Signals and Systems Review. 8/25/15 M. J. Roberts - All Rights Reserved 1

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1 Signals and Sysems Review 8/25/15 M. J. Robers - All Righs Reserved 1

2 g Coninuous-Time Sinusoids ( ) = Acos( 2π / T 0 +θ ) = Acos( 2π f 0 +θ ) = Acos( ω 0 +θ ) Ampliude Period Phase Shif Cyclic Radian g (s) (radians) Frequency Frequency! "!"# Acos 2 f 0 $ A (Hz) (radians/s) ω =2π f T 0 8/25/15 M. J. Robers - All Righs Reserved 2

3 Coninuous-Time Exponenials ( ) = Ae /τ g Ampliude Time Consan (s) 8/25/15 M. J. Robers - All Righs Reserved 3

4 Complex Sinusoids Euler's Ideniy: e jx = cos( x) + j sin( x) cos( x) = e jx + e jx 2, sin x ( ) = e jx e jx j2 8/25/15 M. J. Robers - All Righs Reserved 4

5 The Signum Funcion sgn( ) = 1, > 0 0, = 0 1, < 0 sgn() sgn()!!!! The signum funcion, in a sense, reurns an indicaion of he sign of is argumen. 8/25/15 M. J. Robers - All Righs Reserved 5

6 The Uni Sep Funcion u( ) = 1, > 0 0, < 0 Undefined, = 0 (bu finie) u() 1, u ( ) = 1 2 sgn( ) +1, 0 u() 1 The produc signal g( )u( ) can be hough of as he signal g( ) urned on a ime = 0. 8/25/15 M. J. Robers - All Righs Reserved 6

7 The Uni Ramp Funcion ramp ( ) =, > 0 0, 0 = u( λ)dλ = u ( ) ramp() 1 1 8/25/15 M. J. Robers - All Righs Reserved 7

8 The Impulse The coninuous-ime uni impulse is implicily defined by ( ) = δ g 0 ( )g( )d The uni sep is he inegral of he uni impulse and he uni impulse is he generalized derivaive of he uni sep. 8/25/15 M. J. Robers - All Righs Reserved 8

9 Properies of he Impulse The Sampling Propery g( )δ ( 0 )d = g 0 ( ) The sampling propery exracs he value of a funcion a a poin. (In Ziemer and Traner his is called he "sifing" propery.) The Scaling Propery δ ( a( 0 )) = 1 a δ ( 0 ) This propery illusraes ha he impulse is differen from ordinary mahemaical funcions. 8/25/15 M. J. Robers - All Righs Reserved 9

10 The Uni Periodic Impulse The uni periodic impulse is defined by... δ T!() n= (1) (1) (1) (1) (1) T ( ) = δ nt -2T -T T 2T... ( )..., n an ineger!() T 1-2T -T T 2T... The periodic impulse is a sum of infiniely many impulses uniformly-spaced apar by T. δ ( T a( 0 )) = 1 a δ T /a ( 0 ), n an ineger 8/25/15 M. J. Robers - All Righs Reserved 10

11 The Uni Recangle Funcion Π( ) = rec( ) = 1, 1/ 2 0, > 1/ 2 = u( +1/ 2) u( 1/ 2), 1 2 Π() = rec() 1 Π() = rec() The produc signal g( )rec( ) can be hough of as he signal g( ) urned on a ime = 1/ 2 and urned back off a ime = +1/ /25/15 M. J. Robers - All Righs Reserved 11

12 Random Signals The sinusoid, exponenial, signum, uni sep, uni ramp, and uni recangle are all deerminisic signals. The erm deerminisic means ha heir values are specified a all imes. Signals ha are no deerminisic are random. The exac values of random signals are unpredicable alhough heir general behavior may be known o some degree. x() 8/25/15 M. J. Robers - All Righs Reserved 12

13 Shifing and Scaling Funcions Ampliude Scaling, g( ) Ag( ) 8/25/15 M. J. Robers - All Righs Reserved 13

14 Shifing and Scaling Funcions Time shifing, 0 8/25/15 M. J. Robers - All Righs Reserved 14

15 Shifing and Scaling Funcions Time scaling, / a 8/25/15 M. J. Robers - All Righs Reserved 15

16 Differeniaion -4 x() 4 dx/d x() dx/d x() dx/d x() 1-1 dx/d /25/15 M. J. Robers - All Righs Reserved 16

17 Inegraion 8/25/15 M. J. Robers - All Righs Reserved 17

18 Even and Odd Signals Even Funcions g Odd Funcions ( ) = g( ) g( ) = g( ) Even Funcion g() Odd Funcion g() Even Funcion g() Odd Funcion g() 8/25/15 M. J. Robers - All Righs Reserved 18

19 Even and Odd Pars of Funcions The even par of a funcion is g e ( ) = g( ) + g( ) 2 ( ) = g( ) g( ) The odd par of a funcion is g o. 2 A funcion whose even par is zero is odd and a funcion whose odd par is zero is even. The derivaive of an even funcion is odd and he derivaive of an odd funcion is even. The inegral of an even funcion is an odd funcion, plus a consan, and he inegral of an odd funcion is even.. 8/25/15 M. J. Robers - All Righs Reserved 19

20 Inegrals of Even and Odd Funcions a a Even Funcion g() Area #1 g( )d a = 2 g( )d g( )d = 0 0 Area #2 -a a Area #1 = Area #2 The inegral of an odd funcion, over limis ha are symmerical abou zero, is zero. -a Odd Funcion g() Area #1 a a Area #2 Area #1 = - Area #2 a 8/25/15 M. J. Robers - All Righs Reserved 20

21 Periodic Signals If a funcion g() is periodic, g( ) = g( + nt )where n is any ineger and T is a period of he funcion. The minimum posiive value of T for which g ( ) = g( + T ) is called he fundamenal period T 0 of he funcion. The reciprocal of he fundamenal period is he fundamenal frequency f 0 = 1/ T x() T x() A funcion ha is no periodic is aperiodic. T T x() /25/15 M. J. Robers - All Righs Reserved 21

22 Signal Energy and Power The signal energy of a signal x( ) is E x = x( ) 2 d 8/25/15 M. J. Robers - All Righs Reserved 22

23 Signal Energy and Power Some signals have infinie signal energy. In ha case i is more convenien o deal wih average signal power. The average signal power of a signal x For a periodic signal x 1 P x = lim T T T /2 T /2 P x = 1 T T where T is any period of he signal. ( ) is x( ) 2 d ( ) he average signal power is x( ) 2 d 8/25/15 M. J. Robers - All Righs Reserved 23

24 Signal Energy and Power A signal wih finie signal energy is called an energy signal. A signal wih infinie signal energy and finie average signal power is called a power signal. 8/25/15 M. J. Robers - All Righs Reserved 24

25 Sampling and Discree Time Sampling is he acquisiion of he values of a coninuous-ime signal a discree poins in ime. x discree-ime signal. x n = x nt s ( ) is a coninuous-ime signal, x n ( ) where T s is he ime beween samples is a Sampling Uniform Sampling x() x n x [ ] () x[ n] ω s or f s 8/25/15 M. J. Robers - All Righs Reserved 25

26 Sampling and Discree Time 8/25/15 M. J. Robers - All Righs Reserved 26

27 Exponenials The form of he exponenial is [ ] = Aα n x n Preferred or x n [ ] = Ae βn where α = e β 0 < z < 1 Real α -1 < z < 0 n Re(g[n]) Complex α z < 1 Im(g[n]) n n n z > 1 z < -1 Re(g[n]) z > 1 Im(g[n]) n n n n 8/25/15 M. J. Robers - All Righs Reserved 27

