19 Fourier Series and Practical Harmonic Analysis

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1 9 Fourier Series ad Practica Harmoic Aaysis Eampe : Obtai the Fourier series of f ( ) e a i. a Soutio: Let f ( ) acos bsi sih a a a a a a e a a where a f ( ) d e d e e a a e a f ( ) cos d e cos d ( a cos si ) a a a ae cos ae cos a a a a cos e e a( ) sih a a a a a e b f ( ) si d e si d ( a si cos ) a a a a a ( ) e e e cos e cos a a ( ) a sih a Therefore sih sih ( ) a a a a sih a ( ) f ( ) e cos si a a a Eampe : Prove that ( ) 4 cos,. Hece show that 3 (i) (ii) 6 ( ) 8 (iii) (iv) Soutio: Let a f ( ) a cos bsi... () The by Euer s formuae,

2 a 3 d 3 3 si cos si a ( )cos d 3 cos cos 4( ) ( ) ( ) cos si cos b ( )si d 3 cos cos cos cos ( ) ( ) 3 3 Substitutig a, a ad b i equatio (), we get Put ( ) 4 cos... () 3 i this equatio ( ) ( ) 4 cos 4 ( ) 3 3 or (3) Put i equatio () ( ) 4 3 ( ) or (4) 3 4 Now addig the equatios (3) ad (4), we have or i.e ( ) 8

3 Now mutipy equatio () with 4 ( ) 3 throughout ad itegratig w.r.t. X from to, we get d d 4 cos d 5 3 ( ) si cos si ( ) ( ) Eampe 3: Obtai the Fourier series of i f ( ). a Soutio: Let f ( ) acos bsi... () The Euer s costats a, a ad b are give by 3 a ( ) d 3 3 si cos si a ( )cos d 3 cos cos 4( ) ( ) ( ) cos si cos b ( )si d 3 cos cos cos cos ( ) ( ) ( ) 3 3 Therefore the Fourier series for is f ( ) 3 ( ) ( ) f ( ) 4 cos si cos cos cos3 si si si

4 Puttig, we fid aother iterestig series Eampe 4: Epad f ( ) si as a Fourier series i the iterva (, ). a Soutio: Let f ( ) acos bsi si cos. si The a d a si cos d (cos si ) d si( ) si( ) d cos( ) cos( ) si( ) si( ). ( ) ( ) cos ( ) cos ( ), Whe, a si cos d si d cos si. 4 Fiay, b si si d (si si ) d cos( ) cos( ) d si( ) si( ) cos( ) cos( ). ( ) ( ) cos ( ) cos ( ), ( ) ( ) ( ) ( ) ( ) 4

5 Whe b d d d, si si si cos si cos. 4 Therefore, si si cos cos cos Eampe 5: Epad f ( ) cos, i a Fourier series. Hece evauate Soutio: We have f ( ) cos si, a Let f ( ) acos bsi The a 4 si d cos a si cos d cos si d si si d cos cos cos( ) cos( ) 4 4 b si si d si si d cos cos d 5

6 si si si( ) si( ) 4 Therefore, f ( ) cos cos (4 ) Puttig, we fid 4 cos (4 ) ie CONDITIONS FOR A FOURIER EXPANSION The reader must ot be mised by the beief that the Fourier series epasio of f( ) i each case sha be vaid. The above discussio has merey show that if f( ) has a epasio, the the coefficiets are give by Euer s formuae. The probems cocerig the possibiity of epressig a fuctio by Fourier series ad covergece of this series are may ad cumbersome. Such questios shoud be eft to the curiosity of a pure-mathematicia. However, amost a egieerig appicatios are covered by the foowig we-kow Dirichet s coditios: Ay fuctio f( ) ca be deveoped as a Fourier series where a, a, b are costats, provided: a f a b ( ) cos si (i) f( ) (ii) f( ) (iii) f( ) is periodic, sige vaued ad fiite; has fiite umber of discotiuities i ay oe period; has at the most a fiite umber of maima ad miima. FUNCTIONS HAVING PONT OF DISCONTINUITY I derivig the Euer s formuae for a, a, b it was assumed that f( ) was cotiuous. Istead a fuctio may have a fiite umber of poits of fiite discotiuity i.e. its graph may cosists of a fiite umber of differet curves give by differet equatios. Eve the such a fuctio is epressibe as a Fourier series. For istace, if i the iterva (, ), f( ) is defied by 6

