1、ContentsnDescription of signalsnTransformations of the independent variablenSome basic signalsnSystems and their mathematical modelsnBasic systems properties1.1 Continuous-Time and Discrete-Time Signals1.1.1 Examples and Mathematical Representation(1)A simple RC circuitSource voltage Vs and Capacito
2、r voltage VcA.Examples(2)An automobileForce f from engineRetarding frictional force VVelocity V(3)A Speech Signal(4)A Picture(5)vital statistics(人口统计人口统计)NotenIn this book,we focus on our attention on signals involving a single independent variable.nFor convenience,we will generally refer to the ind
3、ependent variable as time,although it may not in fact represent time in specific applications.0tA)(tx10t)(tx0 ,00 ,)(ttatetxRttAtx),sin()(B.Two basic types of signals t:continuous timex(t):continuum of value1.Continuous-Time signal2.Discrete-Time signal n:discrete timexn:a discrete set of values(seq
4、uence)199020028001600Example1:1990-2019年的某村农民的年平均收入年的某村农民的年平均收入0124.08.01 2 3n4 5 6 7111098nx0124.08.0123t)(txsk81SamplingExample2:xn is sampled from x(t)Why DT?(1)Function Representation Example:x(t)=cos 0t xn=cos 0n x(t)=ej 0t xn=ej 0n(2)Graphical Representation Example:(See page before)(3)Sequenc
5、e-representation for discrete-time signals:xn=-2 1 3 2 1 1 or xn=(-2 1 3 2 1 1)C.Representation3Note:nSince many of the concepts associated with continuous and discrete signals are similar(but not identical),we develop the concepts and techniques in parallel.nThere are many other signals classificat
6、ion:nAnalog vs.DigitalnPeriodic vs.Aperiodic nEven vs.OddnDeterministic vs.Randomn1.1.2 Signal Energy and PowerInstantaneous power:)()(1)()()(22tiRtvRtitvtpLet R=1,so )()()()(222txtvtitp+R_)(tv)(tiEnergy:t1 t t2212121)()()(22ttttttdttxdttvdttpAverage Power:2121)(1)(121212ttttdttxttdttpttTotal Energy
7、Average Power2121211nnnnxnnDefinition:212)(ttdttx21212)(1ttdttxtt212nnnnxContinuous-Time:(t1 t t2)Discrete-Time:(n1 n n2)We will frequently find it convenient to consider signals that take on complex values.whenNNnNnxNP2121limdttxdttxETTT22)()(limntTotal EnergyAverage PowernNNnNnxnxE22limTTTdttxTP2)
8、(21limNote:vIt is important to remember that the terms“Power”and“energy”are used here independently of the quantities actually are related to physical energy.vWith these definitions,we can identify three important class of signalsa.finite total energyb.finite average powerc.infinite total energy,inf
9、inite average powerE02limTEPTP),0(limTTTPEthenPifPE,Read textbook P71:MATHEMATICAL REVIEWHomework：P57-1.21.2.1 Examples of Transformations1.Time Shiftx(t-t0),xn-n0t00 DelayTime Shiftx(t)and x(t-t0),or xn and xn-n0:nThey are identical in shapenIf t00,x(t-t0)represents a delayn00,xn-n0 represents a de
10、laynIf t00,x(t-t0)represents an advance n00)Time Scalingx(at)(a0 )Stretch if a1How about the discrete-time signal?xnGenerally,time scaling only for continuous time signals x2nxnx2n0 1 2 3 4 5 6nThis is also called decimation of signals.（信号的抽取）信号的抽取）xn/2xn2 2 2Example0 11tx(t)Solution 1：Solution 2：So
11、lution 1：Solution 2：0 11tx(t)01tx(t-1/2)1/2 3/20 1tx(3t-1/2)1/6 1/20 11tx(t)0 1/31tx(3t)0 1tx(3t-1/2)1/6 1/2()f t12121 122 1t0(1)f t)1(tf 01 122t)31(tf t0131 232)(tf t01121 2 t1 12012)1(tf)3(tf12t031 32)31(tf 012t3231shiftreversalScalingreversalshiftScalingreversalshiftScalingExamplef(t)f(1-3t)1.2.2
12、 Periodic SignalsA periodic signal x(t)(or xn)has the property that there is a positive value of T(or integer N)for which:x(t)=x(t+T),for all txn=xn+N,for all nIf a signal is not periodic,it is called aperiodic signal.Examples of periodic signalsCT:x(t)=x(t+T)DT:xn=xn+NPeriodic SignalsThe fundamenta
13、l period T0(N0)of x(t)(xn)is the smallest positive value of T(or N)for which the equation holds.Note:x(t)=C is a periodic signal,but its fundamental period is undefined.Examples of periodic signals1.tAtx83sin)(It is periodic signal.Its period is T=16/3.2.0,00,cos)(ttttxIt is not periodic.3.tBtAtx41s
