Transform Techniques

Transcription

1 Trasform Techiques

2 (a)maually derived leas square fi Se of pois smooheed by a had draw curve ad by liear leas square fi. (Show i Fig.4.) Poi X Y Y P P P P P P P P P P P P Table 4. Fig.4.

3 y are correspodig coordiaes as measured o maually draw curve. Differece bewee y ad y represes he degree of smoohess i a subjecive sese. y mus be regarded as he bes fi for a se of empirical daa. Regardless of he curve, goodess of fi is ofe measured by a correlaio coefficie, R. I essece, R is a measure of he usefuless of x as a predicor of y. I saically erms R is compued by subracig from uiy, he raio of he variace of he y daa w.r.. he fied or assumed curve over he daa w.r.. he average of y s.

4 Where ÿ average of y s / Σ (y i -y ) R - (4.) / Σ(y i -ÿ) By subsiuig he give values i Eq.(4.) R.63, R.79. If he curves were o pass hrough each poi, he umeraor of he fracio i Eq.4.. Would become zero ad R would become uiy. Whe he curve y does o improve he fi, he umeraor of he fracio would be he same as he deomiaor ad R would be zero. From his we ca say ha R values of y.,beer he prediced

5 (b)mahemaical leas square (Liear fi) For a bes liear fi o a se of daa, we mus deermie y A + A x (4.) Ad a sum of squared deviaios bewee y ad he y s of daa pois is a miimum. From he saisics, he followig wo liear equaios i A ad A will fulfill hese requiremes. A Σ + A Σx i Σy i A Σx i + A Σx i Σx i y i (4.3) (4.4)

6 From he daa give i he able 4. A + 7A. 7A + 5.A Solvig A -.9, A.4 (4.5) y.9 +.4x (4.6) The compue R usig he appropriae formula whe ÿ ( )/ ad y is deermied for he x s, ha is, for., -.9, ec. The R is foud o be.6 which idicaes ha y is somewha beer i predicig y as a fucio of x ha y (which does o vary wih x).

7 (c)mahemaical leas square (quadraic) fi By usig a secod degree polyomial, he leas square equaio could be represeed as y A +A x (4.7) The A s could be derived from he se of liear equaios: A Σ+A Σx i +A Σx i A Σx i +A Σx i +A Σx i3 Σy i Σx i y i A Σx i +A Σx i3 +A Σx i4 Σx i y i (4.8) (4.9) (4.) By ispecio of he equaios for he A s for he liear fi ad he quadraic fi ad by simple exesio, i is possible o wrie he formaio equaios o ay desired degree. We limi o N- degree polyomial o pass hrough N pois.

8 Higher he seleced degree of polyomial fi, he closer he resulig curve will be o each poi. Bu he curve will wiggle o fulfill he effec. Graphics ca be of cosiderable assisace i deermiig he mos pracical order of polyomial fiig, sice he magiude of R is o a crierio by iself. How leas square polyomial fiig approaches he daa pois wih higher degree fucio is illusraed i he followig example

9 Ex. Four pois are (-,), (,), (,), ad (,-).Draw liear, quadraic ad cubic leas square polyomial fis. The average of he y s y (++-)/4 -/4 Makig use of eq. (4.5) ad (4.6) 4A + A - A + 6A 4 (4.) Solvig Eq. (4.) A -7/ ad A /

10 Fig.4. Now equaio for liear fi ca be wrie as Y / (7/)x (4.) similarly equaios for quadraic ad cubic fi ca be wrie as y 7/ +(/)x (¾)x (4.3) y - x/6 x +x 3 /6 (4.4) The four pois ad he hree polyomials leas squares fis are show i Fig.4..5 (,) (-,) (,) (,-)

11 Fourier Aalysis

12 4. Iroducio Egieerig problems ivolve sysems wih oscillaio or vibraio. Trigoomeric fucios play a fudameal role i modelig such problems. Fourier approximaio represes a sysemaic frame work for usig rigoomeric series for his purpose. The impora aspec of Fourier aalysis is ha i deals wih boh he ime ad he frequecy domais. 4. Curve fiig wih siusoidal fucios Periodic fucios repea afer a ime period T. Commo examples: Waveforms such as square ad saw ooh. Some periodic fucios are show i Fig..

13 f() f( + T) () Fig. Periodic fucios Idealized forms (a) he square wave ad (b) The saw ooh wave. Periodic sigals i aure ca be (c) Noideal (d) Coamiaed by oise. All hese cases ca be represeed ad aalyzed by he rigoomeric fucios.

