Chapter 4Discrete Time System140311 (2)

1、l离散时间系统分类：离散时间系统分类：Linear，shift-invariant，causal，stable，passive and lossless systemQuestionlConsider the linearity and time-invariance properties of the following systems (1) y(n)=x(nn0) (2) y(n)=ax(n)+b上讲回顾上讲回顾(1) y(n)=x(nn0)000 1 111 1nnnx ny nx nnx ny nxy nxny ny n，1110222012102012 x ny nx nnx ny

2、 nx nnax nbx ny nax nnbx nny nay nby n，(2) y(n)=ax(n)+b000 + nx ny nax nbx nnny naxby nny n，11122212121212 2x ny nax nbx ny nax nbx nx ny nax nax nby ny ny nax nax nb，l系统稳定性的时域充要条件：系统稳定性的时域充要条件：h(n)绝对可和，即绝对可和，即l系统因果性的时域充要条件：即系统因果性的时域充要条件：即 上讲回顾上讲回顾 0,0h nnQuestions-判断因果稳定性判断因果稳定性l0.5nu(n) l2nu(n)l(-

3、2)nu(n)l2nu(-n)l0.5nu(-n-1)l2nR10(n)Chapter 4 Discrete-Time Systemsl4.1 Discrete-Time System Examples l4.2 Classification of Discrete-Time Systems l4.3 Impulse and Step Responses l4.4 Time-Domain Characterization of LTI Discrete-Time Systems l4.5 Simple Interconnection Schemes l4.6 Finite-Dimensiona

4、l LTI Discrete-Time Systems l4.7 Classification of LTI Discrete-Time Systems l4.8 Frequency-Domain Representations of LTI Discrete-Time Systems l4.9 Phase and Group Delays4.5 Simple Interconnection SchemeslCascade Connectionnh1nh2nh1nh2 nhn h1nh2nh1*Impulse response hn of the cascade of two LTI disc

5、rete-time systems with impulse responses h1n and h2n is given bynh2nhnh1* 4.5 Simple Interconnection SchemeslNote: The ordering of the systems in the cascade has no effect on the overall impulse response because of the commutative property of convolutionlA cascade connection of two stable systems is

6、 stablelA cascade connection of two passive (lossless) systems is passive (lossless) 4.5 Simple Interconnection SchemeslParallel Connectionnh2nh1 nhn h1nh2nh1Impulse response hn of the parallel connection of two LTI discrete-time systems with impulse responses h1n and h2n is given by hn= h2n + h1n 4

7、.5 Simple Interconnection Schemesh1n= n + 0.5n-1h2n= 0.5n + 0.25n-1h3n= 2nh4n= 2(0.5)n nnh2nh1nh4nh3Consider the discrete-time system where 4.5 Simple Interconnection SchemeslSimplifying the block-diagram we obtainnh2nh143nhnhnh1)(432nhnhnh*)(432nhnhnh*h1n+ 4.5 Simple Interconnection SchemeslOverall

8、 impulse response hn is given bynhnhnhnhnh42321)(nhnhnhnhnh4321*2)1(412132nnnnhnh 121nn*Now, 4.5 Simple Interconnection Schemes)(21nnn 1)()(1212121nnnn 1)()(2121nnnn)(2)1(21412142nnnnhnhn* 1 12121nnnnnnnhThereforeChapter 4 Discrete-Time Systemsl4.1 Discrete-Time System Examples l4.2 Classification o

9、f Discrete-Time Systems l4.3 Impulse and Step Responses l4.4 Time-Domain Characterization of LTI Discrete-Time Systems l4.5 Simple Interconnection Schemes l4.6 Finite-Dimensional LTI Discrete-Time Systems l4.7 Classification of LTI Discrete-Time Systems l4.8 Frequency-Domain Representations of LTI D

