintegrationoftheseriesispermissible. Thisisindeedsobecauseoftheconvergencepropertiesoftheseries. Moreaboutorthogonalityand convergence later. We rst put our attention on determining the Fourier series expansion of a given periodicfunctioni ({) = To obtain the Fourier coef cient dq (q 6= 0),wemultiplythroughoutbycos p${ andintegratewith respectto{ overtheperiod{ = f to{ = f + W ,giving Z f+W Z f+W Z f+W " X 1 cos q${ cos p${g{ cos p${g{ + i ({) cos p${g{ = d0 dq 2 f f f q=1 Z f+W " X cos p${ sin q${g{ + eq q=1

f

the only non-zero integral on the right-hand side is the one that occurs in the rst summation when q = p. Thatis,wehave Z Z f+W i ({) cos p${g{ = dp

f+W

f

f

giving

2 dp = W

which,onreplacingp byq,gives dq =

2 W

Z

Z

1 cos p${ cos p${g{ = dp W 2

f+W

i ({) cos p${g{

f

f+W

i ({) cos q${g{

(q = 0> 1> 2> = = =)

f

Thevalueofd0 maybeobtainedbytakingq = 0 d0 =

2 W

Z

f+W

i ({)g{

f

To obtain the Fourier coef cients eq ,wemultiplythroughoutbysin p${ andintegratewithrespect

4–5

Chapter 4

Fourier Series

to{ overtheperiod{ = f to{ = f + W , Z f+W Z f+W " X 1 Z f+W i ({) sin p${g{ = d sin p${ cos q${g{ sin p${g{ + q d0 f f f 2 q=1 Z f+W " X eq sin p${ sin q${g{ === + f

q=1

Whenp = q,wehave Z f+W

i ({) sin p${g{ = ep

Z

f+W

sin p${ sin p${g{ =

f

f

giving,onreplacingp byq, eq =

2 W

Z

1 ep W 2

f+W

i ({) sin q${g{

(q = 1> 2> 3> = = =)

f

4.5 Summary Aperiodicfunctioni ({) ofperiodW = 2@$ canbeexpandedasaFourier seriesgivenby: "

X 1 i ({) = d0 + (dq cos q${ + eq sin q${) 2 q=1

where the coef cients are given by the Euler formulae Z 2 f+W i ({) cos q${g{ dq = W f Z 2 f+W eq = i ({) sin q${g{ W f

(q = 0> 1> 2> = = =) (q = 1> 2> 3> = = =)

ThelimitsofintegrationinEuler’sformulaemaybe speci ed over any period,thechoiceoff is arbitrary.Sometimes i ({) arespecifyovereithertheperiod 1

0 ? { ? W ,sothatthelimitsofintegrationis 1

2W

2W

? { ? 12 W ortheperiod

to 1 2 W (i.e.f =  12 W )or0 toW (i.e.f = 0).

4.6 Functionsofperiod2 Example4.1 ObtaintheFourierseriesexpansionoftheperiodicfunctioni ({) ofperiod2 de ned by

i ({) = {

(0 ? { ? 2)>

i({) = i ({ + 2)

Solution. Asketchofthefunctioni ({) overtheinterval4 ? { ? 4 isshown.

Sincethefunctionisperiodicweonlyneedtosketchitoveroneperiod,thepatternbeingrepeated 4–6

Chapter 4

Fourier Series

for other periods. Evaluating the Fourier coef cients d0 anddq gives d0 =

1Z



2

i ({)g{ =

0

1Z



2

{g{ =

0

 ¸ 1 {2 2 = 2  2 0

and Z

1 

dq =

Z

1 

=

2

i ({) cos q{g{

(q = 1> 2> = = =)

0 2

{ cos q{g{ 0

¸  cos q{ 2 1 sin q{ + = { q q2 0  ¶ µ cos 0 1 1 2 sin 2q + 2 cos 2q  2 = q q  q = 0

sincesin 2q = 0 andcos 2q = cos 0 = 1. Notetheneedtoworkoutd0 separatelyfromdq .For eq : eq

1 =  =

1 

Z

Z

2

i ({) sin q{g{

(q = 1> 2> = = =)

0 2

{ sin q{g{ 0

¸  sin q{ 2 { 1 =  cos q{ + q q2 0  =

1 2 ( cos 2q)  q

= 

2 q

[sin 2q = sin 0 = 0]

[cos 2q = 1]

TheFourierseriesexpansionofi ({) is i ({) =  

" X 2 q=1

q

sin q{

=   2(sin { +

sin q{ sin 2{ sin 3{ + +=== + + = = =) 2 3 q

Example4.2 Aperiodicfunctioni ({) withperiod2 is de ned by i ({) = {2 + {

( ? { ? )>

i({) = i ({ + 2)

Solution. Sketchagraphofthefunctioni ({) for valuesof{ from{ = 3 to{ = 3 andobtain aFourierseriesexpansionofthefunction.

