EE 224 Signals and Systems I Complex numbers sinusodal signals Complex exponentials e jωt phasor addition

Transcription

1 EE 224 Signals and Sysems I Complex numbers sinusodal signals Complex exponenials e jω phasor addiion 1/28

2 Complex Numbers Recangular Polar y z r z θ x Good for addiion/subracion Good for muliplicaion/division z 1 = x 1 + jy 1 z 2 = x 2 + jy z 1 = r 1 e jθ 1 z 2 = r 2 e jθ 2 2 z z 1 ± z 2 = (x 1 ± x 2 ) + j(y 1 ± y 2 ) 1 z 2 = r 1 r 2 e jθ 1+θ 2 z 1 z 2 = r 1 r 2 e jθ 1 θ 2 z = x + jy = x 2 + y 2 e j an 1 y x z = re jθ = r cos(θ) + jr sin(θ) 2/28

3 Euler s formula e iθ = cos(θ) + i sin(θ) Proof using power series e iθ = 1 + iθ + (iθ)2 2! + (iθ)3 3! + (iθ)4 4! = 1 + iθ θ2 2! iθ3 + θ4 3! 4! + iθ5 ) 5! = (1 θ2 2! + θ4 4! θ6 6! + + i = cos θ + i sin θ + (iθ)5 5! θ6 + (iθ)6 6! + 6! + ) (θ θ3 3! + θ5 5! 3/28

4 Examples/Exercises find (2 + 3j)(3 + 4j) conver 1 + j o polar form conver 5e j π 3 o recangular form find 4e j π 2 + 2e j π 3 Compue 4e j π 2 /(5e j π 6 ) Compue 2 + 3j 3 + 4j 4/28

5 Complex Exponenials z() = e jω : uni magniude, phase linearly increase wih simple bu exremely imporan Using Euler s formula, we have e jω = cos(ω) + j sin(ω) The geomeric picure is very simple: a poin on he uni circle moving a a consan angular speed of ω radians per second. ω rad/sec 1 ω is known as radian frequency angular speed radial frequency circular frequency 5/28

6 Sinusoids The real par of e jω is x() = cos(ω) ω rad/sec 1 6/28 a

7 Sinusoids The real par of e jω is x() = cos(ω) ω rad/sec 1 x() 6/28 b

8 Sinusoids The real par of e jω is x() = cos(ω) x() ω rad/sec 1 In general, we can have a complex exponenial wih magniude A and iniial phase θ: z() = Ae j(ω+θ) A cos(ω + θ) 6/28 c

9 Sinusoids The real par of e jω is x() = cos(ω) x() ω rad/sec 1 In general, we can have a complex exponenial wih magniude A and iniial phase θ: z() = Ae j(ω+θ) A cos(ω + θ) Always use cos form for consisency. sin is special case: sin(ω) = cos(ω π 2 ) 6/28 d

10 Two imporan properies A delayed version of a complex exponenial is a complex exponenial of he same frequency. Ae jω( τ) = Ae jωτ e jω A linear combinaion of wo (or more) complex exponenials of he same frequency yields a complex exponenial of he same freqeuncy. A 1 e jθ 1 e jω + A 2 e jθ 2 e jω = Ae jθ e jω 7/28 a

11 Two imporan properies A delayed version of a complex exponenial is a complex exponenial of he same frequency. Ae jω( τ) = Ae jωτ e jω A linear combinaion of wo (or more) complex exponenials of he same frequency yields a complex exponenial of he same freqeuncy. A 1 e jθ 1 e jω + A 2 e jθ 2 e jω = Ae jθ e jω A 1 e jθ 1 + A 2 e jθ 2 = Ae jθ 7/28 b

12 Two imporan properies A delayed version of a complex exponenial is a complex exponenial of he same frequency. A linear combinaion of wo (or more) complex exponenials of he same frequency yields a complex exponenial of he same freqeuncy. A 1 e jθ 1 e jω + A 2 e jθ 2 e jω = Ae jθ e jω These wo properies make he signal e jω imporan e jω inpu Ae jω( τ) = Ae jωτ e jω Sysem αe jω oupu 7/28 c