28 The Uni Impulse Funcion δ[n] 1 n [ ] = 1, n = 0 δ n 0, n 0 The discree-ime uni impulse (also known as he Kronecker dela funcion ) is a funcion in he ordinary sense (in conras wih he coninuous-ime uni impulse). I has a sampling propery, [ ] Aδ [ n n 0 ]x n = Ax n 0 n= bu no scaling propery. Tha is, δ n [ ] [ ] = δ [ an] for any non-zero, finie ineger a. 8/25/15 M. J. Robers - All Righs Reserved 28

29 The Uni Sequence Funcion [ ] = 1, n 0 u n 0, n < 0 u[n] n 8/25/15 M. J. Robers - All Righs Reserved 29

30 The Signum Funcion 1, n > 0 sgn n = 0, n = 0 1, n < 0 = 2u n δ n 1 sgn[n] n 8/25/15 M. J. Robers - All Righs Reserved 30

31 The Uni Ramp Funcion ramp n [ ] = n, n 0 0, n < 0 = nu n ramp[n] n m= [ ] = u[ m 1] n 8/25/15 M. J. Robers - All Righs Reserved 31

32 The Periodic Impulse Funcion δ N m= [ n] = δ n mn [ ]... [n]! N -N N 2N... n 8/25/15 M. J. Robers - All Righs Reserved 32

33 Scaling and Shifing Funcions Time shifing n n + n 0, n 0 an ineger 8/25/15 M. J. Robers - All Righs Reserved 33

34 Scaling and Shifing Funcions Time compression n Kn K an ineger > 1 8/25/15 M. J. Robers - All Righs Reserved 34

35 Scaling and Shifing Funcions Time expansion n n / K, K > 1 For all n such ha n / K is an ineger, g[ n / K ] is defined. For all n such ha n / K is no an ineger, g[ n / K ] is no defined. 8/25/15 M. J. Robers - All Righs Reserved 35

36 Scaling and Shifing Funcions There is a form of ime expansion ha is useful. Le [ ] = x [ n / m ], n / m an ineger y n 0, oherwise All values of y are defined. This ype of ime expansion is acually used in some digial signal processing operaions. 8/25/15 M. J. Robers - All Righs Reserved 36

37 Differencing 8/25/15 M. J. Robers - All Righs Reserved 37

38 Accumulaion [ ] = h[ m] g n n m= h[n] g[n] n n h[n] g[n] n n 8/25/15 M. J. Robers - All Righs Reserved 38

39 Even and Odd Signals g[ n] = g[ n] g[ n] = g[ n] Even Funcion g[n] Odd Funcion g[n] n n g e [ n] = g [ n ] + g[ n] 2 g o [ n] = g [ n ] g[ n] 2 8/25/15 M. J. Robers - All Righs Reserved 39

40 Symmeric Finie Summaion... Even Funcion g[n] Sum #1 Sum #2 -N N Sum #1 = Sum #2... n... Odd Funcion Sum #1 -N g[n] N Sum #2 Sum #1 = - Sum #2... n N [ ] g n = g 0 n= N N [ ] + 2 g[ n] g n = 0 n=1 n= N [ ] 8/25/15 M. J. Robers - All Righs Reserved 40 N

41 Periodic Funcions A periodic funcion is one ha is invarian o he change of variable n n + mn where N is a period of he funcion and m is any ineger. The minimum posiive ineger value of N for which [ ] = g[ n + N ] is called he fundamenal period N 0. g n 8/25/15 M. J. Robers - All Righs Reserved 41

42 Signal Energy and Power The signal energy of a signal x[ n] is E x = n= x[ n] 2 8/25/15 M. J. Robers - All Righs Reserved 42

43 Signal Energy and Power Some signals have infinie signal energy. In ha case I is usually more convenien o deal wih average signal power. The average signal power of a signal x n 1 P x = lim N 2N For a periodic signal x n P x = 1 N N 1 n= N x[ n] 2 [ ] is [ ] he average signal power is n= N x[ n] 2 The noaion means he sum over any se of n= N consecuive n's exacly N in lengh. 8/25/15 M. J. Robers - All Righs Reserved 43

44 Signal Energy and Power A signal wih finie signal energy is called an energy signal. A signal wih infinie signal energy and finie average signal power is called a power signal. 8/25/15 M. J. Robers - All Righs Reserved 44

45 Lineariy and LTI Sysems If a sysem is boh homogeneous and addiive i is linear. If a sysem is boh linear and ime-invarian i is called an LTI sysem Some sysems ha are non-linear can be accuraely approximaed for analyical purposes by a linear sysem for small exciaions 8/25/15 M. J. Robers - All Righs Reserved 45

46 Response of LTI Sysems An LTI sysem is compleely characerized by is impulse response h( ). The response y( )of an LTI sysem o an exciaion x convoluion of x( ) wih h( ). ( ) is he y( ) = x( ) h ( ) = x λ ( )h( λ)dλ 8/25/15 M. J. Robers - All Righs Reserved 46

47 If g Convoluion Inegral Properies ( ) = g 0 ( ) = x If y Commuaiviy Associaiviy x( ) y Disribuiviy ( ) = Ax( 0 ) x( ) Aδ 0 ( ) δ ( ) hen g( 0 ) = g 0 ( 0 ) δ ( ) = g 0 ( ) δ 0 ( ) h( ) hen y ( ) = x ( ) h( ) = x( ) h ( ) and y( a) = a x( a) h( a) x x( ) y( ) = y( ) x( ) ( ) ( ) + y( ) z( ) = x ( ) y( ) z ( ) z( ) = x( ) z( ) + y( ) z( ) ( ) 8/25/15 M. J. Robers - All Righs Reserved 47

48 The Uni Triangle Funcion ri ( ) = 1, < 1 0, 1 The uni riangle, is he convoluion of a uni recangle wih Iself. 8/25/15 * M. J. Robers - All Righs Reserved 48

49 Sysems Described by Differenial Equaions The ransfer funcion: H( s) = b s M M + b M 1 s M 1 +!+ b 2 s 2 + b 1 s + b 0 a N s N + a N 1 s N 1 +!+ a 2 s 2 + a 1 s + a 0 This ype of funcion is called a raional funcion because i is a raio of polynomials in s. The ransfer funcion encapsulaes all he sysem characerisics and is of grea imporance in signal and sysem analysis. 8/25/15 M. J. Robers - All Righs Reserved 49

50 If he exciaion x( ) is a phasor or complex sinusoid of frequency f 0, of he form hen he response y x( ) = A x e jφ x e j 2π f 0 ( ) is of he form ( ) = H( f 0 )x( ) = H f 0 y The response can also be wrien in he form ( ) = A y e jφ y e j 2π f 0 y Response of LTI Sysems where A y = H f 0 Applying his o real sinusoids, if x y( ) = A y cos( 2π f 0 + φ y ). ( ) A x e jφ x e j 2π f 0. ( ) A x and φ y = φ x +!H( f 0 ). ( ) hen ( ) = A x cos 2π f 0 + φ x 8/25/15 M. J. Robers - All Righs Reserved 50

51 The Convoluion Sum The response y n o an arbirary exciaion x n [ ] of an LTI sysem wih impulse response h[ n] [ ] is [ ] = x m y n m= [ ]h n m [ ] 8/25/15 M. J. Robers - All Righs Reserved 51

