Chapter 4 The Fourier Series and Fourier Transform

Transcription

1 Represenaion of Signals in Terms of Frequency Componens Chaper 4 The Fourier Series and Fourier Transform Consider he CT signal defined by x () = Acos( ω + θ ), = The frequencies `presen in he signal are he frequency ω of he componen sinusoids The signal x() is compleely characerized by he se of frequencies ω, he se of ampliudes A, and he se of phases θ Example: Sum of Sinusoids Consider he CT signal given by x() = Acos() + Acos(4 + π /3) + A3cos(8 + π /), The signal has only hree frequency componens a,4, and 8 rad/sec, ampliudes A and phases, A, A3, π / 3, π / The shape of he signal x() depends on he relaive magniudes of he frequency componens, specified in erms of he ampliudes A, A, A 3 Example: Sum of Sinusoids Con d A =.5 A = A3 = A = A =.5 A3 = A = A = A3 = Example: Sum of Sinusoids Con d A =.5 A = A3 =.5 A = A =.5 A3 =.5 A = A = A3 = Plo of he ampliudes maing up x() vs. ω Example: Ampliude Specrum ω A of he sinusoids

2 Plo of he phases θ of he sinusoids maing up x() vs. ω Example: Phase Specrum ω Complex Exponenial Form jα Euler formula: e = cos( α) + jsin( α) Thus j( ) A cos( ) Ae ω + ω + θ =R θ whence j( ω θ) x () = R Ae +, = real par Complex Exponenial Form Con d And, recalling ha R ( z) = ( z+ z )/ where z = a+ jb, we can also wrie j( ω θ) j( ω θ) x () = Ae + + Ae +, = This signal conains boh posiive and negaive frequencies The negaive frequencies ω sem from wriing he cosine in erms of complex exponenials and have no physical meaning Complex Exponenial Form Con d By defining A j c e θ A j = c e θ = i is also jω jω jω x () = ce c e + = ce, = = complex exponenial form of he signal x() Line Specra The ampliude specrum of x() is defined as he plo of he magniudes c versus ω The phase specrum of x() is defined as he plo of he angles c versus ω = arg( c) This resuls in line specra which are defined for boh posiive and negaive frequencies oice: for =,, c = c c = c c = c arg( ) arg( ) Example: Line Specra x( ) = cos( ) +.5cos(4 + π / 3) + cos(8 + π / )..

3 Fourier Series Represenaion of Periodic Signals Le x() be a CT periodic signal wih period T, i.e., x( + T) = x( ), R Example: he recangular pulse rain The Fourier Series Then, x() can be expressed as () jω, = x = ce where ω = π /T is he fundamenal frequency (rad/sec) of he signal and T / jωo c = x( ) e d, =, ±, ±, T T / is called he consan or dc componen of x() c The Fourier Series Con d The frequencies ω presen in x() are ineger muliples of he fundamenal frequencyω oice ha, if he dc erm c is added o j x() = ce ω = and we se =, he Fourier series is a special case of he above equaion where all he frequencies are ineger muliples of ω Dirichle Condiions A periodic signal x(), has a Fourier series if i saisfies he following condiions:. x() is absoluely inegrable over any period, namely a+ T a x ( ) d<, a. x() has only a finie number of maxima and minima over any period 3. x() has only a finie number of disconinuiies over any period Example: The Recangular Pulse Train Example: The Recangular Pulse Train Con d ( ) / jπ x () = + ( ) e, π = odd From figure, T = whence ω = π /= π Clearly x() saisfies he Dirichle condiions and hus has a Fourier series represenaion 3

4 By using Euler s formula, we can rewrie as Trigonomeric Fourier Series () jω, = x = ce x () = c + c cos( ω + c), = dc componen -h harmonic This expression is called he rigonomeric Fourier series of x() Example: Trigonomeric Fourier Series of he Recangular Pulse Train The expression ( ) / jπ x () = + ( ) e, π = odd can be rewrien as ( )/ π x () = + cos π+ ( ), = π odd Gibbs Phenomenon Given an odd posiive ineger, define he -h parial sum of he previous series Gibbs Phenomenon Con d x3( ) x9( ) ( )/ π x () = + cos π+ ( ), = π odd According o Fourier s s heorem, i should be lim x () x() = Gibbs Phenomenon Con d x x45( ) ( ) overshoo: abou 9 % of he signal magniude (presen even if ) Parseval s Theorem Le x() be a periodic signal wih period T The average power P of he signal is defined as T / P = x () d T T / jω Expressing he signal as x () = ce, = i is also P = c = 4

