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1、1ChapterChapter4 Discrete-time Systems4 Discrete-time SystemsClassification of Discrete SystemsImpulse and Step ResponseLTI Discrete-time SystemsPhase and Group Delay4.1 Discrete-Time Systems ExamplesnAdiscrete-timesystemprocessesagiveninputsequencexntogeneratesanoutputsequenceynwithmoredesirablepro
2、pertiesnInmostapplications,thediscrete-timesystemisasingle-input,single-outputsystem:xnyn=H(xn)Input sequenceOutput sequenceDiscrete-TimeSystem2nAccumulator3In another form4.1 Discrete-Time Systems ExamplesnMoving-Average filter4Exponentially Weighted Running Average filter4.1 Discrete-Time Systems
3、Examples54.1 Discrete-Time Systems ExamplesLinear InterpolatorBilinear/factor-of-2 interpolationBilinear/factor-of-3 interpolationnMedian filter6nThemedianfilterisimplementedbyslidingawindowofoddlengthovertheinputsequencexnonesampleatatime.4.1 Discrete-Time Systems Examples4.2 Classification of Disc
4、rete-Time SystemsnLinear SystemnShift-Invariant SystemnCausal SystemnStable SystemnPassive and Lossless Systems7nLinear System nDefinition For an input signal The response is given byIf superposition property hold for any arbitrary constants, and , and for all possible input signals ,the system can
5、be called as linear system4.2 Classification of Discrete-Time Systems8Is accumulator a linear system?Shift-Invariant SystemnDefinitionIfproperty hold for any n0 can be given, the system is called as shift-invariant systemnExample 4.5: Up-sampler 4.2 Classification of Discrete-Time Systems9Causal Sys
6、temDefinition : In addition to the above two properties, the n0th output sample yn0 depends only on input samples xn for n n0 and does not depend on input samples for nn0.This system is called as causal system4.2 Classification of Discrete-Time Systems10For a causal system If for nN Implies also tha
7、t for nN Note: The definition of causality given above can be applied only to discrete-time systems with the same sampling rate for the input and the output.4.2 Classification of Discrete-Time Systems11StableSystem Definition: If and only if, for every bounded input, the output is also bounded, the
8、system can be called as stable systemThis type of stability is usually referred to as bounded-input, bounded-output (BIBO) stability.4.2 Classification of Discrete-Time Systems12ForaBIBOsystem,iftheresponsetoxnisthesequenceyn.While|xn|Bx,forallvaluesofnThen:|yn|By.forallvaluesofnwhereBxandByarefinit
9、econstants.4.2 Classification of Discrete-Time Systems13Example:M-pointMovingaveragePassiveandLosslessSystemsAdiscrete-timesystemissaidtobepassiveif,foreveryfiniteenergyinputsequencexn,theoutputsequenceynhas,atmost,thesameenergy,i.e.Iftheaboveinequalityissatisfiedwithanequalsignforeveryinputsequence
10、,thediscrete-timesystemissaidtobelossless.4.2 Classification of Discrete-Time Systems14Thepassivityandthelosslessnesspropertiesarecrucialtothedesignofdiscrete-timesystemswithverylowsensitivitytochangesinthefiltercoefficients4.2 Classification of Discrete-Time Systems15Example:4.3 Impulse and Step Re
11、sponses Definition:Theresponseofadigitalsystemtoaunitsamplesequenceniscalledtheunitsampleresponse,orsimply,theimpulseresponse,andisdenotedashn.Theresponseofadiscrete-timesystemtoaunitstepsequencen,denotedassn,isitsunitstepresponseorsimply,thestepresponse.Alineartime-invariantdigitalsystemcanbecomple
12、telycharacterizedinthetime-domainbyitsimpulseresponseoritsstepresponse164.4 Time-Domain Characterization of LTI Discrete-Time SystemInput-OutputRelationshipBecauseTheresponseoftheLTIsystemtoaninputxkn-kwillbexkhn-kwillbe17Thesummationyn=xnhn*iscalledtheconvolutionsumofthesequencesxnandhnandrepresent
13、edcompactlyas4.4 Time-Domain Characterization of LTI Discrete-Time System18Convolution SumnExample 4.13 *19ExampleLength Length 4.4 Time-Domain Characterization of LTI Discrete-Time System20Length 4.4 Time-Domain Characterization of LTI Discrete-Time System21Properties Of Convolution1.Commutativepro