4–7

Chapter 4

Fourier Series

where 1 

d0 =

1 

=

Z



i ({)g{

3 Z 

2 ({2 + {)g{ =  2 3 3

and dq = =

1  1 

1 = 

Z



i ({) cos q{g{

3 Z 

({2 + {) cos q{g{

3  2 {



q

sin q{ +

1 4 = cos q  q2 =

(q = 1> 2> 3> = = =)

4 (1)q q2

¸

¸ 2{ 2 { 1 sin q{ + cos q{  sin q{ + cos q{ q2 q3 q q2 3 [sin q = 0]

[cos q = (1)q ]

and eq

1 =  =

1 

Z



i ({) sin q{g{

3 Z 

(q = 1> 2> 3> = = =)

({2 + {) sin q{g{

3

 2 ¸ 2{ 2 { 1 { 1 cos q{ + 2 sin q{ + 3 cos q{  cos q{ + 2 sin q{ =  q q q q  q 3

2 =  cos q q 2 =  (1)q q

[cos q = (1)q ]

TheFourierseriesexpansionofi ({) is i ({) = =

" " X 2 1 2 X 4 q (1)q sin q{ (  1) cos q{   + 2 q q 3 q=1 q=1 ¶ ¶ µ µ 1 2 sin 3{ sin 2{ cos 2{ cos 3{ +  + = = = + = = = + 2 sin {   + 4  cos { + 2 22 3 32 3

4–8

Chapter 4

Fourier Series

Usingthecomplexform,analternativeapproach,gives(Thismethodwill bedonelaterinthis section) dq + meq =

1 

Z



i ({)hmq{g{ =

3

1 

Z



({2 + {)hmq{g{

3 Z 

µ 2 ¸ ¶ 2{ + 1 mq{ 1 { + { mq{  h h g{  mq mq 3 3 ¸  1 {2 + { mq{ 2{ + 1 mq{ 2hmq{  = h  h + mq (mq)2  (mq)3 3

=

Since hmq = cos q + m sin q = (1)q h3mq = cos q  m sin q = (1)q 1@m = m

Equatingrealandimaginarypartsgives,asbefore, dq =

4 (1)q > q2

2 eq =  (1)q q

Aperiodicfunctioni ({) may be speci ed in a piecewise fashion over a period, or, indeed, it may only be piecewise-continuous over a period. In order to calculate the Fourier coef cients in such cases,itisnecessarytobreakuptherangeofintegrationintheEulerformulaetocorrespondtothe variouscomponentsofthefunction. Example4.3 Aperiodicfunctioni ({) ofperiod2 is de ned within the period 0  {  2 by ; A A A { if 0  {  12  A A ? 1 i ({) = if 12   {   2 A A A A A =   1 { if   {  2 2 Sketchagraphofi ({) for2  {  3 and nd a Fourier series expansion of it.

Solution. Agraphofthefunctioni ({) for2  {  3 isshown,

4–9

Chapter 4

Fourier Series

where d0 =

1Z



2

i ({)g{ =

0

" 1 Z



@2

{g{ +

0

Z



@2

1 2

g{ +

Z

2



( 

1 2

#

5 {)g{ = 8 

and Z

1 

dq =

2

i ({) cos q{g{

Z  Z 2 Z 1 1 1 @2 { cos q{g{ +  cos q{g{ + (  {) cos q{g{] [ 2  0 @2 2  Ã  ¸ ! h i i h   2  { sin q{ cos q{ 2 cos q{ @2 { 1 + +  sin q{ + sin q{ q 2q q2 0 2q2  @2  q 2 ¶ µ 1  1  1 1 1 1 1 1 sin q  2 + 2 cos q sin q + 2 cos q  2  2q  2q 2q q q 2q 2 2 2

= = =

1 1 (2 cos q  3 + cos q) 2 2q2

=

thatis, dq =

and eq =

1 

=

1 

=

1 

=

1 

=

=

Z

; A ?

