Topic 3: Fourier Series (FS)

Transcription

1 ELEC361: Signals And Systems Tpic 3: Furier Series (FS) Dr. Aishy Amer Cncrdia University Electrical and Cmputer Engineering Intrductin t frequency analysis f signals Furier series f CT peridic signals Signal Symmetry and CT Furier Series Prperties f CT Furier series Cnvergence f the CT Furier series Furier Series f DT peridic signals Prperties f DT Furier series Respnse f LTI systems t cmplex expnential Summary Appendix: Applicatins (nt in the exam) Figures and examples in these curse slides are taken frm the fllwing surces: A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Editin, Prentice-Hall, 1997 M.J. Rberts, Signals and Systems, McGraw Hill, 2004 J. McClellan, R. Schafer, M. Yder, Signal Prcessing First, Prentice Hall, 2003

2 Signal Representatin Time-dmain representatin Wavefrm based Peridic / nn-peridic signals Frequency-dmain representatin Peridic signals Sinusidal signals Frequency analysis fr peridic signals Cncepts f frequency, bandwidth, filtering 2

3 Wavefrm Representatin Wavefrm representatin Plt f the signal value vs. time Sund amplitude, temperature reading, stck price,.. Mathematical representatin: x(t) x: variable value T: independent variable 3

4 4 Sample Speech Wavefrm

5 5 Sample Music Wavefrm

6 Sinusidal Signals Sinusidal signals: imprtant because they can be used t synthesize any signal An arbitrary signal can be expressed as a sum f many sinusidal signals with different frequencies, amplitudes and phases Phase shift: hw much the max. f the sinusidal signal is shifted away frm t=0 Music ntes are essentially sinusids at different frequencies 6

7 Cmplex Expnential Signals 7 Cmplex Number : Cartesian representatin : z Magnitude f Phase f Plar representatin : Cmplex Expnential Signal : x( t) = z e z is θ is the phase shift (initial phase) Cmplex Cnjugate : z * ( jω t+ θ ) Euler Frumla : e θ = a + e θ jb 0 z is = Nte : = 2cs( θ ) z = z ( z + z e θ = a + a z e 0 * 2 φ = z = = jφ z cs( ω t + θ ) + e + b 2 1 b tan a = z csφ + ) and θ = jb ( zz j z sin( ω t + θ ) * ) are real j2sin( θ ) j z sinφ 0

8 Real and Cmplex Sinusids 8

9 Peridic CT Signals A CT signal is peridic if there is a psitive value fr which Perid T f : The interval n which x(t) repeats Fundamental perid T 0 : the smallest such repetitin interval T 0 =1/f 0 Fundamental perid: the smallest psitive value fr which the equatin abve hlds Example: x(t) = cs(4*pi*t); T=1/2; T 0 =1/4 Harmnic frequencies f x(t): kf 0, k is integer, Example: is peridic with 9

10 Sums f CT peridic signals The perid f the sum f CT peridic functins is the least cmmn multiple f the perids f the individual functins summed If the least cmmn multiple is infinite, the sum is aperidic 10

11 Peridic DT Signals A DT signal is peridic with perid where is a psitive integer if The fundamental perid f is the smallest psitive value f fr which the equatin hlds Example: is peridic with fundamental perid 11

12 What is frequency f an arbitrary signal? Sinusidal signals have a distinct (unique) frequency An arbitrary signal x(t) des nt have a unique frequency x(t) can be decmpsed int many sinusidal signals with different frequencies, each with different magnitude and phase Spectrum f x(t): the plt f the magnitudes and phases f different frequency cmpnents Furier analysis: find spectrum fr signals Bandwidth f x(t): the spread f the frequency cmpnents with significant energy existing in a signal 12

13 13 Frequency cntent in signals

14 Frequency cntent in signals A cnstant : nly zer frequency cmpnent (DC cmpnent) A sinusid : Cntain nly a single frequency cmpnent Peridic signals : Cntain the fundamental frequency and harmnics : Line spectrum Slwly varying : cntain lw frequency nly Fast varying : cntain very high frequency Sharp transitin : cntain frm lw t high frequency Music: : cntain bth slwly varying and fast varying cmpnents, wide bandwidth 14

