Fundamentals of Electrical Engineering 3

Transcription

1 Fundamenals of Elecrical Engineering 3 Professor Dr.-Ing. Ingolf Willms and Professor Dr. Dr.-Ing. Adalber Beyer based on he scrip of Professor Dr. Dr.-Ing. Ingo Wolff S.

2 Fundamenals of Elecrical Engineering 3 Conens Inroducion Signal heory of deermined signals and applicaions. Prefaces. he Fourier series and applicaions o neworks.3 he Fourier ransform and applicaions o sysems 3 Swiched circuis and he Laplace ransform 3. he Laplace ransform 3. Properies 3.3 Applicaions S.

3 Lieraure Lieraure for he lecure: R. Paul Elekroechnik, Grundlagenbuch Nezwerke Springer-Verlag, Heidelberg 994 I. Wolff Grundlagen der Elekroechnik 4 Vorlesungs-Scrip Alernaive Lieraure : W. Ameling Grundlagen der Elekroechnik II G. Bosse Grundlagen der Elekroechnik IV B.I. Wissenschafverlag Mannheim, Wien Zürich 996 R. UnbehauenGrundlagen der Elekroechnik I Springer-Verlag, Heidelberg 994 S. 3

4 Inroducion GE3 conains predominanly heoreical bases o quesins of informaion echnology Informaion echnology: Developed from compuer science and communicaions I: Efficien daa processing, sorage and ranspor I conains 4 groups: - Bases and echnologies (G) - Srucures, procedure, programs (G) - Devices, mechanisms, plans (G3) - Applicaions (G4) S. 4

5 Signal heory of deermined signals and applicaions Chaper overview:. Prefaces - Model for he informaion ransfer - Signal classes - Descripion and modificaion of signals. Descripion of periodic signals (Fourier Series) - Approximaion of funcions wih he Fourier series - Applicaions o neworks.3 Descripion of aperiodic signals (Fourier ransform) - he Fourier inegral in differen forms -Examples - Properies of he Fourier ransform - Applicaion o sysems (convoluion and he Fourier ransform)- S. 5

6 . Prefaces S. 6

7 .. he Exponenial Signal j s() e cos jsin For volages i holds: ( u ) j u ( ) uˆcos( ) Re uˆe Re ue where uuˆe For increasing/decreasing signals: u j j u e e e e ( j ) j p S. 7

8 .. he Dirac Funcion Definiion: ( ) ( ) ( ) d Properies: ( a) ( ) a If a hen: ( ) ( ) s() ( ) s( ) d See formula above. S. 8

9 .. he Dirac Funcion ( ) ds() where ( ) for () lim rec ( ) S. 9

10 ..3 he Sep Funcion for () for () ( ) d () ( ) ( ) ( ) ( ) d S.

11 ..4 Periodic Signals General formula: s ( ) s ( n) where n,...,,,..., Examples: s () s ( n ) n s () c s ( n ) n n S.

12 ..4 Periodic Signals s ( ) rec n s() rec n n rec n s () n n n S.

13 ..5 Impulse ype Signals rec( x) for x for x rec( x) x S. 3

14 ..5 Impulse ype Signals for s ( ) sign ( ) for for sign() S. 4

15 ..5 Impulse ype Signals s () for oherwise s () S. 5

16 ..6 Adjusmen of ime and Frequency Funcions Case: Compression & expansion s() as b Example: s() urec Case: Shif s s () ( ) S. 6

17 ..6 Adjusmen of ime and Frequency Funcions Example: s () rec s() us u rec u rec( x) s () x S. 7

18 ..6 Adjusmen of ime and Frequency Funcions Case3: Mirroring (b = -) s () s ( ) Example: s () () s () s ( ) ( ) s () S. 8

19 ..6 Adjusmen of ime and Frequency Funcions Expansion & shif: s() rec s() rec 3 s3() s( ) rec 3 s3 3 v Shif & expansion: s s () ( ) s3( ) as Replace by in s( ) b b as b S. 9

20 ..6 Adjusmen of ime and Frequency Funcions Example: s( ) rec ; a u ; b () s rec rec s3() arec urec b u s () 3 v S.