52 Convoluion Sum Properies x[ n] Aδ [ n n ] 0 = Ax[ n n ] 0 Le y[ n] = x[ n] h[ n] hen y[ n n ] 0 = x[ n] h[ n n ] 0 = x[ n n 0 ] h[ n] y[ n] y[ n 1] = x[ n] ( h[ n] h[ n 1] ) = ( x[ n] x[ n 1] ) h n and he sum of he impulse srenghs in y is he produc of he sum of he impulse srenghs in x and he sum of he impulse srenghs in h. [ ] 8/25/15 M. J. Robers - All Righs Reserved 52

53 Sysems Described by Difference The ransfer funcion is H( z) = or, alernaely, H( z) = M k=0 N k=0 M k=0 N k=0 b k z k a k z k b k z k a k z k = b 0 + b 1 z 1 + b 2 z 2 +!+ b M z M a 0 + a 1 z 1 + a 2 z 2 +!+ a N z N = z N M b 0 z M + b 1 z M 1 +!+ b M 1 z + b M a 0 z N + a 1 z N 1 +!+ a N 1 z + a N The ransfer funcion can be wrien direcly from he sysem difference equaion and vice versa. H e jω ( ) is he sysem's frequency response. I is he ransfer funcion H z replaced by e jω. Equaions ( ) wih z 8/25/15 M. J. Robers - All Righs Reserved 53

54 Coninuous-Time Fourier Series Definiion ( ) = X n e j 2πn/T x and X n = 1 T n= 0 +T x( )e j 2πn/T d. The signal and is harmonic funcion form a Fourier series pair x FS ( ) X n where T is he represenaion ime and, T herefore, he fundamenal period of he coninuous-ime Fourier series ( CTFS) represenaion of x( ). If T is also a period of x( ), he CTFS represenaion of x 0 ( ) is valid for all ime. This is, by far, he mos common use of he CTFS in engineering applicaions. If T is no a period of x( ), he CTFS represenaion is generally valid only in he inerval 0 < 0 + T. 8/25/15 M. J. Robers - All Righs Reserved 54

55 CTFS of a Real Funcion I can be shown ha he coninuous-ime Fourier series (CTFS) harmonic funcion of any real-valued funcion x ha X n = X * n. ( ) has he propery One implicaion of his fac is ha, for real-valued funcions, he magniudes of heir harmonic funcions are even funcions and heir phases can be expressed as odd funcions of harmonic number k. 8/25/15 M. J. Robers - All Righs Reserved 55

56 The Sinc Funcion Le x( ) = Arec( / w) δ T0 ( ), w < T 0. Then FS x( ) = Arec( / w) δ T0 ( ) The mahemaical form sin( π x) π x ( ) T 0 X n = A sin πnw / T 0 πn arises frequenly enough o be given is own name, "sinc". Tha is sinc( ) = sin ( π ). π 8/25/15 M. J. Robers - All Righs Reserved 56

57 The Uniqueness Propery If we find a Fourier series represenaion of a signal, i is unique. Tha is, no oher or alernae Fourier series represenaion exiss. Example: Le x ( ) = 3cos( 8π π / 4) + 4sin( 4π) Using rigonomeric ideniies, his can be rewrien as x( ) = 3 cos( 8π)cos( π / 4) sin( 8π)sin( π / 4) + 4sin 4π x( ) = cos( 8π) sin( 8π) + 4sin( 4π) ( ) 8/25/15 M. J. Robers - All Righs Reserved 57

58 The Uniqueness Propery x( ) = x( ) = e j 8π + e j 8π 2 e j 8π e j 8π j2 ( 1+ j)e j 8π + ( 1 j)e j 8π + 4 e j 4π e j 4π j2 j2( e j 4π e j 4π ) x( ) = 3 2 ( 1+ j)e j 8π ( 1 j)e j 8π j2e j 4π + j2e j 4π 4 4 This is THE ( complex)ctfs represenaion of x( ) in which ( ) = X n e j 2πnf 0 x, f 0 = 2, X 2 = n= X 2 = ( 1 j), X 1 = j2, X 1 = j2, ( 1+ j) and all oher CTFS coefficiens are zero. 8/25/15 M. J. Robers - All Righs Reserved 58

59 T [ ], T arbirary FS ( ( ) 1/ T 0 ), n / m an ineger FS 1 δ n δ T0 Some Common CTFS Pairs e j 2πq/T 0 mt 0 0, oherwise FS mt 0 δ [ n mq] FS ( ) mt 0 ( j / 2) δ n + mq FS ( ) ( 1/ 2) δ n mq sin 2πq / T 0 cos 2πq / T 0 rec( / w) δ T0 ri( / w) δ T0 ( [ ] δ [ n mq] ) ( mt 0 [ ]+ δ [ n + mq] ) FS ( ) mt 0 ( w / T 0 )sinc( wn / mt 0 )δ m n mt 0 ( w / T 0 )sinc 2 ( wn / mt 0 )δ m n ( m an ineger) ( ) FS [ ] [ ] 8/25/15 M. J. Robers - All Righs Reserved 59

60 Definiion of he CTFT ( ) = F x( ) X f ( ) = F x( ) X jω Forward f form Inverse ( ) = x ( )e j 2π f d x ( ) = F -1 X( f ) Forward ω form Inverse ( ) = x ( )e jω d x Commonly-used noaion: ( ) F x X( f ) or x ( ) = F -1 X( jω ) ( ) F ( ) = X f ( ) = 1 X( jω ) 2π ( )e + j 2π f df X( jω )e + jω dω 8/25/15 M. J. Robers - All Righs Reserved 60

61 Some CTFT Pairs e α u e α u n e α u ( ) F ( ) F 1/ jω + α 1/ jω + α ( ) F F δ ( ) 1 ( ), α > 0 e α F u( ) ( ) 2, α > 0 e α F u( ) n!, α > 0 n e α u ( jω + α ) n+1 1/ jω + α 1/ jω + α ( ) F ( ), α < 0 ( ) 2, α < 0 n!, α < 0 ( jω + α ) n+1 e α sin( ω 0 )u e α cos( ω 0 )u ( ) F ( ) F ω 0 ( jω + α ) ω 0 jω + α ( jω + α ) ω 0, α > 0 e α sin( ω 0 )u, α > 0 e α cos( ω 0 )u e α F 2α, α > 0 ω 2 + α 2 ( ) F ( ) F ω 0 ( jω + α ) ω 0 jω + α ( jω + α ) ω 0, α < 0, α < 0 8/25/15 M. J. Robers - All Righs Reserved 61

62 More CTFT Pairs δ ( ) F sgn( ) F rec ( ) F ri( ) F δ T0 ( ) F ( ) F cos 2π f 0 F 1 1 δ f F 1/ jπ f u( ) ( 1/ 2)δ f sinc f ( ) sinc( ) F ( ) sinc 2 ( ) F ( ), f 0 = 1/ T 0 T 0 δ T0 ( ) F ( ) δ ( f f 0 ) + δ ( f + f 0 ) sin( 2π f 0 ) F sinc 2 f f 0 δ f0 f 1/ 2 rec f δ f0 f j / 2 ( ) ( ) +1/ j2π f ( ) ri( f ) ( ), T 0 = 1/ f 0 ( ) δ ( f + f 0 ) δ ( f f 0 ) 8/25/15 M. J. Robers - All Righs Reserved 62

