章节目录 (3).ppt

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1、2020/7/17,1,Stability Condition of a Discrete-Time LTI System线性时不变离散系统的稳定条件,BIBO Stability Condition 有界输入有界输出稳定条件- A discrete-time LTI system is BIBO stable if the output sequence yn remains bounded for any bounded input sequencexn 当输入序列Xn有界时,如果输出序列Yn的保持为有界 A discrete-time LTI system is BIBO stable

2、if and only if its impulse response sequence脉冲响应序列hn is absolutely summable绝对可加, i.e.,S,2020/7/17,2,Stability Condition of a Discrete-Time LTI System线性时不变离散系统的稳定条件,Proof证明: Assume hn is a real sequence Since the input sequence xn is bounded we have Therefore,S,2020/7/17,3,Stability Condition of a Di

3、screte-Time LTI System线性时不变离散系统的稳定条件,Thus, S implies indicating that yn is also bounded To prove the converse, assume that yn is bounded, i.e., Consider the input given by,2020/7/17,4,Stability Condition of a Discrete-Time LTI System线性时不变离散系统的稳定条件,where sgn(c) = +1 if c 0 and sgn(c) = if c 0 and Not

4、e: Since , xn is obviously bounded For this input, yn at n = 0 is Therefore, implies S ,S,2020/7/17,5,Stability Condition of a Discrete-Time LTI System线性时不变离散系统的稳定条件,Example - Consider a causal discrete-time LTI system with an impulse response For this system Therefore if for which the system is BIB

5、O stable If , the system is not BIBO stable,S,if,2020/7/17,6,Causality Condition of a Discrete-Time LTI System离散时间LTI系统的因果关系条件,Let and be two input sequences with The corresponding output samples at of an LTI system with an impulse response hn are then given by,for,2020/7/17,7,Causality Condition of

6、 a Discrete-Time LTI System离散时间LTI系统的因果关系条件,2020/7/17,8,Causality Condition of a Discrete-Time LTI System离散时间LTI系统的因果关系条件,If the LTI system is also causal, then As This implies,for,2020/7/17,9,Causality Condition of a Discrete-Time LTI System离散时间LTI系统的因果关系条件,As for the only way the condition will ho

7、ld if both sums are equal to zero, which is satisfied if,for k 0,2020/7/17,10,Causality Condition of a Discrete-Time LTI System离散时间LTI系统的因果关系条件,A discrete-time LTI system is causal if and only if its impulse response hn is a causal sequence因果序列 Example - The discrete-time system defined by is a caus

8、al system因果系统as it has a causal impulse response,2020/7/17,11,Causality Condition of a Discrete-Time LTI System离散时间LTI系统的因果关系条件,Example - The discrete-time accumulator离散时间累加器defined by is a causal system as it has a causal impulse response given by,2020/7/17,12,Causality Condition of a Discrete-Time

9、 LTI System离散时间LTI系统的因果关系条件,Example - The factor-of-2 interpolator因子-2插值defined by is noncausal非因果as it has a noncausal impulse response given by,2020/7/17,13,Causality Condition of a Discrete-Time LTI System离散时间LTI系统的因果关系条件,Note: A noncausal LTI discrete-time system with a finite-length impulse res

10、ponse can often be realized as a causal system by inserting an appropriate amount of delay Example - A causal version of the factor-of-2 interpolator is obtained by delaying the input by one sample period:,2020/7/17,14,2.6 Finite-Dimensional Discrete-Time LTI Systems有限维线性时不变离散系统,An important subclas

11、s of discrete-time LTI systems is characterized by a linear constant- coefficient difference equation（线性常系数差分方程）of the form xn and yn are, respectively, the input and the output of the system and are constants characterizing the system,2020/7/17,15,Finite-Dimensional Discrete-Time LTI Systems有限维离散时间