10、iscrete-Time Systems l4.9 Phase and Group Delays 4.6 Finite-Dimensional LTIDiscrete-Time SystemslAn important subclass of LTI discrete-time systems is characterized by a linear constant coefficient difference equation of the formlxn and yn are, respectively, the input and the output of the systemldk

11、andpk are constants characterizing the systemMkkNkkknxpknyd00 4.6 Finite-Dimensional LTIDiscrete-Time SystemslThe order of the system is given by max(N,M), which is the order of the difference equationl It is possible to implement an LTI system characterized by a constant coefficient difference equa

12、tion as here the computation involves two finite sums of products 4.6 Finite-Dimensional LTIDiscrete-Time SystemslIf we assume the system to be causal, then the output yn can be recursively computed using Provided d00l yn can be computed for all nn0, knowing xn and the initial conditions yn0-1, yn0-

13、2,yn0-NMkkNkkknxdpknyddny0010)(4.7 Classification of LTI Discrete-Time SystemsBased on Impulse Response Length -lIf the impulse response hn is of finite length, i.e., hn不等于不等于0 for N1nN2 and N1N2then it is known as a finite impulse response (FIR) discrete-time systemlThe convolution sum description

14、here is21NNkknxkhny4.7 Classification of LTI Discrete-Time SystemslThe output yn of an FIR LTI discrete-time system can be computed directly from the convolution sum as it is a finite sum of productslExamples of FIR LTI discrete-time systems are the moving-average system and the linear interpolators

15、4.7 Classification of LTI Discrete-Time SystemslIf the impulse response is of infinite length, then it is known as an infinite impulse response (IIR) discrete-time systemlThe class of IIR systems we are concerned with in this course are characterized by linear constant coefficient difference equatio

16、ns4.7 Classification of LTI Discrete-Time SystemslExample - The discrete-time accumulator defined by yn=yn-1+xnis seen to be an IIR system4.7 Classification of LTI Discrete-Time SystemsBased on the Output Calculation Process lNonrecursive System - Here the output can be calculated sequentially, know

17、ing only the present and past input sampleslRecursive System - Here the output computation involves past output samples in addition to the present and past input samplesChapter 4 Discrete-Time Systemsl4.1 Discrete-Time System Examples l4.2 Classification of Discrete-Time Systems l4.3 Impulse and Ste

18、p Responses l4.4 Time-Domain Characterization of LTI Discrete-Time Systems l4.5 Simple Interconnection Schemes l4.6 Finite-Dimensional LTI Discrete-Time Systems l4.7 Classification of LTI Discrete-Time Systems l4.8 Frequency-Domain Representations of LTI Discrete-Time Systems l4.9 Phase and Group De

19、lays4.8.1 The Frequency Response Most discrete-time signals encountered in practice can be represented as a linear combination of a very large, maybe infinite, number of sinusoidal discrete-time signals of different angular frequencies Thus, knowing the response of the LTI system to a single sinusoi

20、dal signal, we can determine its response to more complicated signals by making use of the superposition property4.8.1 The Frequency ResponselIn the above H(ej ) is the frequency response of the LTI systemlThe above equation relates the input and the output of an LTI system in the frequency domain)(

21、)()()(jjjkkjjeXeHeXekheY Hence, we can write4.8.1 The Frequency Response The quantity H(ej ) is called the frequency response of the LTI discrete-time system H(ej ) provides a frequency-domain description of the system H(ej ) is precisely the DTFT of the impulse response hn of the system4.8.1 The Fr

22、equency Response H(ej ), in general, is a complex function of w with a period It can be expressed in terms of its real and imaginary parts H(ej )= Hre(ej ) +j Him(ej ) or, in terms of its magnitude and phase, H(ej )=|H(ej )| e ( )where ( )=argH(ej ) 24.8.1 The Frequency Response The function | H(ej

23、) | is called the magnitude response and the function ( ) is called the phase response of the LTI discrete-time system Design specifications for the LTI discrete-time system, in many applications, are given in terms of the magnitude response or the phase response or both 4.8.1 The Frequency Response

