Fourier Series. Spectral Analysis of Periodic Signals

Transcription

1 Fourier Series. Spectral Analysis of Periodic Signals he response of continuous-time linear invariant systems to the complex exponential with unitary magnitude response of a continuous-time LI system at a certain signal: differential equation convolution product of the input signal with impulse response. input signal periodic: decompose into a series of simpler components, response of the system at each component synthesis of partial responses. frequency domain: Fourier series

2 Response of c.t. LI systems to complex exponential with magnitude one jω () = e x t ω, t R t R h(t) y jω ( t τ) y t h e d () ( τ ) = ω () t e t h( τ ) j τ jω τ = e dτ H(ω ) Fourier transform of h, computed in ω depends on ω and h 3 j () e ω x t h(t) t j t = y( t) = e ω H( ω ) eigenfunction eigenvalue jωτ ( ω) = ( τ) τ = ( ω) H h e d H e jωt ( ) = ( ω ) = ( ω ) jφ ( ω) ( t+φ( ) ) j y t e H H e ω ω 4

3 x () t = a e jω t h(t) () ( ω ) j t y t a H e ω = input = linear combination of complex exponentials the output = a linear combination of complex exponentials j t ( ) as{ e ω } y t = H jω ( ω ) e = ah t j ( ω ) e ω t 5 Orthogonal ransforms Scalar product of two vectors [ x x... x ] ; y [ y y... y ] x= = n * y y xy, = [ x x... x ] x y x y... x y... = * y n * * * * n n n n Scalar product of functions from to L [a,b] b * (), () () () x t y t = x t y t dt a 6 3

4 Properties i) xy, = yx,, ii) xy, + z = xy, + xz,, iii) λxy, = λ xy,, * iv) x, y = x, y C, * λ λ λ n m n m * v) αx, βlyl = αβl x, yl. = l= = l= 7 Proof n m α x, = l= ii) m n * = βl l= = i) n m = α = l= β iii) m n * = βl α l= = β l * l l y l x, y y, α x x, y ii) m n = α l= = iii) * m n * = l l= = l l x, β y i) m n * = βl yl, α l= = β α l l l = x y, x * * = = 8 4

5 he norm x = x, x n = x + x x = x = b = () a x x t dt Rules i)-iv) apply for the space L, so the norms x are finite. 9 Hilbert space A structure frequently used in approximation theory. Space composed by vectors. Each of them has its norm. his norm is defined with the aid of the scalar product of vectors denoted by <>. 5

6 ) Finite dimensional Hilbert space, with dimension n ( ) ( ),,..., n,,,..., n, * y * y n *. * = = ( n) =. = x= x x x y = y y y x, y x y x, x,..., x x y,. * y n x = x, x = x n = he scalar product <x,y>-matrices product of the transposed of the x with the conjugate of the y. he squared norm of x = the scalar product <x,x>. Model for: Discrete time signals on the interval [, n-] or periodic (with period n) discrete time signals. 6

7 ) Finite dimensional Hilbert space of finite energy signal with finite duration xy, L ; [ ab, ] b * () () = () () () = () x : R C b a x () t dt < b x t, y t x t y t dt; x t x t dt. a a Model for: continuous time signals on the interval [ a,b] or periodic (with period =b-a) continuous time 3 signals Orthogonal vectors For two bidimensional vectors x= ix + jx ; y = iy + jy xy, = x ycosα α-angle between vectors cosα = xy, x y 4 7

8 If the two vectors are perpendicular (orthogonal) then <x, y> = If the scalar product is => the two vectors are orthogonal. Orthogonality condition: x, y = x y 5 Orthogonal functions Consider two signals defined on (, ), with =π/ω -- space L [,] () ω ( ) xt = cos t; y t = sinω t he scalar product is ( ω t) cos cos4 cos ω t,sinω t = cos ω t sin ω t dt = sin ω t dt ( ) ( ) ( ) = = = 4ω 4ω π 6 8