14 The mos commo periodic fucios are siusoidal. The erm siusoid represes ay wave ha ca be described as a sie or cosie. There are o guidelies for choosig eiher sie or cosie fucio. I ay case he resuls will be ideical. A cosie fucio ca be geerally expressed as f ( ) A + C cos ( ω + θ ) o o A plo of siusoidal fucio is show i Fig..

15 Fig.. (a) A plo of siusoidal fucio (b) A aleraive for he same curve

16 The followig four parameers serve o characerize he siusoid (i) The mea value, A o ses he average heigh above he abscissa (ii) The ampliude, C, specifies heigh of he oscillaio. (iii) The agular frequecy, ω o, characerizes how ofe he cycle occur. (iv) The phase agle or phase shif, θ, parameerizes he exe o which he siusoid is shifed horizoally. I ca be measured as he disace i radias from o he poi a which he cosie fucio begis a ew cycle. A posiive θ is referred as leadig phase agle ad egaive value is referred as laggig phase agle. The agular frequecy (rad / s) is relaed o frequecy (cycles / s) by ω πf Ad he frequecy i ur is relaed o period T by (3) f T (4)

17 Eq. () is adequae mahemaical characerizaio of a siusoid, is o good from he sadpoi of curve fiig because he phase shif is icluded i he argume of he cosie fucio. This deficiecy ca be overcome by ivokig he rigoomeric ideiy. C cos( ω + θ) C cos( ω )cosθ si( ω )siθ (5) Subsiuig Eq. (5) io Eq. (), gives Where f( ) A + A cos( ω ) + B si( ω ) (6) A C C From Eq. (7) (7) cosθ ; B siθ θ (8) a ( B / A )

18 Ad C A B + (9) Refer Eq (), RHS cosie erm ca be wrie as cos( ω + θ) si( ω + θ + π ) () These wo represe he same fucio wih phase shif 4. Leas square fi of Siusoid Eq. (6) ca be wrie as a liear leas squares model π. y A + A cos( ω ) + B si( ω ) () The sum of he squares of he residuals for his model ca be wrie as N + + r i i i i { [ cos( ω ) si( ω )]} S y A A B

19 This quaiy ca be miimized by akig is parial derivaives w.r. each of he coefficies ad seig he resulig equaios which ca be expressed i marix form as () These equaios ca be employed o solve for he ukow coefficies. Cosider a case wih N observaios equispaced a iervals of Δ ad wih a oal record legh of T (N ) Δ. For his siuaio, he followig average values ca be deermied.

20 si ( ) N ω cos ( ω) N cos( ω)si( ω ) (3) N Thus, for equispaced pois he ormal equaios become N A y N A ycos( ω ) N B ysi( ω) The coefficies A, A, ad B ca be obaied by marix iversio A N y A N ycos( ω) si( ) B N y ω

21 or y A N (4) A ycos( ω) N (5) B ysi( ω) N (6) Example 4.. Leas square fi of a siusoid A curve is described by y.7 + cos ( ). Geerae discree values for his curve a iervals of Δ.5 for he rage o.35. Use his iformaio o evaluae he coefficies of Eq () by leas squares fi.

22 Soluio: The daa required o evaluae he coefficies are y ycos( ) ysi( ω ) ω Σ

23 These resuls ca be used o deermie [Eq (4) o Eq (6)] 7..7 A Thus, he leas-squares fi is A (.5).5 y cos( ω ).866si( ω ) ( 4.33).866 B The model ca also be expressed i he forma of Eq () by calculaig [Eq (8)].866 θ a ( ).47.5 Ad Eq. (9) C (.5) + (.866). o give y.7 + cos( ω +.47) Or aleraively as a sie by usig Eq. () y.7 + si( ω +.68)

24 Exedig he aalysis discussed i he above secio o he geeral model y ( ) A+ Acos( ω ) + Bsi( ω ) + Acos( ω ) + B si( ω ) A cos( mω ) + B si( mω ) m m For equally spaced daa, he coefficies ca be evaluaed by A N y Aj ycos( jω) N Bj ysi( jω) j,, 3,.m. N

25 The above relaioships ca be used o fi daa i he regressio sese (i.e., N>m+). Aleraively hey ca be employed for ierpolaio or collocaio, (i.e.,) o use hem for he case where umber of ukows, m+, is equal o he umber of daa pois, N. This approach is used i he coiuous Fourier series. 4.3 Coiuous Fourier series I he course of sudyig hea flow problems, Fourier showed ha a arbirary periodic fucio ca be represeed by a ifiie series of siusoids of harmoically relaed frequecies. For a fucio wih period T, a coiuous Fourier series ca be wrie as k k k [ ] f( ) a + a cos( kω ) + b si( kω ) (7) π Where ω is called fudameal frequecy ad is cosa muliples ω,3 ω, ec., are T called harmoics.