13 The Phasor Concep A 1 e jθ 1 e jω + A 2 e jθ 2 e jω = Ae jθ e jω Take he real par A 1 cos(ω + θ 1 ) + A 2 cos(ω + θ 2 ) = A cos(ω + θ) Phasor Addiion To obain n i=1 A i cos(ω + θ i ), do Represen each signal A i cos(ω + θ i ) as a phasor A i e jθ i Add all he phasors o obain Ae jθ = n i=1 A i e jθ i The final answer will be A cos(ω + θ) 8/28 a

14 The Phasor Concep A 1 e jθ 1 e jω + A 2 e jθ 2 e jω = Ae jθ e jω Take he real par A 1 cos(ω + θ 1 ) + A 2 cos(ω + θ 2 ) = A cos(ω + θ) Example: To compue 2 cos(10π) + 2 cos(10π + π 2 ) Conver o sum of phasors 2e j0 + 2e j π 2 = 2 + 2j = 2 2e j π 4 Final answer: Re[2 2e j π 4 e j10π ] = 2 2 cos(10π + π 4 ) 8/28 b

15 Phasor Addiion Geomeric View ω rad/sec A 2 e jθ 2 A 1 e jθ 1 9/28

16 Phasor Addiion (con d) A 1 cos(ω + θ 1 ) + A 2 cos(ω + θ 2 ) = A cos(ω + θ) The sum of sinusoids of he same frequency is sill of he same frequency A 2 cos(ω + θ 2 ) A 2 e jθ 2 A 1 cos(ω + θ 1 ) A cos(ω + θ) A 1 e jθ 1 Ae jθ A delayed sinusoid is sill a sinusoid (of he same frequency) inpu LTI Sysem oupu applicaion: phasor in circuis 10/28

17 Summary Complex numbers Euler formula Complex exponenial signal e jω Two properies of e jω Sinusoids A cos(ω + θ) Phasor addiion 11/28

18 EE 224 Signals and Sysems I Basic Signals Discree ime basic funcions Sinusoids (Periodic) Complex exponenials e jω Real exponenials General complex exponenials Uni impulse (Dela funcions) Uni sep funcion 12/28

19 Basic Discree-Time Signals Sinusoids A cos(ωn + θ) 13/28 a

20 Basic Discree-Time Signals Sinusoids A cos(ωn + θ) n General complex exponenials Ae αn Real exponenials e an Pure complex exponenials e jωn 13/28 b

21 Basic Discree-Time Signals Sinusoids A cos(ωn + θ) n General complex exponenials Ae αn Real exponenials e an Pure complex exponenials e jωn Uni Impulse { 1, n = 0 δ[n] = 0, n 0 13/28 c

22 Basic Discree-Time Signals Sinusoids A cos(ωn + θ) n General complex exponenials Ae αn Real exponenials e an Pure complex exponenials e jωn Uni Impulse { 1, n = 0 δ[n] = 0, n 0 n 13/28 d

23 Basic Discree-Time Signals Sinusoids A cos(ωn + θ) n General complex exponenials Ae αn Real exponenials e an Pure complex exponenials e jωn Uni Impulse { 1, n = 0 δ[n] = 0, n 0 n x[n]δ[n] = x[0]δ[n] 13/28 e

24 Decomposion of DT Signals We can view x[n] as sum of delayed impulses. x[n] = k= x[k]δ[n k] 14/28 a

25 Decomposion of DT Signals We can view x[n] as sum of delayed impulses. x[n] = k= x[k]δ[n k] 0 n 14/28 b

26 Decomposion of DT Signals We can view x[n] as sum of delayed impulses. x[n] = k= x[k]δ[n k] 0 n (a) 14/28 c

27 Decomposion of DT Signals We can view x[n] as sum of delayed impulses. x[n] = k= x[k]δ[n k] 0 n (a) (b) 14/28 d

28 Decomposion of DT Signals We can view x[n] as sum of delayed impulses. x[n] = k= x[k]δ[n k] 0 n (a) (b) (c) 14/28 e

29 Decomposion of DT Signals We can view x[n] as sum of delayed impulses. x[n] = k= x[k]δ[n k] 0 n (a) (b) (c) (d) 14/28 f