9 Complete space A system U={u } of orthogonal vectors two by two from a Hilbert space H is complete in H if: there is no other vector x H-U, orthogonal on all the vectors from U only the vector u, x = x=, if x H U. 7 Orthogonal basis of Hilbert space Each complete system U is an orthogonal basis of H. Any element x from H can be expressed lie a linear combination of elements of U uniquely x H, x= au. 8 9

10 Examples of basis he unity vectors {i, j, } are a basis in the three-dimensional space. he set of complex exponential {e jω t } Z with frequency ω is an infinite dimensional basis for the periodic continuous time signals of period 9 Pythagoras theorem in the Hilbert space. Relation between distance and scalar product Consider two vectors in the Hilbert space heir difference is d= x y (, ) d = x y x y

11 For vectors/functions in the Hilbert space (, ) Re {, } d x y = x y = x x y + y Pythagoras theorem in the Hilbert space. If x and y are orthogonal (, ) d x y = x + y Examples for Pythagoras theorem in the Hilbert space L [,] ( ) ( ω t) orthogonal signals cos ω t and sin have the same norm + cos( ωt ) x() t = cos ( ωt) dt = dt = t sin = + ( ω t) = ω Pythagoras theorem d ( cos ωt,sin ω t) = + =.

12 non-orthogonal signals do not satisfy Pythagoras theorem. signals on L [,] are not orthogonal: cos ωt and cos ωt ( ) ( ) t t = tdt = cos ω, cosω cos ω. (, ) Re {, } d x y = x x y + y Should be zero, but it s not 3 Examples for Pithagoras theorem square distance ( x, y) = x Re{ x, y } y d + x = y = / scalar product <x,y>= /. the square distance d ( cos ωt, cosωt) = + =. 4

13 ( ω ω ) < ( ω ω5 ) d cos t,sin t d cos t, cos t. Schwarz s inequality in the Hilbert space (Cauchy- Bunyaovsy-Schwarz inequality) x, y x y with equality holding if and only if x and y are linearly dependent, i.e. y=x, for some scalar 6 3

14 Examples for Schwarz s inequality. Orthogonal signals L [,] ( ) = cos( ω t) y ( t) = sin ( ωt) x t product of the norms xt ( ), yt ( ) = () yt () = = xt 7 Schwarz s inequality is verified < / here is no such that y(t) = x(t) So, in this case the Schwarz s inequality can not become equality 8 4

15 Examples for Schwarz s inequality. Non-orthogonal signals L [,] xt ( ) = cos ( ωt) and yt ( ) = cos( ωt) yt ( ) = xt ( ) there exists a value =- -Schwarz s inequality becomes an equality. = = () ; yt () xt xt yt = = (), () 9 Optimal approximation in Hilbert space n-dimensional Hilbert space, with orthogonal basis U = u, u,..., un { } u, u l l, = u l =, l U is orthonormal u = and c = x, u. 3 5

16 Vector=unique linear combination of vectors from U x = cu = he coefficients c n xu, c =, {,,..., n}, x H. u 3 Optimal approximation in Hilbert space Approximation: represent n-dimensional vector x using only m elements, m<n m ~ x = λ = Best approximation: truncation of its series decomposition u λ = c, =,..., m. increase m (number of terms in the approximation) the error decreases & the approximation becomes better 3 6

17 approximation error e= x x norm (, ) d x x = x x = e minimize the norm of the error e Proof (, ) = d x x 33 m m d ( x, x ) = e = e, e = x λu, x λiui = i= m m m m * * λ, λi, i λλi, i = i= = i= = x u x x u + u u * (,, ) m * m = x λ xu + λ xu + λ u = = 34 7

18 Select coefficients λ to minimize d. Partial derivates of d (function of λ ) = xu, λ = = c, {,,..., m}, m < n, u m xu, m min ( ) = = = u = d x, x x x c u. 35 Projection theorem min (, ) d x x = x x x x = x x x = x + x x x~ the approximation error is orthogonal on so it s orthogonal on the approximation m- dimensional subspace 36 8

