离散时间信号处理PPT_第二章 离散时间信号系统.ppt

1、References 1. 郑君里，信号与系统（下册），高等教育出版社。 2. A.V. 奥本海姆，离散时间信号处理，科学出版社，2000年8月。- 3. 程佩清，数字信号处理教程（第二版），清华大学出版社。 4. Vinay K.Ingle，刘树棠译，数字信号处理及其Matlab实现，西安交通大学出版社。 5. 丁玉美，数字信号处理，西安电子科技大学出版社 .,Digital Signal Processing,The College of Electronics and Informatics Zhejiang Sci-Tech University 2005/06,Digital Sig

2、nal Processing,Theory, method, algorithm,A kind of microprocessor used to implement digital signal processing algorithm,Digital Signal Processor,The foundation of information technology is digitalization. The kernel of digitalization is digital signal processing Most of digital signal processing, es

3、pecially real-time processing are implemented by DSP processor DSP technology becomes hot front edge and is growing up rapidly.,DSP Solution,Mobile phone,The handset is not only used for voice communication,ADSL（Asymmetric Digital Subscriber Line),ADSL是非对称数字用户线路（Asymmetric Digital Subscriber Line）的缩

4、写，有时也作非对称数字用户环路（Asymmetric Digital Subscriber Loop）。它是一种在电话铜缆上进行较高速率数据传输的方法，是DSL 的一种形式。它以普通电话线路做为传输介质，既在普通双绞铜线上实现下行高达8Mbit/s传输速度；上行高达640Kbit/s的传输速度，我们只要在普通线路两端加装ADSL设备，既可使用ADSL提供的高带宽服务，通过一条电话线，便可以比普通MODEM快一百倍,DSL（数字用户线路，Digital Subscriber Line）是以铜质电话线为传输介质的传输技术组合，它包括HDSL、SDSL 、VDSL 、ADSL和RADSL等，一般称之

5、为xDSL。它们主要的区别就是体现在信号传输速度和距离的不同以及上行速率和下行速率对称性的不同这两个方面,Digital Camera,Digital Camera,HDTVHigh Definition TV,PDP (Plasma Display)TV,Home Theater,DVD(Digital Video Disc),GPS(Global Position System),Cruise missile,Smart bomb from F117,Pattern recognization,Fingerprint distinguish,Requirement of DSP Engin

6、eers is growing up rapidly,Development of DSP,Higher processing speed Multi-DSP co-operation More powerful and convenient development environment ASIC based on DSP core Lower price or higher performance price ratio Lower power consumption or longer battery life,CONTENT,2.DISCRETE-TIME SIGNAL AND SYS

7、TEMS 3.THE Z-TRANSFORM 4.SAMPLING OF CONTINUOUS-TIME SIGNALS 6.STRUCTURES FOR DISCRETE-TIME SYSTEMS 7.FILTER DESIGN TECHNIQUES 8.THE DISCRETE FOURIER TRANSFORM 9.COMPUTATION OF THE DISCRETE FOURIER APPENDIX B CONTINUOUS-TIME FILTERS,2 Discrete-Time Signals and Systems,2.0 INTRODUCTION,The term signa

8、l is generally applied to something that conveys information. The independent variable in the mathematical representation of a signal may be either continuous or discrete.,Continuous-time signals are defined along a continuum of times and thus are represented by a continuous independent variable. Co

9、ntinuous-time signals are often referred to as analog signals. .,. Discrete-time signals are defined at discrete times, and thus, the independent variable has discrete values; Digital signals are those for both time and amplitude are discrete.,Introduction,一、数字信号处理的基本概念 信号处理的目的： 对信号进行分析、变换、综合、估值与识别。

10、 信号的分类： 模拟信号：又称连续信号，它的幅度和时间都取连续变量。 数字信号：它的幅度和时间都取离散值。 信号处理的方式： 数字信号处理是采用数值计算的方法，完成对信号的处理，而模拟信号处理则是通过一些模拟器件，例如晶体管、电阻、电容、电感等，完成对信号的处理。当然可以在系统中增加数模转换器和模数转换器，这样数字信号处理系统也可以处理模拟信号，模拟信号处理系统也可以处理数字信号。 二、信号处理的实现方法 三、数字信号处理的特点,模拟高通滤波器与数字高通滤波器的比较,信号处理的实现方法,基本上分为两种方法，一种是软件实现方法，另一种是硬件实现方法。软件实现方法指的是按照原理和算法，自己编写程序

