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1、USSTDigital Signal Processing毛 倩办公室：仪表一馆113Email:1IntroductionDigital Signal Processing(DSP)is used in a wide variety of applications.Telephone&telegramradarAudio signal processingMultimediasystemImage processingMobile telephoneCommunication systemdigitalTVIntroduction3IntroductionSignalsA signal ca
2、n be defined as a function that conveys information.Signals are presented mathematically as functions of one or more independent variables.for example:a speech signal would be represented mathematically as a function of one time variable-f(t);-One-dimensional(1-D)signal 一维信号a picture would be repres
3、ented mathematically as a brightness function of two spatial variables-f(x,y).-Two-dimensional(2-D)signal 二维信号a color video signal(a RGB television signal)is a 3-D signal.-Multidimensional(M-D)signal 多维信号4IntroductionWhat is Digital Signal Processing?Digital Signal Processing is the science to proce
4、ss signals by digital means.This includes a wide variety of goals:filtering,transformation,recognition,enhancement,compression,and much more.DSP is one of the most powerful technologies that will shape science and engineering in the twenty-first century.Suppose we attach an analog-to-digital convert
5、er to a computer,and then use it to acquire a chunk of real world data.5Introduction简单的说，数字信号处理是利用计算机或专用处理设备，以数值计算的方法对信号进行采集、变换、综合、估值与识别等加工处理，借以达到提取信息和便于应用的目的。6IntroductionDigital signal processing includes two meanings:Processing analog signals in a digital way.Processing digital signals.Advantages
6、:High reliability（可靠性）High agility(灵活性好，易于实现系统性能）High precision（高精度）Low cost（成本低）7IntroductionMain tools:Discrete-time signal representations.Discrete transforms and their fast algorithms(z transform,DFT&FFT).Design and implementation of digital filter(IIR（无限长单位冲激响应）&FIR（有限长单位冲激响应）filter).Multirate
7、systems（多率系统）,filter banks（滤波器组）,and wavelets（小波）.Implementation of digital signal processing systems.8References程佩青，数字信号处理教程，清华大学出版社胡广书，数字信号处理理论、算法与实现，清华大学出版社高西全，丁玉美，阔永红，数字信号处理原理、实现及应用，电子工业出版社9Chapter 2 Discrete-Time Signals and System2.1 Discrete-Time Signals:SequencesThe independent variable of a
8、 signal may be either continuous or discrete.Continuous-time signals are those that are defined at continuous times.Discrete-time signals are those that are defined at discrete times.tAmplitudetAmplitudeContinue-time signalDiscrete-time signal112.1 Discrete-time signals notationsA discrete-time sign
9、al can be represented as where T is time interval between samples.Each sample of sequence x(nT)is determined by the amplitude of signal at instant nT.For example where T is 0.03.Another notation is a sequence of numbers.For example,the sequence x can be represented as where Z is the set of integer n
10、umbers（整数集）,and x(n)is referred to as the“nth sample”of the sequence.For example A convenient notation for the sequence x just is x(n).122.1 Discrete-time signals graphDiscrete-time signals are often depicted graphically.x(n)or x(nT)n or nT13unit sample sequence（单位抽样序列）The definition of the unit imp
11、ulse(n)n0114delayed unit sample sequence（延时单位抽样序列）The definition of the delayed unit sample sequence(n m)n01m15unit step sequence（单位阶跃序列）The definition of the unit stepu(n)n01u(n)的后向差分16cosine functionThe definition of the cosine function is ,whose angular frequency（角频率）is rad/sample.n017Exponential
12、 Sequence（实指数序列）The definition of the real exponential function is n0n18unit ramp（单位斜坡序列）The definition of the unit rampr(n)n0192.1 Discrete-time signalsAn arbitrary sequence can be expressed as a sum of scaled,delayed unit impulses.The unit step u(n)can be expressed asAnd the unit ramp r(n)can be e
13、xpressed as 20Example:generate the signal with impulse sequence-3-2-10 1 2 3 4 5x(n)nn00n0n21Periodic sequence A sequence x(n)is defined to be periodic if and only if there is an integer N0 such that x(n)=x(n+N)for all n.In such a case,N is called the period of the sequence.Note,not all discrete cos