24、 In some cases, the magnitude function is specified in decibels as G( ) = 20log10| H(ej ) | dBwhere G(w) is called the gain function The negative of the gain function A( ) = - G( ) is called the attenuation or loss function 4.8.1 The Frequency Response Note: Magnitude and phase functions are real fu

25、nctions of w, whereas the frequency response is a complex function of w If the impulse response hn is real then the magnitude function is an even function of w: |H(ej )| = |H(e - j )| and the phase function is an odd function of w: ( ) = - (- ) 4.8.1 The Frequency ResponselLikewise, for a real impul

26、se response hn, is even and is odd)(jreeH)(jimeH y nh nx n=()()()jjjY eH eX ewww=()()()jjjY eH eX ewww=4.8.2 Frequency-Domain Characterization of the LTI Discrete-Time System4.8.3 Frequency Response of LTI Discrete-Time SystemsFrequency Response of LTI FIR discrete-time systems:Frequency Response of

27、 LTI IIR discrete-time systems:4.8.4 Frequency Response Computation Using Matlabpp.170 Example 4.314.8.4 Frequency Response Computation Using Matlab4.8.4 Frequency Response Computation Using Matlab4.8.4 Frequency Response Computation Using MATLAB The function freqz(h,w) can be used to determine the

28、values of the frequency response vector h at a set of given frequency points w From h, the real and imaginary parts can be computed using the functions real and imag, and the magnitude and phase functions using the functions abs and angleUsing Matlab, we can generate the magnitude and phase response

29、s of an M-point moving average filter as shown below:4.8.6 Response to a Causal Exponential Sequence lWithout any loss of generality, assume xn=0, for n 0lFrom the input-output relation the output is given bykkhknxny we observe that for an input)(nuenxnj)()()(0nuekhnyknjnk4.8.6 Response to a Causal

30、Exponential Sequence lOr,lThe output for n NlHence, ytrn=0, for n N-1lHere the output reaches the steady-state value at n = NnjjsreeHny )( 4.8.7 The Concept of FilteringlOne application of an LTI discrete-time system is to pass certain frequency components in an input sequence without any distortion

31、 (if possible) and to block other frequency componentslSuch systems are called digital filters and one of the main subjects of discussion in this course理想滤波器按幅频响应分类理想滤波器按幅频响应分类 (1)理想低通理想低通c1c0)(jeH(2)理想高通理想高通c1c0)(jeH(3)理想带通理想带通110)(jeH2(4)理想带阻理想带阻110)(jeH2c 1 01 2 通带通带阻带阻带过渡带过渡带理想理想12c ()jH e 实际滤波器

32、实际滤波器以低通为例，幅频响应并非是锐截止的通带和阻带以低通为例，幅频响应并非是锐截止的通带和阻带 4.8.7 The Concept of FilteringlThe key to the filtering process is deeXnxnjj)(21l It expresses an arbitrary input as a linear weighted sum of an infinite number of exponential sequences, or equivalently, as a linear weighted sum of sinusoidal sequenc

33、es 4.8.7 The Concept of FilteringlThus, by appropriately choosing the values of the magnitude function |H(ej )| of the LTI digital filter at frequencies corresponding to the frequencies of the sinusoidal components of the input, some of these components can be selectively heavily attenuated or filte

34、red with respect to the others 4.8.7 The Concept of FilteringlTo understand the mechanism behind the design of frequency-selective filters, consider a real-coefficient LTI discrete-time system characterized by a magnitude function ccjeH, 0, 1)( 4.8.7 The Concept of FilteringlWe apply an input xn to this system lBecause of linearity, the output of this system is of the form )(cos)(111 neHAnyj )(cos)(222 neHBj 21210,coscoscwherenBnAnx 4.8.7 The Concept of FilteringlAs 0)(, 1)(21 jjeHeH )(cos)(111 neHAnyjl Thus, the system acts like a lowpass filterthe output reduces to