章节目录 (1).ppt

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1、2020/7/17,1,Chap 2. Discrete-Time Signals:Time-Domain Representation离散时间信号：时域表现形式,Signals are represented as sequences of numbers, called samples(取样) Sample value of a typical signal or sequence denoted as xn with n being an integer(整数)in the range xn defined only for integer values of n and undefin

2、ed for non-integer values of n Discrete-time signal represented by xn,2020/7/17,2,2.1 Discrete-Time Signals:Time-Domain Representation,Discrete-time signal may also be written as a sequence of numbers inside braces: In the above, etc. The arrow is placed under the sample at time index n = 0,2020/7/1

3、7,3,Discrete-Time Signals:Time-Domain Representation,Graphical representation图示of a discrete-time signal with real-valued samples,In some applications, a discrete-time sequence xn may be generated by periodically sampling周期性取样 a continuous-time signal at uniform intervals of time,2020/7/17,4,Discret

4、e-Time Signals:Time-Domain Representation,Block-diagram representation方框图表示of the sampling process,Here, the n-th sample is given by The spacing间距T between two consecutive samples is called the sampling interval间隔or sampling period采样周期 Reciprocal of sampling interval T, denoted as , is called the sa

5、mpling frequency采样频率:,2020/7/17,5,Discrete-Time Signals:Time-Domain Representation,Unit of sampling frequency is cycles per second, or Hertz (Hz), if T is in seconds Whether or not the sequence xn has been obtained by sampling, the quantity xn is called the n-th sample of the sequence xn is a real s

6、equence实序列, if the n-th sample xn is real for all values of n; Otherwise, xn is a complex sequence复序列,A complex sequence xn can be written as where and are the real and imaginary parts of xn The complex conjugate sequence of xn is given by Often the braces are ignored to denote a sequence if there i

7、s no ambiguity,2020/7/17,6,Discrete-Time Signals:Time-Domain Representation,Example - is a real sequence is a complex sequence We can write where,is the complex conjugate sequence of yn That is,2020/7/17,7,Discrete-Time Signals:Time-Domain Representation,Two types类型of discrete-time signals: - Sample

8、d-data signals采样数据信号in which samples are continuous-valued - Digital signals数字信号in which samples are discrete-valued Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding 凑整 or truncation切断,2020/7/17,8,Discrete-Time Si

9、gnals:Time-Domain Representation,Example -,Sampled-data signal,Digital signal,Amplitude,2020/7/17,9,Discrete-Time Signals:Time-Domain Representation,A discrete-time signal may be a finite-length有限长or an infinite-length sequence无限长序列 Finite-length (also called finite-durationor finite-extent) sequenc

10、e is defined only for a finite time interval: where and with Length or duration of the above finite-length sequence is,Example - is a finite-length sequence of length Example - is an infinite-length sequence,2020/7/17,10,Discrete-Time Signals:Time-Domain Representation,A length-N sequence is often r

11、eferred to as an N-point sequence The length of a finite-length sequence can be increased by zero-padding填充, i.e., by appending it with zeros,Example - is a finite-length sequence of length 12 obtained by zero-padding with 4 zero-valued samples,2020/7/17,11,Discrete-Time Signals:Time-Domain Represen

12、tation,A right-sided sequence右边序列xn has zero-valued samples for If a right-sided sequence is called a causal sequence因果序列,A right-sided sequence,2020/7/17,12,Discrete-Time Signals:Time-Domain Representation,A left-sidedsequence左边序列 xn has zero-valued samples for If a left-sided sequence is called an

13、 anti-causal sequence反因果序列,A left-sided sequence,2020/7/17,13,2.2 Operations on Sequences操作的序列,A single-input, single-output (SISO) discrete-time system operates on a sequence, called the input sequence, according some prescribed rules and develops another sequence, called the output sequence, with

14、more desirable properties,2020/7/17,14,Operations on Sequences,For example, the input may be a signal corrupted with additive noise Discrete-time system is designed to generate an output by removing the noise component from the input In most cases, the operation defining a particular discrete-time s

15、ystem is composed of some basic operations,2020/7/17,15,Basic Operations,Product 乘积(modulation调制) operation: Modulator调制器 An application is in forming a finite-length sequence from an infinite-length sequence by multiplying the latter with a finite-length sequence called a window sequence Process ca

16、lled windowing加窗,2020/7/17,16,Basic Operations,Addition加法operation: Adder加法器 Multiplication乘法,倍增operation Multiplier,2020/7/17,17,Basic Operations,Time-shifting时移operation: where N is an integer If N 0, it is delaying延迟operation Unit delay If N 0, it is an advance提前operation Unit advance,yn,xn,2020/

17、7/17,18,Basic Operations,Time-reversal时间反演(folding折叠) operation: Branching分支operation: Used to provide multiple copies of a sequence,2020/7/17,19,Basic Operations,Example - Consider the two following sequences of length 5 defined for :,New sequences generated from the above two sequences by applying

18、 the basic operations are as follows:,Operations on two or more sequences can be carried out if all sequences involved are of same length and defined for the same range of the time index n,2020/7/17,20,Basic Operations,However if the sequences are not of same length, in some situations, this problem