11、或者采用现成的程序在通用计算机上实现。硬件实现指的是按照具体的要求和算法，设计硬件结构图，用乘法器、加法器、延时器、控制器、存储器以及输入输出接口部件实现的一种方法。前者灵活，但速度慢，达不到实时处理要求；后者速度快，但是不够灵活。 采用专用的数字信号处理芯片（DSP）是目前发展最快、应用最广的一种方法。它内部配有乘法器和累加器，结构上采用流水线工作方式以及并行结构、多总线，且配有适合数字信号处理的指令，这种产品已经进入市场，速度高、体积小、性能优良，价格也在不断下降。,数字信号处理的特点,（1）灵活性可以通过改变数字信号处理系统的参数来改变系统的性能；另外灵活性还表现在数字系统可以分时复用，

12、用一套数字系统部件分时处理几路信号。 （2）高精度数字系统的特性不受环境的变化而变化，计算精度是模拟系统所无法相比的，运算位数由8位提高到16、32、64位。 （3）便于集成数字部件具有高度的规范性，容易大规模集成和大规模生产，数字系统体积小、重量轻、可靠性强。 （4）可存储对数字信号可以存储、运算，系统可以获得高性能指标。,Chapter 1 Discrete Time Signals and Systems,1.0 Introduction 信号通常是一种函数，包括一个自变量或几个自变量。如果仅有一个自变量，则称为一维信号；如果有两个以上的自变量，则称为多维信号。本书仅介绍以时间为自变量的

13、一维信号。针对信号的自变量和函数值的取值，可分为三种信号： （1）模拟信号-自变量和函数值都是连续的，如语音信号、电视信号等。 （2）时域离散信号-自变量取离散值，而函数值连续。这种信号来源于对模拟信号的采样。 （3）数字信号-自变量和函数值均取离散值。它是信号幅度离散化了的时域离散信号。 按照系统的输入输出是哪一类信号，系统也有模拟系统、时域离散系统和数字系统之分。,2.1 DISCRETE-TIME SIGNAL :SEQUENCES(典型序列),Discrete-time signals are represented mathematically as sequences of num

14、bers. A sequence of numbers x, in which the nth number in the sequence is denoted ,is formally written as Where n is an integer,In a practical setting ,such sequences can often arise from periodic sampling of an analog signal. In this case ,the numeric value of the nth number in the sequences is equ

15、al to the value of the analog signal: ,at time nT; i.e., The quantity T is called the sampling period,and its reciprocal is the sampling frequency,2.1.1 Basic Sequences and Sequence Operations,The product and sum of two sequences xn and yn are defined as the sample-by sample product and sum, respect

16、ively. Multiplication of a sequence xn by a number is defined as multiplication of each sample value by .A sequence yn is said to be a delayed or shifted version of a sequence xn if Where is an integer,(基本序列及其运算),1.1 .2 Operation of Sequences序列的运算,1、乘法 2、加法 3、位移 4、翻转 5、尺度变换,序列的加法运算,如 果 两 序 列 分 别 为 x

17、1(n) 和x2(n), 两 序 列 的 和 是 指 同 序 号n 的 序 列 值 逐 次 对 应 相 加 而 构 成 一 个 新 的 序 列z(n), 表 示为z(n)=x1(n)+x2(n)。 如 图：,序列的乘法运算,两 序 列 相 乘 是 指 同 序 号 (n) 的 序 列 值 逐 项 对 应 相 乘. 表 示为x(n)=x1(n)x2(n) 如 图：,序列的位移,设 某 一 序 列 为 x(n), 当 m 为 正 时, 则x(n-m) 是 指 原 序 列x(n) 逐 次 依 次 延 时( 右 移)m 位 而 给 出 的 一 个 新 序 列, 而x(n+m) 则 指 依 次 超 前(左

