III-A. Fourier Series Expansion

Transcription

1 Summer 28 Signals & Sysems S.F. Hsieh III-A. Fourier Series Expansion Inroducion. Divide and conquer Signals can be decomposed as linear combinaions of: (a) shifed impulses: (sifing propery) Why? x() x()δ( )d impulses are orhogonal(disjoin) in ime, leads o convoluion inegral for he LTI sysem oupu signal. (b) complex exponenials: (Fourier series/ransform) Why? periodic: x() X k e jkω X k e jk 2π T synhesis equaion : weighed sums of harmonically relaed sinusoids X k x()e jkω d analysis equaion T T aperiodic: x() X(jω)e jω dω Inverse Fourier ransform 2π : weighed inegrals of sinusoids ha are no all harmonically relaed X(jω) x()e jω d, Fourier ransform Complex exponenials are orhogonal(disjoin) in frequency, Human hearing sysem acs like a specrum analyzer, Daa compression/signal analysis; Sensiiviy of human percepion is frequency-dependen! Complex exponenials are eigen-signals for he LTI sysems. Muliplicaion in specrum is much easier han convoluion in ime..2 Human hearing Human hearing sysem acs like a specrum analyzer. Cochlea is a urn spiral srucure; 35 mm in lengh, 4 nerve fibers connecing o he brain, each nerve fiber has 5 hair cells. III-A -

2 2 Complex exponenials are eigenfuncions of LTI sysems Superposiion of inpu complex exponenials leads o superposiion of oupu complex exponenials wih appropriae complex weighing( H(s)/H(z): ransfer funcion; H(ω)/H(e jω ): frequency response).. Coninuous-ime(CT) Fourier ransform: e jω LTI: h() y() H(ω) e jω y() e jω h() h() e jω h()e jω( ) d e jω h()e jω d } {{ } H(ω) H(ω) e jω H(ω) is no a funcion of. We call {e ω, H(ω)} as he eigenpair of he LTI sysem; e jω is he eigen-funcion/-vecor and H(ω) is is associaed eigenvalue. We call H(ω) he frequency response of he LTI sysem; i is equal o he Fourier ransform of he impulse response. 2. Similarly, discree-ime Fourier ransform(dtft): e jωn LTI: h[n] y[n] H(Ω) e jωn y[n] e jωn h[n] h[n] e jωn h[k]e jω(n k) e jωn H(Ω) e jωn h[k]e jωk } {{ } H(Ω) H(Ω) is he sysem s eigenvalue associaed wih he eigenfuncion e jωn. I is equal o he DT Fourier ransform of he impulse response. We call H(Ω) he frequency response of he LTI sysem. 3. Sum of complex exponenials: From he superposiion propery of an LTI sysem, suppose he inpu is wrien as a sum of eigenfuncions: x() k: a ke jωk. The oupu y() from an LTI sysem wih impulse response h() is y() x() h() h() x() : h() k a k e jω k( ) d : III-A - 2 h()x( )d

3 k k a k e jω k : H(ω k )a k e jω k h()e jω k d k: b k e jω k where b k H(ω k )a k and he complex-valued H(ω k ) : h()e jω k d. linear combinaion of linear combinaion of complex exponenials LTI: h()/h(ω) he same (re-weighed) complex exponenials Similarly, for discree-ime signals and sysems, x[n] k a k e jω kn LTI: h[n]/h(ω) y[n] k a k H(Ω k )e jω kn 3 CT Fourier Series for Periodic Signals Every periodic signal (period /f 2π/ω ) can be expressed as (Fourier series expansion) x() X ke +jkω, synhesis equaion X k + x()e jkω d, analysis equaion. {e jkω, k, ±, ±2,..., } forms a se of Complee OrhoNormal bases. [pf] Inner produc of wo basis funcions e jkω and e jlω (m n) is e jkω If l k, hen 2. X k X k e j X k is complex-valued. If x() is real-valued, x() x (), [ ] e jlω d e j(k l)ω d e j(k l)ω j(k l)ω [ ] e j(k l)2πf (f ) j(k l)2π (orhogonaliy) e jkω e jlω d d (normaliy). (a) X k X k, because X k x ()e +jkω d x()e j( k)ω d X k. (b) Ampliude specrum X n : even-funcion, X k X k. (c) Phase specrum X k : odd-funcion, X k X k, and he phase always falls ino ( π, +π). See Dirichle condiions in Lahi, p 66, for exisence and convergence of he Fourier series. III-A - 3