15 Transfrming Signals X(t) is the signal representatin in time dmain Often we transfrm signals in a different dmain Furier analysis allws us t view signals in the frequency dmain In the frequency dmain we examine which frequencies are present in the signal Frequency dmain techniques reveal things abut the signal that are difficult t see therwise in the time dmain 15

16 Furier representatin f signals The study f signals and systems using sinusidal representatins is termed Furier analysis, after Jseph Furier ( ) The develpment f Furier analysis has a lng histry invlving a great many individuals and the investigatin f many different physical phenmena, such as the mtin f a vibrating string, the phenmenn f heat prpagatin and diffusin Furier methds have widespread applicatin beynd signals and systems, being used in every branch f engineering and science The thery f integratin, pint-set tplgy, and eigenfunctin expansins are just a few examples f tpics in mathematics that have their rts in the analysis f Furier series and integrals 16

17 17 Furier representatin f signals: Types f signals

18 Furier representatin f signals: Cntinuus-Value / Cntinuus-Time Signals All cntinuus signals are CT but nt all CT signals are cntinuus 18

19 Furier representatin f signals: Types f signals 19

20 Furier representatin f signals Fur distinct Furier representatins: Each applicable t a different class f signals Determined by the peridicity prperties f the signal and whether the signal is discrete r cntinuus in time A Furier representatin is unique, i.e., n tw same signals in time dmain give the same functin in frequency dmain 20

21 Overview f Furier Analysis Methds Cntinuus in Time Aperidic in Frequency Discrete in Time Peridic in Frequency Peridic in Time Discrete in Frequency CT Furier Series : a k 1 = T x( t) = T 0 x( t) e jkω t CT Inverse Furier Series : X[ k] = a k k = N 1 n= 0 N 1 k = 0 e jkω t DT Furier Series 0 0 dt jω kn CT - P Inverse DT Furier Series 1 x[ n] = N x[ n] e 0 X[ k] e DT - P jω kn 0 N T DT DT CT - P DT - P DT - P N N T DT - P N Aperidic in Time Cntinuus in Frequency CT Furier Transfrm : X ( jω) = x( t) = DT Furier Transfrm : X ( e jω x[ n] = ) = n= Inverse DT Furier Transfrm : 1 2 π 2 π x[ n] e X ( e jω jωt Inverse CT Furier Transfrm : 1 2π x( t) e X ( jω) e jωn ) e dt jωt jωn dω dω CT CT DT CT + P CT CT CT + P 2π 2π DT 21

22 Furier representatin: Peridic Signals Furier Series Synthesis (inverser transfrm) x( t) = a = 0 + k = a k = 1 k e a 0 k jω kt cs( ω kt + φ ) (single sided, fr real signal nly) 0 k (duble sided, fr bth real and cmplex) Furier Series Analysis(frward transfrm) X[ k] = a a k k 1 = T T is, in general, a 0 x( t) e jω kt 0 dt; cmplex number k = 0, ± 1, ± 2,... Fr real signals : a k = a * k a k = a k (symmetric spectrum) 22 R(z) = z + z 2

23 23 FS: Example Nn - cnstant Cmpnents Cnstant Cmpnent ) ( ) ( ) 2 8cs(500 ) 3 14 cs( ) ( = = = t x e e e e e e e e t x t t t x t j j t j j t j j t j j π π π π π π π π π π π π

24 24 Cncept f Furier analysis

25 25 Cncept f Furier analysis

26 Apprximatin f Peridic Signals by Sinusids Any peridic signal can be apprximated by a sum f many sinusids at harmnic frequencies f the signal (kf 0 ) with apprpriate amplitude and phase The mre harmnic cmpnents are added, the mre accurate the apprximatin becmes Instead f using sinusidal signals, mathematically, we can use the cmplex expnential functins with bth psitive and negative harmnic frequencies 26

27 Apprximatin f Peridic Signals by Sum f Sinusids 27

28 Furier representatin: Peridic CT Signals 28

29 Example: Furier Series f Square Wave 1 The Furier series analysis: 29

30 Example: Spectrum f Square Wave 1 Each line crrespnds t ne harmnic frequency. The line magnitude (height) indicates the cntributin f that frequency t the signal The line magnitude drps expnentially, which is nt very fast. The very sharp transitin in square waves calls fr very high frequency sinusids t synthesize 30

31 31 Negative Frequency?