19、 can be circumvented by appending zero-valued samples to the sequence(s) of smaller lengths to make all sequences have the same range of the time index Example - Consider the sequence of length 3 defined for :,We cannot add the length-3 sequence to the length-5 sequence an defined earlier We therefo

20、re first append with 2 zero-valued samples resulting in a length-5 sequence Then,2020/7/17,21,Combinations of Basic Operations,Example -,2020/7/17,22,Sampling Rate Alteration取样率变换,Employed to generate a new sequence yn with a sampling rate higher or lower than that of the sampling rate of a given se

21、quence xn Sampling rate alteration变换ratio is If R 1, the process called interpolation插补 If R 1, the process called decimation抽取,2020/7/17,23,Sampling Rate Alteration取样率变换,In up-sampling by an integer factor L 1, equidistant zero-valued samples are inserted by the up-sampler between each two consecut

22、ive samples of the input sequence xn:,An example of the up-sampling operation,2020/7/17,24,Sampling Rate Alteration取样率变换,An example of the up-sampling operation,2020/7/17,25,Sampling Rate Alteration取样率变换,In down-sampling下采样 by an integer factor M 1, one every M samples of the input sequence is kept

23、and in-between samples are removed:,2020/7/17,26,Sampling Rate Alteration取样率变换,An example of the down-sampling operation,2020/7/17,27,Classification of Sequences Based on Symmetry对称序列的分类,Conjugate-symmetric sequence共扼对称序列: If xn is real, then it is an even sequence偶序列,An even sequence,2020/7/17,28,C

24、lassification of Sequences Based on Symmetry,Conjugate-antisymmetric sequence共扼反对称序列: If xn is real, then it is an odd sequence奇序列,An odd sequence,2020/7/17,29,Classification of Sequences Based on Symmetry,It follows from the definition that for a conjugate-symmetric sequence xn, x0 must be a real n

25、umber Likewise, it follows from the definition that for a conjugate anti-symmetric sequence yn, y0 must be an imaginary number From the above, it also follows that for an odd sequence wn, w0 = 0,2020/7/17,30,Classification of Sequences Based on Symmetry,Any complex sequence can be expressed as a sum

26、 of its conjugate-symmetric part and its conjugate-antisymmetric part: where,2020/7/17,31,Classification of Sequences Based on Symmetry,Example - Consider the length-7 sequence defined for : Its conjugate sequence is then given by The time-reversed version of the above is,2020/7/17,32,Classification

27、 of Sequences Based on Symmetry,Therefore Likewise It can be easily verified that and,2020/7/17,33,Classification of Sequences Based on Symmetry,Any real sequence can be expressed as a sum of its even part and its odd part: where,2020/7/17,34,Classification of Sequences Based on Symmetry,A length-N

28、sequence xn, can be expressed as where is the periodic conjugate-symmetric part and is the periodic conjugate-antisymmetric part,2020/7/17,35,Classification of Sequences Based on Symmetry,For a real sequence, the periodic conjugate-symmetric part, is a real sequence and is called the periodic even p

29、art For a real sequence, the periodic conjugate-antisymmetric part, is a real sequence and is called the periodic odd part,2020/7/17,36,Classification of Sequences Based on Symmetry,A length-N sequence xn is called a periodic conjugate-symmetric sequence if and is called a periodic conjugate-antisym

30、metric sequence if,A finite-length real periodic conjugate-symmetric sequence is called a symmetric sequence A finite-length real periodic conjugate-antisymmetric sequence is called an antisymmetric sequence,2020/7/17,37,Classification of Sequences Based on Symmetry,Example - Consider the length-4 s

31、equence defined for : Its conjugate sequence is given by To determine the modulo-4 time-reversed version observe the following:,Hence,2020/7/17,38,Classification of Sequences Based on Symmetry,Therefore Likewise,2020/7/17,39,Classification of Sequences Based on Periodicity周期序列的分类,A sequence satisfyi

32、ng is called a periodic sequence with a period N where N is a positive integer and k is any integer The smallest value of N satisfying is called the fundamental period,Example-,A sequence not satisfying the periodicity condition is called an aperiodic sequence,2020/7/17,40,Classification of Sequence

33、s:Energy and Power Signals序列分类：能量和功率信号,Total energy总能量 of a sequence xn is defined by An infinite length sequence with finite sample values may or may not have finite energy A finite length sequence with finite sample values has finite energy,The average power平均功率 of an aperiodic sequence is defined

34、 by,2020/7/17,41,Classification of Sequences:Energy and Power Signals,Define the energy of a sequence xn over a finite interval as,Then,The average power of a periodic sequence with a period N is given by The average power of an infinite-length sequence may be finite or infinite,2020/7/17,42,Classif

35、ication of Sequences:Energy and Power Signals,Example - Consider the causal sequence defined by Note: xn has infinite energy Its average power is given by,2020/7/17,43,Classification of Sequences:Energy and Power Signals,An infinite-energy signal with finite average power is called a power signal功率信