19 For H a Hilbert space, H s a closed subspace of H For each vector x in H there is a vector x~ in H s = the best approximation of x with elements in H s with the properties x to x. he distance from is smaller than the distance from x to each element from H s. he error produced e= x x is orthogonal on the subspace H s 37 x = OA,x ~ = OB,e = u 3 BA A original Hilbert space = 3D space projection Hilbert space = horizontal plane x e= x x au x au u u B 38 9

20 n n n n x = xx, = cu, cu = cc u, u = n = c u l l l l = l= = l= min (, ) d x x = e = x x = min n m n = c u c u = c u = = = m+ 39 Infinite dimensional Hilbert spaces { ( ) } UN = u t, = N, N orthogonal basis in a finite dimensional space, subspace of Hilbert space he decomposition of signal x: () () x t = c u t, with c = = ( ), ( ) u () t x t u t 4

21 he case of infinite dimensional spaces Approximation signal in a finite dimensional Hilbert space of dimension N+: N x N () t = λu () t = N Lie before, we have { } λ = c, N, N +,...,,,..., N for minimum error N () () () () x t x t = x t c u t N = N 4 But: x() t = x() t, x() t = c u () t, cu () t = = () he error becomes: c u t = l l l= Parseval s relation N () () = () () x t x t c u t c u t N = = N = > N () c u t 4

22 he case of infinite dimensional spaces More terms (N high) error decreases We have : () = () Bessel s inequality. N x t c u x t N = N 43 () < because () [ ab, ] x t x t L lim c u = N > N () () lim x t x t = N he approximation signal in mean square to x(t) N x N ( t) converges ( ) = ( ) l.i.m. x N t x t N 44

23 . We have Remars () = () + () () N N x t x t x t x t Pitagora s theorem: orthogonality between the best approximation and the approximation error ( ) ( ) ( ) x N t, x t x N t =. Parseval s relation ( Rayleigh s energy theorem) W = x() t = c u () t = 3. he best approximation is obtained by truncating the series decomposition 45 Fourier Series L, consider in the space basis : jω t u t = e Z ( ), [ ] an orthogonal he elements are orthogonal and the set is complete. j ( l), t l jω t jlω t ω e, e = e =, = l he norm () u t = 46 3

24 Exponential Fourier series For a periodic signal x(t)=x(t+ ) () () x t = c u t = c e = () = jωt, jωt c = = x t e dt x t e e jω t jω t () π x t = c e c = x t e dt = jωt jωt, ω = 47 () () rigonometric Fourier series Euler s relations ωt = e + e ; sin ωt = e e j An orthogonal basis of the same space is: cos jωt jωt jωt jωt ( ) ( ) ( ) ( ) U {, cos( ωt), sin( ω t) } N = the elements are orthogonal and the set is complete. 48 4

25 rigonometric Fourier series he norms of the basis elements are: = dt = ; ( ω ) ( ω ) ( ω ) ( ω ) = = cos t cos t dt ; = = sin t sin t dt ; 49 rigonometric Fourier series So, any periodic signal of period can be expressed in the form: x ( t) = a + ( a cos( ω t) + b ( ω t) ) = sin 5 5

26 rigonometric Fourier series the coefficients are : a ( ) xt, = = () ( ω ) cos( ωt) () ( ω ) sin ( ω t) () x t dt, continuous component. xt,cos t a = = x() t cos ( ωt) dt, x t,sin t b = = x() t sin ( ωt) dt. 5 Remars. a - DC component of the signal x(t). he signal with no DC component(a =) has only oscillatory components: () = ( ω + ω ) x t a cos t b sin t ; = ( ) ( ) x t odd a = ; x t even b = ; 3. For real signals () ( ) x t = x t c = c * * ω () () j t jωt * c = x t e dt = x t e dt = c * 5 6

27 4. the power of the signal x(t) - Parseval s relation : W P = = c P= c = x t () = = another form of the Parseval s relation: a b P= x() t dt a = + + = 53 Harmonic Fourier Series Using the relation: a cos ω t+ b sin ω t = a + b cos ω t+ ϕ the Fourier trigonometric series becomes: harmonic form. ( ) b tg ϕ =. A = a + b x a () t = A cos( ω t + ϕ ) = 54 7