32 32 Negative Frequency?

33 Why Frequency Dmain Representatin f signals? 33 Shws the frequency cmpsitin f the signal Change the magnitude f any frequency cmpnent arbitrarily by a filtering peratin Lwpass -> smthing, nise remval Highpass -> edge/transitin detectin High emphasis -> edge enhancement Shift the central frequency by mdulatin A cre technique fr cmmunicatin, which uses mdulatin t multiplex many signals int a single cmpsite signal, t be carried ver the same physical medium

34 Why Frequency Dmain Representatin f signals? Typical Filtering applied t x(t): Lwpass -> smthing, nise remval Highpass -> edge/transitin detectin Bandpass -> Retain nly a certain frequency range 34

35 Outline Intrductin t frequency analysis Furier series f CT peridic signals Signal Symmetry and CT Furier Series Prperties f CT Furier series Cnvergence f the CT Furier series Furier Series f DT peridic signals Prperties f DT Furier series Respnse f LTI systems t cmplex expnential Summary 35

36 Furier series f CT peridic signals Cnsider the fllwing cntinuus-time cmplex expnentials: T 0 is the perid f all f these expnentials and it can be easily verified that the fundamental perid is equal t Any linear cmbinatin f is als peridic with perid T 0 36

37 Furier series f CT peridic signals fundamental cmpnents r the first harmnic cmpnents The crrespnding fundamental frequency is ω 0 Furier series representatin f a peridic signal x(t): 37 (4.1)

38 38 Furier series f a CT peridic signal

39 39 Furier series f a CT peridic signal

40 Furier series f a CT peridic signal: Example

41 Furier series f a CT peridic signal: Example 4.1 The Furier series cefficients are shwn in the Figures Nte that the Furier cefficients are cmplex numbers in general Thus ne shuld use tw figures t demnstrate them cmpletely: shw real and imaginary parts r magnitude and angle 41

42 Furier series f CT REAL peridic signals If x(t) is real This means that 42

43 Furier series f CT REAL peridic signals 43

44 44 sinc Functin

45 Furier series f a CT peridic signal:example

46 Furier series f a CT peridic signal: Example The Furier series cefficients are shwn in Figures fr T=4T 1 and T=16T 1 Nte that the Furier series cefficients fr this particular example are real

47 47

48 48 Inverse CT Furier Series: Example: Magnitude and Phase Spectra f the harmnic functin X[k]

49 Inverse CT Furier series: Example The CT Furier Series representatin xf(t) f the abve csine signal X[k] is xf(t) is dd The discntinuities make X[k] have significant higher harmnic cntent xf(t) 49

50 Nte: Lg-Magnitude Frequency Respnse Plts 50

51 Nte: Lg-Magnitude Frequency Respnse Plts 51

52 Outline Intrductin t frequency analysis Furier series f CT peridic signals Signal Symmetry and CT Furier Series Prperties f CT Furier series Cnvergence f the CT Furier series Furier Series f DT peridic signals Prperties f DT Furier series Respnse f LTI systems t cmplex expnential Summary 52

53 Effect f Signal Symmetry n CT Furier Series 53

54 Effect f Signal Symmetry n CT Furier Series 54

55 Effect f Signal Symmetry n CT Furier Series 55

56 Effect f Signal Symmetry n CT Furier Series 56

57 Effect f Signal Symmetry n CT Furier Series 57

58 Effect f Signal Symmetry n CT Furier Series: Example 58

59 Outline Intrductin t frequency analysis Furier series f CT peridic signals Signal Symmetry and CT Furier Series Prperties f CT Furier series Cnvergence f the CT Furier series Furier Series f DT peridic signals Prperties f DT Furier series Respnse f LTI systems t cmplex expnential Summary 59