21 ..6 Adjusmen of ime and Frequency Funcions Mirroring & shif: s () s ( ) s ( ) s ( ) s ( ( )) s ( ) 3 New sequence: Shif & mirroring: s4 s () ( ) s () s ( ) s ( ) s () S.

22 ..6 Adjusmen of ime and Frequency Funcions Example wih a ramp funcion r(): r () () s r s() s( ) r s () s () 3 s () s3() s( ) r v S.

23 ..6 Adjusmen of ime and Frequency Funcions s4() s( ) r s5() s5( ) r s5 s4 v v here are 4 cases: v S. 3

24 ..6 Adjusmen of ime and Frequency Funcions All onses can be exended o frequency funcions: f ( x) f ( y) where y f ( x) 3 f ( x) f ( f ( x)) 3 Example: f ( ) rec S. 4

25 . Descripion of non-sinusoidal, periodic funcions.. Approximaion of funcions Moivaion: Deerminaion of characerisical funcions and parameers, daa compression Saring poin: Given is f() Desired is g(), approximaing f() in an inerval wih f() g() n g() igi() by given gi( ) i f() S. 5

26 .. Approximaion of funcions - Requiremen: As small an error of he approximaion as possible - Definiion of error funcion: () f() g() g d min f () g() d mamin min f () g() d mqmin - Mean error: f () () mmin - Mean absolue error: - Mean square error: Advanages/disadvanages of he error measures: Cancelling of errors is possible in he case of mean error Absolue error resuls in nonlineariy Quadraic error measures is mos frequen applicaion min S. 6

27 .. Approximaion of funcions - Deerminaion of coefficen: - Hereby he following seps resul: mq i,,..., n i f() g () d n i i i i n ( ) ( ) ( ) f igi gi d i n f() g () d g () g () d i i i i i his corresponds o a se of equaions, which can be solved for he coefficiens. S. 7

28 .. Approximaion of funcions n f (). g() d g () d g(). g() d... g() g() d... g() g() d n f ( ). g( d ) g( g ) ( d ) g( ). g( d )... g( g ) ( d )... g( g ) ( d ) n... f (). g () d g (). g () d g () g () d... g () g () d... g () g () d n n n n n n n n S. 8

29 .. Approximaion by means of orhogonaler funcion sysems Definiion of orhogonal funcions in inerval (, ) by means of real funcions g i (). hese funcions should be coninous in he inerval. Chronecker s dela funcion is used: Onse for he approximaion: hus follows for he coefficiens: g (). g () d h and suiable h für für () n g igi() i i f g() d g i i () d S. 9

30 .. Approximaion by means of orhogonaler funcion sysems One receives an orhonormal funcion sysem by means of he definiions g () g () g () gn() G ( ), G ( ),..., G ( ),..., Gn ( ) h h h h n o hese applies: G () G () d für für hus a funcion f() in he inerval can be developed ino a se of orhonormal funcions by suiable coefficiens. he resul of he approximaion is hen a funcion G(). In summary i applies: f () G() AG() i i i he coefficiens A i are he so-called generalized Fourier coefficiens: A i f() G () d i S. 3

31 ..3 Approximaion of periodic, non-sinusoidal funcions Example of a funcion wih he period. his funcion is inerpreable as repeiion of one period. f() For one period i applies: f( ) f( ),,,..., Afer Fourier any funcion, for which he Dirichle condiions are fulfilled can be represened in he following rigonomeric form: a f a b () cos( ) sin( ) S. 3