30 Decomposion of DT Signals We can view x[n] as sum of delayed impulses. x[n] = k= x[k]δ[n k] 0 n (a) (b) (c) (d) (e) 14/28 g

31 Decomposion of DT Signals We can view x[n] as sum of delayed impulses. x[n] = k= x[k]δ[n k] 0 n (a) (b) (c)... (d) (e) (f) 14/28 h

32 Decomposion of DT Signals We can view x[n] as sum of delayed impulses. x[n] = k= x[k]δ[n k] 0 n (a) (b) (c)... (d) (e) (f) Imporan when sudying LTI sysems I resuls in he Impulse response 14/28 i

33 DT Uni Sep Funcion u[n] = { 1, n , n < n Relaionship beween δ[n] and u[n] δ[n] = u[n] u[n 1] u[n] = n k= δ[k] Also, u[n] = k=0 δ[n k] Exercise Plo x[n] = u[n] 3u[n 2] + 2u[n 3], and find x[n] x[n 1] 15/28

34 Basic CT Signals There are a number of signals ha are ofen encounered. They are also he building blocks of signals: oher signals can be obained by combining he basic signals. 16/28 a

35 Basic CT Signals There are a number of signals ha are ofen encounered. They are also he building blocks of signals: oher signals can be obained by combining he basic signals. Sinusoids A cos(ω + θ) periodic, finie power, phasor is Ae jθ 16/28 b

36 Basic CT Signals There are a number of signals ha are ofen encounered. They are also he building blocks of signals: oher signals can be obained by combining he basic signals. Sinusoids A cos(ω + θ) periodic, finie power, phasor is Ae jθ Periodic complex exponenials Ae j(ω+θ) an exremely imporan signal 16/28 c

37 Basic CT Signals There are a number of signals ha are ofen encounered. They are also he building blocks of signals: oher signals can be obained by combining he basic signals. Sinusoids A cos(ω + θ) Periodic complex exponenials Ae j(ω+θ) Real exponenials Ae a, a R periodic, finie power, phasor is Ae jθ an exremely imporan signal 16/28 d

38 Basic CT Signals There are a number of signals ha are ofen encounered. They are also he building blocks of signals: oher signals can be obained by combining he basic signals. Sinusoids A cos(ω + θ) Periodic complex exponenials Ae j(ω+θ) Real exponenials Ae a, a R periodic, finie power, phasor is Ae jθ an exremely imporan signal A a > 0 16/28 e

39 Basic CT Signals There are a number of signals ha are ofen encounered. They are also he building blocks of signals: oher signals can be obained by combining he basic signals. Sinusoids A cos(ω + θ) Periodic complex exponenials Ae j(ω+θ) Real exponenials Ae a, a R periodic, finie power, phasor is Ae jθ an exremely imporan signal A A a > 0 a < 0 16/28 f

40 Basic CT Signals There are a number of signals ha are ofen encounered. They are also he building blocks of signals: oher signals can be obained by combining he basic signals. Sinusoids A cos(ω + θ) periodic, finie power, phasor is Ae jθ Periodic complex exponenials Ae j(ω+θ) Real exponenials Ae a, a R an exremely imporan signal A A A a > 0 a < 0 a = 0 16/28 g

41 General Complex Exponenials Le α = a + jω be a complex number. Ae α = Ae a+jω = Ae a e jω 17/28 a

42 General Complex Exponenials Le α = a + jω be a complex number. Ae α = Ae a+jω = Ae a e jω The complex exponenial has a ime-varying magniude, and a linearly changing phase 17/28 b