4 3. Physical meanings of Fourier Series expansion: (a) x() is wrien as a linear combinaion of hese orhonormal bases, {e jkω, k, ±, ±2,...}. (b) The Fourier series coefficien, X k, is i. he projecion of x() on he orhonormal basis e jkω. ii. he ime-average of x()e jkω. iii. measures he specral conen of x() a he k h harmonic. (c) Generalized Fourier series expansion can use any oher orhonormal bases, no necessarily complex exponenials. 4. Role of he phase(iming) specrum in shaping a periodic signal (a) Human hearing is less phase-sensiive! There is almos no difference in hearing x() cos(4π) + cos(6π) and ˆx() cos(4π + θ ) + cos(6π + θ 2 ). (b) Human vision(image) is phase-sensiive!(oppenheim s Signals & sysems, 2ed., p 426-7) 5. Gibbs phenomenon: To approximae a rain of recangular pulses x() by a runcaed rigonomeric Fourier series ˆx() of only he firs n harmonics, he difference beween x() and ˆx() becomes smaller as n increases. However, even for large n, ˆx() exhibis an oscillaory behavior and an overshoo of 9% in he viciniy of he disconinuiy a he firs peak of oscillaion, which is called he Gibbs phenomenon. (See Lahi, Fig 6.7, p 68) 4 Properies of CT Fourier Series. Linear: If x() X k and y() Y k, hen x() + y() X k + Y k. 2. Time Delay: ˆx() x( ) X k e jkω X k. X k x( )e jkω d x(u)e jkω (u+) du e jkω x()e jkω d e jkω X k If a signal is shifed in ime, he magniudes of is Fourier series coefficiens remain unchanged, X k X k and a linear phase shif will be inroduced, X k X k jkω. For example, consider x() A cos(ω + θ ) + A 2 cos(2ω + θ 2 ) wih X k A k /2, X k θ k, k, 2. Now ˆx() x( ) A cos(ω ω + θ ) + A 2 cos(2ω 2ω + θ 2 ). We can see he phase shifs are ω and 2ω for he s and 2nd harmonics, respecively. Higher order harmonics(higher frequency) will experience larger phase-shif for a consan ime delay. 3. Time reversal: x( ) X k. 4. Time scaling: x(α) X ke jk(αω ). If x() is periodic wih period and fundamenal frequency ω 2π/, hen x(α) is periodic wih period T/α and fundamenal frequency αω, and he Fourier series coefficiens remain he same. 5. Muliplicaion: x() y() X k Y k l X k Y k l. 6. Differeniaion: d d x() jkω X k. 7. Inegraion: x()d X k jkω. III-A - 4

5 8. Symmery properies: (a) x () X k. (b) x() : real X k X k : conjugae symmeric. If x() is real, hen X k X k and X k X k. (c) If x() is even, hen X k is real; [Pf] Given x() x( ), hen X k x()e jkω d le : ( ) x( )e T jkω d x()e jkω d Xk (d) If x() is odd, hen X n is imaginary; (e) If x( ± /2) x()(halfwave symmery: wo halves of one period are ani-symmeric), +A T 2 A hen X 2k, k 9. Parseval s heorem: Same average power over ime and specrum, because x() is wrien as a sum of orhogonal componens: P x() 2 d, average power in ime X k 2, average power in specrum [Pf] P x() 2 d x()x ()d X k e jkω Xl T e jlω d k l X k Xl T ej(k l)ω d noe e j(k l)ω if l k k l X k Xk T d X k 2 k n Ex. If z() x() + y() and x() y(), i.e., x()y()d, hen P z P x + P y. 5 Trigonomeric F.S. expansion for a real periodic signal x() X + A k cos kω + B k sin kω k k C k cos(kω + θ k ) k X x()d, dc-value A k 2 x() cos(kω )d III-A - 5