60 Prperties f CT Furier series The prperties are useful in determining the Furier series r inverse Furier series They help t represent a given signal in term f peratins (e.g., cnvlutin, differentiatin, shift) n anther signal fr which the Furier series is knwn Operatins n {x(t)} Operatins n {X[k]} Help find analytical slutins t Furier Series prblems f cmplex signals Example: t FS{ y( t) = a u( t 5) } delay and multiplicatin 60

61 Prperties f CT Furier series Let x(t): have a fundamental perid T 0x Let y(t): have a fundamental perid T 0y Let X[k]=a k and Y[k]=b k The Furier Series harmnic functins each using the fundamental perid T F as the representatin time In the Furier series prperties which fllw: Assume the tw fundamental perids are the same T= T 0x =T 0y (unless therwise stated) The fllwing prperties can easily been shwn using equatin (4.5) fr Furier series 61

62 Prperties f CT Furier series 62

63 Prperties f CT Furier series 63

64 Prperties f CT Furier series 64

65 Prperties f CT Furier series 65

66 Prperties f CT Furier series 66

67 Prperties f CT Furier series 67

68 Prperties f CT Furier series 68

69 Prperties f CT Furier series 69

70 Prperties f CT Furier series 70

71 Prperties f CT Furier series 71

72 Prperties f CT Furier series: Example

73 Prperties f CT Furier series: Example

74 Prperties f CT Furier series: Example

75 Prperties f CT Furier series: Example 75

76 Outline Intrductin t frequency analysis Furier series f CT peridic signals Signal Symmetry and CT Furier Series Prperties f CT Furier series Cnvergence f the CT Furier series Furier Series f DT peridic signals Prperties f DT Furier series Respnse f LTI systems t cmplex expnential Summary 76

77 Cnvergence f the CT Furier series The Furier series representatin f a peridic signal x(t) cnverges t x(t) if the Dirichlet cnditins are satisfied Three Dirichlet cnditins are as fllws: 1. Over any perid, x(t) must be abslutely integrable. Fr example, the fllwing signal des nt satisfy this cnditin 77

78 Cnvergence f the CT Furier series 2. x(t) must have a finite number f maxima and minima in ne perid Fr example, the fllwing signal meets Cnditin 1, but nt Cnditin 2 78

79 Cnvergence f the CT Furier series 3. x(t) must have a finite number f discntinuities, all f finite size, in ne perid Fr example, the fllwing signal vilates Cnditin 3 79

80 Cnvergence f the CT Furier series Every cntinuus peridic signal has an FS representatin Many nt cntinuus signals has an FS representatin If a signal x(t) satisfies the Dirichlet cnditins and is nt cntinuus, then the Furier series cnverges t the midpint f the left and right limits f x(t) at each discntinuity Almst all physical peridic signals encuntered in engineering practice, including all f the signals with which we will be cncerned, satisfy the Dirichlet cnditins 80

81 Cnvergence f the CT Furier series: Summary 81

82 82 Cnvergence f the CT Furier series: Cntinuus signals

83 83 Cnvergence f the CT Furier series: Discntinuus signals

84 Cnvergence f the CT Furier series: Gibb s phenmenn 84

85 Cnvergence f the CT Furier series: Gibbs Phenmenn 85

86 Outline Intrductin t frequency analysis Furier series f CT peridic signals Signal Symmetry and CT Furier Series Prperties f CT Furier series Cnvergence f the CT Furier series Furier Series f DT peridic signals Prperties f DT Furier series Respnse f LTI systems t cmplex expnential Summary 86

87 87 The DT Furier Series

88 88 The DT Furier Series

89 89 The DT Furier Series

90 90 Cncept f DT Furier Series

91 91 The DT Furier Series

92 92 The DT Furier Series

93 93 The DT Furier Series

94 The DT Furier Series: Example

95 The DT Furier Series: Example

96 The DT Furier Series: Example

97 The DT Furier Series: Example

98 Outline Intrductin t frequency analysis Furier series f CT peridic signals Signal Symmetry and CT Furier Series Prperties f CT Furier series Cnvergence f the CT Furier series Furier Series f DT peridic signals Prperties f DT Furier series Respnse f LTI systems t cmplex expnential Summary 98