14、pertyy(n)=x(n)*h(n)=h(n)*x(n)2.Associativepropertyx(n)*h1(n)*h2(n)=x(n)*h1(n)*h1(n)=x(n)*h1(n)*h2(n)=x(n)*h2(n)*h1(n)3.Distributivepropertyx(n)*h1(n)+h2(n)=x(n)*h1(n)+x(n)*h2(n)22nStabilitySystemConditionnIfandonlyiftheoutputsequenceynofthesystemremainsboundedforallboundedinputsequencexn,adiscrete-t
15、imeisdefinedtobeBIBOstable 4.4 Time-Domain Characterization of LTI Discrete-Time System23Causality System Condition IfandonlyiftheresponsesequencehnofaLTIdiscrete-TimesystemissatisfyingtheconditionofEquationbelow,itiscausal:hk=0, for k=n0 knowing xn and initial conditions yn0-1, yn0-2, yn0-N.4.6 Fin
16、ite-dimensional LTI discrete-time system28Total solution calculation We may divided the answer of Eq(4.34) into two parts as below:Here ,yn is called total solution. ycn is the solution of Eq(4.32) with the input xn=0,which is called complementary solution. ypn is the solution of Eq(4.32) with the i
17、nput , which is called particular solution. 4.6 Finite-dimensional LTI discrete-time system29Zero-input response and zero-state responseWe may also divided the answer of Eq(4.33) into the two parts as below:Here ,the zero-input response yZin is obtained by solving Eq(4.33) by setting the input xn=0,
18、the zero-state response yZsn is obtained by solving Eq(4.33) by applying the specified input with all initial conditions set to zero.4.6 Finite-dimensional LTI discrete-time system30Example 4.32 Known the difference equation as bellow: For a step input xn=8n and with initial conditions y-1=1 and y-2
19、=-1,the total response yn are wanted. First, determine the eigen value of the difference equation Setting xn=0 and yn=n in the difference equation we arrive at4.6 Finite-dimensional LTI discrete-time system31And hence its eigen value (roots) are: Therefore we can get the complementary solution as be
20、low:For the particular solution we assume4.6 Finite-dimensional LTI discrete-time system32 The ypn is also satisfies the difference equation, so we arrive at:The total response is the formConsider the initial conditions we get4.6 Finite-dimensional LTI discrete-time system33So we arrive atThus Read
21、and exercise example 4.23 and 4.24 by yourself!4.6 Finite-dimensional LTI discrete-time system34Impulse response calculation We may get the Impulse response by set xn= n, usually it has the form as ycnExample 4.25 Known the difference equation as bellow: its hn is wanted.From example 4.22 we get 4.6
22、 Finite-dimensional LTI discrete-time system35 Substituting from the difference equation we getSo For any other input xn ,we can get the total solution yn according to formula as bellow:4.6 Finite-dimensional LTI discrete-time system36 Impulse and Step Response computation Using Matlab p=0.8 -0.44 0
23、.36 0.02;d=1 0.7 -0.45 -0.6;h,m = impz(p,d,41;s,m = stepz(p,d,41;4.6 Finite-dimensional LTI discrete-time system374.7 Classification of LTI Discrete-time systemBased on impulse response lengthIf hn is finite length,we call it finite impulse response(FIR) system,otherwise, we call it infinite impulse
24、 response(IIR) system.Based on the output calculation process If the output can be calculated just by present and past input the system is called nonrecursive system,otherwise is called recursive system.384.8 Frequency-Domain representations of LTI Discrete-Time systemsnMost discrete-time signals en
25、countered in practice can be represented as a linear combination of sinusoidal discrete-time signals of different angular frequenciesnThus, knowing the response of the LTI system to a single sinusoidal signal, we can determine its response to more complicated signals by making use of the superpositi
26、on property39nAn important property of an LTI system is that for certain types of input signals, called eigen functions, the output signal is the input signal multiplied by a complex constantnWe consider here one such eigen function as the input4.8 Frequency-Domain representations of LTI Discrete-Ti