32 .. Approximaion of periodic, non-sinusoidal funcions (Fourier series) Dirichle condiions (in pracice fulfilled) Funcion f() is eiher coninous in he inerval or has finiely many poins of disconinuiy Finie values of f() exis in he limi, if approaches he poin of disconinuiy from he righ or from he lef he inerval can be divided in such a manner ha f() here is monoonous Senence of Dirichle Wih fulfilmen of he Dirichle condiions he Fourier series converges in he enire inerval he value of he Fourier series is idenical o funcion f() in coninous areas A poins of disconinuiy he value is alike:.5 f( ) f( ) A end poins of he inerval he value is alike:.5 f( ) f( ) S. 3

33 ..3 Fourier series Analogy o he series expansion of orhogonal ransforms o he series expansion applies using orhogonal funcions For he used funcions he orhogonaliy can be shown: sin( )sin( ) d cos( )cos( ) d i () n g igi() g (). g () d h wih i f g() d g i i () d Oherwise applies as given above: hus i is ensured ha Fourier series is he opimum approximaion in he square mean sense (also in a erminaed series) a f a b () cos( ) sin( ) coefficien ses are necessary, so ha even and odd funcion pars can be represened. S. 33

34 ..3 Fourier series hus applies o he deerminaion of he Fourier-coeffiziens of he rigonomeric form: a f() d d f () d his is he DC componen (arihm. average value) a f()cos( ) d f()cos( ) d cos ( ) d b f()sin( ) d f()sin( ) d sin ( ) d S. 34

35 ..4 he polar form of he Fourier series (Fourier Cosinus series) By means of he relaionship Acos( x) Bsin( x) A B cos( xarcan( B/ A)) he Fourier series can be rewrien from a f a b ( ) cos( ) sin( ) o f( ) d d cos( ) wih d a b ; a d and b arcan ( / for negaive a ) a S. 35

36 ..5 Examples for he deerminaion of he Fourier series wih symmerical Funcions B: f() is an even funcion wih f ( ) f( ) and -/ f() f() f() / he Fourier hus has he form: Reason: Represenaion of even funcions is only possible by oher even funcions! 4 a f ()cos( ) d b f()sin( ) d a f a () cos( ) S. 36

37 ..5 Examples for he deerminaion of he Fourier series wih symmerical Funcions B: f() is an odd funcion wih f () f( ) and -/ f() f() f() / a a hus i resuls: 4 b f ()sin( ) d f () b sin( ) S. 37

38 ..5 Examples for he deerminaion of he Fourier series wih symmerical Funcions B3: f() is a compleely symmeric funcion wih f() = -f( + /) f() +/ f(+/) S. 38

39 ..5 Examples for he deerminaion of he Fourier series wih symmerical Funcions For I applies: a f()cos( ) d or afer spli-up of he inerval: for k holds: a f()cos( ) d f()cos( ) d o ak f( )cos( k) d f( )cos( k) d o 4 for k holds: a k f ()cos( k ) d Reason: Even-numbered k give afer / repeaing cosine funcions. Cancellaion of erms resul due o pars of f() being negaive wih respec o / and repea iself. S. 39

40 ..5 Examples for he deerminaion of he Fourier series wih symmerical Funcions - In a similar way he validiy of he following saemens can be seen (also sin funcion repea iself afer /): 4 b k and b k f()sin (k ) d - herefore only odd-number oscillaions occur, for hem in his Fourier series k applies: k f () a cos(k) b sin (k) k k S. 4

41 ..5 Examples for he deerminaion of he Fourier series wih symmerical Funcions B4: hefuncion is compleely symmerically wih f() = f(+ /). From his follows: f() 8 # 4 ak f()cos( k) d and a k / +/ 4 b k f()sin( k) d and b k Reason: Afer / a repeiion of he cos/sin funcions wih he indices k akes place. Cos/sin funcions wih indices k+ have in each case a differen half wave a / disance! he Fourier series of f() hen has he following form wih only even-numbered coefficiens: a f a k b k ( ) k cos ( ) k sin ( ) k S. 4