7 ( ), c f( ) ( ), c i.e. c is the poit of discotiuity, the c a ( ) d ( ) d c c a ( )cos d ( )cos d c c b ( )si d ( )si d c At a poit of fiite discotiuity imit o the eft f( c ) ad the imit o right f( c ) c, there is a fiite jump i the graph of fuctio. Both the eist ad differet. At such a poit, Fourier series gives the vaue of f( ) as the arithmetic mea of these two imits, i.e. at c, f( ) f ( c ) f ( c ) Eampe 6: Fid the Fourier series epasio for f(), if, f( ),. Deduce that a Soutio: Let f ( ) acos bsi Fiay, 7... (i) The a ( ) d d si si cos a ( )cos d cos d ( ) a, a, a, a, a etc

8 cos cos si b ( )si d si d cos cos cos b 3, b, b, b, etc Hece substitutig the vaues of a s ad b s i (i), we get cos cos3 cos5 si 3si3 si 4 f ( ) si (ii) which is the required resut. Puttig = i (ii), we obtai f () (iii) Now f() is discotiuous at =. As a matter of fact f( ) ad f( ) f () f ( ) f ( ) Hece (iii) takes the form whece the resut I Eampe 7: A ateratig curret si, i, after passig through a rectifier, where I is the maimum curret ad period is. Epresss i as a Fourier series ad evauate Soutio: Let I si, a i acos bsi,... (i) Now, I I a Isi d d cos 8

9 I a I si cos d.cos d (si cos ) d I cos( ) cos( ) si( ) si( ) I d I I cos( ) cos( ) cos( ) cos( ) Whe is odd, -, + are eve; cos( ) cos( ) I Hece a Whe is eve, -, + are odd; cos( ) cos( ) Hece a I I, Now I b I si si d.si d (si si ) d I I si( ) si( ) cos( ) cos( ) d, Therefore equatio (i) becomes I si, I I i cos acos bsi, ( ) eve Now, we wi fid a ad b, I I a I si cos d.cos d (si cos ) d si d I cos cos 4 cos I I I b I si si d.si d si si d cos d 9

10 si si I I I I si, I I I i si, ( ) cos eve I I I cos m si m ( m) I I I cos cos 4 cos6 si Put = i this equatio f I I I cos cos 4 cos ( ) si Eampe 8: Fid the Fourier series of the fuctio f( ),, Soutio: 3 3 a d d 3 3 a cos d cos d si cos si 3 si cos si 3 cos cos Ad b si d si d

11 cos si cos 3 cos si cos 3 cos cos cos cos ( ) ( ) 4 ( ) ( ) Case: Whe =eve, b 8 4 Case: Whe =odd, b 3 Therefore, a f a b ( ) cos si f ( ) b si b si b si b si Probems: 4 4 si si si ,. Deveop f( ) i a Fourier series i the iterva (, ), if f( ).,,. Fid the Fourier series epasio for f( ). Deduce that, Fid the Fourier series of the fuctio defied i oe period by the reatios, f( ) ad deduce that ,

12 4. Fid the Fourier series of with period., f( ), which is assumed to be periodic A) EVEN FUNCTION: A fuctio f( ) is said to be eve (or symmetric) fuctio if, f ( ) f ( ) The graph of such a fuctio is symmetrica with respect to y-ais [ f( ) eists]. Here y-ais is a mirror for the refectio of the curve. B) ODD FUNCTION: A fuctio f( ) is said to be odd fuctio if, f ( ) f ( ) Eampe: Fid the Fourier series epasio of the periodic fuctio period. Here, fid the sum of the series f ( ), of Soutio: Give that f ( ), This is a eve fuctio. Therefore b 3 ( ) 3 3 a f d d a f ( )cos d cos d si cos si 3 si cos si 4 3 a Fourier series is f ( ) a cos a cos a3 cos

13 O puttig, we have cos cos cos3 cos or Eampe: Obtai a Fourier series epressio for 3 f ( ),. Soutio: Here f ( ) a ad a 3 is a odd fuctio. 3 b f ( )si d si d cos si cos si cos cos si si si Eampe: If f ( ) cos, epad f( ) as a Fourier series i the iterva,. Soutio: As f ( ) cos( ) cos f ( ), f ( ) cos is a eve fuctio. a f ( ) a cos where a cos d cos d cos d cos is-ve whe 4 si si ad 3