63 Numerical Compuaion of he CTFT I can be shown ha he DFT can be used o approximae samples from he CTFT. If he signal x ( ) is a causal energy signal and N samples are aken from i over a finie ime beginning a = 0, a a rae f s hen he relaionship beween he CTFT of x ( ) and he DFT of he samples aken from i is X( kf s / N ) T s e jπ k/n sinc( k / N )X DFT [ k] For hose harmonic numbers k for which k << N ( ) T s X DFT [ k] X kf s / N As he sampling rae and number of samples are increased, his approximaion is improved. 8/25/15 M. J. Robers - All Righs Reserved 63

64 The Discree-Time Fourier Series The discree-ime Fourier series (DTFS) is similar o he CTFS. A periodic discree-ime signal can be expressed as [ ] = c x k x n [ ]e j 2πkn/N c x k k= N [ ] = 1 N n 0 +N 1 n=n 0 x[ n]e j 2πkn/N where c x [ k] is he harmonic funcion, N is any period of x n and he noaion, means a summaion over any range of k= N consecuive k s exacly N in lengh. [ ] 8/25/15 M. J. Robers - All Righs Reserved 64

65 The Discree Fourier Transform The discree Fourier ransform (DFT) is almos idenical o he DTFS. A periodic discree-ime signal can be expressed as x[ n] = 1 N j 2π kn/n X[ k]e X k k= N n 0 +N 1 n=n 0 [ ] = x n [ ]e j 2π kn/n where X[ k] is he DFT harmonic funcion and N is any period of x[ n]. The main difference beween he DTFS and he DFT is he locaion of he 1/N erm. So X[ k] = N c x [ k]. 8/25/15 M. J. Robers - All Righs Reserved 65

66 The Discree Fourier Transform Because he DTFS and DFT are so similar, and because he DFT is so widely used in digial signal processing (DSP), we will concenrae on he DFT realizing we can always form he DTFS from [ k] = X[ k] / N. c x 8/25/15 M. J. Robers - All Righs Reserved 66

67 The Discree Fourier Transform Noice ha in x[ n] = 1 N k= N j 2π kn/n X[ k]e he summaion is over N values of k, a finie summaion. This is j 2π kn/n because of he periodiciy of he complex sinusoid, e in harmonic number k. If k is increased by any ineger muliple of N he complex sinusoid does no change. e j 2π kn/n j 2π k+mn = e ( )n/n = e j 2π kn/n j 2π mn e =1, m an ineger This occurs because discree ime n is always an ineger. 8/25/15 M. J. Robers - All Righs Reserved 67

68 The Dirichle Funcion The funcional form ( ) ( ) sin π N N sin π appears ofen in discree-ime signal analysis and is given he drcl(,4) drcl(,5) special name Dirichle funcion. Tha is drcl(, N ) = ( ) ( ) sin π N N sin π drcl(,7) drcl(,13) /25/15 M. J. Robers - All Righs Reserved 68

69 Response of LTI Sysems ( ) is X( f ) and he If he Fourier ransform of he exciaion x Fourier ransform of he response y( ) is Y( f ), hen ( ) = H( f ) X( f ) and Y( f ) = H( f ) X( f ) and ( ) =!H( f ) +! X( f ). ( ) is an energy signal (finie signal energy) hen, from Parseval's Y f! Y f If x heorem E x = ( ) 2 df X f and E y = Y f = H f ( ) 2 df ( ) 2 X( f ) 2 df 8/25/15 M. J. Robers - All Righs Reserved 69

70 Signal Disorion in Transmission Disorion means changing he shape of a signal. Two changes o a signal are no considered disorion, muliplying i by a consan and shifing i in ime. The impulse response of an LTI sysem ha does no disor is of he general form h ( ). where K and d are consans. The corresponding ( ) = Kδ d ( ) = Ke j 2π f d. ( ) = K and!h( f ) = 2π f d. If H( f ) K he sysem has ( ) 2π f d he sysem has delay frequency response of such a sysem is H f H f ampliude disorion. If!H f or phase disorion. Boh of hese ypes of disorion are classified as linear disorions. 8/25/15 M. J. Robers - All Righs Reserved 70

71 If!H f if!h f Signal Disorion in Transmission ( ) = 2π f d, hen d =!H( f ) 2π f and d is a consan ( ) = Kf (K a consan). If d is no a consan, H( f ) phase disorion resuls. A Frequency response of a disorionless LTI sysem H( f ) f -2π f = 1 d f 8/25/15 M. J. Robers - All Righs Reserved 71

72 Signal Disorion in Transmission Mos real sysems do no have simple delay. They have phases ha are no linear funcions of frequency. H( f ) H( f ) f f 8/25/15 M. J. Robers - All Righs Reserved 72

73 For a bandpass signal wih a small bandwidh W compared o is cener frequency f c, we can model he frequency response phase variaion as approximaely linear over he frequency ranges f c W < f < f c + W, and he frequency response magniude as approximaely consan, of he form ( ) Ae j 2π f g e jφ 0, f c W < f < f c + W H f Signal Disorion in Transmission where φ 0 =!H( f c ). e jφ 0, f W < f < f + W c c H( f ) 2W f c φ 0 φ 0 f c 2W f 8/25/15 M. J. Robers - All Righs Reserved 73

74 Signal Disorion in Transmission If we now le he bandpass signal be x( ) = x 1 ( )cos 2π f c Is Fourier ransform is ( ) + x 2 ( ) ( )sin 2π f c X( f ) = X 1 ( f ) ( 1/ 2) δ ( f f c ) + δ ( f + f c ) + X 2 ( f ) ( j / 2) δ ( f + f c ) δ ( f f c ) X( f ) = ( 1/ 2) { X 1 ( f f c ) + X 1 ( f + f c ) + j X 2 ( f + f c ) X 2 ( f f c ) } The frequency response is modeled by e H( f ) jφ 0, f Ae j 2π f g c W < f < f c + W e jφ 0, f W < f < f + W c c hen he Fourier ransform of he response y Y( f ) H( f ) X( f ) = A / 2 ( ) X 1 f f c ( ) is ( )e j ( 2π f g φ 0 ) + X1 ( f + f c )e j ( 2π f g +φ 0 ) + j X 2 ( f + f c )e j ( 2π f g +φ 0 ) j X2 ( f f c )e j ( 2π f g φ 0 ) 8/25/15 M. J. Robers - All Righs Reserved 74

75 Y( f ) H( f ) X( f ) = ( A / 2) X 1 ( f f c )e j ( 2π f g φ 0 ) + X1 ( f + f c )e j ( 2π f g +φ 0 ) + j X 2 ( f + f c )e j ( 2π f g +φ 0 ) j X2 ( f f c )e j ( 2π f g φ 0 ) Inverse Fourier ransforming, using he ime and frequency shifing properies, y( ) ( A / 2) e jφ0 x 1 ( g )e j 2π fc + e jφ 0 x 1 ( g )e j 2π f c + je jφ 0 x 2 ( g )e j 2π fc je jφ 0 x 2 ( g )e j 2π f c y y Signal Disorion in Transmission y( ) ( A / 2) x 1 g ( ) A x 1 g ( ) e j 2π f c+φ 0 ( ) j e j 2π f c+φ 0 ( ) + e j( 2π f c +φ 0 ) ( ) + x 2 g e j( 2π f c +φ 0 ) ( )cos( 2π f c + φ 0 ) + x 2 ( g )sin( 2π f c + φ 0 ) ( )cos( 2π f c ( d )) + x 2 ( g )sin 2π f c d { } { ( ( ))} ( ) A x 1 g ( ) where d = φ 0 =!H f c 2π f c 2π f c is known as he phase or carrier delay. 8/25/15 M. J. Robers - All Righs Reserved 75

