9/9/99 (T.F. Weiss) Signals and systems This subject deals with mathematical methods used to describe signals and to analyze and synthesize systems.

Transcription

1 9/9/99 (T.F. Weiss) Lecure #: Inroducion o signals Moivaion: To describe signals, boh man-made and naurally occurring. Ouline: Classificaion ofsignals Building-block signals complex exponenials, impulses Signals and sysems This subjec deals wih mahemaical mehods used o describe signals and o analyze and synhesize sysems. Signals are variables ha carry informaion Sysems process inpu signals o produce oupu signals. Today SIGNALS; Nex ime SYSTEMS. Conclusions 2 Demonsraion of differen ypes of signals EKG Microphone CD Oscillaor Pulse generaor AM/FM generaor Swich box Oscilloscope Audio Amplifier Speaker Classificaion of signals Ideniy of he independen variable Time is ofen he independen variable for signals. For example, he elecrical aciviy ofhe hear recorded wih elecrodes on he surface of he ches he elecrocardiogram (ECG or EKG). ECG ampliude (mv) Time (secs) 3 4

2 Generic ime The erm ime is ofen used generically o represen he independen variable ofa signal. The independen variable may be a spaial variable as in an image. Here color informaion is specified as a funcion of posiion. Dimensionaliy of he independen variable The independen variable can be -D (ime in he EKG signal) or 2-D (space x, y in he image), 3-D, or N-D. ECG ampliude (mv) Time (secs) In 6.3 we shall consider largely -D signals, bu signals in many applicaions (e.g., radio asronomy, medical imaging, seismomery) have muliple dimensions. 5 6 Coninuous ime (CT) and discree ime (DT) signals CT signals ake on real or complex values as a funcion of an independen variable ha ranges over he real numbers and are denoed as x(). DT signals ake on real or complex values as a funcion of an independen variable ha ranges over he inegers and are denoed as x[n]. Noe he suble use ofparenheses and square brackes o disinguish beween CT and DT signals. x() x[n] n 7 For example, consider he image shown on he lef and is DT represenaion shown on he righ 2 3 m M 2 3 n N The image on he lef consiss of picure elemens (pixels) each ofwhich is represened by a riple ofnumbers {R,G,B} ha encode he color. Thus, he signal is represened by c[n, m] where m and n are he independen variables ha specify pixel locaion and c is a color vecor specified by a riple ofhues {R,G,B} (red, green, and blue). 8

3 Real and complex signals Signals can be real, imaginary, or complex. An imporan class ofsignals are he complex exponenials: Real and complex signals, con d For boh exponenial CT (x() =e s ) and DT (x[n] =z n ) signals, x is a complex quaniy. To plo x, we can choose o plo eiher is magniude and angle or is real and imaginary pars whichever is more convenien for he analysis. he CT signal x() =e s where is s is a complex number, he DT signal x[n] =z n where z is a complex number. Q. Why do we deal wih complex signals? A. They are ofen analyically simpler o deal wih han real signals. For example, suppose s = jπ/8 and z = e jπ/8, hen he real pars are: R{x()} = R{e jπ/8 } = cos(π/8), R{x[n]} = R{e jπn/8 } = cos[πn/8] x() x[n] n 9 Periodic and aperiodic signals Periodic signals have he propery ha x( + T ) = x() for all. The smalles value of T ha saisfies he definiion is called he period. Shown below are an aperiodic signal (lef) and a periodic signal (righ). Causal and ani-causal signals A causal signal is zero for < and an ani-causal signal is zero for > x() x() x() x() Righ- and lef-sided signals A righ-sided signal is zero for <T and a lef-sided signal is zero for >T where T can be posiive or negaive. T x() x() T T 2

4 Bounded and unbounded signals x() Causal Ani-causal x() Even and odd signals Even signals x e () and odd signals x o () are defined as x e () =x e ( ) and x o () = x o ( ). Unbounded Unbounded x e () x o () Bounded Bounded 3 4 Any signal is a sum ofunique odd and even signals. Using yields x() =x e ()+x o () and x( ) =x e () x o (), x e () = 2 (x()+x( )) and x o() = (x() x( )). 2 x() x( ) /2 /2 x e () x o () -/2 Building-block signals We will represen signals as sums ofbuilding-block signals. Imporan families of building-block signals are he eernal, complex exponenials and he uni impulse funcions. Eernal, complex exponenials These signals have he form x() =Xe s for all and x[n] =Xz n for all n, where X, s, and z are complex numbers. We illusrae he richness ofhis class offuncions for CT signals; DT signals are similarly rich. In general s is complex and can be wrien as s = σ + jω, where σ and ω are he real and imaginary pars of s. 5 6