42 ..5 Examples for he deerminaion of he Fourier series wih symmerical Funcions B5: f() is even and compleely symmerical: f() = - f( + /) : Resul: Only odd-number cosine oscillaions occur. f() / a a and b k 4 8 a k f ()cos[( k ) ] d hus he appropriae Fourier series can be wrien as: f () ak cos k k S. 4

43 ..5 Examples for he deerminaion of he Fourier series wih symmerical Funcions B6: f() is odd and compleely symmeric wih f() = -f( + /): Here only odd-number sine oscillaion occur in he Fourier series f() / a and b a 4 8 b k f()sin[( k ) ] d k he Fourier series can hus be wrien as: f() bk sin k k S. 43

44 ..5 Examples for he deerminaion of he Fourier series wih symmerical Funcions B7: f() is shifed on he ime axis: If he shif amouns o hen i applies wih ' : a g f a b ( ') ( ) cos[ ( )] sin[ ( )] A simpler expression resuls for he complex coefficiens: f c e jv ( ) resuls o n his expression makes i possible, o deermine he Fourier series for arbirary shifs. I ofen is of advanage o shif he origin e.g. if hereby symmerical characerisics of he funcion resul. S. 44

45 ..6 Fourier analysis I exiss he possibiliy o represen a periodical non-sinusoidal funcion regarding is "informaion conen" in wo versions: ) In he ime inervall (s. he following picure) f() d -/4 / /4 3/4 A ) In he specral region (frequency range): Represenaion of he ampliudes a, b and or he cos ampliude d and he phase as a funcion of he frequency. S. 45

46 ..6 Fourier analysis Furher examples Fourier analysis of he rapezoidal funcion his funcion is even and compleely symmerically wih (s. B5) 4 a, a 8 k b and a k f()cos[( k ) ] d o f() applies in he firs quarer of he period: A cons für d 4 f() A für d d d hus i resuls: d 4 4 a 8 A cos[( k A k ) ] d ( )cos[( ) ] 4 k d d d 4 S. 46

47 ..6 Fourier analysis he final resul hen is: 4A a k cos[(k ) ( d)] (k) d 4 or a 4Asin[(k) d] sin[( k ) ] (k) d k he Fourier series of he rapezoidal funcion is hereby: 4A f( ) [sin( d)cos( ) sin(3 d)cos(3 ) sin(5 d)cos(5 )......) d 9 5 S. 47

48 ..6 Fourier analysis Special case of he rapezoidal funcion : he riangle funcion (d = /4) : f() d A -/4 / /4 3/4 S. 48

49 /..6 Fourier analysis For his funcion he following ampliudes and phase specrum resuls: c v 8A ² v² v k Endergebnisse : 8A ak, b, k k, k k / (k ) 3 S. 49

50 ..6 Fourier analysis Special case of he rapezoidal funcion : Recangle funcion wih d f() A d -/4 /4 / 3/4 S. 5

51 ..6 Fourier analysis Limiing performed by means of he rule of Bernoulli L'Hospial: c v 3 c v 4A v v ' sin[(k ) d] sin[(k ) d] lim lim lim si((k ) d) si() d ' (k ) d d d (k ) d hus i follows: v - 4 A ak sin[(k ) ], bk (k ) 4 A ck, k sin[(k ) ] (k ) v S. 5

52 ..7 he complex form of he Fourier series Generally i applies o he Fourier series: a f a b () cos( ) sin( ) In addiion i applies: cos( ) j j e e sin( ) e e j j j hus i is obained a e e e e j j j j f() a b j or: j j f e e () a a jb a jb S. 5

53 ..7 he complex form of he Fourier series Now also negaive values for are included. Wih he abbreviaions c c c, a a one receives pairs of coefficiens. a jb jb his can be wrien in he very compac represenaion of he Fourier series in complex form: for posiive for negaive Also: c f () c * j c e S. 53