26 Eq (7) expresses f() as a liear combiaio of he basic fucios:, cos (ω ), si (ω ), cos (ω ), si (ω ). The coefficies i Eq. (7) ca be compued usig he relaios T ak f ( )cos( kω) d T (8) T bk f ( )si( kω) d T (9a) For k,,. ad a T f () d T (9b)

27 Example 4.. Use he coiuous Fourier series o approximae he square or recagular wave fucio as show i Fig.3 T / < <T /4 f( ) T /4 < < T /4 T /4 < < T / Fig.3. A square or recagular wave form wih a heigh of ad a period T π / ω

28 Soluio: Because he average heigh of he wave is zero, a value of a ca be obaied direcly. The remaiig coeffiecies ca be evaluaed as Eq. (8) T / ak f ( )cos( kω) d T T T / T /4 T /4 T / cos( kω) d + cos( kω) d cos( kω ) d T / T /4 T /4 The iegrals ca be evaluaed o give 4 / kπ for k,5,9,... ak 4 / kπ for k 3,7,,... for k eve iegers Similarly, i ca be deermied ha all he b s. Therefore, he Fourier series approximaio is f( ) cos( ω) cos(3 ω) + cos(5 ω) cos(7 ω) +... π 3π 5π 7π The resuls up o he firs hree erms are show i Fig. 4.

29 Fig. 4. The Fourier series approximaio of he square wave from Fig. 3. The series of plos shows he summaio up o ad icludig he (a) firs, (b) secod, (c) hird erms. The idividual erms ha were added a each sage are also show.

30 RESPONSE UNDER A GENERAL PERIODIC FORCE

31 Whe he exeral force F() is periodic wih period i ca be expaded i a Fourier series π / ω F( ) a + a cos + j jω b j j j si jω () where a F( ) d, () a j F( )cos jω d, j,,... (3)

32 ad b j F( )si jω d, j,,... (4) The equaio of moio for a sprig, mass damper sysem ca be expressed as.. m x. + c x + kx F( ) a + j a j cos jω + j b j si jω (5)

33 The righ-had side of his equaio is a cosa plus a sum of harmoic fucios. Usig he priciple of superposiio, he seady sae soluio of Eq. (5) is he sum of he seady sae soluios of he followig equaios:.. m x. + c x + kx a (6).. m x. + c x + kx a j cos jω (7).. m x. + c x + kx b j si jω (8)

34 Noig ha he soluio of Eq.5 is give by a x p ( ) k ad we ca express he soluio of Eqs.(7) ad (8) respecively, as x p ( ) ( j ( a / k) r j ) + (ζjr) cos (9) ( jω φ ) () j x p ( ) ( b / k) ( j r ) j + (ζjr) si ( jω φ ) () j

35 where () a r j jr j ζ φ ad (3) r ω ω Thus he complee seady sae soluio of Eq. (5) is give by ( ) ( ) ( ) ( ) ) (4 si ) ( ) ( / cos ) ( ) ( / ) ( j j j j j j p j jr r j k b j jr r j k a k a x φ ω ζ φ ω ζ

36 EXAMPLE : Valve Periodic Vibraio of a Hydraulic I he sudy of vibraios of valves used i hydraulic corol sysems, he valve ad is elasic sem are modeled as a damped sprig mass sysem, as show i Fig (a). I addiio o he sprig force ad dampig force, here is a fluid pressure force o he valve ha chages wih he amou of opeig or closig of he valve. Fid he Fourier series erms of he valve whe he pressure i he chamber varies as idicaed i Fig.. Assume k 5 N/m, c N-s/m, ad m.5kg.

37 Soluio: The valve ca be cosidered as a mass coeced o sprig ad a damper o oe side ad subjeced o a forcig fucio F() o he oher side. The forcig fucio ca be expressed as F()Ap() (E.)