5 Eernal, complex exponenials real s If s = σ is real and X is real hen x() =Xe σ, and we ge he family of real exponenial funcions. Eernal, complex exponenials imaginary s If s = jω is imaginary and X is real hen x() =Xe jω = X(cos ω + j sin ω), and we ge he family of sinusoidal funcions. Eernal, complex exponenials complex s If s = σ + jω is complex and X is real hen For x() =Xe s, R{x()} = Xe σ cos ω is ploed for differen values of s superimposed on he complex s-plane. jω Complex s-plane σ x() =Xe (σ+jω) = Xe σ (cos ω + j sin ω), and we ge he family of damped sinusoidal funcions. 7 8 For x() = Xe s, I{x()} = Xe σ sin ω is ploed for differen values of s superimposed on he complex s-plane. Eernal complex exponenials why are hey imporan? jω Complex s-plane Almos any signal ofpracical ineres can be represened as a superposiion (sum) ofeernal complex exponenials. σ The oupu ofa linear, ime-invarian (LTI) sysem (o be defined nex ime) is simple o compue ifhe inpu is a sum ofeernal complex exponenials. Eernal complex exponenials are he eigenfuncions or characerisic (unforced, homogeneous) responses of LTI sysems. 9 2

6 Building-block signals Uni impulse definiion The uni impulse δ(), aka he Dirac dela funcion, is no a funcion in he ordinary sense. I is defined by he inegral relaion f()δ() d = f(), and is called a generalized funcion. The uni impulse is no defined in erms ofis values, bu is defined by how i acs inside an inegral when muliplied by a smooh funcion f(). To see ha he area ofhe uni impulse is, choose f() = in he definiion. We represen he uni impulse schemaically as shown below; he number nex o he impulse is is area. Uni impulse δ() 2 Uni impulse narrow pulse approximaion To obain an inuiive feeling for he uni impulse, i is ofen helpful o imagine a se of recangular pulses where each pulse has widh and heigh / so ha is area is. p () The uni impulse is he quinessenial all and narrow pulse! 22 Uni impulse inuiing he definiion To obain some inuiion abou he meaning ofhe inegral definiion ofhe impulse, we will use a all recangular pulse ofuni area as an approximaion o he uni impulse. p () f() f() f()p () f() Uni impulse he shape does no maer There is nohing special abou he recangular pulse approximaion o he uni impulse. A riangular pulse approximaion is jus as good. As far as our definiion is concerned boh he recangular and riangular pulse are equally good approximaions. Boh ac as impulses. Area = As he recangular pulse ges aller and narrower, lim f()p () d f() = f()

7 Uni impulse he values do no maer The values ofhe approximaing funcions do no maer eiher. The funcion on he lef has uni area and akes on he arbirary value A for =. The funcion on he righ, which we shall encouner frequenly in laer lecures, has he propery ha i has non-zero values a mos ofis values, all bu a counably infinie number ofpoins, bu sill acs as a uni impulse. Uni impulse wha do we need i for? The uni impulse is a valuable idealizaion and is used widely in science and engineering. Impulses in ime are useful idealizaions. A Area = sin(π/) π Impulse ofcurren in ime delivers a uni charge insananeously o a nework. A 2 2 Impulse offorce in ime delivers an insananeous momenum o a mechanical sysem. Wha all hese approximaions have in common is ha as ges small he area ofeach funcion occupies an increasingly narrow ime inerval cenered on = Impulses in space are also useful. Impulse ofmass densiy in space represens a poin mass. Impulse ofcharge densiy in space represens a poin charge. Impulse ofligh inensiy in space represens a poin ofligh. Uni sep Inegraion ofhe uni impulse yields he uni sep funcion u() = δ(τ) dτ, which is defined as { if < u() = if. We can imagine impulses in space and ime. Uni impulse δ() Uni sep u() Impulse ofligh inensiy in space and ime represens a brief flash ofligh a a poin in space

8 Uni impulse as he derivaive of he uni sep As an example of he mehod for dealing wih generalized funcions consider he generalized funcion x() = d d u(). Since u() is disconinuous, is derivaive does no exis as an ordinary funcion, bu i does as a generalized funcion. To see wha x() means, pu i in an inegral wih a smooh esing funcion y() = f() d u() d, d and apply he usual inegraion-by-pars heorem y() =f()u() u() d f() d, d o obain y() =f( ) d d f() d = f(). 29 Uni impulse as he derivaive of he uni sep, con d The resul is ha f() d u() d = f(), d which, from he definiion of he uni impulse, implies ha δ() = d d u(). Tha is, he uni impulse is he derivaive ofhe uni sep in a generalized funcion sense. 3 Successive inegraion of he uni impulse Successive inegraion ofhe uni impulse yields a family offuncions. Inegraion on Uni impulse Uni sep Uni ramp Uni parabola δ() u() u() 2 2! u() n (n )! u() Building-block signals can be combined o make a richpopulaion of signals Eernal complex exponenials and uni seps can be combined o produce causal and ani-causal decaying exponenials. x() =e σ cos(ω)u() x() =e +σ cos(ω)u( ) Laer we will alk abou he successive derivaives of δ(), bu hese are oo horrible o conemplae in he firs lecure. 3 32

9 Conclusions Uni seps and uni ramps can be combined o produce pulse signals. u() u( ) u() 2( )u( )+( 2)u( 2) 2 We are awash in a sea ofsignals. Signal caegories ideniy ofindependen variable, dimensionaliy, CT & DT, real & complex, periodic & aperiodic, causal & ani-causal, bounded & unbounded, even & odd, ec. Building block signals eernal complex exponenials and impulse funcions are a rich class of signals ha can be superimposed o represen virually any signal ofphysical ineres