76 Signal Disorion in Transmission From he approximae form of he sysem frequency response we ge ( ) Ae j 2π f g e jφ 0, f c W < f < f c + W H f e jφ 0, f W < f < f + W c c!h f 2π f g φ 0, f c W < f < f c + W If we differeniae boh sides w.r.. f we ge or d df ( ) 2π f g + φ 0, f c W < f < f c + W (!H( f )) 2π g, f c W < f < f c + W g 1 d 2π df g is known as he group delay. (!H( f )), f c W < f < f c + W 8/25/15 M. J. Robers - All Righs Reserved 76

77 Signal Disorion in Transmission Phase and Group Delay x() 1 Exciaion Modulaed Carrier Modulaion Exciaion -1 y() 1 Response Phase Delay Modulaion Group Delay -1 Modulaed Carrier Response 8/25/15 M. J. Robers - All Righs Reserved 77

78 Signal Disorion in Transmission Linear disorion can be correced (heoreically) by an equalizaion nework. If he communicaion channel's frequency response is H C f and i is followed by an equalizaion nework wih frequency response ( f ) hen he overall frequency response is H( f ) = H C ( f )H eq ( f ) H eq and he overall frequency response will be disorionless if H( f ) = H C ( f )H eq ( f ) = Ke jω d. Therefore, he frequency response of he equalizaion nework should be H eq ( f ) = Ke jω d f H C ( ) ( ). I is very rare in pracice ha his can be done exacly bu in many cases an excellen approximaion can be made ha grealy reduces linear disorion. 8/25/15 M. J. Robers - All Righs Reserved 78

79 Signal Disorion in Transmission Communicaion sysems can also have nonlinear disorion caused by elemens in he sysem ha are saically nonlinear. In ha case he exciaion and response are relaed hrough a ransfer characerisic of he form y( ) = T( x( ) ). For example, some amplifiers experience a "sof" sauraion in which he raio of he response o he exciaion decreases wih an increase in he exciaion level. y() x() 8/25/15 M. J. Robers - All Righs Reserved 79

80 Signal Disorion in Transmission The ransfer characerisic is usually no a simple known funcion bu can ofen be closely approximaed by a polynomial curve fi of he form y ransform of y ( ) = a 1 x( ) + a 2 x 2 ( ) + a 3 x 3 ( ) +!. The Fourier ( ) is Y( f ) = a 1 X( f ) + a 2 X( f ) X( f ) + a 3 X( f ) X( f ) X( f ) +! In a linear sysem if he exciaion is bandlimied, he response has he same band limis. The response canno conain frequencies no presen in he exciaion. Bu in a nonlinear sysem of his ype if X( f ) conains a range of frequencies, X( f ) X( f ) conains a greaer range of frequencies and X( f ) X( f ) X f a sill greaer range of frequencies. ( ) conains 8/25/15 M. J. Robers - All Righs Reserved 80

81 Signal Disorion in Transmission If X( f ) X f ( ) conains frequencies ha are all ouside he range of ( ) hen a filer can be used o eliminae hem. Bu ofen X f X f ( ) X f conains frequencies boh inside and ouside ha range, and hose inside he range canno be filered ou wihou affecing he specrum of X( f ). As a simple example of he kind of nonlinear disorion ha can occur le x( ) = A 1 cos ω 1 ( ) = x 2 ( ). Then ( ) = A 1 cos ω 1 y ( ) + A 2 cos( ω 2 ) and le y ( ) + A 2 cos( ω 2 ) ( ) + A 2 2 cos 2 ( ω 2 ) + 2A 1 A 2 cos ω 1 ( ) + A 2 2 / 2 ( ) 2 = A 1 2 cos 2 ω 1 ( ) 1+ cos 2ω 1 ( ) 1+ cos 2ω 2 + A 1 A 2 cos( ( ω 1 ω 2 )) + cos( ( ω 1 + ω 2 )) = A 1 2 / 2 ( )cos ω 2 ( ) ( ) 8/25/15 M. J. Robers - All Righs Reserved 81

82 Signal Disorion in Transmission ( ) 1+ cos( 2ω 1 ) + ( A 2 2 / 2) 1+ cos 2ω 2 (( )) + cos( ( ω 1 + ω 2 )) y( ) = A 2 1 / 2 + A 1 A 2 cos ω 1 ω 2 ( ) ( ) conains frequencies 2ω 1, 2ω 2, ω 1 ω 2 and ω 1 + ω 2. The y frequencies ω 1 ω 2 and ω 1 + ω 2 are called inermodulaion disorion producs. When he exciaion conains more frequencies (which i usually does) and he nonlineariy is of higher order (which i ofen is), many more inermodulaion disorion producs occur. All sysems have nonlineariies and inermodulaion disorion will occur. Bu, by careful design, i can ofen be reduced o a negligible level. 8/25/15 M. J. Robers - All Righs Reserved 82

83 Transmission Loss and Decibels Communicaion sysems affec he power of a signal. If he signal power a he inpu is P in and he signal power a he oupu is P ou, he power gain g of he sysem is g = P ou / P in. I is very common o express his gain in decibels. A decibel is one-enh of a bel, a uni named in honor of Alexander Graham Bell. The sysem gain g expressed in decibels would be g db = 10 log 10 ( P ou / P in ). g , , 000 g db Because gains expressed in db are logarihmic, hey compress he range of numbers. If wo sysems are cascaded, he overall power gain is he produc of he wo individual power gains g = g 1 g 2. The overall power gain expressed in db is he sum of he wo power gains expressed in db, g db = g 1,dB + g 2,dB. 8/25/15 M. J. Robers - All Righs Reserved 83

84 Transmission Loss and Decibels The decibel was defined based on a power raio, bu i is ofen used o indicae he power of a single signal. Two common ypes of power indicaion of his ype are dbw and dbm. dbw is he power of a signal wih reference o one wa. Tha is, a one wa signal would have a power expressed in dbw of 0 dbw. dbm is he power of a signal wih reference o one milliwa. A 20 mw signal would have a power expressed in dbm of dbm. Signal power gain as a funcion of frequency is he square of he magniude of frequency response H( f ) 2. Frequency response magniude is ofen expressed in db also. H f ( ) db = 10log 10 H( f ) 2 ( ) = 20 log 10 H f ( ) ( ). 8/25/15 M. J. Robers - All Righs Reserved 84

85 Transmission Loss and Decibels A communicaion sysem generally consiss of componens ha amplify a signal and componens ha aenuae a signal. Any cable, opical or copper, aenuaes he signal as i propagaes. Also here are noise processes in all cables and amplifiers ha generae random noise. If he power level ges oo low, he signal power becomes comparable o he noise power and he fideliy of analog signals is degraded oo far or he deecion probabiliy for digial signals becomes oo low. So, before ha signal level is reached, we mus boos he signal power back up o ransmi i furher. Amplifiers used for his purpose are called repeaers. 8/25/15 M. J. Robers - All Righs Reserved 85