14、in31cos)(8,621TTx(t)is periodic.Its period is24TThe smallest multiples of T1 and T2 in common4.tttx2coscos)(21,2 TTIt is aperiodic,too.There is no the smallest multiples of T1 and T2 in common5.nnx4cosx(t)is aperiodic.6.nnx83cosIt is periodic with period N=16.-20-15-10-505101520-101-20-15-10-5051015
15、20-101-20-15-10-505101520-202CostCos2tcost+cos2t1.2.3 Even and Odd SignalsNote:An odd signal must necessarily be 0 at t=0,or n=0.ie.x(t)=0,or xn=0.Even signal:x(-t)=x(t)or x-n=xn Odd signal:x(-t)=-x(t)or x-n=-xnEven-Odd Decomposition Any signal can be expressed as a sum of Even and Odd signals.)()(2
16、1)()(txtxtxtxEve)()(21)()(txtxtxtxOdo21nxnxnxnxEve21nxnxnxnxOdox(t)=xeven(t)+xodd(t)xn=xevenn+xoddnExample of the even-odd decompositon Example of the even-odd decompositon Homework：P57-1.9 1.10 1.21(a)(b)(c)(d)1.22(a)(b)(c)(g)1.23 1.241.3 Exponential and Sinusoidal Signals1.3.1 Continuous-time Comp
17、lex Exponential and Sinusoidal SignalsThe continuous-time complex exponential signal is of the formatCetx)(where C and a are,in general,complex numbers.A.Real Exponential Signalsx(t)=Ceat (C,a are real value)a0a0growingdecayingB.Periodic Complex Exponential and Sinusoidal Signalsx(t)=ej 0tx(t)is per
18、iodic for x(t)=x(t+T),and its fundamental period is .x(t)=Ceat,C=1,a=j 0(purely imaginary)002wT(1)For e j 0t00200001TdtdteETTtjwperiod110periodperiodETPif 0=0,x(t)=1,then it is periodic for any T0.(2)x(t)=Acos(0t+)002 fw 0rad/s f0HzEulers Relation:e j 0t =cos 0t+j sin 0t and cos 0t=(e j 0t+e-j 0t)/2
19、 sin 0t=(e j 0t -e-j 0t)/2j We havetjjtjjeeAeeAtA0022)cos(0Im)sin(Re)cos()(0)(000twjtwjeAtwAeAtwAif c is a complex number,Rec denotes its real part;Imc denotes the imaginary part.-20-15-10-505101520-101-20-15-10-505101520-101-20-15-10-505101520-101 1 2 0r10 1-1 0-1xn=C nB.Sinusoidal Signals Complex
20、exponential:xn=e j 0n =cos 0n+jsin 0n Sinusoidal signal:xn=cos(0n+)12/2cos(nnx)31/8cos(nnx)6/cos(nnxC.General Complex Exponential SignalsIf let C and in polar formviz.C=|C|ej And =|ej 0,then xn=C n =|C|ncos(0n+)+j|C|nsin(0n+)Real or Imaginary of Signal|1growingdecaying1.3.3 Periodicity Properties of
21、 Discrete-time Complex ExponentialsContinuous-time:ej 0t ,T=2/0Discrete-time:ej 0n,N=?Two properties of continuous-time signal ej 0t:(1)ej 0t is periodic for any value of 0.(2)the lager the magnitude of 0,the higher is the rate of oscillation in the signal.Periodicity Properties Calculate period:By
22、definition:e j 0n=e j 0(n+N)thus e j 0N=1 or 0N=2 m So N=m(2/0)Condition of periodicity:0/2 is rationalFundamental periodFrom these figures,we can conlude:Signals oscillate rapidly when 0=,3,(high-frequency);signals oscillate slowly when 0=0,2,4,(low-frequency)0002on the most occasions we will use t
23、he intervalHarmonically related complex exponentialsNote:,2,1,0,)/2(kennNjkknnkNkNnNjNNnjNnjenenenn/)1(21/42/210,1Comparison of the signals e j 0t and e j 0n,see P28 Table 1.1So,Only N distinct periodic exponentials in the setForExamples:Determine the following equations fundamental period:(1)nnxttx
24、318cos318cos)(T=31/4N=31(2)6cosnnxIt is not period.(3)njnjeenx)43()32(N1=3,N2=8N=N1N2=24The smallest multiple of N1 and N2 in commonHomework:P61-1.26 *1.25(d)(e)(f)1.4 The Unit Impulse and Unit Step Functions1.4.1 The Discrete-time Unit Impulse and Unit Step Sequences(1)Unit Sample(Impulse):0,10,0nn
25、n(2)Unit Step Function:0,10,0nnnu(3)Relation Between Unit Sample and Unit Stepnmmnununun 1or0kknnurunning sumfirst difference0nm1I In nt te er rv va al l o of f s su um mm ma at ti io on n0n0nm1I In nt te er rv va al l o of f s su um mm ma at ti io on n0n(4)Sampling Property of Unit Samplekknkxnxnnn