38 Fig. Periodic vibraio of a hydraulic valve

39 where A is he cross secioal area of he chamber, give by A π 5) 4 ( 65π mm.65π m ( E.) ad p() is he pressure acig o he valve a ay isa. Sice p() is periodic secods ad A is a cosa, F() is also a periodic fucio of period secods. The frequecy of he forcig fucio is ω ( π / ) π rad / s. F() ca be expressed i a Fourier series as

40 F( ) a + a cos ω + a cos ω b si ω + b si ω +... ( E.3) where a j ad b j are give by Eqs. (3) ad (4). Sice he fucio F() is give by 5 A for F( ) ( E.4) 5 A( ) for

41 he Fourier coefficies a j ad b j ca be compued wih he help of Eqs. (3) ad (4): a 5 A d + 5 A( ) d 5 A ( E.5) a 5 A cosπ d + 5 A( ) cos π d π 5 A ( E.6)

42 .9) ( si ) ( 5 si 5.8) ( cos ) ( 5 cos 5.7) ( si ) ( 5 si 5 E d A d A b E d A d A a E d A d A b π π π π π π.) ( 9 3 cos ) ( 5 cos E A d A d A a π π π +

43 b 3 5 A si 3π d + 5 A( ) si 3π d ( E.) Likewise, we ca obai a 4 a 6.. b 4 b 5 b 6. By cosiderig oly he firs hree harmoics, he forcig fucio ca be approximaed: 5 ( A F ) 5 A cosω π 9π 5 A cos3ω ( E.)

44 Respose Uder a Periodic Force of Irregular Form I some cases, he force acig o a sysem may be quie irregular ad may be deermied oly experimeally. Examples of such forces iclude wid ad earh quake iduced forces. I such cases, he forces will be available i graphical form ad o aalyical expressio ca be foud o describe F(). Someimes, he value of F() may be available oly a a umber of discree pois,,.., N. I all hese cases, i is possible o fid he Fourier coefficies by usig a umerical iegraio procedure.

45 If F, F,,F N deoe he values of F() a,,,.., N respecively, where N deoes a eve umber of equidisa pois i oe ime period ( N ), as show i Fig., he applicaio of rapezoidal rule gives a N N F i i () a j N Fi N i cos jπ i, j,,... ()

46 b j N N i F i si jπ i, j,,... (3) Oce he Fourier coefficies a o, a j, ad b j are kow, he seady sae respose of he sysem ca be foud usig Eq.(4) wih r π ω (4)

47 Respose uder a Periodic Force of Irregular form Example : The pressure flucuaios of waer i a pipe, measured a. secod iervals, are give i Table below. These flucuaios are repeiive i aure. Make a harmoic aalysis of he pressure flucuaios ad deermie he firs wo harmoics of he Fourier series expasio. Time, i (secod s) p i p( i ) (kn/m )

48 Fig. A irregular forcig fucio

49 Soluio: Sice he give pressure flucuaios repea every. s, he period is. s ad he circular frequecy of he firs harmoic is π radias per. s or ω π / rad/s. As he umber of observed values i each wave (N) is, We obai a 49 N pi pi ( E.) N i i a N N i p i cos πi 6 i p i cos π. i ( E.) b πi si N N i p i 6 i p i si π. i ( E.3)

50 i i p i P i cos πi. P i πi P i i i ( ( ) ) si cos 4πi. P i si πi.

51 From hese calculaios, he Fourier series expasio of he pressure flucuaios p() (E.4) ca be obaied as p( ) cos si cos si N / m

52 FOURIER SERIES IN EXPONENTIAL FORM

53 Complex Fourier Series The Fourier series ca also be represeed i erms of complex umbers. By oig, e i ω ad cosω + i siω () e iω cosω i siω () cos ω ad si ω ca be expressed as e cosω iω + e iω (3)

54 ad (4) si i e e i i ω ω ω Fourier expasio ca be wrie as ( ) i i i i i e e b e e a a x ω ω ω ω ( ) (5) i i i ib a e ib a e ib a e ω ω ω

55 where b. By defiig he complex Fourier coefficies c ad c - as c a ib (6) ad c a + ib (7) Eq(5) ca be expressed as x ( ) c e iω (8)

56 The Fourier coefficies c ca be deermied as c a ib x ( )[ cos ω i si ω] d x( ) e iω d (9) Frequecy Specrum The harmoic fucios a cos ω or b si ω are called he harmoics of order of he periodic fucio x(). The harmoic of order has a period /. These harmoics ca be ploed as verical lies o a diagram of ampliude (a ad b or d ad φ ) versus frequecy (ω), called he frequecy specrum or specral diagram. Figure shows a ypical frequecy specrum.