86 Transmission Loss and Decibels On a signal cable of 100's or 1000's of kilomeers many repeaers will be needed. How many are needed depends on he aenuaion per kilomeer of he cable and he power gains of he repeaers. Aenuaion will be symbolized by L = 1/ g = P in / P ou or L db = g db = 10 log10 P in / P ou ( ), (L for "loss".) For opical and copper cables he aenuaion is ypically exponenial and P ou = 10 αl/10 P in where l is he lengh of he cable and α is he aenuaion coefficien in db/uni lengh. Then L = 10 αl/10 and L db = αl. 8/25/15 M. J. Robers - All Righs Reserved 86

87 Filers and Filering An ideal bandpass filer has he frequency response H( f ) = Ke jω d, f l f f h 0, oherwise where f l is he lower cuoff frequency and f h is he upper cuoff frequency and K and d are consans. The filer's bandwidh is B = f h f l. An ideal lowpass filer has he same frequency response bu wih f l = 0 and B = f h. An ideal highpass filer has he same frequency response bu wih f h and B. These filers are called ideal because hey canno acually be buil. They canno be buil because hey are non-causal. Bu hey are useful ficions for inroducing in a simplified way some of he conceps of communicaion sysems. 8/25/15 M. J. Robers - All Righs Reserved 87

88 Filers and Filering Sricly speaking a signal canno be boh bandlimied and imelimied. Bu many signals are almos bandlimied and imelimied. Tha is, many signals have very lile signal energy ouside a defined bandwidh and, a he same ime, very lile signal energy ouside a defined ime range. A good example of his is a Gaussian pulse ( ) = e π 2 F x X( f ) = e π f 2 Sricly speaking, his signal is no bandlimied or imelimied. The oal signal energy of his signal is 1/ 2. 99% of is energy lies in he ime range 0.74 < < 0.74 and in he frequency range 0.74 < f < So in many pracical calculaions his signal could be considered boh bandlimied and imelimied wih very lile error. 8/25/15 M. J. Robers - All Righs Reserved 88

89 Filers and Filering Real filers canno have consan ampliude response and linear phase response in heir passbands like ideal filers. H( f ) Passband Ripple Sop Band Pass Band Sop Band f H( f ) db Transiion Bands Minimum Sopband Aenuaion Sop Band f 8/25/15 M. J. Robers - All Righs Reserved 89

90 Filers and Filering There are many ypes of sandardized filers. One very common and useful one is he Buerworh filer. The frequency response of a lowpass Buerworh filer is of he form H( f ) = 1 ( ) 2n where 1+ f / B n is he order of he filer. As he order is increased, is magniude response approaches ha of an ideal filer, consan in he passband and zero ouside he passband. (Below is illusraed he magniude frequency response of a normalized lowpass Buerworh filer wih a corner frequency of 1 radian/s.) H (jω) a n = 2 n = 1 n = 8 n = ω 8/25/15 M. J. Robers - All Righs Reserved 90

91 Filers and Filering The Buerworh filer is said o be maximally fla in is passband. I is given his descripion because he firs n derivaives of is magniude frequency response are all zero a f = 0 (for a lowpass filer). The passband of a lowpass Buerworh filer is defined as he frequency a which is magniude frequency response is reduced from is maximum by a facor of 1/ 2. This is also known as is half-power bandwidh because, a his frequency he power gain of he filer is half is maximum value. 1 H (jω) a 1 2 n = 2 n = 1 n = 8 n = ω 8/25/15 M. J. Robers - All Righs Reserved 91

92 The sep response of a filer is (g h 1 Filers and Filering ( ) = h( λ)u( λ)dλ = h( λ)dλ ( ) in he book). Tha is, he sep response is he inegral of he impulse response. The impulse response of a uniy-gain ideal lowpass filer wih no delay is h ( ) = 2Bsinc( 2B) where B is is bandwidh. Is sep response is herefore 0 h 1 ( ) = 2Bsinc( 2Bλ)dλ = 2B sinc( 2Bλ)dλ + sinc( 2Bλ)dλ 0 This resul can be furher simplified by using he definiion of he sine inegral funcion θ ( )! sin( α ) Si θ α dα = π sinc( λ)dλ 0 θ /π 0 8/25/15 M. J. Robers - All Righs Reserved 92

93 Filers and Filering The Sine Inegral Funcion π/ Si() π/ /25/15 M. J. Robers - All Righs Reserved 93

94 Filers and Filering h 1 0 ( ) = 2B sinc( 2Bλ)dλ + sinc( 2Bλ)dλ 0 Le 2Bλ = α. Then h 1 ( ) = sinc( α )dα + sinc( α )dα. Using he fac ha sinc is an even funcion, Then, using Si θ h 1 0 ( ) = π sinc α ( ) ( ) = Si π θ /π 2 B 0 0 sinc( α )dα = sinc( α )dα. ( )dα and Si 0 ( ) = π / 2,we ge + 1 Si ( 2π B ) π = Si ( 2π B ) π 0 8/25/15 M. J. Robers - All Righs Reserved 94

95 Filers and Filering h 1 ( ) = Si ( 2π B ) π This sep response has precursors, overshoo, and oscillaions (ringing). Riseime is defined as he ime required o move from 10% of he final value o 90% of he final value. For his ideal lowpass filer he rise ime is 0.44/B. The rise ime for a single-pole, lowpass filer is 0.35/B Sep response of an Ideal Lowpass Filer wih B = 1 h -1 () /25/15 M. J. Robers - All Righs Reserved 95

96 Filers and Filering The response of an ideal lowpass filer o a recangular pulse of widh τ is y( ) = h 1 ( ) h 1 ( τ ) = 1 π ( ) ( ) Si 2π B ( τ ) Si 2π B. From he graph (in which B = 1) we see ha, o reproduce he recangular pulse shape, even very crudely, requires a bandwidh much greaer han 1/τ. If we have a pulse rain wih pulse widhs τ and spaces beween pulses also τ and we wan o simply deec wheher or no a pulse is presen a some ime, we will need a leas B 1/ 2τ. If he bandwidh is any lower he overlap beween pulses makes hem very hard o resolve. y() τ = 1/4 τ = 1/2 τ = /25/15 M. J. Robers - All Righs Reserved 96

97 Pulse Widh and Bandwidh Pulses (and heir Fourier ransforms) can have many shapes Infinie pulse widh and bandwidh Finie pulse widh and infinie bandwidh Infinie pulse widh and finie bandwidh x() x() x() (s) (s) (s) X( f ) X( f ) X( f ) f (Hz) f (Hz) f (Hz) 8/25/15 M. J. Robers - All Righs Reserved 97

98 Pulse Widh and Bandwidh We need a pracical general relaionship beween pulse widh and bandwidh. x() X( f ) Equal Areas x(0) Equal Areas X(0) T 2 T 2 Le he recangular pulse approximae he general pulse wih he same heigh and area. Then T x 0 ( ) = x( ) d W x( )d = X( 0). Le he recangular bandwidh approximae he general pulse bandwidh wih he same heigh and area. Then 2W X 0 W ( ) = X( f ) df X( f )df = x( 0). 8/25/15 M. J. Robers - All Righs Reserved 98 f

99 Pulse Widh and Bandwidh Now we have he relaionships ( ) ( ) 1 T x 0 X 0 ( ) ( ) x 0 and 2W X 0 which combine o 2W 1 T or W 1. This is a handy, pracical 2T rule of humb for he approximae bandwidh of a pulse. 8/25/15 M. J. Robers - All Righs Reserved 99