14、ine functions are periodic.If 2/is an integer（整数）or a rational number（有理数),this sequence will be periodic;If 2/is an irrational number（无理数）,this cosine function will not be periodic at all.22Example Determine whether following discrete signal is periodic or not.If it is,calculate the period of the s
15、ignal.Solution:if the signal is periodic,then we have Then so system is periodic,its period is 40(when k=17).232.2 Discrete-time systemsDefinition:A system is defined mathematically as a unique transformation or operator that maps an input sequence x(n)into an output sequence y(n).This can be denote
16、d as y(n)=Tx(n)where T expresses a discrete-time system.Discrete-time systemy(n)x(n)ExcitationResponseT242.2.1 Memoryless SystemsA system is referred to as memoryless if the output y(n)at every value of n depends only on the input x(n)at the same value of n.252.2.2 Linearity SystemsLinearity（线性）If y
17、1(n)and y2(n)are the responses when x1(n)and x2(n)are the inputs respectively,then a system is linear if and only ifTa x(n)=a T x(n)and T x1(n)+x2(n)=T x1(n)+T x2(n)for any constants a and b.Example:y(n)=Tx(n)=3x(n)+4 Ta x(n)=3a x(n)+4 aTx(n)=3a x(n)+4a Ta x(n)aT x(n)So it is not a linearity system.
18、262.2.3 Time-Invariant SystemsTime invariance（时不变性）A discrete-time system is time invariant if and only if,for any input sequence x(n)and integer n0,thenT x(n-n0)=y(n-n0)with y(n)=T x(n).Note:another name of time invariance is shift invariance（移不变性）.Example:y(n)=3x(n)+4 Tx(n-n0)=3x(n-n0)+4 y(n-n0)=3
19、x(n-n0)+4 y(n-n0)=Tx(n-n0)So this system is time invariance.272.2.4 CausalityCausality（因果性）A discrete-time system is causal if and only if,when x1(n)=x2(n)for n n0,then T x1(n)=Tx2(n),for n n0A causal system is one for which the output at instant n does not depend on any input occurring after n.Usua
20、lly,in the case of a discrete-time signal,a noncausal system is not implementable in real time.However,in some cases a discrete signal does not consist of time samples,a noncausal system can be easily implemented.282.3 Linear Time-Invariant SystemAn input sequence x(n)can be expressed as a sum of sc
21、aled,shifted unit impulses.The output can be expressed as29Impulse responses and convolution sumsIf the system is linear,we can obtain Since x(k)in the above equation is just a constant（常数）,the output iswhere we define30Impulse responses and convolution sumsh(n)=T(n)is referred to as the impulse res
22、ponse（冲激响应）of the system.The equationis called a convolution sum or a discrete-time convolution（卷积和，又称离散卷积和、线性卷积和）.This equation indicates that a linear time-invariant system is completely characterized by its unit impulse response h(n).(1.37)31Impulse responses and convolution sumsThe convolution s
23、um can also be written asA shorthand notation for the convolution is and where“*”represents the convolution sum.32ExampleCompute the linear convolution y(n)=x(n)*h(n),andSolution:33Cont.34Cont.Solution II:352.4 Properties of Linear Time-Invariant SystemSuppose now that there are two linear time-inva
24、riant system which are in cascade.That is to say,the output of a system with impulse response h1(n)is the excitation for a system with impulse response h2(n).h1(n)h2(n)y(n)x(n)36Systems in cascadeThe output of the first system h1(n)is And the output of the second system h2(n)is 37Systems in cascadeB
25、y exchange the summing sequence,we getThen we can express its output as38Systems in cascadeConclusion:Two linear time-invariant systems in cascade form a linear time-invariant system with an impulse response which is the convolution sum of the two impulse responses.h(n)=h1(n)*h2(n)y(t)x(n)h1(n)h2(n)
26、y(n)x(n)39*Discrete-time systems stability（稳定性）A system is referred to as bounded-input bounded-output(BIBO)stable if,for every input limited in amplitude,the output signal is also limited in amplitude.（有界输入产生有界输出）If x(n)is bounded,i.e.,|x(n)|xmax for all n,thenLinear shift invariant systems are sta