14 a cos cos d cos cos d cos cos d cos( ) cos( ) d cos( ) cos( ) d si ( ) si ( ) si ( ) si ( ) si ( ) si ( ) si ( ) si ( ) cos cos 4cos, I particuar, a cos d cos d Hece 4 cos cos cos Eampe: Obtai the Fourier series for f ( ) si i the iterva(, ). Soutio: Sice f ( ) si is a eve fuctio of, therefore b. a Let f ( ) si a cos si ( cos ).( si ) ( cos ) The a d a si cos d (cos si ) d si( ) si( ) d cos( ) cos( ) si( ) si( ). ( ) ( ) cos( ) cos( ), 4

15 cos( ) cos( ) Whe is odd,, - ad + are eve a Whe is eve, - ad + are odd a Whe, we have a si cos d (si cos ) d cos si si d. 4 cos cos cos3 cos 4 cos5 f ( ) si cos Puttig, we get Eampe: Fid the Fourier series of the fuctio f( ),, Soutio: Sice,, f ( ) f ( ),, Therefore, it is a odd fuctio ad hece a, a are zero ad b si d 5

16 cos si cos 3 cos cos 3 3 ( ) ( ) 4 ( ) ( ) ( ) ( ) ( ) si Therefore, f 3 Probems. Obtai the Fourier series for the fuctio f ( ) cos i the iterva (, ). si a si si si 3. Show that for (, ), si a a a 3 a 3. Fid the Fourier series of the fuctio Fid the Fourier series to represet the fuctio deduce that Fid the Fourier series for the fuctio 6. Fid the Fourier series for the fuctio 6, f( ). ad hece show that,, f ( ),., Fid a Fourier series for f ( ) from c to c. 8. Fid a Fourier series to represet for 9. Fid the Fourier series for the fuctio f ( ) k, f( ). Aso k,, f( ) ad deduce that, i the iterva (, )., f ( ) k,.,

17 Haf Rage Series: Sometimes it is required to epad a fuctio f( ) i the rage (, ) i Fourier series of period or more geeray i the rage (, ) i a Fourier series of period. If it is required to epad f( ) i the iterva (, ), the it is immateria what the fuctio may be outside the rage. We are free to choose it arbitrariy i the iterva (, ). If we eted the fuctio f( ) by refectig it i the y-ais so that f ( ) f ( ), the the eteded fuctio is eve for which b. The Fourier epasio of f( ) wi cotai oy cosie terms. If we eted the fuctio f( ) by refectig it i the origi so that f ( ) f ( ), the the eteded fuctio is odd for which a, a. The Fourier epasio of f( ) wi cotai oy sie terms. Hece a fuctio f( ) defied over the iterva is capabe of two distict haf-rage a series. The haf-rage cosie series is f ( ) a cos Where a f ( ) d; a ( )cos f d The haf-rage sie series is f ( ) b si, whereb f ( )si d. Cor:. If the rage is, the a (i) The haf rage cosie series is f ( ) a cos, where (ii) a f ( ) d; a ( )cos f d The haf-rage sie series is f ( ) b si, where b f ( )si d 7

18 Eampe: Epress f ( ) Soutio: Let us eted the fuctio f ( ) as a haf rage sie series i. i the iterva so that the ew fuctio is symmetrica about the origi ad, therefore, represets a odd fuctio i (, ). Hece the Fourier series for f( ) over the fu period (,) f ( ) b si wi cotai oy sie terms give by where b f ( )si d si d 4 4( ) cos si 4, 4, 4, 4, etc. 3 4 Thus b b b3 b4 Hece the Fourier sie series for f( ) over the haf rage is f( ) si si si si Eampe: Obtai the haf-rage sie series for f ( ) e i. Soutio: Let us eted the fuctio f ( ) e i the iterva so that the ew fuctio is symmetrica about the origi ad, therefore, represets a odd fuctio i (, ). Hece the Fourier series for f( ) over the fu period (, ) wi cotai oy sie terms give by f ( ) b si, where e b e si d (si cos ) e ( cos ) ( ) e( ) e( ) Hece e [ e( ) ] si e e e si si 3 si

19 Eampe: Obtai the haf-rage cosie series for k whe f( ) k( ) whe i. Deduce the sum of the series 3 5 a Soutio: Let f ( ) a cos The a f ( ) d k d k( ) d k k k k k k k a f ( )cos d k.cos d k( ). cos d k. si k. cos k( ). si k. cos k si k cos k cos k si k cos k k k k cos cos cos cos Whe is odd, cos ad cos a a a3 a5... k 8k cos cos ; Whe is eve, a k a4 cos cos 4 ; 4 9

