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1、离散时间信号处理DSPIntroduction Still waters run deep.流静水深流静水深,人静心深人静心深 Where there is life,there is hope。有生命必有希望。有生命必有希望IntroductionDigital Signal Processing(DSP)is used in a wide variety of applications.Telephone&telegramradarAudio signal processingMultimediasystemImage processingMobile telephoneCommunica
2、tion systemdigitalTVIntroduction3IntroductionSignalsA signal can 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
3、-f(t);-One-dimensional(1-D)signal 一维信号a picture would be represented 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 Sign
4、al Processing?Digital Signal Processing is the science to process 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 twen
5、ty-first century.Suppose we attach an analog-to-digital converter 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
6、 signals in a digital way.Processing digital signals.Advantages: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
7、 digital filter(IIR（无限长单位冲激响应）&FIR（有限长单位冲激响应）filter).Multirate 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
8、.1 Discrete-Time Signals:SequencesThe independent variable of a 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
9、signal112.1 Discrete-time signals notationsA discrete-time signal 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 s
10、equence x can be represented as where Z is the set of integer numbers（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
11、 nT13unit sample sequence（单位抽样序列）The definition of the unit impulse(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 functio
12、n is ,whose angular frequency（角频率）is rad/sample.n017Exponential 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
13、unit step u(n)can be expressed asAnd the unit ramp r(n)can be expressed 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
14、 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 an irrational number（无理数）,this cosine function will not be periodic at all.22Example Determine whether following discrete s
15、ignal is periodic or not.If it is,calculate the period of the signal.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 inp
16、ut sequence x(n)into an output sequence y(n).This can be denoted 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
17、 the same value of n.252.2.2 Linearity SystemsLinearity（线性）If y1(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 a
18、Tx(n)=3a x(n)+4a Ta x(n)aT x(n)So it is not a linearity system.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 inva
19、riance（移不变性）.Example:y(n)=3x(n)+4 Tx(n-n0)=3x(n-n0)+4 y(n-n0)=3x(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 a
20、t instant n does not depend on any input occurring after n.Usually,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-Invari
21、ant SystemAn input sequence x(n)can be expressed as a sum of scaled,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
22、 and convolution sumsh(n)=T(n)is referred to as the impulse response（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).
23、(1.37)31Impulse responses and convolution sumsThe convolution sum 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-
24、Invariant SystemSuppose now that there are two linear time-invariant 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 An
25、d the output of the second system h2(n)is 37Systems in cascadeBy 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
26、of the two impulse responses.h(n)=h1(n)*h2(n)y(t)x(n)h1(n)h2(n)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.,
27、|x(n)|xmax for all n,thenLinear shift invariant systems are stable 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 so it i
28、s time varying;3.Because i.e.the output for a certain time t=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.S
29、o the system is unstable.Therefore the system is nonlinear,time 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，
30、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 typeDescription 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,yie
31、ldingSo the output y(n)is dependent both on samples of the input 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
32、=a2=aN=0,then the output at sample n depends only on values 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 discre
33、te-time system are often referred to as finite-duration impulse-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-d
34、uration impulse-response(IIR)（无限长单位冲激响应）filters.47Review 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 cosine functions are periodic.If 2/is an integer（整数）or a
35、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.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
36、 any constants a and b.time invariance:If y(n)=H x(n),thenH 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
37、expressed as where h(n)=H(n)is the impulse response of the system.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-
38、duration impulse-response(FIR)filters.A recursive digital system 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(
39、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