27、me systems404.8.1 Frequency ResponsenConsider the LTI discrete-time system with an impulse response hn shown belowIts input-output relationship in the time-domain is given by the convolution sum41According to Eq(4.66),if we setThe output of an LTI system isWhere Definition4.8.1 Frequency Response42
28、Obviously , for a complex exponential input signal ejn, the output of an LTI discrete-time system is also the same signal multiplied by a complex constant H(ej). Here ,ejn is called an eigen function of the LTI discrete-time system. H(ej) is called the frequency response of the LTI discrete-time sys
29、tem , it provides a frequency-domain description of the system and is precisely the DTFT of the impulse response hn of the system.4.8.1 Frequency Response43 Obviously, that H(ej) can completely characterizes the LTI discrete-time system in the frequency domain. Just like any other DTFT , H(ej) is us
30、ually also a complex function of with a period 2 and can be expressed in terms of its real and imaginary parts or its magnitude and phase. Where 4.8.1 Frequency Response44 The quantity |H(ej)| is called the magnitude response and the quantity () is called the phase response of the LTI discrete-time
31、system. The gain function:Attenuation or loss function. 4.8.1 Frequency Response45In engineering, The gain function commonly use another definition as below: For a discrete-time system characterized by a real impulse response hn (real system) should note:1. |H(ej)| and Hre (ej) are even functions.2.
32、 () and Him (ej) are odd functions.4.8.1 Frequency Response464.8.2 Frequency-Domain characterization LTI Discrete-Time systemsIf we applying the convolution theorem to the Eq.(4.66), we arrive at Where the Y(e-j) and X(e-j) denote the DTFT of yn and xn.So 474.8.3 Frequency Response of LTI Discrete-T
33、ime systemsFrequency Response of LTI FIR Discrete-Time SystemsFrequency Response of LTI IIR Discrete-Time Systems484.8.4 Frequency Response Computation Using MATLABExample 4.31 ForThe frequency response is wanted.49Solution:4.8.4 Frequency Response Computation Using MATLAB50nThe function freqz(h,1,w
34、) can be used to determine the values of the frequency response vector h at a set of given frequency points wnFrom 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 angleProgram 4_3.m can be used to ge
35、nerate the magnitude and phase responses of an M-point moving average filter as shown on next-page4.8.4 Frequency Response Computation Using MATLAB51Magnitude responses of the moving-average filters of length 5 and 14 Phase responses of the moving-average filters of length 5 and 14 4.8.4 Frequency R
36、esponse Computation Using MATLAB52nThe phase response of a discrete-time system when determined by a computer may exhibit jumps by an amount 2p p caused by the way the arctangent function is computednThe phase response can be made a continuous function of by unwrapping the phase response across the
37、jumps4.8.4 Frequency Response Computation Using MATLAB53nTo this end the function unwrap can be used, provided the computed phase is in radiansnThe jumps by the amount of 2p p should not be confused with the jumps caused by the zeros of the frequency response as indicated in the phase response of th
38、e moving average filter4.8.4 Frequency Response Computation Using MATLAB544.8.5 Steady-state and transient Response For a periodic signal xn,we should represented it as So the response of an LTI system is55 That is shown, for every frequency-part of xn, the response of an LTI system is also a comple
39、x exponential signal of the same frequency multiplied by a complex constant H(ejlo). The response of an LTI system of a periodic input is commonly called the steady-state response.The transient Frequency response of the LTI system:4.8.5 Steady-state and transient Response564.8.6 Response to a Causal