43 General Complex Exponenials Le α = a + jω be a complex number. Ae α = Ae a+jω = Ae a e jω The complex exponenial has a ime-varying magniude, and a linearly changing phase a > 0 17/28 c

44 General Complex Exponenials Le α = a + jω be a complex number. Ae α = Ae a+jω = Ae a e jω The complex exponenial has a ime-varying magniude, and a linearly changing phase a > 0 a = 0 17/28 d

45 General Complex Exponenials Le α = a + jω be a complex number. Ae α = Ae a+jω = Ae a e jω The complex exponenial has a ime-varying magniude, and a linearly changing phase a > 0 a = 0 a < 0 17/28 e

46 Uni Impulse I is an absracion of a very narrow pulse ha has uni srengh 18/28 a

47 Uni Impulse I is an absracion of a very narrow pulse ha has uni srengh Consider he signal { A, 0 1 x() = A 0, oherwise 0 A 1 A 18/28 b

48 Uni Impulse I is an absracion of a very narrow pulse ha has uni srengh Consider he signal { A, 0 1 x() = A 0, oherwise As A increases, he area under he curve remains 1 x()d = 1 0 A 1 A 18/28 c

49 Uni Impulse I is an absracion of a very narrow pulse ha has uni srengh Consider he signal { A, 0 1 x() = A 0, oherwise As A increases, he area under he curve remains 1 x()d = 1 0 A 1 A The pulse ges narrower bu higher, as A increases 18/28 d

50 Uni Impulse (con d) In he limi, as w 0, he funcion is call he uni impulse I is also known as he Dirac s Dela Funcion A δ() 1 A Properies of δ() δ()d = 1 δ() = 0, 0 19/28

51 Decomposiion of a Signal ino Impulses δ( τ) x() x() δ( τ)d = x(τ)δ( τ) 20/28 a

52 Decomposiion of a Signal ino Impulses δ( τ) x() x() δ( τ)d = x(τ)δ( τ) x() δ( τ)d = x(τ) 20/28 b

53 Decomposiion of a Signal ino Impulses δ( τ) x() x() δ( τ)d = x(τ)δ( τ) x() δ( τ)d = x(τ) Example: e δ( 1)d = e 1 20/28 c

54 Decomposiion of a Signal ino Impulses δ( τ) x() x() δ( τ)d = x(τ)δ( τ) x() δ( τ)d = x(τ) Exchange and τ 20/28 d

55 Decomposiion of a Signal ino Impulses δ( τ) x() x() δ( τ)d = x(τ)δ( τ) x() δ( τ)d = x(τ) Exchange and τ x() = x(τ) δ(τ )dτ 20/28 e

56 Decomposiion of a Signal ino Impulses δ( τ) x() x() δ( τ)d = x(τ)δ( τ) x() δ( τ)d = x(τ) Exchange and τ x() = x(τ) δ(τ )dτ δ() is an even funcion: δ( τ) = δ(τ ) 20/28 f

57 Decomposiion of a Signal ino Impulses δ( τ) x() x() δ( τ)d = x(τ)δ( τ) x() δ( τ)d = x(τ) Exchange and τ x() = x(τ) δ(τ )dτ δ() is an even funcion: δ( τ) = δ(τ ) x() = x(τ) δ( τ)dτ 20/28 g

58 Decomposiion of a Signal ino Impulses δ( τ) x() x() δ( τ)d = x(τ)δ( τ) x() δ( τ)d = x(τ) Exchange and τ x() = x(τ) δ(τ )dτ δ() is an even funcion: δ( τ) = δ(τ ) x() = x(τ) δ( τ)dτ We can view x() as a sum of delayed and scaled impulses 20/28 h

