Sigals ad Systems Sigals ad Systems Sigals are variables that carry iformatio Systemstake sigals as iputs ad produce sigals as outputs The course deals with the passage of sigals through systems T-6.4 / Chapter Sigals Sigalsdescribe a wide variety of (physical) pheomea Sigals may be represeted i may ways Iformatio i a sigal is cotaied i a patter of variatios of some form, i.e. variatio of voltages over time i a circuit applied force ad resultig velocity of a car fluctuatios of acoustic pressure i speech productio by huma vocal mechaism Examples of Sigals A simple RC circuit with source voltage v s ad capacitor voltage v c A automobile respodig to a applied force f from the egie ad to a fractioal force ρv proportioal to the velocity v T-6.4 / Chapter 3 T-6.4 / Chapter 4 Examples of Sigals Examples of Sigals Example of a recordig of speech: The sigal represets acoustic pressure variatios as a fuctio of time for the spoke words: should we chase A moochromatic picture T-6.4 / Chapter 5 T-6.4 / Chapter 6
Represetatio of Sigals Sigals are represeted mathematically as fuctios of oe or more idepedet variables We will geerally refer the idepedet variable as time Two basic types of sigals: Cotiuous-time (CT) sigals ad Discrete-time (DT) sigals Cotiuous-Time ad Discrete-Time Sigals Symbol t is used to deote the idepedet variable of cotiuous-time sigals Symbol is used to deote the idepedet variable of discrete-time sigals Cotiuous-time sigal: x(t) Discrete-time sigal: x[ x[ is a sequece, defied oly for iteger values of T-6.4 / Chapter 7 T-6.4 / Chapter 8 Cotiuous-Time ad Discrete-Time Sigals Digital Image Two-dimesioal (digital) sigal: Itesity is a fuctio of spatial coordiates T-6.4 / Chapter 9 T-6.4 / Chapter Sigal Eergy ad Power Sigals are directly related to physical quatities capturig power ad eergy i a physical system Istataeous power, e.g., p ( t) = v( t ) i( t) = v R ( t) where v(t) ad i(t) are the voltage ad curret, respectively, across the resistor of resistace R Sigal Eergy ad Power Total eergy expeded over the time iterval t < t < t t p( t) dt = t t R v ( t ) dt Average power over this time iterval t t p t dt t t ( ) = t t R v t dt t ( ) t t T-6.4 / Chapter T-6.4 / Chapter
Sigals with Complex Values Total eergy over time iterval t < t < t of a cotiuous-time sigal x(t) t t x( t ) dt where x is the magitude of the (possibly complex) umber x Similarly, the total eergy of a discrete-time sigal x[ over time iterval t < t < t is x[ = T-6.4 / Chapter 3 Total Eergy over Ifiite Time Iterval I may systems we are iterested i examiig power ad eergy i sigals over ifiite time iterval ad T + E = lim x ( t ) dt = x ( t ) dt T T + N E = lim x[ N = N + = T-6.4 / Chapter 4 Total Power over Ifiite Time Iterval The time-averaged power over ifiite time iterval is defied as ad P P T = x t T T = lim ( ) T N N = N + N lim + dt x[ Three Importat Classes of Sigals (3) Sigals with fiite total eergy, E Such a sigal must have zero average power E P = lim = t T Example: A sigal that takes the value for < t< ad otherwise. I this case E ad P = = T-6.4 / Chapter 5 T-6.4 / Chapter 6 Three Importat Classes of Sigals (3) Sigals with fiite average power P If P > the, of ecessity, E = If there is ozero average eergy per uit time (i.e. ozero power), the itegratig or summig this over a ifiite time iterval yields a ifiite amout of eergy Three Importat Classes of Sigals 3(3) There are also sigals for which either or are ifiite P E A simple example is the sigal x(t) = t Example: Costat sigal x[ = 4 has ifiite eergy, but average power = 6 P T-6.4 / Chapter 7 T-6.4 / Chapter 8
