信号与系统信号与系统信号与系统 (2).pdf

BEIJING JIAOTONG UNIVERSITYThe Course Group of Signals and Systems,Beijing Jiaotong University.P.R.CHINA.Copyright 2020Signals and Systems Deterministic signal and random signal Continuous-time signal and discrete-time signal Periodic signal and nonperiodic signal Energy signal and power signalClassification of Signals The deterministic signal is a signal about which there is no uncertainty to its value at any time.)t(x1t x tttt=()sin,R)t(x11t0tx tt t=()esin,0Ok k xkx kk k=2 sin,IDeterministic signal and random signalOtNoise signalsAudio signals The random signal is a signal about which there is uncertainty before it occurs.Deterministic signal and random signalDiscussion：Are these signals in the practical applications(such as ECG signals and audio signals)are generally deterministic signals or random signals?Deterministic signal and random signal Deterministic signal and random signal Continuous-time signal and discrete-time signal Periodic signal and nonperiodic signal Energy signal and power signalClassification of Signals The continuous-time signal is defined for all time t.It is represented as x(t),tRt is a continuous variableThere exist finite discontinuous points1102123t)t(x310v011v011t)t(xContinuous-time signal and discrete-time signal The discrete-time signal is defined only at discrete instants of time.It is represented as xk,kI.k xOk k xOkk is an integer variableContinuous-time signal and discrete-time signal Relation between the C-T signals and D-T signalsSampling=x kx tt kT()T0t)t(xT2T0t)t(xContinuous-time signal and discrete-time signal Deterministic signal and random signal Continuous-time signal and discrete-time signal Periodic signal and nonperiodic signal Energy signal and power signalClassification of Signals Continuous-time periodic signal x(t)：if x(t),a function of time,satisfies the condition+=tx tTx t for all)()(Where T is a positive constant.Clearly,if this condition is satisfied for T=T0,then it is also satisfied for T=2T0,3T0,4T0,.The smallest positive value T is termed as the fundamental period of x(t).The fundamental frequency and angular frequency of x(t)are defined as Periodic signal and nonperiodic signal=fT1/(Hz)=fT/(rad/s)22)t(xO 0T0TtAPeriodic signal of rectangular wavePeriodic signal of triangular wave=T(s)=f/2(rad/s)2=f1/(Hz)=/(rad/s)26842862t042)t(x=T4(s)=f1/4(Hz)Periodic signal and nonperiodic signal Discrete-time periodic signal xk：if the discrete-time signal xk satisfies the condition+=kx kNx k r integer)foWhere N is a positive integer.The smallest positive integer N is termed as the fundamental period of xk.The fundamental angular frequency of xk is defined as=N/(rad)2measured in radians.Periodic signal and nonperiodic signal k x002k2010188 k x0kDiscrete rectangular waveDiscrete triangular wave=N(rad)/0.12=N20=N10=N(rad)/0.22Periodic signal and nonperiodic signal nonperiodic signal：Any signal x(t)for which no value of T satisfies the periodic condition is termed as an aperiodic,or nonperiodic signal.Any signal xk for which no value of N satisfies the periodic condition is termed as an aperiodic,or nonperiodic signal.12132 k x02321kPeriodic signal and nonperiodic signalto)(txtuning forks testsignals generated by the tuning forksPeriodic signal and nonperiodic signalExample 1.1：Is the signal x(t)=sin(0t)periodic signal?Solution:+=+x tTtT0()sin()0T=m2，m is an integer=Tm20Therefore,sin(0t)is a periodic signal.C-T sinusoidal signal is always periodic.=+tT00sin()+=x tTx t()()If there existsThen=T|20Periodic signal and nonperiodic signal+=+x kNkN0sin()N=m2,integer m=Nm20,integer m,N0/2 can be represented as a rational number.=+kN00sin()Example 1.2：Is the signal xk=sin(0k)periodic signal?Solution:+=x kNx k If there existsThen Therefore,sin(0k)may be a periodic signal,only if D-T sinusoidal signal may be periodic.Periodic signal and nonperiodic signalProblem 1.1:Is periodic?If yes,find its period N.A.Nonperiodic signalB.Periodic signal.N=6C.Periodic signal.N=24=x kk34 sin()=Nm62100=/3Periodic signal and nonperiodic signal Deterministic signal and random signal Continuous-time signal and discrete-time signal Periodic signal and nonperiodic signal Energy signal and power signalClassification of SignalsEnergy signal and power signalFor C-T signal x(t),its energy E and power P are defined as For D-T signal xk,its energy E and power P are defined as=Ex ttx ttTTT|()|dlim|()|d22=TPx ttTTT2lim|()|d12=Ex kx kkkNNN|lim|22=NPx kkNNN2lim|12A signal is termed as an energy signal if and only if its energy Esatisfies the condition A signal is termed as a power signal if and only if its power Psatisfies the condition Energy signal and power signalE0P0E()=P(0)Power signalEnergy signalEnergy signal and power signalE0=P(0)P0E()The periodic signal x(t)=Acos(0t)is a power signal.Example 1.3:Determine the energy and power of signal x(t)=Acos(0t),(A0).Solution:As there are infinite periods for the periodic signal x(t),the energy E of the periodic signal x(t)is infinite().Acos(0t)is a periodic signal,and its fundamental period is T0=2/|0|.Obviously,the energy E0in one period 0,T0 is=Ex ttAttTT|()|dcos()d000022200=A T202The power P of the periodic signal x(t)is calculated as=TPAE2002Energy signal and power signalThe energy E is infinite,and its power P is a positive number(4).Therefore,the direct current signal x(t)is a power signal.Example 1.4:Categorize the signal x(t)=2 as an energy signal or a power signal.Its energy E of the direct current signal x(t)is calculated asAll the periodic signals and direct current signals are power signals=22=2lim|()|dTTTEx tt=tTTTlim2 d2=2lim12|()|dTTTPTx tt=TTT2lim222=TtTTT2lim2 d12Solution:Its power P of the direct current signal x(t)is calculated asEnergy signal and power signalExample 1.5:Categorize the signal xk=3kas an energy signal or a power signal.Because its energy E and power P of the signal are infinite,the signal xk is not an energy signal,neither a power signal.lim|2Ex kNkNN=kNNkNlim(3)2+=NPkNNkN21lim(3)12=Its energy E of the signal xk is calculated asSolution:Its Power P of the signal xk is calculated asEnergy signal and power signal Deterministic signal and random signal Continuous-time signal and discrete-time signal Periodic signal and nonperiodic signal Energy signal and power signalClassification of SignalsAcknowledgmentsMaterials used here are accumulated by authors for years with helpfrom colleagues,media or other sources,which,unfortunately,cannotbe noted specifically.We gratefully acknowledge those contributors.Classification of Signals