27、ble if and only if （充要条件）40Example Characterize following system as being either linear or nonlinear,time invariant or time varying,causal or noncausal,stable or not stable.Solution:1.For So it is not a linear system.41Cont.2.For 3.so it is time varying;3.Because4.i.e.the output for a certain time t
28、=n of this system not only depends on the input at time t=n,but also depends on the time after n(i.e.t=n+2).So the system is noncausal;42Cont.4.for a special bounded input we have then the output for system i.e.systems output is unbounded.So the system is unstable.Therefore the system is nonlinear,t
29、ime invariant,noncausal and unstable.432.5 Linear Constant-Coefficient Difference EquationsThe input x(n)and the output y(n)of a system described by a linear difference equation（线性差分方程）are generally related by线性线性：指方程中各y(n-i)和x(n-l)项都只有一次幂，且不存在它们的相乘项。输出序列y(n)变量序号的最高值与最低值之差N称为差分方程的阶数阶数。System typeDes
30、cription of the systemthe continuous-time systemthe differential equation（微分方程）the discrete-time systemthe difference equation（差分方程）(2.1)44IIR and FIR filtersEquation(2.1)can be rewritten,without loss of generality,considering that a0=1,yieldingSo the output y(n)is dependent both on samples of the i
31、nput x(n),x(n-1),x(n-M),and on previous samples of the output y(n-1),y(n-2),y(n-N).(2.2)45IIR and FIR filtersSince in order to compute the output,we need the past samples of the output itself,we say that the system is recursive（递归的）.When a1=a2=aN=0,then the output at sample n depends only on values
32、of the input signal.In such case,the system is called nonrecursive（非递归的）.It is(2.3)46IIR and FIR filtersIf we compare the above equation with equation we see that the system in equation(2.3)has a finite-duration impulse response.Such discrete-time system are often referred to as finite-duration impu
33、lse-response(FIR)（有限长单位冲激响应）filters.In contrast,when y(n)depends on its past values,as shown in equation(2.2),the impulse response of the system might not be zero when n.Therefore,recursive digital system are often referred to as infinite-duration impulse-response(IIR)（无限长单位冲激响应）filters.47Review A s
34、equence x(n)is defined to be periodic if and only if there is an integer N0 such that x(n)=x(n+N)for all n.In such a case,N is called the period of the sequence.Note,not all discrete cosine functions are periodic.If 2/is an integer（整数）or a rational number（有理数),this sequence will be periodic;If 2/is
35、an irrational number（无理数）,this cosine function will not be periodic at all.48ReviewThe characteristics of the discrete-time system y(n)=H x(n):Linearity:If y1(n)=H x1(n),y2(n)=H x2(n),then H ax(n)=aH x(n)and H x1(n)+x2(n)=H x1(n)+H x2(n)for any constants a and b.time invariance:If y(n)=H x(n),thenH
36、x(n-n0)=y(n-n0)Causality:If,when x1(n)=x2(n)for n n0,then H x1(n)=H x2(n),for n n0Stability:For every input limited in amplitude,the output signal is also limited in amplitude.49ReviewThe output y(n)of a linear time-invariant system can be expressed as where h(n)=H(n)is the impulse response of the s
37、ystem.Two linear time-invariant systems in cascade form a linear time-invariant system with an impulse response which is the convolution sum of the two impulse responses.50ReviewA nonrecursive system such as are often referred to as finite-duration impulse-response(FIR)filters.A recursive digital sy
38、stem such as are often referred to as infinite-duration impulse-response(IIR)filters.51Exercises1.Characterize the systems below as linear/nonlinear,causal/noncausal and time invariant/time varying.y(n)=(n+a)2x(n+4)y(n)=ax(nT+T)y(n)=x(n)/x(n+3)2.For each of the discrete signals below,determine whether they are period or not.Calculate the periods of those that are periodic.52Exercises3.Compute the convolution sum of the following pairs of sequences.4.Discuss the stability of the systems described by the impulse responses below:h(n)=2-nu(n)h(n)=0.5nu(n)-0.5nu(4-n)53