40、 Exponential SequenceWhile the input of the LTI system is a causal exponential sequence :57The steady-state response: The transient response: 4.8.6 Response to a Causal Exponential Sequence584.8.7 Concept of FilteringConcept of Filtering One application of an LTI discrete-time system is to pass cert
41、ain frequency components in an input sequence without any distortion (if possible) and to block other frequency components. Such systems are called digital filters and are one of the main subjects of discussion in this text.59nThe key to the filtering process is It expresses an arbitrary input as a
42、linear weighted sum of an infinite number of exponential sequences, or equivalently, as a linear weighted sum of sinusoidal sequences4.8.7 Concept of Filtering60nThus, by appropriately choosing the values of the magnitude function |H(ej )| of the LTI digital filter at frequencies corresponding to th
43、e frequencies of the sinusoidal components of the input, some of these components can be selectively heavily attenuated or filtered with respect to the othersExample (4.33) known 4.8.7 Concept of Filtering61 Design a Filter to pass the high-frequency component(x2n) but block the low-frequency compon
44、ent(x1n) of xn.Solution :For simplicity we assume the filter to be an FIR filter of length 3 with an impulse response :4.8.7 Concept of Filtering62The magnitude functionThe phase function In order to stop the low-frequency component from appearing at the output of the filter, the magnitude function
45、at =0.1 should be equal to zero. Similarly, the magnitude function at =0.4 must equal to 1. So4.8.7 Concept of Filtering63nThus the output-input relation of the FIR filter is given by yn = - 6.76195(xn+xn-2)+13.456335xn-1 where the input is xn = cos(0.1n) + cos(0.4n) nnProgram 4_4.m can be used to v
46、erify the filtering action of the above system4.8.7 Concept of Filtering644.8.7 Concept of Filtering65nThe first seven samples of the output are shown below4.8.7 Concept of Filtering66nFrom this table, it can be seen that, neglecting the least significant digit, yn = cos(0.4(n-1) for n 2nComputation
47、 of the present value of the output requires the knowledge of the present and two previous input samplesnHence, the first two output samples, y0 and y1, are the result of assumed zero input sample values at n = -1 and n = -24.8.7 Concept of Filtering67nTherefore, first two output samples constitute
48、the transient part of the outputnSince the impulse response is of length 3, the steady-state is reached at n = N = 2nNote also that the output is delayed version of the high-frequency component cos(0.4n) of the input, and the delay is one sample period4.8.7 Concept of Filtering684.9 Phase and Group
49、Delays Definitions: If the input is a sinusioidal signal of frequency but lagging in phase by radians, as demonstrated in Eq.(4.86):69From example 4.32 we can see that the output signal yn of an LTI discrete-time system exhibits some delay relative to the input signal xn caused by the nonzero phase
50、response of the system. We can rewrite Eq.(4.86) as indicating a time delay, more commonly known as phase delay at = 0 given by 4.9 Phase and Group Delays70While input isWhere xn is a narrowband signal . In this situation ,we can derivate the output signal as below: group delay or envelope delay 4.9
51、 Phase and Group Delays71phase delay An example of group delay and phase delay is shown in Figure 4.15 The waveform of the underlying continuous-time output shows distortion when the g of the LTI system is not constant over the bandwidth of xn. A delay equalizer is usually cascaded with the LTI syst
52、em4.9 Phase and Group Delays72nThe figure(4.16) below illustrates the effects of the two delays on an amplitude modulated sinusoidal signal4.9 Phase and Group Delays734.10 Phase Delay and Group Delay Computation using MATLABRead and exercise by yourself !Phase delay can be computed using the function phasedelayGroup delay can be computed using the function grpdelay7475HomeworknProblems 4.1, 4.3, 4.5, 4.8, 4.10, 4.15, 4.18,4.22, 4.30, 4.41, 4.51, 4.61, 4.64,4.69, 4.77M4.1, M4.2, M4,5