100 Quadraure Filers and Hilber Transforms A quadraure filer is an allpass nework ha shifs he phase of posiive frequency componens by 90 and negaive frequency componens by Is frequency response is herefore H Q ( f ) = j, f > 0 j, f < 0 = j sgn( f ). Is magniude is one a all frequencies, herefore an even funcion of f and is phase is an odd funcion of f. The inverse Fourier ransform of H Q ( f ) is he impulse response h Q ( ) = 1 / π. The Hilber ransform ˆx ( ) of a signal x( ) is defined as he response of a quadraure filer o x( ). Tha is ˆx ( ) = x( ) h Q ( ) = 1 x( λ) π F ( ˆx ( ) ) = j sgn( f ) X( f ) λ dλ. 8/25/15 M. J. Robers - All Righs Reserved 100

101 Quadraure Filers and Hilber Transforms The impulse response of a quadraure filer h Q ( ) = 1 / π is non-causal. Tha means i is physically unrealizable. Some imporan properies of he Hilber ransform are 1. The Fourier ransforms of a signal and is Hilber ransform have he same magniude. Therefore he signal and is Hilber ransform have he same signal energy. ( ) is he Hilber ransform of x( ) hen x( ) is he Hilber 2. If ˆx ransform of ˆx ( ). 3. A signal x ( ) and is Hilber ransform are orhogonal on he enire real line. Tha means for energy signals 1 power signals lim T 2T T x( ) ˆx ( )d = 0. T x( ) ˆx ( )d = 0 and for 8/25/15 M. J. Robers - All Righs Reserved 101

102 Quadraure Filers and Hilber Transforms a 1 g 1 g( ) ĝ( ) ( ) + a 2 g 2 ( ); a 1,a 2! a 1 ĝ 1 ( ) + a 2 ĝ 2 ( ) h( 0 ) ĥ( 0 ) h( a);a 0 sgn( a)ĥ( d d ( h( ) ) δ ( ) e j e j cos d d ( ĥ( ) ) 1 π je j je j ( ) sin( ) rec ( ) ln π 2 1 sinc( ) ( π / 2)sinc 2 ( / 2) = sin( π / 2)sinc( / 2) /25/15 M. J. Robers - All Righs Reserved 102

103 Analyic Signals and Complex Envelopes An analyic signal x p ( ) corresponding o a real signal x( ) is defined by x p ( ) = x( ) + j ˆx ( ). The envelope of a signal x( )is defined as he magniude of he analyic signal x p ( ). I follows ha ( f ) = X f X p ( ) + j j ( )sgn( f )X f ( ), f > 0 ( ) = X( f ) 1+ sgn( f ) Therefore X p ( f ) = 2 X f. Similarly, 0, f < 0 x n = 2 X( f )u( f ) ( ) = x( ) j ˆx ( ) and X n ( f ) = 2 X( f )u( f ) = 0, f > 0 2 X( f ), f < 0. 8/25/15 M. J. Robers - All Righs Reserved 103

104 Analyic Signals and Complex Envelopes ( ) is defined as The complex envelope of a real signal x!x ( ) = x p ( )e j 2π f 0 where f 0 is a reference frequency chosen for convenience. Therefore x p and x( ) = Re!x ( )e j 2π f 0 ( ) = Re!x ( ) cos 2π f 0 x ( ) =!x ( )e j 2π f 0 ( ) and ˆx ( ( ) + j sin( 2π f 0 ) ) ( ) = Im!x ( ) = x( ) + j ˆx ( ) ( )e j 2π f 0 ( ). Re (!x ( ) )cos( 2π f 0 ) + j Im (!x ( ) )cos( 2π f 0 ) x( ) = Re + jre (!x ( ) )sin( 2π f 0 ) + " j j Im (!x ( ) )sin 2π f 0 = 1 x( ) = x R ( )cos( 2π f 0 ) x I ( )sin( 2π f 0 ) where x R ( ) = Re (!x ( ) ) and x I ( ) = Im (!x ( ) ),!x ( ) ( ) = x R ( ) + j x I ( ), x R ( ) is he "in-phase" componen of x( ) ( ) is he "quadraure" componen of x( ). and x I 8/25/15 M. J. Robers - All Righs Reserved 104

105 Analyic Signals and Complex Envelopes I can be shown (page 89 in he ex) ha if a sysem has a ( ) and i is bandpass response wih impulse response h excied by a bandpass signal x( ), ha he complex envelope of he sysem response is!y ( ) =!x ( ) h! ( ) = F 1 X! ( f ) H! ( f ) and he sysem response is y ( ) = 1 2 Re!y ( ) ( ( )e j 2π f 0 ). (The erm "bandpass" means ha here is a finie-widh band of frequencies, including f = 0, in which he Fourier magniude specrum is zero or, as a pracical maer, small enough o be considered negligible.) 8/25/15 M. J. Robers - All Righs Reserved 105

106 Analyic Signals and Complex Envelopes In he previous wo slides a real signal x( )was relaed o is complex envelope!x ( ) by x( ) = Re!x ( )e j 2π f 0 response h ( ) = Re! h h ( ) and a real sysem impulse ( ) was relaed o is complex envelope h! ( ) by ( ( )e j 2π f 0 ). Bu hen whenx ( ) is applied o he sysem and he response is y( ), we found!y ( ) =!x ( ) h! ( ) and relaed i o y( ) by y( ) = 1 2 Re!y ( ( )e j 2π f 0 ). Where did he facor of 1 2 come from? I can be seen in he derivaion on page 89. Bu i can also be seen in concep by looking a wha happens when we convolve a bandpass signal and a bandpass impulse response and compare ha o convolving he corresponding complex envelopes. 8/25/15 M. J. Robers - All Righs Reserved 106

107 Le x Analyic Signals and Complex Envelopes ( ) = Π( )sin( 8π) and h( ) = Π( )sin 8π ( ) has wice he signal energy of x envelope of x rue for h( ). As a resul, he complex envelope of y ( ) has four imes he signal energy of y( ). x() ( ). The complex ( ). The same is Bandpass Signals and Impulse Response Complex Envelopes h() ~ x() ~ h() y() 0 ~ y() /25/15 M. J. Robers - All Righs Reserved 107

108 Analyic Signals and Complex Envelopes Example Le x ( ) = sinc Then X( f ) = rec f X f ( )cos( 4π). ( ) 1 2 δ f 2 ( ) + δ ( f + 2) ( ) = 1 2 rec( f 2) + rec( f + 2) and ˆX ( f ) = j sgn( f )X( f ). ˆX ( f ) = j 2 rec( f 2) + j 2 rec( f + 2) = rec( f ) j 2 δ f + 2 ˆx ( ) = sinc( )sin( 4π). ( ) δ ( f 2) 8/25/15 M. J. Robers - All Righs Reserved 108

109 Analyic Signals and Complex Envelopes x p ( ) = sinc( )cos( 4π) ˆx ( ) = sinc ( ) = sinc( ) cos( 4π) + j sin( 4π) x x p ( )sin 4π ( ) ( ) is he envelope of x( ). The concep of an envelope will be very useful laer in he exploraion of modulaion echniques x() x() x() (s) (s) x() 0 x p () (s) (s) 8/25/15 M. J. Robers - All Righs Reserved 109