6 B k 2 x() sin(kω )d C k A 2 k + B2 k, θ k an ( B k /A k ). {cos kω, sin kω, k, ±, ±2,...} also forms an orhogonal basis. In paricular, cos kω sin kω, which is useful in quadraure muliplexing(qm) and QAM of communicaion sysems. 2. x() X }{{} + 2 X cos(ω + /X ) + 2 X 2 cos(2ω + /X 2 ) + 2 X 3 cos(3ω + /X 3 ) + }{{}}{{}}{{} dc fundamenal 2nd harmonic 3rd harmonic [Pf] x() X + X + [X ] k e jkω + X k e jkω k [ X ] k e j X k e j(kω) + X k e j X k e j(kω ) k : if x() is real, hen X k X k and X k X k X + X k [e ] j(kω + X k ) + e j(kω + X k ) X + k 2 X k cos(kω + X k ) k 3. Average of real-valued signal: P x x() 2 d X k 2 X Exponenial Fourier specra(2-sided) and rigonomeric specra(-sided) For a real-valued periodic signal: [ 2 X k 2 or X 2 + ] A 2 k 2 + B2 k k k X k D k 2 C ke jθ k, and D k 2 C ke jθ k 6 Examples of CT Fourier Series 6. Impulse rain [Lahi, Ex 6.7] Given an impulse rain: x() m X k T T/2 T/2 δ() e jkω d T δ( mt ). The Fourier series coefficiens of he impulse rain are idenical! Can you explain is physical meaning graphically? III-A - 6

7 6.2 A recangular pulse rain ( mt 2 A ) m k: [ ] A sinc(kf )e jπkf e j2πkf A x() 2. F.S. coefficiens: X k T x()e jkω d Ae jkω d A e jkω jkω A e jk2πf jk2πf A e jkπf [e jkπf e jkπf ] jk2πf rick A e jkπf [ 2j sin kπf ] jk2πf A e jkπf sin kπf kπf A sinc(kf )e jπkf where sinc(z) sin πz πz. Sinc(z) is also defined in Lahi s book as sinc(z) sin z z. sinc(z) Ampliude and phase specra: X k A sinc(kf ) X k πkf (±2lπ), so ha he phase falls wihin ±π 3. Inverse relaionship beween Time and Frequency: (a) As he pulse widh reduces, he specra becomes wider, because he mainlobe has a null a /. Local emporal behavior is inversely relaed wih he global specral shape. (b) As he period increases, he line specra becomes denser, because he specral spacing is /. Global emporal behavior is inversely relaed wih he local specral shape. (c) The phase specra has a slope of /2, which will be shown laer ha consan delay in ime resuls in linear phase shif. III-A - 7

8 (d) Plos of specra: / f f f 7 Fourier series and LTI sysems. Frequency response H(ω) is he eigenvalue corresponding o he e jω -eigenvecor: i.e., linear combinaion of linear combinaion of complex exponenials LTI: h()/h(ω) he same (re-weighed) complex exponenials x() k a k e jkω n LTI: h()/h(ω) y() k a k H(kω )e jkω where ω 2π T, T is he fundamenal period, and H(ω) h()e jω d. Similarly, for discree-ime signals and sysems, x[n] k a k e jkω n LTI: h[n]/h(ω) y[n] k a k H(kΩ )e jkω n where Ω 2π N, N is he fundamenal period, and H(Ω) n h[n]e jωn. 2. Frequency response and differeniaion equaion: Consider an LTI sysem d dy() + ay() bx(). hen is frequency response is H(ω) b a+jω. (Pf) If x() e jω, he oupu mus be y() H(ω)e jω. Thus, he differeniaion equaion becomes d [ H(ω)e jω ] + ah(ω)e jω be jω d and we have (jω) H(ω) e jω + ah(ω)e jω be jω Afer cancelling ou e jω, we ge H(ω) b a+jω. (Noe, a similar approach using he Laplace ransform echnique is more general and useful o find he ransfer funcion.) III-A - 8

9 3. Filering: For a real-valued periodic signal x() X }{{} + 2 X cos(ω + X ) + 2 X }{{} 2 cos(2ω + X 2 ) + 2 X }{{} 3 cos(3ω + X 3 ) + }{{} dc fundamenal 2nd harmonic 3rd harmonic LTI: H(ω) H(ω) e j H(ω) ( ) y() H() X + 2 H(ω ) X cos ω + X + H(ω ) ( ) +2 H(2ω ) X 2 cos 2ω + X 2 + H(2ω ) ( ) +2 H(3ω ) X 3 cos 3ω X 3 + H(3ω ) + 4. In summary, mah bases: {e jnω } X n : projecion How? Fourier Series x() X n e jnω X n x()e jnω d Why? (a) hearing (b) LTI: eigenfuncion Wha ime specra e jω LTI:h() H(ω) e jω III-A - 9