5 Fourier Transform We have seen ha periodic signals can be represened wih he Fourier series Can aperiodic signals be analyzed in erms of frequency componens? Yes, and he Fourier ransform provides he ool for his analysis The major difference w.r.. he line specra of periodic signals is ha he specra of aperiodic signals are defined for all real values of he frequency variable ω no jus for a discree se of values xt () Frequency Conen of he Recangular Pulse x() x() = lim x () T T Frequency Conen of he Frequency Conen of he Since xt () is periodic wih period T, we can wrie where jω T(), = x = c e T / jωo c = x( ) e d, =, ±, ±, T T / Wha happens o he frequency componens of xt () as T? For = c = T For ω ω c = sin = sin, =±, ±, ω T π ω = π /T plos of T c vs. ω = ω for T =,5, Frequency Conen of he Frequency Conen of he I can be easily shown ha ω lim Tc = sinc, ω T π where sin( πλ sinc( λ) ) πλ 5

6 Fourier Transform of he Recangular Pulse The Fourier ransform of he recangular pulse x() is defined o be he limi of Tc as T, i.e., ω X( ω) = limtc = sinc, ω T π X ( ω ) arg( X ( ω )) Fourier Transform of he The Fourier ransform X ( ω) of he recangular pulse x() can be expressed in erms of x() as follows: jωo c = x() e d,,,, T = ± ± x () = for < T/and > T/ whence jωo Tc = x( ) e d, =, ±, ±, Fourier Transform of he ow, by definiion X( ω) = limtc and, T since ω ω as T jω X( ω) = x( ) e d, ω The inverse Fourier ransform of X ( ω) is jω x () = X( ω) e dω, π The Fourier Transform in he General Case Given a signal x(), is Fourier ransform X ( ω) is defined as jω X( ω) = x( ) e d, ω A signal x() is said o have a Fourier ransform in he ordinary sense if he above inegral converges The Fourier Transform in he General Case Con d The inegral does converge if. he signal x() is well-behaved. and x() is absoluely inegrable, namely, x ( ) d < oe: well behaved means ha he signal has a finie number of disconinuiies, maxima, and minima wihin any finie ime inerval Example: The DC or Consan Signal Consider he signal x() =, Clearly x() does no saisfy he firs requiremen since x( ) d = d = Therefore, he consan signal does no have a Fourier ransform in he ordinary sense Laer on, we ll see ha i has however a Fourier ransform in a generalized sense 6

7 Example: The Exponenial Signal b Consider he signal xt ( ) = e u ( ), b Is Fourier ransform is given by b jω X( ω) = e u( ) e d ( b+ jω) ( b+ jω) = e d = e b+ jω = = Example: The Exponenial Signal Con d If b <, X ( ω) does no exis If b =, x() = u() and X ( ω) does no exis eiher in he ordinary sense If b >, i is X ( ω) = b + jω ampliude specrum phase specrum X ( ω) = ω arg( X ( ω)) = arcan b + ω b Example: Ampliude and Phase Specra of he Exponenial Signal x() = e u () Recangular Form of he Fourier Transform Consider jω X( ω) = x( ) e d, ω Since X ( ω) in general is a complex funcion, by using Euler s formula X( ω) = x( )cos( ω) d+ j x( )sin( ω) d R( ω ) I ( ω ) X( ω) = R( ω) + ji( ω) Polar Form of he Fourier Transform X( ω) = R( ω) + ji( ω) can be expressed in a polar form as X( ω) = X( ω) exp( jarg( X( ω))) where X( ω) = R ( ω) + I ( ω) I( ω) arg( X ( ω)) = arcan R( ω) Fourier Transform of Real-Valued Signals If x() is real-valued, i is X( ω) = X ( ω) Moreover Hermiian symmery X ( ω) = X( ω) exp( jarg( X( ω))) whence X( ω) = X( ω) and arg( X( ω)) = arg( X( ω)) 7