28 Relations between coefficients For real signals we have c = a + b = A, c = c, ; arg c = ϕ, ; arg c = ϕ, ; c = a ; arg c =. 55 Spectrum diagrams represent periodic signals in the frequency domain. () xt, t < = <, t 56 8

29 DC component: a = x t dt = dt = ; A = a () he oscillatory component is odd x () t, t < = <, t a = 57 the other coefficients or b = ( ) 4 cos ω t 4 b = x() t sin tdt ; ω = = ω ω b = 4 ; =,,3,... π ( ) 58 9

30 4 x() t = + sin ( ) ωt π = ( ) 4 π xt () = + cos ( ) ωt = π ( ) A ( ) th order harmonics of frequency ( ) ω, 59 Amplitude spectrum (ω, A ) Fundamental frequency π/ DC component 3 rd harmonic nd harmonic Harmonic Fourier series 6 3

31 Phase spectrum (ω, ϕ ) Harmonic Fourier series 6 Amplitude spectrum (ω, c ) obtained also with the complex exponential form of the Fourier series. he coefficients c : c = x t dt = a = () jω t jωt () ( ) c = x t e dt = e dt = ; jπ π π j j c = e ; ; c = e ; π ( ) π c =, 6 3

32 Magnitude spectrum (ω, c ) c = π EVEN FUNCION negative frequencies 63 Phase spectrum (ω, ϕ ) Another representation in the frequency domain. For the square wave we have: π ϕ = sgn ( ) ODD FUNCION 64 3

33 Other forms of Parseval s relation complex exponential Fourier series : P = xt () dt= c = c + c trigonometric & harmonic = = P= a + a b + = x t dt = A + A () = = 65 he power of this square wave () P = x t dt = 4dt = 4 / 66 33

34 Power spectrum the association of the frequencies of its components with their powers harmonic form (ω, A /) complex exponential form (ω, c ) 67 Power spectrum with the harmonic Fourier series square wave 68 34

35 Power spectrum with the exponential Fourier series 69 non-band-limited signal: the signal has infinite frequency bandwidth. he power decreases as the frequency increases; it approaches zero only at infinite frequency effective bandwidth of the signal = positive frequency range with a significant percentage of the power of the signal. For this case, in the bandwidth 9ω we find 96,5% of the power of the signal 7 35

36 Gibbs Phenomenon he physicist Albert Michelson tried to construct a spectrum analyzer in 898. He observed that the spectrum analyzer was not woring properly for non band-limited signals. He ased to Gibbs to explain this phenomenon. 7 Gibbs considered the following non bandlimited signal: a square wave with duty factor.5 with no DC component 7 36

37 Fourier expansion, non band-limited signal 4 x() t = sin ω t+ sin 3ω t+ sin 5 ω t+... π 3 5 truncation in frequency : non band-limited input signal was approximated with a band-limited signal, n odd harmonics 4 x () t = sin ω t+ sin 3ω t+ sin 5 ω t sin ( n ) ωt π 3 5 n 73 nωt sinu Si () = = ( ω ) x t du n t π u π Si(x) sine integral odd function x sin u Si ; Si Si u ( x) = du ( x) = ( x) π limsi( x) = x 74 37

38 nr sin cos cos ( )... cos ( ) n α + α + r + + α + n r = cos α + r r sin α = ωτ and r = α + ( ) + + ( n ) ( nωτ ) ( nωτ ) cos( nωτ) cosωτ cos 3 ωτ... cos ωτ sin sin = = sinωτ ωτ sin Proof t 4ω x t = ωτ+ ωτ+ ωτ+ + n ωτ dτ () cos cos3 cos5... cos( ) π 75 he truncated Fourier series t 4ω y () t = cosωτ + cos3 ωτ cos( n ) ωτ dτ = π t τ sin n π dτ π τ approximated sin x = x (very small x). ( τ ) t sin τ < τ < << π π 76 38