12、LTI系统,If we assume the system to be causal, then the output yn can be recursively computed using provided yn can be computed for all , knowing xn and the initial conditions初始条件,2020/7/17,16,The output response yn of the LTI system described by can be computed as where,Total Solution Calculation全解（to

13、tal solution）的计算,is the complementary solution齐次解 to the homogeneous difference equation obtained by setting,is the particular solution特解 resulting from the specified input signal xn,2020/7/17,17,We assume that it is of the form By substitution in the homogeneous equation齐次方程, it is,Computing the Co

14、mplementary Solution计算齐次解,2020/7/17,18,The polynomial is called the characteristic polynomial of the given LTI system Let denote its N roots根,Characteristic Polynomial特征多项式,2020/7/17,19,If the N roots are distinct, the complementary solution is expressed by where are constants determined by the spec

15、ified initial conditions of the DT system,Complementary Solution齐次解,2020/7/17,20,If there are multiple roots, the complementary solution takes on a different form. For instance, if has multiplicity L, and the remaining roots are distinct, the complementary solution is,Complementary Solution齐次解,2020/

16、7/17,21,For the computation of , the procedure consists in assuming that it is of the same form as the specific input signal xn. E.g., if xn is a sinusoidal signal, so is If xn is a constant signal, so is , etc.,Particular Solution特解,2020/7/17,22,An alternate approach to the solution of consists in

17、computing where,Zero-Input Response and Zero-State Response零输入响应和零状态响应,is the zero-input solution零输入响应obtained by solving the difference equation by setting,is the zero-state solution零状态响应obtained by solving the difference equation by applying xn and setting all initial conditions to zero,2020/7/17,

18、23,The impulse response hn of a causal discrete-time LTI system is simply the zero-state response with Assuming that all the roots of the characteristic polynomial are distinct, the impulse response can be expressed as,Impulse Response of a Causal Discrete-Time LTI System因果离散时间LTI系统的脉冲响应,2020/7/17,2

19、4,Locations of the Roots of the Characteristic Equation for BIBO Stability特征方程的根的位置,It follows that Therefore, if for all values of i, it is implying the BIBO stability of the system,2020/7/17,25,Classification of Discrete-Time LTI Systems离散时间LTI系统的分类,Based on the Impulse Response Length基于脉冲响应长度: If

20、 the impulse response hn is of finite length, i.e., then it is known as a finite impulse response (FIR)有限脉冲响应（FIR）discrete-time system The convolution sum卷积和description here is,2020/7/17,26,Classification of Discrete-Time LTI Systems离散时间LTI系统的分类,The output yn of an FIR LTI discrete-time system can b

21、e computed directly from the convolution sum as it is a finite sum of products Examples of FIR LTI discrete-time systems are the moving-average system and the linear interpolators,2020/7/17,27,Classification of Discrete-Time LTI Systems离散时间LTI系统的分类,If the impulse response is of infinite length, then

22、 it is known as an infinite impulse response (IIR)无限冲激响应（IIR）discrete-time system The class of IIR systems we are concerned with in this course are characterized by linear constant coefficient difference equations,2020/7/17,28,Classification of Discrete-Time LTI Systems离散时间LTI系统的分类,Example - The dis

23、crete-time accumulator defined by is an IIR system,2020/7/17,29,Classification of Discrete-Time LTI Systems离散时间LTI系统的分类,Example - The familiar numerical integration formulas that are used to numerically solve integrals of the form can be shown to be characterized by linear constant coefficient diffe

24、rence equations, and hence, are examples of IIR systems,2020/7/17,30,Classification of Discrete-Time LTI Systems离散时间LTI系统的分类,If we divide the interval of integration into n equal parts of length T, then the previous integral can be rewritten as where we have set t = nT and used the notation,2020/7/1

25、7,31,Classification of LTI Discrete-Time Systems离散时间LTI系统的分类,Using the trapezoidal method梯形法we can write Hence, a numerical representation of the definite integral is given by,2020/7/17,32,Numerical Integration with the Trapezoidal Method梯形法数值积分,2020/7/17,33,Classification of Discrete-Time LTI Syste