54 ..7 he complex form of he Fourier series In addiion i applies: a Re( c ) b Im( c ) For he complex coefficiens hereby he condiional equaions resul: c a f() d, a jb j c ( )[cos( ) sin( )] ( ), f j d f e d c j f( ) e,,,,... S. 54

55 ..8 Inerpreaion of Fourier-coefficiens Usually he following represenaions of he Fourier series are used: a f a b () cos( ) sin( ) or f() or f() d d cos( ) c e Appoinmen off here: j c insead of c for beer overview of formulas. S. 55

56 ..8 Inerpreaion of Fourier-coefficiens Summarizing he Fourie series gives he following ses of parameers: a - DC componen of he signal: c d - Peak values or ampliudes of he Fourier componens: a, b and - Zero-phase (or phase) of he cosine oscillaions: d - Basic oscillaion a he fundamenal frequency: d cos( ) S. 56

57 ..9 Applicaion of he Fourier series o a nework Given is a cosine-ype volage A complex poiner is usually assigned like his: j u uˆ ue ˆ Now one can se: u () uˆ cos( ) j u () ue v wih uv * uˆ v, für uv u für uˆ v für Also for elecrical neworks one uses he represenaion (for volages and currens) in cosine form: u () u uˆ cos( ) i () i iˆ cos( ) v v i u S. 57

58 ..9 Applicaion of he Fourier series o a nework Example of a nework: Series oscillaor circui u () i() R u () u () L û C / Here is given: From his follows: u ( ) u sin( ) mi = / ˆ uˆ 4uˆ u () cos( k) (4 ) k k o deermine are: and u () i () L S. 58

59 ..9 Applicaion of he Fourier series Soluion : o a nework - Use of impedance for each frequency k : - Rewriing of he Fourier series of u () u () uˆ k k k ˆ u e (4 ) 4uˆ k (4 ) jk Z k uˆ iˆ k k in complex form wih: S. 59

60 ..9 Applicaion of he Fourier series o a nework Impedance and/or curren of he series oscillaor circui for a cerain frequency k : Z R jx wih k hen applies o he curren: k Xk kl k C and / i u Z k k k uˆ i () i. e k k k (4k ) R jxk jk Using he Euler' formula i resuls: uˆ i ( )..[cos( k) jsin( k)] k R jx k (4 ) k S. 6

61 ..9 Applicaion of he Fourier series o a nework Afer rewriing he equaion, i resuls: i () uˆ Rcos( k ) X sin( k ) Rsin( k ) X cos( k ) k k j k (4k ) R Xk R Xk If he characerisics of he funcions (cos, sin ) for +/- k are used, hen i applies wih X X : k k i () 4uˆ Rcos( k) X sin( k) k k (4k ) R Xk S. 6

62 ..9 Applicaion of he Fourier series o a nework () One receives he volage a he coil by means of : () di ul L d u L () 4uˆ kl[ Rsin( k) Xk cos( k)] (4k ) R X k k he resuls can be represened also in polar form: 4uˆ kl R ul( ). cos[( k) arcan( )] (k ) X k R Xk k 4uˆ k ( ). cos[( k) arcan( )] k (k ) R X R k i X S. 6

63 .. Formulaion of Parseval's equaion - wo generally non-sinusoidal periodic funcions are regarded - he funcions f () and f () have he same period : - he appropriae Fourier series formulas hen look as follows: j j () wih () f C e C f e d and j j () wih () f D e D f e d - o he produc of boh funcions applies: and a he same ime as hey are periodical: f () f () C e. D e jk jk () () k wih k () () k f f E e E f f e d j j S. 63

64 .. Formulaion of Parseval's equaion Furher applies: j j jk Ek Ce. De e d j( k) Ek C D e d and/or j( k) Ek CI wih I D e d One can show ha I is differen from zero only in case of: (due o orhogonaliy of cos(nx) and sin(nx)) hus he inegrand becomes idenical o and i applies: I D D k S. 64

