1、 Chap 4 Discrete-Time Systems Time-Domain Characterization of LTI Discrete- Times Systems Classification of discrete-time systems Impulse and step responses Input-output relationship and Difference equation; Stability and Causality of LTI DT system Frequency-Domain Characterization of LTI Discrete-T

2、imes Systems Phase and Group DelayDSP group 2007-chap2-ed12 4.1 Discrete-Time System ExamplesDiscrete-Time SystemxnynInput sequenceOutput sequenceExample: A DT system is composed of adder, multiplier, and delayerDSP group 2007-chap2-ed13 4.1 Discrete-Time System Examples Accumulator: Initial conditi

3、on: y1DSP group 2007-chap2-ed14 Moving Average Filter: 4.1 Discrete-Time System ExamplesOr DSP group 2007-chap2-ed15Noise + signal 4.1 Discrete-Time System ExamplesNoise cancellationFiltered signalP138DSP group 2007-chap2-ed16 4.1 Discrete-Time System Examplesfreqz(1/3 1/3 1/3, 1)freqz(1/6*ones(1,6)

4、, 1)DSP group 2007-chap2-ed17 4.1 Discrete-Time System Examples Exponentially Weighted Running Average Filter Linear Interpolator - linear factor-of-2 interpolatorP138-140 - bilinear interpolationFigure4.5Median FilterFigure4.5Figure4.6DSP group 2007-chap2-ed18 4.2 Classification of Discrete-Time Sy

5、stems Linear system - superposition always holdsare the responses to the inputsequencesrespectively, for input, the response is Above property must hold for any arbitrary constants a and b and for all possible inputs x1n and x2nDSP group 2007-chap2-ed19 4.2 Classification of Discrete-Time Systems -

6、ExamplesExample4.3: AccumulatorSolution:Assume:Let Then Linear systemDSP group 2007-chap2-ed110 4.2 Classification of Discrete-Time Systems - ExamplesExample: systemSolution:Assume:Let Then Nonlinear systemDSP group 2007-chap2-ed111 4.2 Classification of Discrete-Time Systems Shift-invariant (time-i

7、nvariant) systemis the responses to the input sequence , for input, the response isExample4.5: up-samplerSolution:Assume:DSP group 2007-chap2-ed112 4.2 Classification of Discrete-Time Systems - ExamplesLet Then While Time-varying systemDSP group 2007-chap2-ed113 4.2 Classification of Discrete-Time S

8、ystems Causal systemDSP group 2007-chap2-ed114 4.2 Classification of Discrete-Time Systems Stable system If and only if for every bounded input, the output is also bounded - stable system BIBO (bounded-input, bounded output)DSP group 2007-chap2-ed115A discrete-time system is defined to be passive if

9、, for every finite-energy input xn, the output yn, has, at most, the same energy, i.e. For a lossless system, the above inequality is satisfied with an equal sign for every input. 4.2 Classification of Discrete-Time Systems Passive and Lossless SystemsDSP group 2007-chap2-ed116yn= xnN, with N a posi

10、tive integerIts output energy is given by Hence, it is a passive system if | 1 and is a lossless system if | =1 4.2 Classification of Discrete-Time SystemsExample4.8: Discrete-time systemDSP group 2007-chap2-ed117 4.3 Impulse and step response Impulse response: the response of a digital filter to a

11、unit sample sequence n. denotation: hn. Step response: the response of a digital filter to a unit step sequence un. denotation: sn. DSP group 2007-chap2-ed118 4.3 Impulse and step responseOr Example4.9: LTI discrete-time systemImpulse responseDSP group 2007-chap2-ed119 4.4 Time-domain Characterizati

12、on of LTI Discrete-Time System Linear time-invariant (LTI) Discrete-time (DT) system: satisfies both the linearity and the time-invariance properties. interconnection of simple subsystems.4.4.1 Input-output relationship A consequence of the linear, time-invariance property is that an LTI DT system i

13、s completely characterized by its impulse response;DSP group 2007-chap2-ed120 4.4.1 Input-output relationshipKnowing the impulse response, we can compute the output of the system to any arbitrary input.Because hn is the impulse response of LTI DT system.i.e.,Using time-invarianceUsing linearityDSP g

14、roup 2007-chap2-ed121 4.4.1 Input-output relationship Convolution sum: Definition Commutative operation DSP group 2007-chap2-ed122 4.4.1 Input-output relationship - Convolution sum distributive operation associative operation DSP group 2007-chap2-ed123 4.4.3 Stability Condition in terms of the impul

15、se response Meaning of stable system: BIBO An LTI digital filter is BIBO stable if and only if absolutely summableP151DSP group 2007-chap2-ed124 4.4.4 Causality Condition in terms of the impulse response An LTI digital filter is causal if and only ifDSP group 2007-chap2-ed125Cascade ConnectionImpuls

16、e response hn of the cascade of two LTI discrete-time systems with impulse responses h1n and h2n is given by 4.5 Simple interconnection schemesDSP group 2007-chap2-ed126Note: The interconnection order of the systems in the cascade has no effect on the overall impulse response.A cascade connection of

17、 two stable systems is stable.A cascade connection of two passive (lossless) systems is passive (lossless) . 4.5 Simple interconnection schemesDSP group 2007-chap2-ed127Parallel ConnectionImpulse response hn of the parallel connection of two LTI discrete-time systems with impulse responses h1n and h