Trasformatios of the Idepedet Variable Time shift: x[ - ] Time reversal: x[- obtaied from x[ Time scalig: x(t); x(t); x(t/) Trasformatio: x(t) -> x(αt + β) preserves the shape of x(t) ; liear stretchig if α < or liear compressio if α > time reversal ifα < time shift if β is ozero Time-Shift of a Discrete-Time Sigal Origial sequece x[ Delayed sequece x[- ] T-6.4 / Chapter 9 T-6.4 / Chapter Time-Reversal of a Discrete-Time Sigal Time-Scalig of a Cotiuous-Time Sigal T-6.4 / Chapter T-6.4 / Chapter Examples of Operatios Sigal x(t) Advace x(t+) (shiftto the left) Reversed versio of x(t+): x(-t+) Compressed versio of x(t): x((3/)t) Liearlycompressed ad advaced sigal: x((3/)t+) Periodic Sigals A sigal x(t) is periodic with period T if x(t) = x(t+t) for all values of t The fudametal period T of x(t) is the smallest positive value of T for which the above equality holds A sigal x(t) that is ot periodic is referred to as a aperiodic sigal T-6.4 / Chapter 3 T-6.4 / Chapter 4
Examples of Periodic Sigals x[ + 3] Eve ad Odd Sigals Cotiuous ad discrete eve sigals: x(-t) = x(t) or x[- Cotiuous ad discrete odd sigals: x(-t) = -x(t) or x[- = -x[ A odd sigal must be ecessarily zero at t = or = T-6.4 / Chapter 5 T-6.4 / Chapter 6 Examples of Eve ad Odd Sigals Eve-Odd Decompositio of a Sigal Eve part of x(t): Ev{x(t)} = [x(t)+x(-t)] / Odd part of x(t): Od {x(t)} = [x(t)-x(-t)] / T-6.4 / Chapter 7 T-6.4 / Chapter 8 Cotiuous-Time Complex Expoetial ad Siusoidal Sigals Complex expoetial sigal: x(t) = C e at where C ad a are i geeral complex umbers Real expoetial sigals: C ad a are real Growig expoetial: a> Decayig expoetial: a< T-6.4 / Chapter 9 Periodic Complex Expoetial ad Siusoidal Sigals jωt x( t) = e Number a is purelyimagiary: x(t) is periodic with period T : Sice e jω ( t + T ) j ωt jω T e = e e j ω t j ( t+ T ) = e ω j It follows that for periodicity, we must have e ω T = If ω = the x(t)= which is periodic for ay value of T. If ω is ozero, the the fudametal period T of x(t) is π T = ω T-6.4 / Chapter 3
Periodic Complex Expoetial ad Siusoidal Sigals A sigal closelyrelated to the periodic complex expoetialis the siusoidal sigal x( t) = Acos( ω t + φ) It is commo to write ω = πf f has uits of cycles per secod or Hertz (Hz) ω has uits of radias per secod Fudametal period: T Fudametal frequecy: ω =/T Illustratio: Siusoidal Sigal T < T < T3 ω > ω > ω 3 T-6.4 / Chapter 3 T-6.4 / Chapter 3 Discrete-Time Complex Expoetial Sigals Siusoidal Sigals (Sequeces) x[ β = Cα = Ce ; α = e β x[ = cos(π /) α > <α < < α < x[ = cos(8π /3) α < x [ = cos( /6) T-6.4 / Chapter 33 T-6.4 / Chapter 34 Geeral Complex Expoetial Sigals θ C = C e j, α = α e j ω ( ω + θ ) + j C α ( ω θ ) Cα = C α cos si + α > α < Periodicity Properties of Discrete-Time Complex Expoetials Cotiuous-time exp(jω t):. Icreasig ω icreases the rate of oscillatio. exp(jω t) is periodic for ayvalueof ω CosiderDT complex expoetial with frequecy ω +π: j ( ω j j j e + π ) ω e e π ω = = e The expoetial at frequecyω +π is the same as that of frequecy ω I CT case, the expoetial sigalsexp(jω t) are all distict for distict values of ω T-6.4 / Chapter 35 T-6.4 / Chapter 36