39 π/ -π/ 77 Gibbs proved that Gibbs Phenomenon truncating the square wave y(t) duty factor.5 preserving only n odd harmonic components 4 x () t = sin ω t+ sin 3ω t+ sin 5 ω t sin ( n ) ωt π 3 5 n We have nωt sinu Si () = = ( ω ) x t du n t π u π 78 39

40 Gibbs phenomenon for a square wave, with =s(duty factor.5) 79 Gibbs Phenomenon he approximation error is high in the neighborhood of the discontinuity. It has a damped oscillatory waveform. he maximum of the oscillation :.8 V, appears at the moment t m. he rise time. π tr tm = = ω f M M 8 4

41 runcated signals for and 45 harmonics, respectively 8 maximum amplitude of the oscillations does not decrease the oscillation is compressed in time (its frequency increases). convergence in mean square. Gibbs phenomenon proves that the non band-limited signals can t be perfectly approximated with band-limited signals 8 4

42 Periodic distributions Example: the Dirac periodic distribution, period, δ (t) δ t = t c = () δ ( ) = 83 π j t c () = t e dt () t dt δ = δ = δ ω () t = δ ( t ) = e = for [-/,/], δ (t)= δ(t). he product of a c.t. function with δ(t) ( ) δ ( ) = ( ) δ ( ) x t t x t j t = 84 4

43 Exponential Fourier Series Properties Fourier coefficients of signal x, period () F { } x t c x jωt c = x() t e dt Fourier decomposition a.e. : jω t x() t = ce a.e.w. = 85. Linearity If the signals x(t) and y(t) are periodic with period : F x F y () { }, ( ) { } F x y ax() t + by () t { ac + bc} x t c y t c the Fourier decomposition - linear

44 . ime shifting ime shifting modulation with complex exponential. jω t x ( ) F { } x t t e c jω ( τ+ t ) = ( ) ( ) = τ τ= ω ω c x t t e dt x e d e c j t j t x Complex conjugation Complex conjugation reversal in frequency x * x ( t) c 88 44

45 4. ime reversal ime reversal reversal in frequency. ω j( ) τ = ( ) ( ) = τ τ= x t x t c ω c x t e dt x e d c j t x F x () = ( ) { } ime scaling x(t) - period x(at), period /IaI. he time scaled version has the same Fourier coefficients lie the initial version. jω t π c = x( at) e dt; a ω = = ω / a a ( ) jωτ x c = x τ e dτ= c F x ( ) { c } x at 9 45

46 6. Signal s Modulation Modulation in time frequency shifting ω ω c = x() t e e dt x() t e dt c = = j t j t j( ) ωt x jωt F x () { c } x t e 9 ime-frequency duality operation in time another operation in frequency : modulation shifting nd operation in time first operation in frequency. time shifting modulation his behavior is named duality. Reversal is an auto-dual operation 9 46

47 7. Product of signals discrete convolution of the Fourier coefficients sequences. x t y t c c c c n= F x y x y () () n n = { } Periodic convolution periodic signals do not finite energy their convolution can not be defined. circular convolution - for periodic signals. F x y () = ( τ) ( τ) τ= () () { } z t x y t d x t y t c c dual operations: multiplication convolution 94 47

48 Circular convolution for square waves, different duty factors he circularity effect can be observed Signal Differentiation After differentiation, DC component =. ime differentiation multiplication with jω. dx( t) dt F x { jωc } 96 48

49 . Signal s Integration Periodic signal with no DC component ime integration multiplication with /jω. t x F c x ( ) x τ dτ c = jω 97. Real Signal s Case. he Series of the Even and Odd Parts x(t)- real signal; even x e (t) and odd part x o (t) spectrum of real even signal x e (t) real xt ( ) + x( t) x xe () t = F { Rec } spectrum of a real odd signal x o (t) - pure imaginary ( ) ( ) xt x t xo () t = F j c x { Im } 98 49