57 Fig. Frequecy Specrum of a ypical periodic fucio of ime Fig. Represeaio of a fucio i ime ad frequecy domais

58 Fourier Aalysis Ay periodic fucio x(), of period, ca be expressed i he form of a complex Fourier series x ( ) c e iω () where ω is he fudameal frequecy give by π ω ()

59 ad he complex Fourier coefficies c ca be deermied by im ω muliplyig boh sides of Eq.() wih e ad iegraig over oe ime period: / / x( ) e imω d / c / e i( m) ω d : c / / [ cos( m) ω + i si ( m) ω ] () d Equaio () ca be simplified o obai c / iω / x( ) e d (3)

60 Equaio () shows ha he fucio x () of period ca be expressed as sum of a ifiie umber of harmoics. The harmoics have ampliudes give by Eq. (3) ad frequecies which are muliples of he fudameal frequecy ω. The differece bewee ay wo cosecuive frequecies is give by π ω + ω ( + ) ω ω ω ω (4) Thus he larger he period, he deser he frequecy specrum becomes. Equaio (3) shows ha he Fourier coefficies c are, i geeral, complex umbers. However, if x() is a real ad eve fucio, he c will be real. If x() is real, he iegrad of c i Eq. (3) ca be also be ideified as he complex cojugae of ha of c -. Thus

61 (5) * c c The mea square value of x () - ha is, he ime average of he square of he fucio x () ca be deermied as d e c d x x i / / / / ) ( ) ( ω ( ) d c e c e c d e c c e c i i i i * / / / / ω ω ω ω

62 / / * cc + c d c + c c (6) Complex Fourier Series Expasio Example:Fid he complex Fourier series expasio of he fucio show i Fig. 3 Soluio: The give fucio ca be expressed as

63 .) (,, ) ( E a A a A x + where he period () ad he fudameal frequecy (ω ) are give by a a π π ω ad (E.)

64 Figure 3 Complex Fourier series represeaio.

65 The Fourier coefficies ca be deermied as.3) ( ) ( / / / / E d e a A d e a A d e x c i i i + + ω ω ω Usig he relaio.4) ( ) ( E k k e d e k k

66 c ca be evaluaed as [ ] / / ) ( ω ω ω ω ω + i i e a A e i A c i i [ ] + / / ) ( ω ω ω ω ω i i e a A e i A i i (E.5)

67 This equaio ca be reduced o A i + A A c iπ iπ iπ e e e ω a ω iω a ω a ω A A e iπ + A a ω A ω iπ iπ ( iπ ) e ( iπ ) e (.6) E a Noig ha iπ e or e i π,,,,3,5,...,4,6,... ( E.7)

68 Eq. (E.6) ca be simplified o obai.8) (,4,6,...,,3,5,..., 4, E A a A A c π ω The frequecy specrum is show i Fig.3(b)

69 Figure 4 Noperiodic fucio

70 Fourier Iegral A operiodic fucio, such as he oe show by he solid curve i Fig. 4, ca be reaed as a periodic fucio havig a ifiie period ( ). The Fourier series expasio of a periodic fucio is give by Eqs.(), () ad (3) iω x( ) c e (7) wih ω π (8)

71 ad c / iω / x( ) e d (9 ) ( ), As he frequecy specrum becomes coiuous ad he fudameal frequecy becomes ifiiesimal. Sice he fudameal frequecy ω is very small, we ca deoe i as Δω, ω as ω, ad rewrie Eq. (9) as lim c lim / / x( ) e iω d x( ) e iω d ()

72 By defiig X (ω) as iω X ( ω) lim( c ) x( ) e d () we ca express x() form Eq. (7) as x( ) iω lim c e π π ( ) lim c e π iω π

73 i ( ) ω e ω X dω π () This equaio idicaes he frequecy decomposiio of he operiodic fucio x() i a coiuous frequecy domai, similar o Eq. (7) for a periodic fucio i a discree frequecy domai. The equaios x( ) i ω ω X ( ) e d π ω (3)

74 ad X ( ω ) x( ) e iω d (4) are kow as he (iegral) Fourier rasform pair for a operiodic fucio x(), similar o Eqs. (7) ad (9) for a periodic fucio x() The mea square value of a operiodic fucio x() ca be deermied from Eq. (6)

