1、School of Computer Science and Information School of Computer Science and Information 1.1 SignalsSignals are functions of independent variables that carry information.The independent variables can be continuous or discrete.The independent variables can be 1-D,2-D,n-D.For this course:Focus on a singl
2、e(1-D)independent variable which we call“time”.Continuous-Time signals:x(t),t-continuous values.Discrete-Time signals:x(n),n-integer values only.School of Computer Science and Information ExamplesElectrical signals voltages and currents in a circuit.Acoustic signals audio or speech signals.Video sig
3、nals intensity variations in an image.Biological signals sequence of bases in a gene.School of Computer Science and Information A Simple RC CircuitThe patterns of variation over time in the source voltage Vs and capacitor voltage Vc are examples of signals.School of Computer Science and Information
4、A Speech SignalSchool of Computer Science and Information A PictureSchool of Computer Science and Information Vertical Wind ProfileSchool of Computer Science and Information 1.2 SystemsFor the most part,our view of systems will be from an input-output perspective.A system responds to applied input s
5、ignals,and its response is described in terms of one or more output signals.School of Computer Science and Information ExampleRLC circuitSchool of Computer Science and Information By interconnecting simpler subsystems.We can build more complex systems.School of Computer Science and Information 1.3 T
6、ypes of Signals1.Certain Signal and Random SignalCertain Signal Can be represented mathematically as a function of certain time.Random Signal Cant be represented mathematically as a function of certain time.We only know the probability of certain value.School of Computer Science and Information Exam
7、pleNoise Signal and Interfere SignalSchool of Computer Science and Information 2.Periodic Signal and Aperiodic Signal)()()()(kTnfnfnTtftf uPeriodic Signal Has the property that it is unchanged by a time shift of T.For example,A periodic continuous-time or discrete-time signal can be represented as:u
8、Aperiodic Signal Has not the property that it is unchanged by a time shift of T.Notice:When T,then Periodic Signal Aperiodic Signal.School of Computer Science and Information Exampletf(t)A ATTof(t)240246kPeriodic SignalSchool of Computer Science and Information 3.Continuous-time Signal and Discrete-
9、time SignaluContinuous-time Signal The independent variable is continuous,and thus these signals are defined for a continuum of values of the independent variable.uDiscrete-time Signal The independent variable takes on only a discrete times,and thus these signals are defined only at discrete times.S
10、chool of Computer Science and Information Example01212A Af1(t)to1tf2(t)oAtf3(t)t0(a)(b)(c)Continuous-time SignalSchool of Computer Science and Information ExampleDiscrete-time Signal0 12345678 2 4 6 8AAkf1(k)1 3102 34 1 310234 10132f2(k)f3(k)kk56A(a)(b)(c)School of Computer Science and Information 4
11、.Energy and Power Signals uEnergy(Continuous-time)Instantaneous power:)()(1)()()(22tiRtvRtitvtp Let R=1,so)()()()(222tftitvtp Energy over t1 t t2:2121)()(2ttttdttfdttpSchool of Computer Science and Information Total Energy:21)(lim2ttTdttxEAverage Power:TTTdttxTP)(21lim2uEnergy(Discrete-time)Instanta
12、neous power:)(2nxnp School of Computer Science and Information Energy over n1 n n2:212nnnnxETotal Energy:nnxE2Average Power:NNnNnxNP121lim2School of Computer Science and Information uFinite Energy and Finite Power SignalFinite Energy Signal(P 0):dttfE)(2 nnxE2Finite Power Signal(E ):NNnNnxNP121lim2