26、ms离散时间LTI系统的分类,Let yn = y(nT) and xn = x(nT) Then reduces to which is the difference equation representation of a first-order IIR discrete-time system,2020/7/17,34,Classification of LTI Discrete-Time Systems离散时间LTI系统的分类,Based on the Output Calculation Process基于输出计算处理: Nonrecursive System非递归系统- Here

27、the output can be calculated sequentially, knowing only the present and past input samples Recursive System递归系统- Here the output computation involves past output samples in addition to the present and past input samples,2020/7/17,35,Classification of LTI Discrete-Time Systems离散时间LTI系统的分类,Based on th

28、e Coefficients基于系数: Real Discrete-Time System实离散时间系统- The impulse response samples are real valued Complex Discrete-Time System复离散时间系统- The impulse response samples are complex valued,2020/7/17,36,2.7 Correlation of Signals信号的相关,There are applications where it is necessary to compare one reference s

29、ignal参考信号with one or more signals to determine the similarity相似度between the pair and to determine additional information based on the similarity,2020/7/17,37,Correlation of Signals信号的相关,For example, in digital communications, a set of data symbols are represented by a set of unique discrete-time seq

30、uences If one of these sequences has been transmitted, the receiver has to determine which particular sequence has been received by comparing the received signal with every member of possible sequences from the set,2020/7/17,38,Correlation of Signals信号的相关,Similarly, in radar and sonar applications,

31、the received signal reflected from the target is a delayed version of the transmitted signal and by measuring the delay, one can determine the location of the target The detection problem gets more complicated in practice, as often the received signal is corrupted by additive random noise,2020/7/17,

32、39,Correlation of Signals信号的相关,Definitions定义 A measure of similarity between a pair of energy signals, xn and yn, is given by the cross-correlation sequence序列的互相关 defined by The parameter called lag时延, indicates the time-shift between the pair of signals,2020/7/17,40,Correlation of Signals信号的相关,yn i

33、s said to be shifted by samples to the right with respect to the reference sequence xn for positive values of , and shifted by samples to the left for negative values of The ordering of the subscripts xy in the definition of specifies that xn is the reference sequence which remains fixed in time whi

34、le yn is being shifted with respect to xn,2020/7/17,41,Correlation of Signals信号的相关,If yn is made the reference signal and xn is shifted with respect to yn, then the corresponding cross-correlation sequence is given by Thus, is obtained by time-reversing,2020/7/17,42,Correlation of Signals信号的相关,The a

35、utocorrelation自相关sequence of xn is given by obtained by setting yn = xn in the definition of the cross-correlation sequence Note: , the energy of the signal xn,2020/7/17,43,Correlation of Signals信号的相关,From the relation it follows that implying that is an even function for real xn An examination of r

36、eveals that the expression for the cross-correlation looks quite similar to that of the linear convolution,2020/7/17,44,Correlation of Signals信号的相关,This similarity is much clearer if we rewrite the expression for the cross-correlation as Thus, the cross-correlation of yn with the reference signal xn

37、 can be computed by processing xn with a discrete-time LTI system of impulse response,2020/7/17,45,Correlation of Signals信号的相关,Likewise, the autocorrelation of xn can be computed by processing xn with a discrete-time LTI system of impulse response,2020/7/17,46,Properties of Autocorrelation and Cross

38、-correlation Sequences自相关和互相关序列的特性,Consider two finite-energy sequences xn and yn The energy of the combined sequence is also finite and nonnegative, i.e.,2020/7/17,47,Properties of Autocorrelation and Cross-correlation Sequences自相关和互相关序列的特性,Thus where and We can rewrite the equation on the previous