99 Prperties f DT Furier Series 99

100 Prperties f DT Furier Series 100

101 Outline Intrductin t frequency analysis Furier series f CT peridic signals Signal Symmetry and CT Furier Series Prperties f CT Furier series Cnvergence f the CT Furier series Furier Series f DT peridic signals Prperties f DT Furier series Respnse f LTI systems t Cmplex Expnential Summary 101

102 Respnse f LTI systems t Cmplex Expnential Eigen-functin f a linear peratr S: a nn-zer functin that returns frm the peratr exactly as is except fr a multiplicatr (r a scaling factr) Eigenfunctin f a system S: characteristic functin f S System applied n a functin x( t) : S{ x( t)} = λx( t) λ :eigen - value (a nn - null vectr) x( t) :eigen - functin 102

103 Respnse f LTI systems t Cmplex Expnential Eigenfunctin f an LTI system = the cmplex j k expnential e ω t Any LTI system S excited by a cmplex sinusid respnds with anther cmplex sinusid f the same frequency, but generally a different amplitude and phase The eigen-values are either real r, if cmplex, ccur in cmplex cnjugate pairs 103

104 Respnse f LTI systems t Cmplex Expnential 104 Cnvlutin represents: The input as a linear cmbinatin f impulses The respnse as a linear cmbinatin f impulse respnses Furier Series represents: x ( t) = x( τ ) δ ( t τ ) dτ y ( t) = x( τ ) h( t τ ) dτ a peridic signal as a linear cmbinatin f cmplex sinusids jωkt x( t) = ake ωk = 2π fk = 2π k f0 k =

105 Respnse f LTI systems t Cmplex Expnential : Linearity and Superpsitin If x(t) can be expressed as a sum f cmplex sinusids the respnse can be expressed as the sum f respnses t jω t cmplex sinusids k y( t) e ω = 2πf = bk k = k k 105

106 Respnse f LTI systems t Cmplex Expnential Let a cntinuus-time LTI system be excited by a cmplex expnential f the frm, The respnse is the cnvlutin f the excitatin with the impulse respnse r The quantity 106 will later be designated the Laplace transfrm f the impulse respnse and will be an imprtant transfrm methd fr CT system analysis

107 Respnse f LTI systems t Cmplex Expnential CT system: x( t) = e st = e ( σ + jω) t ; s = σ + jω This leads t the fllwing equatin fr CT LTI systems: 107

108 Respnse f LTI systems t Cmplex Expnential H(s) is a cmplex cnstant whse value depends n s and is given by: Cmplex expnential e st are eigenfunctins f CT LTI systems H(s) is the eigenvalue assciated with the eigenfunctin e st 108

109 Respnse f LTI systems t Cmplex Expnential DT systems: This leads t the fllwing equatin fr DT LTI systems: 109

110 Respnse f LTI systems t Cmplex Expnential H(z) is a cmplex cnstant whse value depends n z and is given by: Cmplex expnential z n are eigenfunctins f DT LTI systems H(z) is the eigenvalue assciated with the eigenfunctin z n 110

111 Respnse f LTI systems t Cmplex Expnential Frm superpsitin: 111

112 Respnse f LTI systems t Cmplex Expnential: Example 3.3 Cnsider an LTI system whse input x(t) and utput y(t) are related by a time shift as fllws: Find the utput f the system t the fllwing inputs: 112

113 113 Example Slutin

114 114 Example Slutin

115 jωt Respnse f LTI systems t Cmplex Expnential : x( t) = e ; s = 0 + jω 115

116 Respnse f LTI systems t Cmplex Expnential: Example 116

117 Respnse f LTI systems t Cmplex Expnential: Example 117

118 Outline Intrductin t frequency analysis Furier series f CT peridic signals Signal Symmetry and CT Furier Series Prperties f CT Furier series Cnvergence f the CT Furier series Furier Series f DT peridic signals Prperties f DT Furier series Respnse f LTI systems t peridic signals Summary 118

119 Furier series: summary 119 Sinusid signals: Can determine the perid, frequency, magnitude and phase f a sinusid signal frm a given frmula r plt Furier series fr peridic signals Understand the meaning f Furier series representatin Can calculate the Furier series cefficients fr simple signals (nly require duble sided) Can sketch the line spectrum frm the Furier series cefficients