65 .. Formulaion of Parseval's equaion For he fourier-coefficiens of he produc f () f () hen resuls E C D C D k k due o k or k v E Deerminaion of he DC componen of he produc f () f () using k = : E f (). f () d C. D here are various applicaions of his relaionship (deerminaion of he inegral in he ime or frequency range)! S. 65

66 .. Formulaion of Parseval's equaion All complex Fourier-coefficiens possess he characerisic: C C and D D * * hus he following formula can be rewrien based on E f (). f () d C. D C. D v o: * * E Re C D Re C D his is Parseval's equaion! S. 66

67 .. Power of non-sinusoidal periodic nework funcions he elecrical energy E per period (or he power P) concerning an Ohm's resisance amouns o: / ( ) ( ) P E u d R i d R Applicaion of Parseval's equaion for he special case f () f () f () i() wih f f f i () () () () resuls in: * () Re. E f d C C C C P ER RC ( C ) S. 67

68 .. Power of non-sinusoidal periodic nework funcions a Wih, a jb c C c C i follows: a av bv f () d c c If f() has characer of a volage or a curren, he appropriae specral parameers a, b and c describe he effecive power condiions of he appropriae nework elemens. S. 68

69 .. Power of non-sinusoidal periodic nework funcions Now a wo-pole wih non-sinusoidal periodic nework funcions u() and i() is regarded: i() U() wopole ' For he power i applies: and u( ) f ( ) C e i( ) f ( ) D e j j * P uid ()() f() f() dcd ReC D CD Re CD * S. 69

70 .. Power of non-sinusoidal periodic nework funcions Wih consideraion of he relaions C U ˆ, D I, C u, D i ˆ follows: * * ˆ ˆ Reˆ ˆ v v v v P U I Re u i U I u i hus he oal power is o be deermined over he sum of all individual powers for each specral line! S. 7

71 .. Assessing of deviaions from he sinusoidal form of periodic funcions Definiion of he rms value of a periodic funcion: () eff () f f d Parseval's equaion permis he deerminaion of he rms value using Fourier coefficiens (and/or he associaed rms values): f() eff c a a b S. 7

72 .. Assessing of deviaions from he sinusoidal form of periodic funcions - he rms value for a periodic volage u() amouns o: U : DC componen of u() û Ueff : Peak value uˆ eff Ueff U U uˆ / : Rms value of he componen (a frequency: ) - he rms value for a periodic curren amouns o: ˆ i eff eff eff I I I I I I S. 7

73 .. Assessing of deviaions from he sinusoidal form of periodic funcions For pure alernaing currens (AC), wihou any DC componen applies: a Oherwise: f() conains boh (DC and AC componens) hus a he oscillaion conen s describes he amoun of AC in he oal signal. s Ueff Ueff U eff U eff S. 73

74 .. Assessing of deviaions from he sinusoidal form of periodic funcions he deviaion from he sinusoidal form can be described by he basic oscillaion amoun g: g U U eff eff eff U U eff he harmonic conen k or disorion facor k amouns o: k U eff U eff U eff U eff g k S. 74

75 .. Assessing of deviaions from he sinusoidal form of periodic funcions Addiionally here are furher definiions named shape facor and ampliude facor : Form facor: k f U eff u ( ) d Cres facor for signals wihou DC componen: k a u () max U eff For purely sinusoidal form one finds: kf, and ka,4 S. 75

76 ..3 Addiional properies of he Fourier series Lineariy ime-shif k s( ) gives a series wih kcv a s ( ) bs () gives a series wih acv bcv s ( ) gives a series wih ce v jv Reflecion * s( ) gives a series wih c v S. 76

77 Fundamenals of Elecrical Engineering 3 Chaper.3 Descripion of aperiodic ime operaions by means of he Fourier ransform S. 77