18、2n is given by hn = h1n + h2n 4.5 Simple interconnection schemes28h1n= n + 0.5 n 1h2n= 0.5 n 0.25 n 1h3n= 2 n h4n= 2(0.5)nunConsider the discrete-time system where 4.5 Simple interconnection schemesDSP group 2007-chap2-ed129Overall impulse response hn is given bySimplifying the block-diagram we obta

19、inh1n= n + 0.5 n 1h2n= 0.5 n 0.25 n 1h3n= 2 n h4n= 2(0.5)nun 4.5 Simple interconnection schemesDSP group 2007-chap2-ed130 4.6 Finite-Dimensional LTI Discrete-Time Systems -Linear Constant Coefficient Difference Equation Order: max(N,M) Input: xn Output: ynSolution is referred to “Signals and Systems

20、”DSP group 2007-chap2-ed131 4.6 Finite-Dimensional LTI Discrete-Time Systems -Linear Constant Coefficient Difference EquationExample 4.22Solution:Homogenous response:forcing response:Total response:P158DSP group 2007-chap2-ed132 4.6 Finite-Dimensional LTI Discrete-Time Systems -Linear Constant Coeff

21、icient Difference Equation Output Computation Using Matlaby=filter (p,d,x)y, sf=filter (p,d,x,si)h,m=impz (p,d,n)s, m=stepz (p,d,n) Impulse and Step response computation Using MatlabDSP group 2007-chap2-ed133 4.7 Classification of LTI DT Systems Classification based on impulse response length Finite

22、 impulse response (FIR) Infinite impulse response (IIR)DSP group 2007-chap2-ed134 Nonrecursive: The output sample can be calculated sequentially, knowing only the present and past input samples. 4.7.1 Classification of LTI DT Systems-FIRDSP group 2007-chap2-ed135 For a causal IIR DT system with a ca

23、usal input xn, the convolution sum is expressed: 4.7.1 Classification of LTI DT Systems-IIRDSP group 2007-chap2-ed1364.7.2 Classification based on Output calculation process application for model-based spectral analysis, statistical analysis based on the form of the difference equation- Moving avera

24、ge (MA) model- Autoregressive(AR) model- ARMA model37 4.8 The Frequency Response of LTI DT System Eigenfunction : e j n Let xn = ej n, LTI system with impulse response hn, the output of the LTI system is:Or rewritten as DSP group chap3-ed138 4.8.1 Definition of frequency response -Property of Freque

25、ncy Response Frequency response H(e j) : is the DTFT of the impulse response hn; is a continuous function of ; is a periodic function of with a period 2; is a complex function of real variable .DSP group chap3-ed139 4.8.1 Definition of frequency response - Gain and Attentuation Gain function : Atten

26、uation (loss) function : H(e j) Provides a frequency-domain description of the systemDSP group chap3-ed140 4.8.2 Frequency-Domain Characterization of the LTI DT SystemExample Input sequence xn = anun, |a|1,LTI system with impulse response:hn = bnun, |b|1. Find the output sequence yn .Solution:DSP gr

27、oup chap3-ed142 4.8.3 Frequency Response of LTI DT Systems LTI FIR DT Systems :DSP group chap3-ed143 4.8.3 Frequency Response of LTI DT Systems LTI IIR DT Systems :DSP group chap3-ed144 4.8.3 Frequency Response of LTI DT SystemExample 4.31Determine the frequency response of moving-average filter:Sol

28、ution:DSP group chap3-ed145H=freqz(h,1,w); 4.8.4 Frequency Response Computation Using MatlabDSP group chap3-ed146 4.8.5 The Concept of Filtering pass certain frequency components in an input sequence without any distortion (if possible) block other frequency components. digital filters: Systems that

29、 complement filtering functions The key to the filtering process is : Filtering DSP group chap3-ed147 4.8.5 The Concept of Filtering - Lowpass digital filter Real coefficient LTI DT system characterized by a magnitude function : input sequence: output sequence: Lowpasscc|H( e j )|10 Ex.4.32 : Design

30、 a simple digital filterDSP group chap3-ed148 4.9 Phase and Group delay 4.9.1 Definition: The input xn of LTI DT system H(e j) is a sinusoidal signal of frequency 0, i.e., The output ynDSP group chap3-ed149 4.9.1 The Definition of Phase delay phase delay The output yn is a time-delayed version of th

31、e input xn when phase delay is an integer; Phase delay has a physical meaning only with respect to the underlying continuous-time functions associated with the input xn and output sequence yn ;DSP group chap3-ed150 4.9.1 The Definition of - Group delay When the input xn contains many sinusoidal comp

32、onents with different frequencies that are not harmonically related, each component will go through different phase delays. Group delay is defined to describe the signal delay:DSP group chap3-ed151 In defining the group delay, It is assumed that the phase function is unwrapped so that its derivative

33、 exists. Group delay also has a physical meaning only with respect to the underlying continuous-time functions associated with yn and xn. 4.9.1 The Definition of - Group delayDSP group chap3-ed152 4.9.1 Definition -Comparison between phase delay and group delayDSP group chap3-ed153 4.9.1 Definition - Examples of group delayExample-A1FIR system with input-output relation:Phase function:Group delay:DSP group chap3-ed154 4.9.1 Definition - Examples of group delayExample-A2moving-average filter:Phase fu