18、 移) m位。 如 图：,序列的翻转,如 果 序 列 为 x(n), 则 x(-n) 是 以n=0 的 纵 轴 为 对 称 轴 将 序 列 x(n) 加 以 翻 转。 如 图：,序列的尺度变换,如 果 序 列 为 x(n), 则 x(mn) 是x(n)序列每隔m点取一个点形成的，相当于时间轴n压缩了m倍。当m=2时，其波形如图：,1.1 Discrete Time Signal,时域离散信号是对模拟信号 进行等间隔采样获得的，采样间隔为T，得到： 这里n取整数。对于不同的n值， 是一个有序的数字序列，该数字序列就是时域离散信号。注意，这里的n取整数，非整数时无定义，另外，在数值上它等于信号的采

19、样值，即 时域离散信号的表示方法：公式表示法 图形表示法 集合符号表示法，如,The unit sample sequence(单位采样序列 ) The unit sample sequence (Figure 2.3a) The unit step sequence(单位阶跃序列) The unit step sequence u (n) ( Figure 2.3b) The unit step is related to the impulse by That is the value of the unit step sequence at (time) index n is equal

20、 to the accumulated sum of the value at index n and all previous values of the impulse sequence.,An alternative representation of the unit step in the terms of the impulse is obtained by interpreting the unit step in Figure 2.3(b) in terms of a sum of delayed impulses as in Eq.(2.6). In this case ,t

21、he nonzero values are all unity ,so Or Conversely ,the impulse sequence can be expressed as the first backward different of the unit step sequence ,i.e.,1.1 .1 Typical Sequences 典型序列,1、单位采样序列(unit-sample sequence) 注：任意序列，常用单位采样序列的位移加权和表示。即 例如,The exponential sequence (指数序列) The general form of an ex

22、ponential sequence is If and are real numbers ,then the sequence is real . If and is positive ,then the sequence values are positive and decrease with n, as in Figure2.3(c) For ,the sequence values alternate in sign ,but again decrease in magnitude with increasing n. If then the sequence grows in ma

23、gnitude as n increase. The sinusoidal sequence (正弦序列) A sinusoidal sequence has the general form for all n. With A and real constants, and is illustrated in Figure2.3(d).,Figure 2.3 Some basic sequences . The sequences shown play important roles in the analysis and representation of discrete-time si

24、gnals and systems.,The exponential sequence with complex has real and imaginary parts that are exponentially weighted sinusoids .Specifically ,if and ,the sequence can be expressed in any of the flowing ways:,the sequence oscillates with an exponentially growing envelop if or with an exponentially d

25、ecaying envelope if When |a|=1 ,the sequence is referred to as a complex exponential sequence and has the form . that is ,the real and imaginary parts of vary sinusoidally with n.,The quantity is called the frequency of the complex sinusoid or complex exponential, and is called the phase. An importa

26、nt difference between continuous time and discrete-time complex sinusoids is seen when we consider a frequency ( ). In this case , .,Another important difference between continuous-time and discrete-time complex exponentials and sinusoids concerns their periodicity . In the continuous-time case, a s

27、inusoids signal and a complex exponential signal are both periodic, with the period equal to 2divided by the frequency. In the discrete-time case, a period sequence is a sequence for which where the period N is necessarily an integer. If this condition for periodicity is tested for the discrete-time

28、 sinusoid, then which requires that where k is an integer.,2、单位阶跃序列 3、矩形序列 4、实指数序列 5、正弦序列 6、复指数序列 7、周期序列,单位采样序列 unit-sample sequence,(unit-sample sequence),unit-step sequence单位阶跃序列,unit-step sequence,Rectangle Sequence矩形序列,real exponential sequence 实指数序列,(real exponential sequence),sinusoidal sequen

29、ce 正弦序列,(sinusoidal sequence),complex exponential sequence 复指数序列,复指数序列 式中w0为数字频率。,(complex exponential sequence),periodic sequence with period N 周期序列,周期为N 的周期序列 (periodic sequence with period N),Example 2.1 Combining Basic Sequences,We often combine basic sequences to form simple representations of

30、others sequences. If we want an exponential sequence that is zero for ,we can write this as the somewhat cumbersome expression A much simpler expression is,For the discrete-time sinusoidal signal as increase from toward , x n oscillates more and more rapidly, however , as increases from to , the osc

31、illations become slower . This is illustrated in Figure 2.5,Figure 2.5,2.2 DISCRETE-TIME SYSTEMS (离散时间系统),Representation of a discrete-time system ,i.e., a transformation that maps an input sequence into a unique output sequence,Example 2.3 The Ideal Delay System,The ideal delay system is defined by