20 k 6 8 k a cos3 cos 6 ad so o. 6 6 k 8k 6 f( ) cos cos cos () Puttig, f ( ) k 8k from (), we have Hece Probems:. a) Obtai cosie ad sie series for f ( ) i the iterva. Hece show that b) Prove that for, cos cos cos Fid the haf rage cosie series for the fuctio 3. Fid the haf rage cosie series for the fuctio Hece show that (i) (ii) (iii) Epress f ( ) as a haf rage (i) Sie series i (ii) Cosie series i. 5. Fid the Fourier sie ad cosie series of 6. Fid the haf rage sie series for the fuctio f ( ) i the rage. i the rage f ( ) ( ), f( ).,, t f () t t t..

21 7. Prove that for t, cos t cos 4t cos6t t( t) Fid the haf rage sie series for, f( ) 4. 3, 4 9. Let f ( ), whe ad f ( ) ( ), whe. Show that 4 ( ) ( ) f( ) si. Hece obtai the sum of the series ( ) si, for. If f( ) 4. Epad the fuctio i the series of sie. cos, for 4 Assimet-. Obtai a Fourier series to represet e a from = πto = π. Hece derive series for π sihπ.. Prove that = π cos + 4 ( ) 3 =, π < < π. Hece show that (i) = π 6 (ii) = π ( ) 8 (iii) = π (iv) = π If f = π i the rage to π, show that f = π 4. Prove that i the rage π < < π, cosha = a sih aπ π + cos =. a + ( ) = +a cos. 5. f = + for π < < π ad f = π for = ±π. Epad f()i Fourier series. Hece Show that + = π + 3 ( ) 4 cos = si ad = π 8.

22 Assimet- State givig reasos whether the foowig fuctios ca be epaded i Fourier series i the iterva π π.. cosec. si (/) m + 3. f =, π < π m m + m, m =,, 3.. Assimet-3. Fid the Fourier series to represet the fuctio f() give by Deduce that f = for π π for π π. A ateratig curret after passig through a rectifier has the form i = I Si for π for π π wherei is the maimum curret ad the period is π Epress ias a Fourier series ad evauate Draw the graph of the fuctio f =, π < <, < < π. Iff π + = f(), obtai Fourier series of f(). 4. Fid the Fourier series of the foowig fuctio: f =, π, π. 5. Fid a Fourier series for the fuctio defied by

23 f =, π < <, =, < < π Hece prove that = π 4. Assimet-4. Obtaied the Fourier series for f = π i.. (i) Fid the Fourier series to represet i the iterva (, a). (ii) Fid a Fourier series for f t = t whe t. 3. If f = i, show thatf = 4 3 = π cos π., 3 4. Fid the Fourier series for f = 6, A siusoida votage E si ωt is passed through a haf wave rectifier which cips the egative portio of the wave. Deveop the resutig periodic fuctio U t =, T < t E si ωt, t T ad T = π/ω, i a Fourier series. 6. Fid the Fourier series of the fuctio f = π, < <, = π, < < Hece show that π 4 = Assimet-5. Obtai the Fourier series epasio of f = i (, a). Hece show that π 6 = Show that for π < < π, sia = siaπ si si 3si 3 + π a a 3 a 3. Epad the fuctio f = si as a Fourier series i the iterva π π Deduce that = 4 π. 4. Prove that i the iterva π < < π, cos = si + ( ) si. 3 =

24 5. For a fuctio f()defied by f =, π < < π, obtai a Fourier series. Deduce that π 8 = Fid the Fourier series to represet the fuctio (i) f = si, π < < π. (ii) f = cos π i the iterva (-,). + for π 7. Give f = + for π. Is the fuctio eve or odd? Fid the Fourier series for f() ad deduce the vaue of Fid the Fourier series of the periodic fuctio k, π < < f = adf + π = f. Sketch the graph of f() ad the two k, < < π partia sums. (See fig..7). Deduce that = ( ) + = π A fuctio is defied as foows:f =, for π < <, for < < π Show that f = π 4 π cos + 3 cos3 + 5 cos5 ad deduce that = π =. ( ) 8 4

CHAPTER 4 FOURIER SERIES

CHAPTER 4 FOURIER SERIES CONTENTS PAGE 4. Periodic Fuctio 4. Eve ad Odd Fuctio 3 4.3 Fourier Series for Periodic Fuctio 9 4.4 Fourier Series for Haf Rage Epasios 4.5 Approimate Sum of the Ifiite Series