System Processes input signal (excitation) and produces output signal (response)

Transcription

1 Signal A funcion of ime Sysem Processes inpu signal (exciaion) and produces oupu signal (response) Exciaion Inpu Sysem Oupu Response

2 1. Types of signals 2. Going from analog o digial world 3. An example of a sysem 4. Mahemaical represenaion of signals

3 Types of Signals Time Value 1 Coninuous Coninuous 2 Coninuous Discree Analog Digial 3 Discree Coninuous 5 Discree Discree

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5 Types of Signals Time Value 1 Coninuous Coninuous 2 Coninuous Discree 3 Discree Coninuous Analog Digial 5 Discree Discree Role of Noise!

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9 Advanage of Digial World

10 Going from Analog o Digial World Three Sep Process Sampling Quanizaion Encoding

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13 Example of Sysem A simple sysem example Sound Recording Sysem Wha consiues a sound recording sysem?

14 Recorded Sound as a Signal Example s i gn al

15 Represenaion of Signals Time Domain Frequency Domain

16 Anoher Example of Sysem A very complex sysem example Human Brain

17 Sampling a CT Signal o Creae a DT Signal Sampling is acquiring he values of a CT signal a discree poins in ime x() is a CT signal --- x[n] is a DT signal xn xnt s where T s is he ime beween samples

18 Mahemaical Represenaion of Signals Coninuous Time x ( ) A sin( 0 ) A sin( 2 f 0 ) Discree Time x[ nts ] Asin[ onts ] A sin[ 2 f nt o s ]

19 Coninuous Time Signals Sinusoidal Signal: g( ) Asin( ) o Asin( 2f o 2 Asin( ) T o ) General Form: g( ) Asin(2 f ) o

20 Review of Euler s Ideniy Complex valued sinusoidal signals ) sin(2 ) cos(2 2 f j f e F F f j F Euler s Ideniy ) sin(2 ) cos(2 2 f j f e F F f j F 2 ) cos(2 2 2 f j f j F F F e e f j e e f f j f j F F F 2 ) sin(2 2 2 and Also

21 Review of Euler s Ideniy Complex valued sinusoidal signals ) sin(2 ) cos(2 ) (2 f jc f C Ce F F f j F Euler s Ideniy 2 ) cos(2 ) (2 ) 2 ( f j f j F F F Ce Ce f C and Also ) sin(2 ) cos(2 ) (2 f jc f C Ce F F f j F f j j f j j F F F e Ce e Ce f C ) cos(2

22 Exponenial Funcions g( ) Ae g( ) Ae Complex valued exponenial signal: g( ) Ae ( j) Ae [cos j sin ] Where do hese funcions occur in real life?

23 Disconinuiy of a funcion Definiion: lim 0 g( ) lim 0 g( ) Simple words: If he value of funcion is differen a ime 0 when approached a 0 by decreasing and increasing ime, hen he funcion is disconinuous a ime 0 Examples:

24 Uni Sep Funcion Definiion: u( ) 1 1/ Real Physical Phenomenon: Swiching

25 Signum Funcion Definiion: sgn( ) u( ) 1

26 Ramp Funcion Definiion: ramp( ) u() u ( x) dx Can you generae his funcion?

27 Uni Impulse Funcion 1 a 1 a 1 a a lim a 0 Area 1 a ( a) 1

28 Definiion: ( ) 0 ( ) d 1 0 Can you represen u() in erms of uni impulse funcion? u ( ) ( x) dx

29 Anoher Imporan Fac abou Uni Impulse Funcion! ( ) d 1 g ( ) ( ) d g(0) Isn i Sampling?

30 Uni Comb n n comb ( ) ( n) comb()

31 Recangular Funcion rec ( ) 1 1/ 0 2 1/ 2 1/ 2 1/

32 Triangular Funcion ri( )

33 Uni Sinc Funcion sin c( ) sin( )

34 Combinaions of Funcions g( ) sin c( )cos(20 ) g( ) Ae cos 20 g( ) u( ) ramp( ) g( ) sgn( )sin(2 )

35 Some More Examples

36 Ampliude Transformaions of Funcions Ampliude Shifing g( ) A g( ) Ampliude Scaling g( ) Ag( )

37 Time Transformaions of Funcions Time Shifing g( ) g( a) Time Scaling g( ) g( a )

38 Muliple Transformaions Case 1 g( ) Ag( a o ) g() Ampliude Scaling Time Time Scaling Shifing o Ag () Ag ( ) Ag( ) a a Case 2 g ( ) Ag ( b o) Ampliude Scaling Time Shifing g () Ag () Ag ) Ag b ) ( o Time Scaling ( o

39 Example of Case 1

40 Example of Case 2

41 Some More Examples

42 Differeniaion and Inegraion of Funcions Differeniaion: Slope of he funcion a ime dg ( ) d Inegraion: Accumulaive area under he curve g ( x) dx