75 / x ( ) d c / c c * ω ω c c * ω π ω ( )( * c ) c π (5) Sice c * * X ( ω), c X ( ω), ad ω dω as, Eq. (5) gives he mea square value of x() as

76 x / ( ) lim x ( ) d / X ( ω) π dω (6) Equaio (6) is kow as Parseval s formula for operiodic fucios Fourier Trasform of a Triagular Pulse Example: Fid he Fourier rasform of he riagular pulse show i Fig. 5

77 Soluio: The riagular pulse ca be expressed as x( ) A, a, a oherwise (E.) The Fourier rasform of x() ca be foud, usig Eq.(4), as X ( ) iω ω A e d a iω A + e d + a A a e iω d (E.)

78 Fig. 5 Fourier rasform of a riagular Pulse

79 Sice x() for >, Eq.(E.) ca be expressed as d e a A d e a A X i a i a ω ω ω + + ) ( [ ] ) ( a i a i i i e a A e i A + ω ω ω ω ω [ ] a i a i i i e a A e i A ) ( + ω ω ω ω ω (E.3)

80 Equaio (E.3) ca be simplified o obai X A ( ) + iωa iωa ω e e aω A aω + A aω A aω A aω A aω ( cosωa + i siωa) ( cosωa i siωa) A 4A ( cos ) si ωa ωa ( E.4) aω aω Equaio (E.4) is ploed i Fig. 5 (b). Noice he similariy of his figure wih he discree Fourier specrum show i Fig.3 (b).

81 Fig. 5 (a) A depicio of how a siusoid ca be porrayed i he ime ad he frequecy domais. The ime projecio is reproduced i (b), whereas he ampliudefrequecy projecio is reproduced i (c). The phase-frequecy projecio is show i (d)

82 Fig. 5 shows a hree dimesioal graph of a siusoidal fucio f( ) C cos( + π / ) I he above equaio he magiude or ampliude of he curve f() is he depede variable. Time,, ad frequecy f ω /π are idepede variables. Thus, (i) The ampliude ad ime axis form a ime plae. Whe we speak abou he behavior of he siusoid i he ime domai, i is he projecio of he curve oo he ime plae (Fig. 5b). Fig. 5c shows a projecio of he measure of he siusoid s maximum posiive ampliude C. The full peak-o-peak swig is uecessary because of he symmery. Hece, Fig. 5c defies he ampliude ad frequecy of he siusoid. This iformaio is sufficie o produce he shape ad size of he curve i he ime domai.

83 (ii) The ampliude ad frequecy axis form a frequecy plae. The behavior of he siusoid i he frequecy domai is merely is projecio oo he frequecy plae. The phase agle is deermied as he disace (i radias) from zero o he poi a which posiive peak occurs. If he peak occurs afer zero, i is said o be delayed, ad by coveio, he phase agle is give a egaive sig. Coversely, a peak before zero is said o be advaced ad he phase agle is posiive. I Fig. 5d, he peak leads zero ad he phase agle is ploed as + π /. Fig. 5c ad d provide a aleraive way o prese he perie feaures of he siusoid i Fig. 5a. They are referred o a lie specra. Fig. 6 shows some oher possibiliies.

84 Fig. 6 Various phases of a siusoid showig he associaed phase lie specra.

85 Fig. 7 shows he ampliude ad phase lie specra for he square wave fucio from he Example 4.. The origial square wave show i Fig. 4,( of CONTINUOUS FOURIER SERIES secio) ells us ohig abou he siusoids ha comprise i. The aleraive o display hese siusoids (4 / π)cos( ω ), -(4 / 3 π)cos(3 ω ), (4 / 5 π)cos(5 ω ), ec. Fig. 7 (a) ampliude ad (b) phase lie specra for he square wave from Fig. 3.

86 This aleraive does o provide a adequae visualizaio of he srucure of hese harmoics. I coras, Figs. 7(a) ad (b) provide a graphic display of his srucure. The lie specra represe figerpris ha ca help o characerize ad udersad a complicaed waveform. They are paricularly useful for oidealized cases where hey someimes allow o discer srucure i oherwise obscure sigals. Fourier rasform will allow o exed such aalysis o operiodic waveforms.