13、TTTdttfTP)(21lim2School of Computer Science and Information Example 010)10(1)(PEttfothers(Finite Energy Signal)164)(PEnx(Finite Power Signal)PEttf)(Signals with neither finite total energy nor finite average power)School of Computer Science and Information 1.Sinusoidal Signal)sin()(tAtfAAtf(t)oT1.4
14、Typical SignalsSchool of Computer Science and Information Property:The Differential or Integral of f(t)is also a sinusoidal signal with the same frequency.important formulas:fT12 School of Computer Science and Information 2.Real Exponential Signalvalue)realare(C,tCetf )(of(t)a1 000tSchool of Compute
15、r Science and Information Property:The Differential or Integral of f(t)is also a real exponential signal.Notice:When 0,f(t)is a growing function with t.When 0,f(t)is a decaying function with t.When 0,f(t)is a constant function with t.School of Computer Science and Information 3.Complex Exponential S
16、ignal)()(jsCetfts )sin(cos)()(tjtCeeCeCetfttjttj Properties:The real and imaginary of complex exponential signal are sinusoidal.For 0 they correspond to sinusoidal signal multiplied by a growing exponential.For 0 they correspond to sinusoidal signal multiplied by a decaying exponential.School of Com
17、puter Science and Information ottoot(a)(b)(c)0)(0)(0)(cba School of Computer Science and Information 4.Sampling SignaltttSasin)(0)(1)(0)()()(tSakttSattSatSadttSasignal)(evenPropertiesSchool of Computer Science and Information 5.Unit Step Signal 0,10,0)(tttu 000,1,0)(ttttttuSchool of Computer Science
18、 and Information 0,0,0)(ttt 6.Unit Impulse Signaltop(t)to(t)(1)(a)(b)1)(dtt School of Computer Science and Information )(|1)();(|1)()()()();0()()()()()()();()0()()(0000000attatattaattxdttttxxdtttxtttxtttxtxttx Properties:Relation Between Unit-Impulse and Unit-Step.tdtudttdut )()()()(Sampling Propert
19、ies of(t).School of Computer Science and Information)()(tdtdt 7.Unit Impulse Even Signal)()()()(0)(ttdttdttt ot(1)(1)(t)Properties:School of Computer Science and Information )0()0(0)(ttttR8.Unit Triangle Signal)()(tutRdtd Properties:School of Computer Science and Information 1.5 Basal Operation of S
20、ignals1.Plus and multiplication)()()(21tftftf )()()(21tftftf 2.Time Inversal)()(tftf School of Computer Science and Information School of Computer Science and Information School of Computer Science and Information 3.Time Shift)()(0ttftf School of Computer Science and Information 4.Flexibility)()(atf
21、tfSchool of Computer Science and Information Example).21()(tftf signal the carefully label and Sketch Figure,in shown is signal time-continuousA School of Computer Science and Information School of Computer Science and Information 1.6 The Representation of Continuous-time Signals in Terms of Impulse
22、sSchool of Computer Science and Information otherwise,00,1)(tt Define:School of Computer Science and Information We have the expression kktkftf)()()(Therefore kktkftf)()(lim)(0 or dtftf)()()(School of Computer Science and Information 1221121)()()(.)()()()()()(iiinnntgctfntgctgctgctft,g,t,gtgFor:expr
23、ession the have Weaggregate,cross integrated a compose If:Signals of ionDecomposit Cross:NoticeSchool of Computer Science and Information 1.7 Discrete-time Signal1.The Concepts of Discrete-time Signal,.)2,1()()()(kkfkTftfThe independent variable takes on only a discrete times,and thus these signals
24、are defined only at discrete times.School of Computer Science and Information 2.Basal Operation of Sequences)()()()()()()()()()(2121mkfkfkfkfkfkfkfkfkfkf :Shift Time:Inversal Time:tionMultiplica:PlusSchool of Computer Science and Information School of Computer Science and Information School of Compu
25、ter Science and Information School of Computer Science and Information 3.Typical of Sequences(1)Unit Sample 0,10,0nnn(2)Unit Step Sequence 0,10,0nnnuSchool of Computer Science and Information 01kknnununun Step-Unit and Sample-Unit Between Relation:NoticeSchool of Computer Science and Information(3)U