36、号 Example - A periodic sequence which has a finite average power but infinite energy A finite-energy signal with zero average power is called an energy signal能量信号 Example - A finite-length sequence which has finite energy but zero average power,2020/7/17,44,Other Types of Classifications其他类型的分类,A se

37、quence xn is said to be bounded有界的if Example - The sequence is a bounded sequence as,A sequence xn is said to be absolutely summable绝对可加的if,Example - The sequence is an absolutely summable sequence as,2020/7/17,45,Other Types of Classifications,A sequence xn is said to be square-summable平方可加的if Exam

38、ple - The sequence is square-summable but not absolutely summable,2020/7/17,46,Basic Sequences基本序列,Unit sample sequence单位取样序列- Unit step sequence单位阶跃序列-,The unit sample and unit step sequences relate as follows:,2020/7/17,47,Basic Sequences,Real sinusoidal sequence实正弦序列- where A is the amplitude, is

39、 the angular frequency, and is the phase of xn Example -,2020/7/17,48,Basic Sequences,Exponential sequence指数序列- where A and are real or complex numbers If we write then we can express where,2020/7/17,49,Basic Sequences,and of a complex exponential sequence are real sinusoidal sequences with constant

40、 , growing , and decaying amplitudes for n 0,2020/7/17,50,Basic Sequences,Real exponential sequence - where A and a are real numbers,2020/7/17,51,% Program 2_2 % Generation of real exponential sequence % a = input(Type in exponent = ); K = input(Type in the gain constant = ); N = input (Type in leng

41、th of sequence = ); n = 0:N; x = K*a.n; stem(n,x); xlabel(Time index n);ylabel(Amplitude); title(alpha = ,num2str(a);,2020/7/17,52,clc; clear; close all; % Generate the basic signals of common length 16. N = 16; n = 0:N-1; b = ones(1,N); % Block of ones. r = n; % Ramp function. P = 8; % Triangular w

42、ave period. n1 = 0:P/2-1; n2 = P/2:P-1; P1 = P*ones(1,length(n2); A = 1; tri_block = 2*A*n1/P 2*A*(P1-n2)/P ; t = tri_block tri_block; % % Periodic Triangular wave e = (5/6).n; % One sided exponential. % a. Create and display sn = rn-6*un. figure(Name,Exercise 2.1.3. Elementary Signals); subplot(3,1

43、,1); stem(n,r); grid; hold on; stem(n+6,r,r*); title(rn (blue) and sn=rn-6*un (red);,Exercise 2.1.3. Elementary Signals.,2020/7/17,53,% b. Create and display tn=bn-bn-6. a = b(1:6) zeros(1,10); subplot(3,1,2); stem(n,b); grid; hold on; stem(n,a,r*); title(bn (blue) and tn=bn-bn-6 (red); % c. Create

44、and display yn=en+10*bn. y = e(11:16) zeros(1,10).*b; subplot(3,1,3); stem(n,e); grid; hold on; stem(n-10,e,r*); stem(n,y,g); title(en (blue), en+10 (red) and yn=en+10*bn (green); % d. Create and display zn=tn*(un-un-10). z = t(1:10) zeros(1,6); figure(Name,Exercise 2.1.3. Elementary Signals); subpl

45、ot(3,1,1); stem(n,t); grid; hold on; stem(n,z,r*); title(tn (blue) and zn=tn*(un-un-10) (red);,2020/7/17,54,% e. Create and display e_en=evenen*(un+5-un-5). m = -N+1:N-1; % The whole duration of the signal is 2*(N-1) + 1. e_ext = zeros(1,N-1) e; % figure(3); % stem(m,e_ext); % grid on; e_even = (e_e

46、xt + fliplr(e_ext)/2; e_odd = (e_ext - fliplr(e_ext)/2; subplot(3,1,2); stem(m,e_even); grid; hold on; stem(-5:4,e_even(11:20),r*); title(e_e_v_e_nn (blue) and e_en=e_e_v_e_nn*(un+5-un-5) (red); % f. Create and display e_on=odden*(un+5-un-5). subplot(3,1,3); stem(m,e_odd); grid; hold on; stem(-5:4,e

47、_odd(11:20),r*); title(e_o_d_dn (blue) and e_on=e_o_d_dn*(un+5-un-5) (red);,2020/7/17,55,Result,figure 1,2020/7/17,56,figure 2,2020/7/17,57,Basic Sequences,Sinusoidal sequence and complex exponential sequence are periodic sequences of period N if where N and r are positive integers Smallest value of

48、 N satisfying is the fundamental period of the sequence To verify the above fact, consider,2020/7/17,58,Basic Sequences,Now which will be equal to only if and These two conditions are met if and only if or,2020/7/17,59,Basic Sequences,If is a noninteger rational number, then the period will be a multiple of Otherwise, the sequence is aperiodic非周期的 Example - is an aperiodic sequence,2020/7/17,60,Here Equation is satisfied for r = 0 and any positive integer N; N=1 is the smallest,Basic Sequences,2020/7/17,61,Basic Sequences,Here Hence for r = 1,2020/7/17,62,Basic Sequen