32、 the equation Where is a fixed positive integer called the delay of the system . In a words, the ideal delay system simply shifts the input sequence to the right by samples to form the output.,Example 2.4 Moving Average,The general moving-average system is defined by the equation This system compute

33、s the nth sample of the output sequence as the average of samples of the input sequence around the nth sample.,2.2.1 Memoryless System (无记忆系统),A system is referred to as memoryless if the output at every value of n depends only on the input at the same value of n.,Example 2.5,An example of a memoryl

34、ess system for which and are related by for each value of n. The system in Example 2.3 is not memoryless unless .,2.2.2 Linear System （线性系统）,The class of linear system is defined by the principle of superposition . If and are the responses of a system when and are the respective inputs ,then the sys

35、tem is linear if and only if and where is an arbitrary constant.,The first property is called the additivity property ,and the second is called the homogeneity or scaling property. These two properties can be combined into the principle of superposition ,stated as for arbitrary constant and .This eq

36、uation can be generalized to the superposition of many inputs. Specifically, if then the output of a linear system will be where is the system response to the input,Example 2.6 The Accumulator System,The accumulator system 2.29 since the output at time n is just the sum of the present and all previo

37、us input samples. The accumulator system is a linear system.,Proving :define two arbitrary inputs and and their corresponding outputs When the input is ,the superposition principle requires the output for all possible choices of and .We can show this by staring from Eq.(2.29):,Example 2.7 A Nonlinea

38、r System,Consider the system defined by This system is not linear. in order to prove this, we only need to find one counterexample that is , one set of inputs and output which demonstrates that the system violates the superposition principle, Eq.(2.27). The inputs x1n=1 and x2n=10 are a counterexamp

39、le. The output for the first signal is w1n=0, while for the second , w2n=1. the scaling property of linear systems requires that , since x2n=10 x1n, if the system is linear, it must be true that w2n=10w1n. Since this is not so far Eq.(2.36) for this set of inputs and outputs, the system is not linea

40、r.,2.2.3 Time-Invariant System（时不变系统）,A time-invariant system is a system for which a time shift or delay of the input sequence causes a corresponding shift in the output sequence. suppose that a system transforms the input sequence with values into the output sequence with values , for all ,the inp

41、ut sequence with values Produces the output sequence with values,Example 2.8 The Accumulator as a Time-Invariant System,We define x1n=xn-n0 to show time invariance, we solve for both yn-n0 and y1n and compare them to see whether they are equal . First Next, we find Substituting the change of variabl

42、es k1=k-n0 into the ummation gives Thus the accumulator is a time-invariant system.,一 个 时域离 散 系 统 是 将 输 入 序 列 x(n) 变 换 成 输 出 序 列 y(n) 的 一 种 运 算， 以 T. 表 示 为： y(n)=T x(n) 我 们 所 关 心 与 讨 论 的 主 要 是 线 性 系统（Linear Systems)和时不变系统(Shift-Invariant Systems )， 内 容 包 括 它 的 概 念 表 征 和 性 质。另 外 还 将 解 释 与 它 有 关 的 系 统

43、 因 果 性(Causality) 和 稳 定性(Stability)。,1.2 Linear Shift-Invariant Systems(线性时不变系统）,Example 2.9 The Compressor System,The system defined by the relation with M a positive integer ,is called a compressor. This system is not time invariant. We can show that it is not by considering the response y1n to th

44、e input x1n=xn-n0. In order for the system to be time invariant , the output of the system when the input is x1n must be equal to yn-n0. The output y1n that results from the input x1n can be directly computed from Eq.(2.41) to be Delaying the output y n by n0 samples yields Comparing these two outpu

45、ts , we see that yn-n0 is not equal to y1n for M and n0, and therefore, the system is not time invariant.,2.2.4 Causality（因果性）,A system is causal if ,for every choice of ,the output sequence value at the index depends only on the input sequence values for .This implies that if for ,then for The syst

46、em is nonanticipative .,Example 2.10 The Forward and backward Difference System,Consider the forward difference system defined by the relationship The system is not causal, since the current value of the output depends on the future value of the input, is not causal. The backward difference system ,

47、defined as has an output that depends on the present and past values of the input ,is causal.,2.2.5 Stability（稳定性）,A system is stable in the bounded-input, bounded-output (BIBO) sense if and only if every bounded input sequence produces a bounded output sequence. The input is bounded if there exists

48、 a fixed positive finite value such that for all Stability requires that ,for every bounded input ,there exist a fixes positive finite value ,such that for all,Example 2.11 Testing for Stability or Instability,The system of Example 2.5 is stable. Consider the case when which is clearly bounded by .