43 Differeniaion A kind of Transformaion of a Signal

44 Inegraion A kind of Transformaion of a Signal

45 Even and Odd Funcions Funcion is Even if g( ) g( ) Example: cos( ) Funcion is Odd if g( ) g( ) Example: sin( )

46 If funcion is neiher even nor odd, hen ) ( ) ( ) ( g g g o e Even and Odd Componens of a Funcion Where 2 ) ( ) ( ) ( g g g e 2 ) ( ) ( ) ( g g g o

47 Producs of Even and Odd CT Funcions Even x Even = Even

48 Producs of Even and Odd CT Funcions Even x Odd = Odd

49 Producs of Even and Odd CT Funcions Even x Odd = Odd

50 Producs of Even and Odd CT Funcions Odd x Odd = Even

51 Inegrals of Even and Odd CT Funcions a a g d a 0 2 g d a g d 0 a

52 Coninuous Time Periodic Funcions Funcion is periodic wih period T, if g( ) g( nt) Wha is he effec on periodic funcion of ime shifing by nt?

53 Examples of Periodic Signals g( ) 3sin(400 ) g( ) 2 2 g( ) sin(12 ) sin(6 ) g( ) sin( ) sin(6 )

54 Discree Time Signals Coninuous Time x ( ) A sin(2 f 0 ) Discree Time x[ nts ] Asin[2 fonts ] Asin[ 2f T o s n] 2f Asin[ f s o n]

55 2f o x[ nts ] Asin[ n] f fo x[ n] Asin[2 n] f Ts x[ n] Asin[2 n] T s s o x[ n] Asin[2 Kn] p x[ n] Asin[2 n] q To be periodic, Kn has o be an ineger for some n => K has o be a raio of inegers Period = q

56 Discree-Time Sinusoids Periodic Sinusoids

57 How Many CT periodic Cycles are Presen in One DT Periodic Cycle p x[ n] Asin[2 n] q Period = q x[ n] Asin[2p n q ] x[ n] Asin[2 p] When n = q one DT period => There are p cycles of CT periodic sinusoidal funcion per one cycle of DT periodic sinusoidal funcion

58 Examples g [ n] 2 cos[ 5 1 n ] Period = 5 g [ n] 12 cos[ 5 2 n ] Period = 5

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60 Two Discree Sinusoids could be similar? Two differen-looking DT sinusoids, g 1 n Acos 2K 1 n g 2 n and Acos 2K 2 n may acually be he same. If K 2 K 1 m, where m is an ineger hen (because n is discree ime and herefore an ineger), Acos2K 1 n Acos2K 2 n (Example on nex slide)

61 Examples g [ n] 2 cos[ 5 1 n ] Period = 5 g [ n] 12 cos[ 5 2 n ] Period = 5 g [ n] 16 cos[ 5 3 n ] Period = 5

62 2 cos[ n] 5 12 cos[ n] 5 16 cos[ n] 5

63 1 2 cos[ n] cos[ n] 5 16 cos[ n]

64 Oher Discree Funcions Uni Impulse Funcion [ n] 1 0 n n 0 0 Please noe: [ n] [ an] n x[ n] [ n] x[0] Discree Sampling n x[ n] [ n n0 ] x[ n0 ]

65 Uni Sep Funcion u[ n] 1 0 n 0 n 0 Uni Ramp Funcion ramp[ n] n 0 n n 0 0

66 w w N N n N n n rec w 0 1 ] [ Recangular Funcion 1] [ ] [ ] [ w w N N n u N n u n rec w Please noe ha

67 Transformaions on Discree Time Funcions Ampliude Shifing g[ n] A g[ n] Ampliude Scaling g[ n] Ag[ n]

68 Time Shifing g[ n] g[ n a] Same as coninuous ime Time Scaling g[ n] n g[ a ] Tricky! Isn i?

69 Example of Time Shifing

70 Example of Time Compression

71 Discree Time Even and Odd Funcions ] [ ] [ n g n g ] [ ] [ n g n g Funcion is Even if Funcion is Odd if If funcion is neiher even nor odd, hen ] [ ] [ ] [ n g n g n g o e Where 2 ] [ ] [ ] [ n g n g n g e 2 ] [ ] [ ] [ n g n g n g o

72 Differencing and Accumulaion g[ n] g[ n 1] g[ n] n g [ n]

73 Energy of a Signal For Coninuous Time Signals 2 E x x( ) d For Discree Time Signals E x n x[ n] 2

74 Visual Example of Energy of a Signal CT Signal

75 Visual Example of Energy of a Signal DT Signal

76 Power of a Signal Some signals have infinie signal energy. In ha case I is more convenien o deal wih average signal power. For Coninuous Time Signals P x lim T 1 T T / 2 T / 2 x( ) 2 d For Discree Time Signals P x lim N 1 2N N 1 nn x[ n] 2

77 Power of a Periodic Signal For a periodic CT signal, x(), he average signal power is P x 1 T T x 2 d where T is any period of he signal. For a periodic DT signal, x[n], he average signal power is P x 1 N n N xn 2 where N is any period of he signal.

78 Energy and Power Signals A signal wih finie signal energy is called an energy signal. A signal wih infinie signal energy and finie average signal energy is called a power signal.