Periodicity Properties of Discrete-Time Complex Expoetials DT Siusoidal Sequeces I DT case, the sigalsare otdistict, as the sigal with frequecy ω is idetical to the sigalsto the sigalswith frequeciesω + π, ω + 4π etc. Cosiderig complex expoetialswe eed oly cosider a frequecy iterval of legth π, i.e., ω < π or -π ω < π T-6.4 / Chapter 37 T-6.4 / Chapter 38 Some Basic Sequeces The Uit Impulse ad the Uit Step Fuctios Uit sample sequece, δ[ =, = Uit step sequece, µ [ =, <... T-6.4 / Chapter 4 Relatios betwee Basic Sequeces Uit sample ad uit step sequeces are related as follows: δ [ ] = µ [ µ [ ] µ[ ] = δ[ m] m= Relatios betwee Basic Sequeces The uit sample is the first differece of the uit step: δ [ ] = µ [ µ [ ] µ[ µ [ ]...... µ[ D δ[ + - The above relatios ca be implemeted with simple computatioal structures cosistig of basic arithmetic operatios δ[ µ [ ] Realizatio T-6.4 / Chapter 4 T-6.4 / Chapter 4
Relatios betwee Basic Sequeces Uit step is the ruig sum of the uit step: µ [ ] = δ[ m] = δ [ m] + δ [ = µ [ ] + δ [ m= m= δ [ + D Realizatio µ[ µ [ ] Relatios betwee Basic Sequeces By chagig the variable of summatio i the ruig sum from m to k=-m, the discrete-time uit step ca be writte i terms of the uit sample as µ [ = = k = k= δ[ k] δ [ k] T-6.4 / Chapter 43 T-6.4 / Chapter 44 Cotiuous-Time ad Discrete-Time Systems A system ca be viewed as a process i which iput sigals are trasformed by the system resultig i other sigals as outputs x(t) Cotiuous-time system y(t) Examples Example.8: A RC circuit Example.9: A forces affectig the car Example.: A balace i a bak accout Example.: Digital simulatio of the differetial equatio x[ Discrete-time system y[ T-6.4 / Chapter 45 T-6.4 / Chapter 46 Mathematical Descriptios of Systems Classes of systems that have two importat characteristics: ) The systems have properties ad structures that ca be exploited to gai isight ito their behavior ad to develop effective tools for their aalysis ) May systems of practical importace ca be accurately modeled usig these systems Tools are developed for a particular class of systems referred to as liear ad time-ivariat systems T-6.4 / Chapter 47 Itercoectios of Systems Series or cascade itercoectio Iput System System Output Parallel itercoectio System Iput + Output System T-6.4 / Chapter 48
Itercoectios of Systems Combiatio of parallel ad cascade itercoectios Iput System System 3 System Feedback itercoectio Iput + System System + System 4 Output Output Basic System Properties Systems with ad without memory Ivertibility ad iverse systems Causality Stability Time ivariace Liearity Covolutio T-6.4 / Chapter 49 T-6.4 / Chapter 5 Memoryless Systems Idetity System Output for each value of the idepedet variable at a give time is depedet oly o the iput at the same time Example : y[ = ( x[ x [ ) Cotiuous time : y( t) = x( t) Discrete time : y[ A idetity system is a simple memoryless system whose output is idetical to its iput T-6.4 / Chapter 5 T-6.4 / Chapter 5 Accumulator y [ x[ k] = k = A accumulator is a discrete-time system with memory Delay y[ ] The output is the delayed versio of the iput Realizatio usig a memory locatio or register with delay T x [ T x[ ] T-6.4 / Chapter 53 T-6.4 / Chapter 54