59 Decomposiion of a Signal ino Impulses (con d) x() 21/28 a

60 Decomposiion of a Signal ino Impulses (con d) x() 21/28 b

61 Decomposiion of a Signal ino Impulses (con d) x() w x( i ) i 21/28 c

62 Decomposiion of a Signal ino Impulses (con d) x() w x( i ) area x( i )w x( i )wδ( i ) i τ 21/28 d

63 Decomposiion of a Signal ino Impulses (con d) x() w x( i ) area x( i )w x( i )wδ( i ) i τ x() i x( i)wδ( i ) 21/28 e

64 Decomposiion of a Signal ino Impulses (con d) x() w x( i ) area x( i )w x( i )wδ( i ) i τ x() i x( i)wδ( i ) x() = x(τ) δ( τ)dτ 21/28 f

65 Decomposiion of a Signal ino Impulses (con d) x() w x( i ) area x( i )w x( i )wδ( i ) i τ This viewpoin of decomposing a signal as a sum of impulses is useful when we sudy he response of a sysem o a signal. 21/28 g

66 Signal Decomposiion All funcions (of pracical ineres) can be expressed as linear combinaions of delayed impulses x() = x(τ) δ( τ)dτ 22/28 a

67 Signal Decomposiion All funcions (of pracical ineres) can be expressed as linear combinaions of delayed impulses x() = x(τ) δ( τ)dτ All funcions (of pracical ineres) can also be expressed as linear combinaions of periodic complex exponenials: x() = 1 2π X(ω) ejω dω 22/28 b

68 Signal Decomposiion All funcions (of pracical ineres) can be expressed as linear combinaions of delayed impulses x() = x(τ) δ( τ)dτ All funcions (of pracical ineres) can also be expressed as linear combinaions of periodic complex exponenials: x() = 1 2π X(ω) ejω dω This is known as he Fourier ransform 22/28 c

69 Signal Decomposiion All funcions (of pracical ineres) can be expressed as linear combinaions of delayed impulses x() = x(τ) δ( τ)dτ All funcions (of pracical ineres) can also be expressed as linear combinaions of periodic complex exponenials: x() = 1 2π X(ω) ejω dω This is known as he Fourier ransform δ() and e jω are he wo mos basic signals 22/28 d

70 Comparing δ() and e jω δ( 0 ) e jω 0 waveform 0 Re[e jω 0 ] Im[e jω 0 ] 23/28 a

71 Comparing δ() and e jω δ( 0 ) e jω 0 waveform 0 Re[e jω 0 ] Im[e jω 0 ] ime localized all imes 23/28 b

72 Comparing δ() and e jω δ( 0 ) e jω 0 waveform 0 Re[e jω 0 ] Im[e jω 0 ] ime localized all imes Fourier Transform e jω 0 2πδ(ω ω 0 ) 23/28 c

73 Comparing δ() and e jω δ( 0 ) e jω 0 waveform 0 Re[e jω 0 ] Im[e jω 0 ] ime frequency Fourier Transform localized all imes all freq. localized e jω 0 2πδ(ω ω 0 ) 23/28 d

74 Comparing δ() and e jω δ( 0 ) e jω 0 waveform 0 Re[e jω 0 ] Im[e jω 0 ] ime frequency Fourier Transform localized all imes all freq. localized e jω 0 2πδ(ω ω 0 ) role: Signals x() = 1 x(τ) δ( τ)dτ x() = 2π X(ω) ejω dω 23/28 e

75 Comparing δ() and e jω δ( 0 ) e jω 0 waveform 0 Re[e jω 0 ] Im[e jω 0 ] ime frequency Fourier Transform localized all imes all freq. localized e jω 0 2πδ(ω ω 0 ) role: Signals x() = 1 x(τ) δ( τ)dτ x() = 2π X(ω) ejω dω role: Sysems Impulse Response Frequency Response 23/28 f

76 Comparing δ() and e jω δ( 0 ) Two Basic Signals e jω 0 waveform 0 Re[e jω 0 ] Im[e jω 0 ] ime frequency Fourier Transform localized all imes all freq. localized e jω 0 2πδ(ω ω 0 ) role: Signals x() = 1 x(τ) δ( τ)dτ x() = 2π X(ω) ejω dω role: Sysems Impulse Response Frequency Response 23/28 g