49、For this input ,the output of the accumulator is There is no finite choice for such that for all ,thus the system is unstable.,2.3 LINEAR TIME-INVARIANT SYSTEM（线性移不变系统）,A particularly important class of systems consists of those that are linear and time invariant .These two properties in combination

50、 lead to especially convenient representations for such system. Most important ,this class of system has significant signal-processing applications.,1.2 .1 线性系统（Linear Systems),若 系 统 满 足 可加 性 与 比例 性, 则 称 此 系 统 为 离 散 时 间 线 性 系 统。 这 就 是 说, 若 输 入 序 列 为 x1(n) 与 x2(n), 输 出 序 列 为 y1(n) 与 y2(n)。 如 果 用 T 表示

51、系统的运算 即 y1(n)=Tx1(n) y2(n)=Tx2(n) 则 Ta1 x1(n) + a2 x2(n) = a1 y1(n) + a2y2(n)，其 中a1、a2 为 任 意 常 数。 所 以 线 性 系 统 的 数 学 式 表 示 为 ： Ta1x1(n)+a2x2(n)= a1 Tx1(n)+ a2 Tx2(n)=a1y1(n)+a2y2(n)。 例 1.3.1,1.2.2 时不变系统(Shift-Invariant Systems ),若 系 统 的 响 应 与 激 励 加 于 系 统 的 时 刻 无 关, 则 称 该 系 统 为 时 不 变 系 统。 也 就 是 说, 若 输

52、 入 x(n) 产 生 输 出 为 y(n), 则 输 入 x(n-m) 相 应 地 产 生 输 出 为 y(n-m), 即 输 入 移 动 任 意 位, 其 输 出 也 移 动 相 同 位 数, 并 且 其 幅 值 不 变。 若 用 T 表 示 系 统 的 运 算 即 y(n)=Tx(n), 则 时不变 系 统 的 数 学 式 表 示 为 Tx(n-m)=y(n-m), 其 中 m 为 任 意 整 数。 例 1.3.2,1.2.3 Linear Shift-Invariant Systems,一、单位取样响应 设系统的输入为单位采样序列 ,即 ，系统输出 的初始状态为零，在这种条件下系统输出

53、称为系统的单位取样响应。用公式表示为： 用来表征系统的时域特征。 二、线性时不变系统输入与输出之间的关系 线性时不变系统的输出等于输入序列和该系统的单位取样响应的卷积。用公式表示为： （证明）,A linear system can be completely characterized by its impulse response. Specifically ,let be the response of the system to ,an impulse occurring at ,then from Eq.(2.6) From the principle of superpositio

54、n in Eq.(2.27),we can write,The property of time invariance implies that if is the response to ,then the response to is .With this additional constraint , Eq.(2.51) becomes (2.52) It shows that a linear time-invariant system is completely characterized by its impulse response in the sense that ,give

55、n ,it is possible to use Eq.(2.52) to compute the output due to any input It commonly called the convolution sum. We say that is the convolution of with and represent this by the notation,The derivation of Eq.(2.52)suggests the interpretation that the input sample at represented as is transformed by

56、 the system into an output sequence for ,and that ,for each ,these sequences are superimposed to form the overall output sequence. This interpretation is illustrated in Figure 2.8.,Figure2.8 representation of the output of a linear time-invariant system as the superposition of responses to individua

57、l samples of the input,Example 2.12 Computation of the Convolution,Suppose is the sequence shown in Figure 2.9(a) and we wish to find Define to be ,which is shown in Figure 2.9(b).Next ,define to be ,delayed by n samples on the k axis, i.e., Figure 2.9(c) shows the sequence that results from delaying the sequence in Figure 2.9(b) by n samples. From Example 2.3, it should be clear that ,in general ,the sequence is obtained by 1. reflecting about the origin to obtain 2.shifting the origin of the reflected sequence to,Figure 2.9 Forming