26、nit Rectangle Sequence NnnNnnGN,0,010,110NnununGnununknnuNk NoticeSchool of Computer Science and Information(5)Exponential Sequence)()(nunxn Notice:For|1,x(n)is a growing sequence.For|1,x(n)is a decaying sequence.For|1,x(n)is a constant sequence.School of Computer Science and Information(a)1；(b)0 1；
27、(c)-1 0；(d)-1School of Computer Science and Information(6)Sinusoidal Sequence)sin()(0nnx )cos()(0nnx k(a)(b)kk(c)School of Computer Science and Information(7)Complex Exponential Sequence)sin()cos()(000njnenxnj Property:The real and imaginary of complex exponential sequences are sinusoidal sequence.S
28、chool of Computer Science and Information(8)Complex Exponential SignalthennCnx )(in which jjeeCC|,|)sin(|)cos(|)(00 nCjnCnxnnSchool of Computer Science and Information(a)Growing sinusoidal sequence；(b)Decaying sinusoidal sequenceSchool of Computer Science and Information 4.The Representation of Disc
29、rete-time Signal in Terms of Unit Samples mmnmxnx)()(dtxtx)()()(:AnalogySchool of Computer Science and Information 1.8 Calculation of Convolution1.Calculation of Convolution Integral dtxxtxtx)()()()(2121 Definition dtxxtxxtxxxxxtx)()()()()()()()()()(21212222:Integral :tionMultiplica:ShiftTime:Invers
30、alTime:variabletIndependenChange:StepsSchool of Computer Science and Information Example).()()().()(),3()()(2121tftftytuetftututft Calculate To signals time-continuous IfSchool of Computer Science and Information 01234f1()1o1f1()1f2()t0t310t3f2()t(c)t 0(d)0 t 310t3t03y(t)y(3)(e)t 3(f)f2()(a)(b)f1()f
31、1()f2()tSchool of Computer Science and Information 0)()()(21 tffty When t0ttttttedeededtueuutffty 1)()3()()()()(00)()(21 When 0t3School of Computer Science and Information 2.Calculation of Convolution Sum kknxkxnxnx)()()()(2121Definition kknxkxknxkxknxkxkxkxkxnx)()()()()()()()()()(212122:Integral :t
32、ionMultiplica:ShiftTime:InversalTime:variabletIndependenChange:StepsSchool of Computer Science and Information Example 1).()()().()(),()(2121nfnfnynunfnuenfn Calculate To signals time-Discrete If1)1(110211111)()()()()(eeeeeeinuiuenfnfnynnniiiiSolution:School of Computer Science and Information Examp
33、le 2If Discrete-times signals:others 0221301)(1kkkkf others 03,2,1,04)(2kkkfTo calculate).()()(21nfnfnf School of Computer Science and Information 1023if1(i)13214(a)1023if2(i)3214(b)45110f2(i)321(c)4i123410f2(1i)321(d)4i1234510f2(1i)321(e)4i123421023kf(k)151914(f)51347267School of Computer Science a
34、nd Information In order to calculate the convolution of two limited long sequences,we can put the two sequence in two lines,and then to calculate according to the common multiplication apart from the intermediate results not carrying.Finally,the results of convolution can be gotten by locating the i
35、ntermediate together at the same list.Another way:School of Computer Science and Information School of Computer Science and Information 3.The Properties of Convolution)()()()()()()4()()()()()()()3()()()()()()()()2()()()()()1(21212121212131213211221tfdfdftfdfftfdtdtftftfdtdtftfdtdtftftftftftftftftftf
36、tfttt CT SignalsSchool of Computer Science and Information)()()()()()()4()()()()()()()3()()()()()()()()2()()()()()1(21212132132131213211221nxnxnxnxnxnxnxnxnxnxnxnxnxnxnxnxnxnxnxnxnxnxnxninini DT SignalsSchool of Computer Science and Information tdftutftfttftttfttttfttftttftfttftfttf )()(*)()6()()(*)()5()()(*)()4()()(*)()3()()0()()()2()()(*)()1(2121004.Several important formulasCT SignalsSchool of Computer Science and Information niixnunxmmnxmnmnxmnxmnnxnxnnxnxnnx)()(*)()5()()(*)()4()()(*)()3()()0()()()2()()(*)()1(2121 DT Signals