1 q@2 q2 [(1)

A =  2 q2

 1]

even q odd

q

2

"0Z

i ({) sin q{g{

(q = 1> 2> 3> = = =)

# Z 2 1 1  sin q{g{ + (  {) sin q{g{ { sin q{g{ + 2 0  @2 2 Ã   ¸2 ! ¸@2 h i  {  2 1 1 { +  cos q{ + cos q{ + 2 sin q{ cos q{  2 sin q{  2q q 2q @2 2q q  0 µ ¶  1 1 1   1  cos q  cos q + 2 sin q  cos q + cos q + 2q 2q 2 q 2 2q 2q 2 Z

@2

1 1 sin q q2 2 ; A ? 0 A =

(q = 1> 2> 3> = = =)

0

(31)(q1)@2 q2



; evenq ; oddq

TheFourierseriesexpansionofi ({) is i ({) =

5 2 cos 3{ cos 5{ + = = =)   (cos { + + 52 16  32 2 cos 2{ cos 6{ cos 10{ + = = =)  ( 2 + + 102  2 62 1 sin 3{ sin 5{ sin 7{ + (sin {  + = = =) +  72  32 52

4–10

Chapter 4

Fourier Series

4.7 Periodicfunctionswithperiodsotherthan2 If| = i ({) is de ned in the range o too andnotfrom to ,ithasaperiod2o i.e. i ({ + 2o) = i ({) = Wecanconvertthistoanintervalof2 bychangingthevalueoftheunitsofthe

independentvariable. Example4.4 DeterminetheFourierseriesfor ; A ? { for 4 ? { ? 0 i ({) = A = { for 0 ? { ? 4

andi ({ + 8) = i ({)

Solution. Here d0

1 =  4

Z

0

1 {g{ + 4 34

Z

4

{g{ 0

1 1 £ ¤4 1 1 £ 2 ¤0 =  · { 34 + · {2 0 4 2 4 2 = 2+2 =4

and Z ³ q{ ´ 1 4 cos g{ (q = 1> 2> 3> = = =) g{ + 4 0 4 4 34 ( ) Z 0 ³ q{ ´ ³ q{ ´¸0 1 4{ 4 =  sin g{  sin q q 34 4 4 4 34 ( ) Z 4 ³ q{ ´¸4 ³ q{ ´ 1 4{ 4 +  sin sin g{ q 4 q 0 4 4 0   ³ q{ ´¸4 ³ q{ ´¸0 1 16 1 16 cos cos + =  4 q2  2 4 4 4 q2  2 0 34

dq = 

1 4

Z

0

{ cos

³ q{´

8

(cos q  1) ; A A ?  16 q odd q2  2 = A A = 0 q even =

q2  2

4–11

Chapter 4 andeq = 0

Fourier Series

(FKHFN this!!!)

TheFourierseriesexpansionofi ({) is i ({) = 2 +

" X

8

(cos q  1) cos

³ q{ ´

4 q2  2 µ ¶ 5{ 1 3{ 1 { 16 cos + + cos +··· cos = 2 25 4 9 4 4 2 q=1

Exercise 2 Obtain the Fourier series for the functions de ned by: 1. i ({) = {2 >

0 ? { ? 2 andi ({ + 2) = i ({)

Sketchthegraphfor{ 5 [4> 4] ; A ? { +  for  ? { ? 0 2. i ({) = A =0 for 0 ? { ?  Sketchthegraphfor{ 5 [3> 3] ; A ? 1 for 1 ? { ? 0 3. i ({) = A = { for 0 ? { ? 1

Sketchthegraphfor{ 5 [3> 3] ;  A ?{ for 0 ? { ? 2 4. i ({) = A =   { for  ? { ?  2 Sketchthegraphfor{ 5 [> 2]

andi ({ + 2) = i ({)

andi ({ + 2) = i ({)

andi ({ + ) = i ({)

4.8 Evenandoddfunctions Aparticularfunctionpossessescertainsymmetricalpropertiesenablesusbothtotellwhichterms areabsentfromaFourierseriesexpans...