26、xnnnxnxnnx0000Illustration of Sampling1.4.2 The Continuous-time Unit Step and Unit Impulse Functions(1)Unit Step Function:0,10,0)(tttu)(0ttu(2)Unit Impulse Function:dttdut)()(But,u(t)is discontinuous at t=0 and consequently is formally not differentiable.So,how can we get?)(tfirst derivative()0,0,()
27、1.tfor tandt dttdtu)()(running integralAnalogous to the relationship between un and nConsidering:)(tudttdut)()(0)()(tutu)()(lim0ttthen:When:That means,has no duration but unit area.)(tWe can get:1)(dtt(3)Relation Between Unit Impulse and Unit Steptdtu)()(running integral0()I In nt te er rv va al l o
28、 of f i in nt te eg gr ra at ti io on n0t 0t0t()tdttdut)()(first derivative(4)Sampling Property of (t)()()()0()()()()()()()()0()()(00000txdttttxxdtttxtttxtttxtxttx(5)The transformation of (t)(1)()()(taatttProof:11(/)()(0),0()()11(/)()(0),0 xadxaaax tat dtxadxaaa 11()()(0)x tt dtxaaSo,1()()attaLet ()
29、()tt1,a?)()cos(ttExample?)()12(355dtttt)(t1sin()()0tt?323(1)(1)(4)2tttdt?02TT21t)(tx)(aTT21t)(tx)(bSignal representation using step functions)2()()(TtuTtutx)(Ttu)2(TtuExample )()()(0ttutuetxtx(t)tt00te01231f(t)Signal representation using step functions Example 1-1 1 x(t)t(-1)2-2 1 x1(t)t(-2)(1)?)(1t
30、x?)(2dxt 1-1 1 x(t)t?)(tx 2-2 1 x1(t)t 2-2 2 x2(t)t 2 2 x2(t)t1Homework:P57 1.6 1.22(e)(f)1.?)2(sindttt2.?)2(sin11dttt3.2-2 1 g(t)tsketch(1)()(3)(1)2(1)(2)()(2)Caculate:sin(2)?2kg ttttg ttkttdt4.sketch?)(dxt1.5 Continuous-time and Discrete-time System(1)A continuous-time system is a system in which
31、continuous-time input signals are applied and result in continuous-time output signals.Continuous-time system x(t)y(t)()(tytx(2)A discrete-time systemthat is,a system that transforms discrete-time inputs into discrete-time outputs.Discrete-time system xnynnynx1.5.1 Simple Example of systems1.Example
32、 1.8(p39)(1)(1)(tvRCtvRCdttdvsccRC Circuit(system)vs(t)vc(t)Rtvtvtics)()()(From Ohms lawdttdvctic)()(andWe can get2.Example 1.10(p40)Balance in a bank account from month to month:balance -yn（余额）余额）net deposit -xn(净存款）净存款）interest -1%so yn=yn-1+1%yn-1+xn or yn-1.01yn-1=xnBalance in bank(system)xnynCo
33、nclusion:The mathematical descriptions of systems as the preceding examples are the first-order linear differential or difference equation of forms:)()()(tbxtaydttdy 1nbxnayny1.5.2 Interconnections of SystemMany real system are built as interconnections of several subsystems.(1)Series(cascade)interc
34、onnection(2)Parallel interconnection(3)Series-Parallel interconnection(4)Feed-back interconnection1.6 Basic System Properties1.6.1 Systems with and without MemoryMemoryless system:Its output for each value of the independent variable at a given time is dependent only on the input at the same time.Fe
35、atures:No capacitor,no conductor,no delayer.Examples:1)(1)()()(kxnynxnydxctytxdtdtynkt)()()2(22nxnytxtynxnxnywithmemorywithoutmemoryidentity system1.6.2 Invertibility and Inverse SystemsNote:(1)If system is invertible,then an inverse system exists.(2)An inverse system cascaded with the original syst
36、em,yields an output equal to the input.Invertible systemdistinct inputs lead to distinct outputs.Examples of invertible systemsExamples of noninvertible systems)()(2txty1.6.3 CausalityA system is causal If the output at any time depends only on values of the input at the present time and in the past
37、.(nonanticipative 不超前不超前)Note：For causal system,if x(t)=0 for tt0,there must be y(t)=0 for tt0.Memoryless systems are causal.)1cos()()()(1)()()()(ttxtykxnydxctytbxtaytydtdnkt121 1nxnyknxmnynxnxnymmkcausalnoncausalExamples of causal systems1.6.4 StabilityThe stable systemSmall inputs lead to responses that don not diverge.Bounded input lead to Bounded output(BIBO）if|x(t)|M,then|y(t)|0Solution:tcetytytydtd2)(,0)(2)(Then 0,3)(2tetytcWhich is the zero-input response of the system