Arbitrary Delay y[ k] A arbitrary delay of k time istats ca be realized usig a shift register of legth k x [ T T T x[ k] Accumulator or Ruig Sum y[ = k = x[ k] + x[ y [ = y[ ] + x[ The accumulator must remember the ruig sum of previous iput values to obtai the output at curret time T-6.4 / Chapter 55 T-6.4 / Chapter 56 Ivertible Systems Ivertible Systems x[ System y[ Iverse system w [ δ[ System u[ Iverse system δ[ If a system is ivertible, the a iverse system exists that whe cascaded with the origial system yields a output w[ equal to iput x[ u [ = δ [ k] δ[ = u[ u[ ] k = Accumulator is a ivertible discretetime system T-6.4 / Chapter 57 T-6.4 / Chapter 58 Causality A system is causal if the output at ay time depeds oly o the values of the iput at the same time ad i the past Example: Accumulator ad delay are causal systems All memoryless systems are causal Nocausality A system is ocausal if the output at ay time depeds also o the future values of the iput Nocausal systems are physically ot realizable Example : y[ x[ + ] T-6.4 / Chapter 59 T-6.4 / Chapter 6
Nocausality A ocausal averagig filter M y [ = x[ k] M + k = M The filter ca be realized with a delay of M samples Stability Iformally, a system is stable if small iputs lead to resposes that do ot diverge y(t) x(t) Pedulum x (t ) y(t) Iverted pedulum T-6.4 / Chapter 6 T-6.4 / Chapter 6 Time Ivariace A system is time ivariat if a time shift i the iput sigal results i a idetical time shift i the output sigal y [ = T ( x[ ) ( x[ ]) y[ ] = T The system properties do ot chage with time Time Ivariace A time ivariat cotiuous-time system [ x( )] y ( t) = si t A time variat discrete-time system y [ Coefficiet is chagig with time T-6.4 / Chapter 63 T-6.4 / Chapter 64 Liearity A liear system is a system that possesses the importat property of superpositio Additivity: The respose to x (t)+x (t) is y (t)+y (t) Scalig or homogeeity: The respose to ax (t) is ay (t) where a is ay complex costat Liearity Combiig the two properties of superpositio ito a sigle statemet Discrete-time: ax [ + bx[ ay [ + by[ where a ad b are ay complexcostats The superpositio property holds for liear systems i cotiuous ad discrete time T-6.4 / Chapter 65 T-6.4 / Chapter 66
Liearity Basic Operatios o Sequeces a x [ ] T[ ] y [ ] [ ] ay b + x [ ] T[ ] y [ ] by [ ] a x ax [ ] [ ] x [ ] bx [ ] ay + by [ ] [ b + T[ ] ay [ + by[ ax + bx [ ] [ Additio: Multiplicatio: Uit delay: x [ + x [ + x[ x [ a x [ ax[ x [ D x[ ] T-6.4 / Chapter 67 T-6.4 / Chapter 68 Arbitrary Sequece x[ x[-3] x[] Arbitrary Sequece x[ x[-3] x[] -7-6 -5-4 -3 - - 3 4 5 6 7 x[4] -7-6 -5-4 -3 - - 3 4 5 6 7 x[4] A arbitrary sequece x[ ca be expressed as a superpositio of scaled versios of shifted uit impulses, δ[-k] T-6.4 / Chapter 69 [ 3] δ [ + 3] x[ ] δ [ ] x[ 4] δ[ 4] x + I geeral: + k= x [ k] δ [ k] T-6.4 / Chapter 7 - Covolutio x[ is represeted as a superpositio of scaled versios of shifted uit impulses, δ[-k] Liearity: The respose of a liear system to x[ will be the superpositio of the scaled resposes of the system to each of these shifted impulses Time ivariace: The resposes of a time-ivariat system to time-shifted uit impulses are the timeshifted versios of oe aother Covolutio The uit impulse respose of a system is h[ T( ) δ () h() T-6.4 / Chapter 7 T-6.4 / Chapter 7
Covolutio y [ = T ( x[ ) = T x[ k] δ [ k] k = Additivity: Homogeeity : Shift ivariace : T k = x[ k= x k= ( ) y [ k] δ[ k] ( δ[ k ) y [ = k] T ] y [ = [ k] h[ k] T-6.4 / Chapter 73