Sigal Processig i Mechatroics Summer semester, 1 Lecture 3, Covolutio, Fourier Series ad Fourier rasform Dr. Zhu K.P. AIS, UM 1
1. Covolutio Covolutio Descriptio of LI Systems he mai premise is that the respose of a liear time-ivariat (LI) system to ay iput ca be determied from its respose h[] to the uit sample sequece δ[], usig a formula kow as covolutio summatio. he sequece h[], which is kow as impulse respose, ca also be used to ifer all properties of a liear time-ivariat system. Represetatio of geeral sequece as a liear combiatio of delayed impulse, [ ] = [ ] δ [ ] x xk k k =
he left had side represets the sequece x[] as a whole whereas the right had side summatios represet the sequece as a superpositio of scaled ad shifted uit sample sequeces. he, from the superpositio property for a liear system, the respose y[] to the iput x[] is the same liear combiatio of the basic resposes h k [], that is, y= xkhk [ ] [ ] [ ] k = y H xk k xkh k k= k= { } [ ] = [ ] δ [ ] = [ ] δ [ ] = k = [ ] [ ] = [ ] [ ] xkh k x h 3
Equatio y[ ] = x[ k] h[ k] = x[ ] h[ ] k = is commoly called the covolutio sum or simply covolutio is deoted usig the otatio y[] = x[] * h[]. herefore, the respose of a liear time ivariat system to ay iput sigal x[] ca be determied from its impulse respose h[] usig the covolutio sum. Covolutio describes how a liear time-ivariat system modifies the iput sequece to produce its output. he covolutio sum Whe = 1,, 1,, 3:, 4
Example: x[] = {1,, 3, 4, 5} ad h[] = { 1,, 1}, 5
Covolutio as a superpositio of scaled ad shifted replicas Each colum is a shifted impulse respose sequece h[ k], < <, multiplied by the value x[] of the iput at = k. he sum of all these scaled ad shifted sequeces produces the output sequece y[]. 6
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More examples y[ ] = x[ ] h[ ] = x[ k] h[ k] k = x(k) 1 3 4 5 6 k 1 3 4 5 6 h(k) k 1 3 4 5 6 h( k) k 1 3 4 5 6 1 3 4 5 6 1 3 4 5 6 x(k) h( k) h(1 k) k compute y() k compute y(1) k 8
8563A SPECRUM ANALYZER 9 khz - 6.5 GHz. Fourier rasforms of sigals Why Fourier methods: Sigal aalysis i the frequecy domai he sigal aalysis i the frequecy domai is to apply the Fourier methods to trasform the time sigal x(t)to frequecy sigal X(f), so as to provide iformatio from aother domai. x(t)= si(πft) Fourier rasform t f 9
amplitude frequecy time ime aalysis Frequecy aalysis he sigal frequecy spectrum X(f) represet the sigal s amplitude variatio at differet frequecy compoets, ad provide more iformatio tha the waveforms i time domai. 1
Cotiuous-time Siusoids he goal of Fourier aalysis of sigals is to break up all sigals ito summatios of siusoidal compoets. xt () = Acos( π Ft + θ ), < t< where A is the amplitude, θ is the phase i radias, ad F is the frequecy. ± jθ Euler s formula, e = cosωθ ± jsiωθ. More coveiet way is to use the agular or radia frequecy measured i radias per secod, Ω = π F. A iθ jωt A iθ jωt xt () = Acos( Ω t+ θ ) = e e + e e 11
Discrete-time Siusoids Obtaied by samplig the cotiuous time siusoid at equally spaced poits t =. F x [ ] = x ( ) = Acos( π F + θ) = Acos( π + θ) F s x [ ] = Acos( π f+ θ) = Acos( ω + θ) 1
Defiitio: x[] is periodic if x[ + N] = x[] for all. x [ + N] = Acos(π f+ π fn+ θ) = Acos( π f+ θ) = x [ ] his is possible if ad oly if π fn = π k, where k is a iteger. Hece: he sequece x [ ] = Acos( π f + θ ) is periodic if ad oly if f = k/n, that is, f is a ratioal umber. If k ad N are a pair of prime umbers, the N is the fudametal period of x[]. 13
Fourier series represetatio of sigal x(t), Fourier Series or, a xt a t b t, ( = 1,,,3,...) () = + ( = 1 cos ω + si ω ) a () = + si( ω + φ) = 1 = xt A t, ( 1,,,3,...) Fourier coefficiets: a / / / / / / x() t dt; a = x()cos t ω tdt; b = x()si t ω tdt; = A = a + b a ta φ = ; b ; 14
Example 1: Expad the rectagle sigal f(t) with its Fourier Series. 1 f(t) - -/ -1 / t a = f()cos( t ω t) dt ω = ( 1)cos( t) dt+ (1)cos( ω t) dt 1 1 = [ si( ωt)] + [si( ωt)] ω ω π As ω =, the a =. 15
b = f()si( t ωt) dt ω = si( t) dt + si( ω t) dt = 1 1 cos( ωt) [ cos( ωt)] ω + ω = [1 cos( π )] π, =, 4,6, = 4 = 1,3,5,, = 1,3,5, π, 4 1 1 1 f( t) = [si( ωt) + si(3 ωt) + si(5 ωt) + + si( ωt) + ]. π 3 5 16
f 1 (t) f (t) t t f t 4 π 4 1 f () t [si( t) si(3 ω t)] π 3 1() = si( ωt) = ω + f 3 (t) f 4 1 1 3( t ) = [si( ω t ) + si(3 ω ) si(5 )] 3 t + ω π 5 t t 17
Example : with the x(t) ad its figure, compute its Fourier Series represetatio. A A + t t xt () = A A t t x(t) A / / t 1) ) 3) 4) A a = xtdt () = ( A tdt ) = A 4. A a = x()cos t tdt = ( A t)cosω tdt 4 ω 4A 4A π = 1, 3, 5, = si = π π =, 4, 6, b = x()si t tdt ω = 4A A = a + b = ; π 18