87 4.5 Fourier Iegral ad Trasforms Fourier series is a useful ool for ivesigaig he specrum of a periodic fucio. Bu, here are may wave forms ha do o repea hemselves regularly (ex: ligheig bold, disurbace from he radom sigals of machies ec). Aleraive o Fourier series, Fourier iegrals is he primary ool available for his purpose. This ca be derived from expoeial form of he Fourier series. f() k ik k ~ Ce ω () T ~ () ikω where C k f e d (3) T T where ω π / T ad k,,,...

88 The rasiio from a periodic o a operiodic fucio ca be effeced by allowig he period o approach ifiiy. I oher words, as T becomes ifiie, he fucio ever repeas iself ad hus becomes aperiodic. If his is allowed o occur, Fourier series reduces o () ( ) ω π (4) iω f Fiω e d ad he coefficies become a coiuous fucio of he frequecy variable ω as i iω F iω f e d ( ) () (5) The fucio Fiω ( ) as defied i Eq (5) is called Fourier iegral of f(), Eqs (4) ad (5) are collecively referred o as he Fourier rasform pair. Fiω ( ) is also called he Fourier rasform of f(). f() is called iverse Fourier rasform of Fiω ( ).

89 Differece bewee Fourier series ad rasform. Fourier series Fourier rasform Applied o periodic waveforms. Applied o operiodic waveforms. Covers a coiuous, periodic imedomai fucio o frequecy domai magiudes a discree frequecies Covers a coiuous ime-domai fucio o a coiuous frequecydomai fucio. The discree frequecy specrum geeraed by he Fourier series is aalogous o a coiuous frequecy specrum geeraed by he Fourier rasform.

90 The shif from a discree o coiuous specrum is illusraed i Fig. 8. Fig. 8. Illusraio of how he discree frequecy specrum of a Fourier series for a pulse rai (a) approaches a coiuous frequecy specrum of a Fourier iegral (c) as he period is allowed o approach ifiiy.

91 Fig. 8a shows a pulse rai of recagular wave wih pulse widhs equal o oe-half he period alog wih is associaed discree specrum. I Fig. 8b, a doublig of he pulse rai s period has wo effecs o he specrum. Firs, wo addiioal frequecy lies are added o eiher side of he origial compoes. Secod, he ampliudes of he compoes are reduced. As he period is allowed o approach ifiiy, hese effecs coiue as more ad more specral lies are packed ogeher uil he spacig bewee lies goes o zero. A he limi, he series coverges o o he coiuous Fourier iegral as show i Fig. 8c. Sigal is rarely characerized as a coiuous fucio of he ype eeded o impleme Eq (5). The daa is ivariably i a discree form. The compleio of Fourier rasform for such a discree measureme is show i he ex secio.

92 4.6 Discree Fourier Trasform Fucios are ofe represeed by fiie ses of discree values as show i Fig. 9. A ierval from o T is divided io N equispaced subiervals wih widh ΔT T/N. Fig. 9. The samplig pois of he discree Fourier series.

93 - discree imes a which he sample is ake f - value of fucio f() ake a where daa pois are specified a,,,..n-, a discree Fourier rasform for daa show ca be wrie as N for o (6) ikω F fe k N k ad he iverse Fourier rasform as N k for o (7) N ikω f Fe N k where ω π /N

94 Pseudocode for compuig he DFT: DOFOR k o N- DOFOR o N- agle kω real k real k + f cos(agle)/n imagiary k imagiary k f si(agle)/n ENDDO ENDDO 4.7 Fas Fourier Trasform (FFT) The DFT algorihm explaied i he above secio ca serve he purpose, bu is compuaioally expesive (N operaios are required). The Fas Fourier Trasform (FFT) is a algorihm developed o compue DFT i a exremely ecoomical fashio. I uilizes he resuls of previous compuaios o reduce he umber of operaios. I explois he periodiciy ad symmery of rigoomeric fucios o compue he rasform wih approximaely N log N operaios. (For N5 samples, FFT is abou imes faser ha DFT. N, i is imes faser).