8 Fourier Transforms of Signals wih Even or Odd Symmery Even signal: x() = x( ) X( ω) = x( )cos( ω) d Odd signal: x() = x( ) X( ω) = j x( )sin( ω) d Example: Fourier Transform of he Recangular Pulse Consider he even signal I is τ / = τ / ωτ X( ω) = ()cos( ω) d = [ sin( ω) ] = sin = ω ω ωτ = τ sinc π Example: Fourier Transform of he ωτ X ( ω) = τsinc π Example: Fourier Transform of he ampliude specrum phase specrum Bandlimied Signals A signal x() is said o be bandlimied if is Fourier ransform X ( ω) is zero for all ω > B where B is some posiive number, called he bandwidh of he signal I urns ou ha any bandlimied signal mus have an infinie duraion in ime, i.e., bandlimied signals canno be ime limied Bandlimied Signals Con d If a signal x() is no bandlimied, i is said o have infinie bandwidh or an infinie specrum Time-limied signals canno be bandlimied and hus all ime-limied signals have infinie bandwidh However, for any well-behaved signal x() i can be proven ha lim X ( ω) = ω whence i can be assumed ha X( ω) ω > B B being a convenien large number 8

9 Inverse Fourier Transform Given a signal x() wih Fourier ransform X ( ω ), x() can be recompued from X ( ω) by applying he inverse Fourier ransform given by jω x () = X( ω) e dω, π Transform pair x() X( ω) Properies of he Fourier Transform Lineariy: x() X( ω) y () Yω ( ) α x() + βy() αx( ω) + βy( ω) Lef or Righ Shif in Time: Time Scaling: x ( ) X( ω) e ω xa ( ) j ω X a a Properies of he Fourier Transform Time Reversal: x( ) X( ω ) Muliplicaion by a Power of : n n n d x () ( j) X( ω) n dω Muliplicaion by a Complex Exponenial: xe jω () X( ω ω) Properies of he Fourier Transform Muliplicaion by a Sinusoid (Modulaion): j x ()sin( ω) X( ω + ω) X( ω ω) x ()cos( ω) X( ω + ω) + X( ω ω) [ ] [ ] Differeniaion in he Time Domain: d n x n () ( j ω) X ( ω) n d Properies of he Fourier Transform Inegraion in he Time Domain: x( τ ) dτ X( ω) + πx() δ( ω) jω Convoluion in he Time Domain: x() y() X( ω) Y( ω) Muliplicaion in he Time Domain: x() y() X( ω) Y( ω) Properies of he Fourier Transform Parseval s Theorem: x() y() d X ( ω) Y( ω) dω π if y () = x () Dualiy: ( ) ( ω) π x d X d X() π x( ω ) ω 9

10 Properies of he Fourier Transform - Summary Example: Lineariy x() = p () + p () 4 ω ω X ( ω) = 4sinc + sinc π π Example: Time Shif x() = p ( ) Example: Time Scaling p( ) sinc ω π p( ) ω sinc π ω X( ω) = sinc π j e ω a > ime compression < a < ime expansion frequency expansion frequency compression Example: Muliplicaion in Time x() = p () Example: Muliplicaion in Time Con d ω cosω sinω X( ω) = j ω d ω d sinω ωcosω sinω X( ω) = j sinc j j dω π = = dω ω ω

11 Example: Muliplicaion by a Sinusoid x() p ()cos( ω ) = sinusoidal τ burs Example: Muliplicaion by a Sinusoid Con d τ ( ω+ ω) τ( ω ω) X ( ω) = τsinc τsinc + π π ω = 6 rad / sec τ =.5 τ ( ω+ ω) τ( ω ω) X ( ω) = τsinc τsinc + π π Example: Inegraion in he Time Domain v () = pτ () τ dv() x () = d Example: Inegraion in he Time Domain Con d The Fourier ransform of x() can be easily found o be τω τω X( ω) = sinc jsin 4π 4 ow, by using he inegraion propery, i is τ τω V( ω) = X( ω) + πx() δ( ω) = sinc jω 4π Example: Inegraion in he Time Domain Con d τ τω V ( ω) = sinc 4π Generalized Fourier Transform Fourier ransform of δ () jω δ () e d= Applying he dualiy propery x() =, πδ( ω) δ () generalized Fourier ransform of he consan signal x () =,

12 Generalized Fourier Transform of Sinusoidal Signals [ ] cos( ω ) π δ( ω + ω ) + δ( ω ω ) [ ] sin( ω ) jπ δ( ω + ω ) δ( ω ω ) Fourier Transform of Periodic Signals Le x() be a periodic signal wih period T; as such, i can be represened wih is Fourier ransform x () j = ce ω ω = π /T = j Since e ω πδω ( ω ), i is X( ω) = πc δ( ω ω ) = Fourier Transform of he Uni-Sep Funcion Common Fourier Transform Pairs Since u () = δ ( τ) dτ using he inegraion propery, i is u () = δ ( τ) dτ + πδ( ω) jω