78 .3. Prefaces Onse: Developmen of he Fourier ransform from he Fourier series by ransfer of periodic funcions o aperiodic impulses Example : A periodic recangular impulse is regarded u() is an even funcion f() - U for oherwise i i () u A Fourier analysis of his signal is o be accomplished. S. 78

79 .3. Prefaces Soluion: c i i U d U i i sin( ) cos( ) i a jb a U c U d b wih v i v i sin sin i sin U U i c i i i S. 79

80 .3. Prefaces and: i sin U U a c si i i i ( ) i hus i applies: U i e i () si u j C C C C 3 C 4 C C 5 sin( x) x C5 5 Skech of he specrum of u() for he case 5 i, S. 8

81 .3. he Fourier inegral In he following he Fourier series of a periodic funcion f() is examined. Here he period is: f() -o/ / he following condiions are assumed: f () ) f() shall be coninous ) In each finie period mees he Dirichle' condiions c e j he funcion shall 3) Wih infinie period f() shall be absoluely inegrable S. 8

82 .3. he Fourier inegral he following represenaion sars from a periodic signal which is ransformed ino a non-periodical of signal. Noe: Enlargemen of he period is made by: lim Each erm in he complex Fourier series corresponds o a line in specrum. Disances beween lines amoun o: / In an inerval around any poin of frequency hereby lie in he following number of lines: m Wih sufficien small inervals hen only small difference of he m individual erms of he complex Fourier series resul. Consequence: Summarizing of hese erms is permied! S. 8

83 For.3. he Fourier inegral In each inerval wih m lines applies hereby applies concerning is conribuion o he row: jv jv mc ve c ve one can selec infinie small inervals (if m should remain unchanged) hus arises for he conribuion of each inerval o he row: jv j d c ve wih v in which i holds: c ve d In addiion can be wrien shorening: c v F ( ) Alogeher i resuls : j f () ( ) F e d S. 83

84 .3. he Fourier inegral hus i also applies: j * j f ( ) F( ) e d F ( ) e d or j f() F( ) e d mi c () f e Now i consequenly follows because of he Fourierspekrum and/or he Fourier ransform F) represens he funcion f() as follows: F j ) F( ) f ( e ) d c v F ( ): j he corresponding symbol: f() F( ) S. 84

85 .3.3 he Fourier inverse ransform he funcion f() can hus be represened by means of is specrum: j f () ( ) F e d he inverse ransform (from frequncy domain o he ime domain is in shor wrien as follows: F ( ) F( ) f () has no he propery of an ampliude (as in he Fourier series), bu i is an ampliude densiy funcion wih he dimension: Ampliude x ime or Ampliude Frequency he exisence of he Fourier inegral is ensured if f() is absoluely inegrable: f () d S cons S. 85

86 .3.4 Inerpreaion and summary Now a signal s() is considered, from which by ideal BP filering only porions wihin a cerain frequency band are exraced. he filering akes place so narrow-banded ha herein he specrum (and he exponenial funcion) changes only insignificanly. For his exraced porion g() follows: j j j g( ) S( ) e d S( ) e d S( ) e d j j j ( S( ) e S( ) e ) Re{ S( ) e } jarg( S( )) j * Re{ S( ) e e } wegen S( ) S ( ) S( ) cos( arg( S( )) S. 86

87 .3.4 Inerpreaion and summary he Fourier ransform (under given condiions) hus is a measure for he ampliude and he phase of a signal componen wih reference o he regarded bandwih concerning he regarded frequency. he mehod of he Fourier ransform permis: ) o describe a signal known in he ime domain equivalenly in he frequency domain ) o deermine he funcion in he ime domain from a known Fourier ransform he Fourier ransform is an imporan and one of he mos powerful ools of elecrical engineering, conrol engineering, physics (opics, mechanics and many more). his mehod forms a he same ime he basis of he Laplace ransform, Z- ransform and he discree Fourier ransform. S. 87