b 5) θ = arcta = a Amplitude spectrum ( ω A), ad phase spectrum (ω θ ). A A ϕ θ ω 3ω 5ω 7ω ω ω 3ω 5ω 7ω ω he properties of period sigal spectrum: 1) he spectrum is discrete. (discrete) ) Every spectrum lie appears oly at multiples of the fudametal frequecy (harmoics) 3) he height at differet frequecies represets the amplitudes of the harmoics ad phase. For the period sigals i egieerig applicatios, the harmoics amplitude decreases with the icrease of the order. (covergece) 19
Fourier series represetatio of as complex expoetials Recall Euler s formula, For the sigal, ± jωt e = cosωt± jsiωt 1 jωt jωt cos ωt = ( e + e ) j jωt jωt si ωt = ( e e ) a xt a t b t () = + ( cos ω + si ω ) = 1 1 1 ake c = a, c = ( a jb), c = ( a + jb) ω ω the, xt () = c + c e + ce j t j t = 1 = 1
I the compact form, xt ce jωt ( ) =,( =, ± 1, ±,...) = jωt c x t e dt j c = cr + jci = c e φ 1 = () ( c is complex geerally.) A c = cr + ci = c φ arcta I = = θ c R he plot of c k is kow as the magitude spectrum of x(t), while the plot of c k is kow as the phase spectrum of x(t). If c k is real-valued, we ca use a sigle plot, kow as the amplitude spectrum. c k is called the power spectrum. What is the differece betwee the complex expoetial ad geometric represetatio? 1
Parseval s relatio he average power i oe period of x(t) ca be expressed i terms of the Fourier coefficiets usig Parseval s relatio he value of c k provides the portio of the average power of sigal x(t) that is cotributed by the kth harmoic of the fudametal compoet. he graph of c k as a fuctio of F = kf is kow as the power spectrum of the periodic sigal x(t). Because the power is distributed at a set of discrete frequecies, we say that periodic cotiuous-time sigals have discrete or lie spectra.
he Spectrum of Aperiod Sigals he aperiod sigal is the sigal that ot repeats. It is geerally time limited sigal ad has fiite eergy. he frequecy aalysis method is Fourier trasform. We ca thik of a aperiodic sigal as a periodic sigal with ifiite period, π, ω, = ω,cotiuous spectrum, 1 xt () = ce = xte () dte = = jω t jω t jω t Whe, = 1 ω dω π π = ω Δω ω 3
dω jωt he xt () = xte () dte π 1 j t j t = xte () dte d π ω ω ω jωt Fourier trasform pair jωt X ( ω) = xte ( ) dt (Fourier trasform) 1 jωt xt () = X( ω) e dω π (Iverse Fourier trasform) If replace the ω with f = ω/π i X(ω), the Fourier trasform pair is, xt () = X( f) e j π ft df jπ ft X ( f) = x( t) e dt he Fourier represetatio F xt X f () ( ), or F[ xt ( )] = X( f) 4
Questio: What is the meaig of spectrum of aperiod sigal? j For period sigal, xt () = ce ω = j xt () c = c e φ, the t c uit is the same as x(t). For aperiod sigal, () ( ) j π = ft xt X f e df xt () X( f) df c df amplitude Hz he X(f) is the amplitude of per frequecy bad, is the desity fuctio of, the uit is amplitude/hz. 5
Discrete-time Fourier rasform (DF) Aalysis: Sythesis: = jω jω X( e ) = x( ) e Fourier rasform (F) = 1 π jω jω x ( ) = X( e ) e dω π Iverse Fourier rasform (IF) π DF j x ( ) X( e ω ) jω jω jω X ( e ) = X ( e ) + jx ( e ) is complex-valued. X( e ) = X( e ) e R jω jω j X( e ) I jω X I ( e jω ) X ( e jω X ( e ) j ω ) X ( e X ( e R jω jω ) ) 6
Fourier rasform Pairs Sequece Fourier rasform δ() 1 j δ( d ) e ω d 1 a u( ) ( a < 1) jω 1 ae 1 u() + πδ( ω + πk) jω 1 ae k = 1 ( + 1) a u( ) jω (1 ae ) 1 M si[ ω( M + 1) / ] jωm / x( ) = e otherwise si( ω/ ) si ω 1 ω < ω c jω c X ( e ) = π ωc < ω π jω πδ( ω ω + πk) e k = 7
Properties of the DF Liearity: he DF is a liear operator, that is, Frequecy shiftig: Accordig to this property ime shiftig: Covolutio of sequeces: Covolvig two sequeces is equivalet to multiplyig their Fourier trasforms: 8
he frequecy shiftig property of the DF for ω c >. For ω c <, the spectrum is shifted to the left (at lower frequecies). 9
he covolutio theorem of the DF. 3
Summary of the DF properties 31
Summary of Fourier represetatio of sigals he represetatios of magitude ad phase of each compoet as a fuctio of frequecy are kow as the magitude spectrum ad phase spectrum of the sigal. he spectrum from the sigal (Fourier aalysis equatio) ad the sigal from its spectrum (Fourier sythesis equatio) deped o whether the time is cotiuous or discrete ad whether the sigal is periodic or aperiodic. 3
Real example: Micro-machiig 6 Normal coditio 4 Force (N) -.1..3 time (s) 15 power spectrum 1 5 5 1 15 5 3 frequecy 33
1 Faulty coditio 8 Force (N) 6 4 -.1..3 time (s) 5 Power spectrum 15 1 5 5 1 15 5 3 frequecy 34