120 120 Furier series: summary

121 Furier series: summary Steps fr cmputing Furier series: 1. Identify perid 2. Write dwn equatin fr x(t) 3. Observe if the signal has any summitry (even r dd) 4. Use the expnential equatin (1) in previus slide, and if needed use Eq. (2) fr the trignmetric cefficients 121

122 FS Summary: a quiz Prblem: Find the Furier series cefficients f the peridic π cntinuus signal: x( t) = cs( t), 0 t < 3 3 and T = 3 is the perid f the signal. Plt the spectrum (magnitude and phase) f x(t). What is the spectrum f x(t-4)? Slutin: Using the definitin f Furier cefficients fr a peridic cntinuus signal, the cefficients are: a k = π π 1 T 3 2π 3 j t j t π 1 jk t 1 e + e jk t jkω 3 3 ( ) 0 t π x t e dt = cs( t) e dt = e T 2 dt which wuld be simplified after sme manipulatins t: a k 4kj = π (1 4k 2 ) 122

123 Outline 123 Intrductin t frequency analysis Furier series f CT peridic signals Signal Symmetry and CT Furier Series Prperties f CT Furier series Cnvergence f the CT Furier series Furier Series f DT peridic signals Prperties f DT Furier series Respnse f LTI systems t cmplex expnential Summary Appendix: Applicatins (nt in the exam)

124 Applicatins f frequency- dmain representatin 124 Clearly shws the frequency cmpsitin a signal Can change the magnitude f any frequency cmpnent arbitrarily by a filtering peratin Lwpass -> smthing, nise remval Highpass -> edge/transitin detectin High emphasis -> edge enhancement Can shift the central frequency by mdulatin A cre technique fr cmmunicatin, which uses mdulatin t multiplex many signals int a single cmpsite signal, t be carried ver the same physical medium Prcessing f speech and music signals

125 Typical Filters Lwpass -> smthing, nise remval Highpass -> edge/transitin detectin Bandpass -> Retain nly a certain frequency range 125

126 Lw Pass Filtering (Remve high freq, make signal smther) 126

127 High Pass Filtering (remve lw freq, detect edges) 127

128 Filtering in Tempral Dmain (Cnvlutin) 128

129 Cmmunicatin: Signal Bandwidth 129 Bandwidth f a signal is a critical feature when dealing with the transmissin f this signal A cmmunicatin channel usually perates nly at certain frequency range (called channel bandwidth) The signal will be severely attenuated if it cntains frequencies utside the range f the channel bandwidth T carry a signal in a channel, the signal needed t be mdulated frm its baseband t the channel bandwidth Multiple narrwband signals may be multiplexed t use a single wideband channel

130 Signal bandwidth Highest frequency estimatin in a signal: Find the shrtest interval between peak and valleys 130

131 131 Signal Bandwidth

132 Estimatin f Maximum Frequency 132

133 Prcessing Speech & Music Signals 133 Typical speech and music wavefrms are semi-peridic The fundamental perid is called pitch perid The fundamental frequency (f 0 ) Spectral cntent Within each shrt segment, a speech r music signal can be decmpsed int a pure sinusidal cmpnent with frequency f 0, and additinal harmnic cmpnents with frequencies that are multiples f f 0. The maximum frequency is usually several multiples f the fundamental frequency Speech has a frequency span up t 4 KHz Audi has a much wider spectrum, up t 22KHz

134 134 Sample Speech Wavefrm 1

135 Numerical Calculatin f CT Furier Series The riginal signal is digitized, and then a Fast Furier Transfrm (FFT) algrithm is applied, which yields samples f the FT at equally spaced intervals Fr a signal that is very lng, e.g. a speech signal r a music piece, spectrgram is used. Furier transfrms ver successive verlapping shrt intervals 135

136 Sample Speech Spectrgram 1 136

137 137 Sample Speech Wavefrm 2

138 138 Speech Spectrgram 2

139 139 Sample Music Wavefrm

140 140 Sample Music Spectrgram