77 Do we have Dirac Dela Funcion in real applicaions? = 0 i() V C 24/28 a

78 Do we have Dirac Dela Funcion in real applicaions? = 0 = 0 R i() i() V C V C 24/28 b

79 Do we have Dirac Dela Funcion in real applicaions? = 0 = 0 R i() i() V C V C V R i() = V R e RC 24/28 c

80 Do we have Dirac Dela Funcion in real applicaions? = 0 = 0 R i() i() V C V C V R i() = V R e RC area Q = V C 24/28 d

81 Do we have Dirac Dela Funcion in real applicaions? V R V = 0 i() C V = 0 R i() C V R i() = V R e RC area Q = V C area Q = V C 24/28 e

82 Do we have Dirac Dela Funcion in real applicaions? = 0 = 0 R i() i() V C V C Qδ() V R i() = V R e RC In he limi R 0 area Q = V C 24/28 f

83 Do we have Dirac Dela Funcion in real applicaions? = 0 = 0 R i() i() V C V C Qδ() V R i() = V R e RC In he limi R 0 δ() models a pulse ha is very narrow. Is srengh is he inegral of he pulse. 24/28 g

84 How Narrow is Narrow When can a pulse be approximaed as a dela funcion? 25/28 a

85 How Narrow is Narrow When can a pulse be approximaed as a dela funcion? 25/28 b

86 How Narrow is Narrow When can a pulse be approximaed as a dela funcion? 2 25/28 c

87 How Narrow is Narrow When can a pulse be approximaed as a dela funcion? /28 d

88 How Narrow is Narrow When can a pulse be approximaed as a dela funcion? /28 e

89 How Narrow is Narrow When can a pulse be approximaed as a dela funcion? /28 f

90 How Narrow is Narrow When can a pulse be approximaed as a dela funcion? I depends /28 g

91 How Narrow is Narrow When can a pulse be approximaed as a dela funcion? I depends We can approximae using dela funcion if he pulse widh has lile effec on wha are ineresed in, as long as we fix he pulse srengh (is inegral) 25/28 h

92 Uni Sep Funcion u() = { 1, > 0 0, < 0 26/28 a

93 Uni Sep Funcion 1 u() = u w () { 1, > 0 0, < 0 26/28 b

94 Uni Sep Funcion u() = { 1, > 0 0, < 0 u() = lim w 0 u w () 1 u w () w 0 1 u() 26/28 c

95 Uni Sep Funcion u() = { 1, > 0 0, < 0 u() = lim w 0 u w () u w () u() 1 w 0 1 w d d 1 δ w () 1/w w 26/28 d

96 Uni Sep Funcion u() = { 1, > 0 0, < 0 u() = lim w 0 u w () 1 u w () w d d w 0 1 u() δ w () 1/w 1 w 0 δ() d d w 26/28 e

97 Example x() = δ( 1) 2δ( 2) + δ( 3) Draw x() and find y() = x()d 27/28 a

98 Example x() = δ( 1) 2δ( 2) + δ( 3) Draw x() and find y() = x()d x() /28 b

99 Example x() = δ( 1) 2δ( 2) + δ( 3) Draw x() and find y() = x()d y() = u( 1) 2u( 2) + u( 3) x() /28 c

100 Example x() = δ( 1) 2δ( 2) + δ( 3) Draw x() and find y() = x()d y() = u( 1) 2u( 2) + u( 3) x() y() d d /28 d

101 Summary Basic Coninuous-Time Signals Dirac Dela funcion (uni impulse) δ() Exponenial signals, especially e jω Sinusoids cos(ω + θ) Uni sep funcion u() Boh δ() and e jω are imporan signals ha can be used o build oher signals Basic Discree-Time Signals Uni Impulse δ[n] Exponenial signals, especially e jωn Sinusoids cos(ωn + θ) Uni sep funcion u[n] Boh δ[n] and e jωn are imporan signals ha can be used o build oher signals 28/28