95 I FFT algorihms, a DFT of legh N is decomposed or decimaed io successively smaller DFT s. Cooley-Tukey algorihm is a decimaio-i-ime echique ad Sade-Tukey algorihm is decimaio-i-frequecy echique. (a) Cooley-Tukey Algorihm This algorihm assumes ha N is a iegral power of, i.e. N m, where m is a ieger. The basic idea of his algorihm is o decompose he N-poi DFT io wo N/-poi DFTs, he decompose each of he N/-poi DFTs io wo N/4-poi DFTs ad coiuig his process uil N/ wo-poi DFTs are obaied. The umber of seps required o achieve his is clearly m. For easy udersadig of his algorihm, N8 is cosidered ad i ca be easily geeralized. The firs sep (or sage) of he algorihm is described below:

96 Le f, f, f,... f 7 be a sequece of values of f(). The DFT for f is give by k 7 k 8,,,,...7 () F fw k where W e πi /8 8 () The summaio o he righ side of Eq () is spli io wo equal pars of legh 4, oe coaiig he eve-idexed values of f() ad he oher of he oddidexed values. They ca be wrie as (3) F fw + fw k k k 8 8 ( eve) ( odd ) Puig r i he firs sum ad r+ i he secod sum of Eq (3), we obai 3 3 kr (r ) k k r 8 + r+ 8 r r (4) F f W f W +

97 Bu ad W e e W (5a) kr πi( kr )/8 πikr /4 kr 8 4 W W W WW (5b) (r+ ) k rk k k kr Usig Eq (5) i Eq (4), we ge 3 3 rk k rk k r r + 4 r r (6) F f W W f W I is easily see ha he wo sums o he righ side of Eq (6) represe 4-poi DFTs. Seig F e k 3 f W (7a) r kr r 4 o k Eq (6) becomes: 3 F f W (7b) r kr r + 4 e k o F F + W8 F, k,,,3, (8) k k k

98 where e F k ad o F k are he 4-poi DFTs of he eve ad odd-idexed sequeces defied by Eq (7). This complees he firs sage of decomposig he 8-poi DFT io wo 4-poi DFTs. Furher, o compue Eq (8) for k4,5,6,7 we use he formula: e k o F F, 4,5,6,7 4 + W8 F 4 k (9) k k k The compuaios ivolvig equaios Eqs (8) ad (9) for he firs sage of he 8- poi DIT-FFT are show i flow-graph i Fig..

99 Fig.. Firs sage of he 8-poi DIT-FFT.

100 I he secod sage, each of he 4-poi rasforms i Eq (8) is decomposed io wo -poi rasforms. We he wrie F e k 3 r f W kr r 4 sk k sk f4sw W4 f4s+ W s s + F + W F ee k eo k 4 k () where Similarly, we obai ee sk eo sk k 4s ad k 4s + s s () F f W F f W where F F + W F () o oe k oo k k 4 k oe lk oo lk k 4l+ ad k 4l+ 3 l l (3) F f W F f W

101 This complees he secod sage of decomposiio where each of he 4-poi rasforms is broke io wo -poi rasforms. The flow-graph of he secod sage is show i Fig.. Fig.. secod sage of he decomposiio.

102 From Eq (), we have ee sk o k k 4s + 4 s F f W fw fw (4a) eo sk o k k 4s+ + 6 s F f W fw fw (4b) Eqs (4a) ad (4b) show ha a he hird sage (which is he fial sage, sice N8), we obai F f, F f, F f, F f (5) eee eeo eoe eoo k k 4 k k 6 I follows ha, for he 8-poi compuaio, we sar wih he ipu sequece f, f, f, f, f, f ad f, ad he compue he various Fourier coefficies Fig.. Flowgraph of a 8-poi DIT-FFT

103 A close ispecio of Fig. eables us o make he observaios: (i) The ipu daa is shuffled ad are i he order f, f4, f, f6, f, f5, f3,a f d. 7 hey are i he bi-reserved order, as show i Table. Table. Ipu Daa i he Reserved Bis Ipu posiio Biary equivale Reverse bis Idex of he sequece

104 (ii) The oupu das for he Fourier coefficies F is i he aural order. (iii) The compuaios are carried ou i erms of a fudameal molecule called buerfly. A ypical buerfly is show i Fig. 3, where i ad j represe he posiio umbers i he sage ad m represes he sage of he compuaio. Fig. 3. A ypical buerfly. The oupus g + ad g + are give by m m i j g g + W g, g g W g (6) m+ m r m m+ m r m i i N j j i N j where r is a variable depedig o he posiio of he buerfly. The mehod of compuaio is illusraed i he followig umerical example.

105 Example 4.3. Usig he Cooley-Tukey algorihm, fid he DFT of sequece {,,3,4,4,3,,} f. We have W, W e ( i) /, W ( e ) i, πi/8 πi/ W ( + i) /, W, W ( i) /, W i, W ( + i) / The DIT-FFT flowgraph for DFT compuaio is give below i Fig. 4.

106 Fig. 4. Flowgraph for Example 4.3