110 Example 2.32 in he ex : Le x ( ) = Π τ cos 2π f 0 ( ) and le h Find he sysem oupu signal y x p x p ( ) = x( ) + j ˆx ( ) = Π τ ( ) = Π τ e j 2π f 0!x ( ) = αe α u ( )cos( 2π f 0 ). ( ) using complex envelope echniques. cos 2π f 0 ( ) = Re x p ( ) + jπ τ ( )e j 2π f 0 sin 2π f 0 ( ) = Π τ!h ( ) = αe α u( ). Therefore!y ( ) =!x ( ) h! ( ) = Π τ ( ) = Π λ τ!y Analyic Signals and Complex Envelopes αe ( ) u( λ)dλ α λ ( ). Similarly, αe α u 8/25/15 M. J. Robers - All Righs Reserved 110 ( )

111 ( ) = α!y!y α λ u( λ + τ / 2)e ( ) u( λ)dλ α λ u( λ τ / 2)e ( ) u( λ)dλ ( α ( +τ /2 ) α τ /2 )u( + τ / 2) ( 1 e ( ) )u τ / 2 ( ) = 1 e ( ) = 1 2 Re!y y Analyic Signals and Complex Envelopes y( ) = 1 2 y( ) = 1 2 ( )e j 2π f 0 ( ) ( ) ( ) Re (( +τ /2 1 ( ) e α )u( + τ / 2) ( α τ /2 1 e ( ) )u( τ / 2) )e j 2π f 0 +τ /2 ( 1 ( ) e α )u + τ / 2 ( ) 1 e ( α ( τ /2 ) )u τ / 2 ( ) ( ) cos 2π f 0 8/25/15 M. J. Robers - All Righs Reserved 111

112 According o Parseval's Theorem, he signal energy of a signal can be found direcly from is Fourier ransform, E x = X( f ) 2 df. X( f ) indicaes he variaion of he ampliudes of he complex sinusoidal componens of x ( ) as a funcion of heir frequencies, f. So he unis are V Hz (if x ( ) is a volage signal). Therefore he unis of X( f ) 2 mus be V Hz 2. When we inegrae X( f ) 2 over all frequencies we ge signal energy whose unis are signal energy. Energy Specral Densiy V Hz 2 Hz or V 2 s, he unis of 8/25/15 M. J. Robers - All Righs Reserved 112

113 Energy Specral Densiy Therefore X( f ) 2 is he densiy of signal energy as a funcion of frequency. I is known as Energy Specral Densiy (ESD). The erm "specral densiy" means "variaion wih respec o frequency". A common symbol for ESD is G f ( ) = X( f ) 2. 8/25/15 M. J. Robers - All Righs Reserved 113

114 Power Specral Densiy Energy Specral Densiy is he variaion of signal energy wih frequency. I applies o energy signals. The corresponding quaniy ha applies o power signals is Power Specral Densiy (PSD). Power specral densiy S f ( ) is defined by P = S f ( )df. Tha is, is inegral over all frequency yields oal average signal power. Therefore S f a funcion of frequency. ( ) indicaes he variaion of average signal power as 8/25/15 M. J. Robers - All Righs Reserved 114

115 Suppose we ake he inverse Fourier ransform of energy specral densiy F ( 1 G ( f )) = F 1 X f So F 1 ( ) 2 ( ( )) = F ( 1 X( f )X * ( f )) ( ( )) = F 1 X f!#"# $ F 1 X * f! #" ## $ =x( ) =x( ) ( G ( f )) = x ( ) x ( ) = x τ ( )x( + τ )dτ. We can exchange he meanings of and τ o form φ( τ ) = x( τ ) x( τ ) = x Auocorrelaion ( )x( + τ )d This inegral is he area under he produc of a funcion x and a version of x ha has been shifed o he lef by τ, as a funcion of he shif amoun. This funcion is called auocorrelaion. 8/25/15 M. J. Robers - All Righs Reserved 115

116 The auocorrelaion funcion indicaes how similar a signal is o iself when shifed. When he shif is zero (τ =0), φ 0 ( ) = x 2 ( ) which is he signal energy. If he shif τ is small and he value of ( ) does no change much, we say here is a srong correlaion φ τ beween x and he shifed version of x for small shifs. So a slowly changing φ τ Auocorrelaion ( ) indicaes ha he signal sill looks like iself even when shifed a significan amoun. A quickly changing φ( τ ) indicaes ha even a small shif makes he signal look very differen. d 8/25/15 M. J. Robers - All Righs Reserved 116

117 The definiion φ τ ( ) = x ( )x( + τ )d applies o energy signals. For power signals, he definiion of auocorrelaion is F R( τ ) = x( )x( + τ ) and R( τ ) S( f ). Some properies of auocorrelaion are 1. R 0 2. R 0 3. R τ ( ) = S( f )df = oal average signal power ( ) R( τ ), auocorrelaion can never exceed he signal power ( ) is always an even funcion, ha is R( τ ) = R( τ ) ( ( )) is everywhere non-negaive ( ) is periodic hen R x ( τ ) is also, wih he same period ( ) conains no periodic componens lim ( τ ) = x( ) 2 4. F R τ 5. If x 6. If x Auocorrelaion R τ x 8/25/15 M. J. Robers - All Righs Reserved 117

118 Auocorrelaion Examples Auocorrelaions of a cosine and sine "burs". They are very similar bu no exacly he same. Noice ha boh are even funcions, even hough cosine is even and sine is odd.

119 Auocorrelaion Examples Three random power signals wih differen frequency conen and heir auocorrelaions. 8/26/15 M. J. Robers - All Righs Reserved 119

120 Auocorrelaion Examples Four Differen Random Signals wih Idenical Auocorrelaions 8/26/15 M. J. Robers - All Righs Reserved 120

121 Auocorrelaion Examples Four Differen Random Signals wih Idenical Auocorrelaions 8/26/15 M. J. Robers - All Righs Reserved 121

122 Sampling Uniform sampling of a coninuous-ime signal can be represened by muliplying he signal by a periodic impulse, forming a signal consising only of impulses. x δ n= ( ) = x( nt s )δ nt s ( ) where T s is he ime beween samples. When sampling a signal, he salien quesion is always wheher he original coninuous-ime signal can be recovered from he samples. The sampling heorem says ha if he signal is sampled for all ime a a rae greaer han wice he highes frequency in he signal, he original signal can be recovered exacly from he samples. 8/26/15 M. J. Robers - All Righs Reserved 122

123 Sampling Sampling Theorem: If he signal is sampled for all ime a a rae greaer han wice he highes frequency in he signal, he original signal can be recovered exacly from he samples. Pracically speaking, a signal can never be sampled for all ime. Also if a signal is no bandlimied is highes frequency is infinie, requiring an infinie sampling rae, and no real signal can be bandlimied because all real signals are ime limied. Therefore he sampling heorem can never quie be saisfied. The sampling heorem really jus serves as a limiing requiremen o be approached bu never reached in pracice. Pracically, sampling always yields an approximaion, bu one which can ofen be very good. 8/26/15 M. J. Robers - All Righs Reserved 123

124 Sampling The sampling rae ha is wice he highes rae in a signal is called he Nyquis rae in honor of Harry Nyquis, one of he earlies conribuors o sampling heory. There is a variaion on he sampling heorem for signals ha are narrowband. Tha is, signals whose cener frequency is much greaer han he bandwidh. If he bandwidh is W and he highes frequency is f u and he signal is sampled a a rae f s = 2 f u / m where m is he greaes ineger in f u / W, hen he signal can be recovered from he samples by passing i hrough a bandpass filer. 8/26/15 M. J. Robers - All Righs Reserved 124