39、 slide as for any finite value of a,2020/7/17,48,Properties of Autocorrelation and Cross-correlation Sequences自相关和互相关序列的特性,Or, in other words, the matrix is positive semidefinite半正定 This implies that or, equivalently,2020/7/17,49,Properties of Autocorrelation and Cross-correlation Sequences自相关和互相关序列

40、的特性,The last inequality on the previous slide provides an upper bound for the cross-correlation samples If we set yn = xn, then the inequality reduces to Thus, at zero lag ( ), the sample value of the autocorrelation sequence has its maximum value,2020/7/17,50,Properties of Autocorrelation and Cross

41、-correlation Sequences自相关和互相关序列的特性,Now consider the case where N is an integer and b 0 is an arbitrary number In this case,2020/7/17,51,Properties of Autocorrelation and Cross-correlation Sequences自相关和互相关序列的特性,Therefore Using the above result in we get,2020/7/17,52,Correlation Computation Using MATL

42、AB,The cross-correlation and autocorrelation sequences can easily be computed using MATLAB Example - Consider the two finite-length sequences,2020/7/17,53,Correlation Computation Using MATLAB,The cross-correlation sequence computed using Program 2_7 of textbook (with a correction!) is plotted below,

43、2020/7/17,54,Correlation Computation Using MATLAB,The autocorrelation sequence computed using Program 2_7 is shown below Note: At zero lag, is the maximum,2020/7/17,55,Correlation Computation Using MATLAB,The plot below shows the cross-correlation of xn and for N = 4 Note: The peak of the cross-corr

44、elation is precisely the value of the delay N,2020/7/17,56,Correlation Computation Using MATLAB,The plot below shows the autocorrelation of xn corrupted with an additive random noise generated using the function randn Note: The autocorrelation still exhibits a peak at zero lag,2020/7/17,57,Computati

45、on of Autocorrelation of a% Noise Corrupted Sinusoidal Sequence,% Program 2_8 % Computation of Autocorrelation of a % Noise Corrupted Sinusoidal Sequence % N = 96; n = 1:N; x = cos(pi*0.25*n); % Generate the sinusoidal sequence d = rand(1,N) - 0.5; % Generate the noise sequence y = x + d; % Generate

46、 the noise-corrupted sinusoidal sequence r = conv(y, fliplr(y); % Compute the correlation sequence k = -28:28; stem(k, r(68:124); xlabel(Lag index); ylabel(Amplitude);,2020/7/17,58,Correlation Computation Using MATLAB,The autocorrelation and the cross-correlation can also be computed using the funct

47、ion xcorr In Matlab, the cross-correlation is in fact defined as,2020/7/17,59,Exercise 2.4.6. Autocorrelation.,clc; clear; close all; % a. Signal Definitions. N = 240; n = 0:N-1; x = 1/2*sin(pi/12*n) + sin(pi/24*n); y = x + 3*(rand(1,N) - 1/2); figure(Name,Exercise 2.4.6. Autocorrelation); subplot(2

48、,2,1); stem(n,x,b.); grid on; axis tight; title(xn = 1/2*sin(pin/12) + sin(pin/24); subplot(2,2,2); stem(n,y,r.); grid on; axis tight; title(yn = xn + 3*(randn-1/2);,2020/7/17,60,% b. m = -N+1:N-1; subplot(2,2,3); % Calculate the autocorrelation function as convolution: % Acorrn = xn*x-n. stem(m,con

49、v(x,fliplr(x),.); % stem(m,xcorr(x); grid on; axis tight; title(Autocorrelation Function A_xn); % c. subplot(2,2,4); % Calculate the autocorrelation function as convolution: % Acorrn = yn*y-n. stem(m,conv(y,fliplr(y),g.); % stem(m,xcorr(y); grid on; axis tight; title(Autocorrelation Function A_yn);,2020/7/17,61,result,2020/7/17,62,Exercise 2.4.7. Cross-Correlation.,clc; clear; close all; % Signal Definition. N =