奥本海姆版信号与系统ppt.ppt

Chapter1Signals And Systems崔琳莉崔琳莉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 Capacitor 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 independent variable as time, although it may not in fact represent time in specific applications.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 (sequence)Example1: 1990-2002年的某村农民的年平均收入年的某村农民的年平均收入SamplingExample2: 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) Sequence-representation for discrete-time signals:xn=-2 1 3 2 1 1 or xn=(-2 1 3 2 1 1)C. RepresentationNote: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 classification:nAnalog vs. DigitalnPeriodic vs. Aperiodic nEven vs. OddnDeterministic vs. Randomn1.1.2 Signal Energy and PowerInstantaneous power:Let R=1, so +R_Energy : t1 t t2Average Power:Total EnergyAverage PowerDefinition:Continuous-Time:(t1 t t2 )Discrete-Time:(n1 n n2 )We will frequently find it convenient to consider signals that take on complex values.whenTotal EnergyAverage PowerNote: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, infinite average powerRead 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 delaynIf 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：Solution 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 1/2shiftreversalScalingreversalshiftScalingreversalshiftScalingExamplef(t) f(1-3t)1.2.2 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 fundamental 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. It is periodic signal. Its period is T=16/3.2. It is not periodic.3. x(t) is periodic. Its period isThe smallest multiples of T1 and T2 in common4. It is aperiodic, too. There is no the smallest multiples of T1 and T2 in common5. x(t) is aperiodic. 6. It is periodic with period N=16. CostCos2tcost+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.x(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 Complex Exponential and Sinusoidal SignalsThe continuous-time complex exponential signal is of the formwhere 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 periodic for x(t) = x(t+T) , and its fundamental period is .x(t)= Ceat, C=1, a=j 0 (purely imaginary)(1)For e j 0tif 0=0, x(t)=1, then it is periodic for any T0.(2)x(t) = Acos( 0t+ ) 0rad/s f0HzEulers Relation: e j 0t = cos 0t + j sin 0t and cos 0t = (e j 0t + e -j 0t ) / 2 sin 0t = (e j 0t - e -j 0t ) / 2j We haveif c is a complex number, Rec denotes its real part; Imc denotes the imaginary part. 1 2 0r10 1-1 0 -1xn = C nB. Sinusoidal Signals Complex exponent。