1、Chapter 2 Main Tasks2.1 Discrete-Time Signals Time-Domain Representation Basic Operations on Sequences Classification of Sequences2.2 Typical Sequences and Sequence Representation Some Basic Sequences Sequence Generation Using MATLAB Representation of an Arbitrary SequenceMain Tasks2.3 The Sampling
2、Process2.4 Discrete-Time Systems Examples of Simple Discrete-Time Systems Classification of Discrete-Time Systems2.5 Time-Domain Characterization of LTI Discrete-Time Systems The Convolution Sum Representation Simple Interconnection Schemes Stability and Causality in terms of the Impulse ResponseMai
3、n Tasks2.6 Finite-Dimensional LTI Discrete-Time Systems Difference Equations and Their Solutions Classification of LTI Discrete-Time SystemsHomework2.1 Discrete-Time Signals2.1.1 Time-Domain RepresentationThere are three ways to represent a discrete-time signal(or sequence):1.A discrete-time signal
4、is represented as a sequence of numbers,called samples.2.Using a graph to represent a sequence3.Using a mathematical function to represent a sequenceExamples2.1 Discrete-Time SignalsIn some applications,a discrete-time sequence xn may be generated by periodically sampling a continuous-time signal xa
5、(t)at uniform intervals of time:xn=xa(t)|t=nT=xa(nT)=,-2,-1,0,1,22.1 Discrete-Time Signalsxn=xa(t)|t=nT=xa(nT)=,-2,-1,0,1,2The spacing T between two consecutive samples is called the sampling interval or sampling periodFT=1/T HzS=2/T radians/s2.1 Discrete-Time SignalsNote:Whether or not the sequence
6、 xn has been obtained by sampling,the quantity xn is called the nth sample of the sequence.xn may be real valued or complex valued.It is a real sequence,if the nth sample xn is real for all values of n.Otherwise,xn is a complex sequence2.1 Discrete-Time SignalsComplex sequenceA complex sequence xn c
7、an be expressed as a sum of its real and imaginary parts:xn=xren+jximnIf xn is complex sequence,then it has its conjugate counterpart given by:x*n=xren-jxim n2.1 Discrete-Time SignalsSampled-Data Signals and Digital Signals(Page43)Sampled-Data Signals in which the samples are continuous-valued.Digit
8、al Signals in which the samples are discrete-valued.Quantization:by rounding or truncation.2.1 Discrete-Time SignalsFinite-length and infinite-length sequences Finite-length sequenceA finite-length sequence xn has a start at n=N1 and an end at n=N2.The length of xn N=N2-N1+1A finite-length sequence
9、is often called an N-point sequence.2.1 Discrete-Time SignalsA finite-length sequence can also be considered as an infinite-length sequence by assigning zero values to samples whose arguments are outside the above range(outside the range from N1 to N2).The process of lengthening a sequence by adding
10、 zero-valued samples is called appending with zeros or zero-padding.2.1 Discrete-Time SignalsFor a finite-length sequence,if N1=-and/or N2=Then the sequence is called an infinite-length sequence.If N1=-and N2=,then the sequence xn is called two-sided sequence.2.1 Discrete-Time SignalsIf N1=-and N2-a
11、nd N2=,then the sequence xn is called right-sided sequence.2.1 Discrete-Time SignalsFurther more,for a right-sided sequence,if N1 0,then the sequence xn is called causal sequence.For a left-sided sequence,if N2 0,then the sequence xn is called anti-causal sequence.nxn0A causal sequencenxn0An anti-ca
12、usal sequence2.1 Discrete-Time Signals2.1.2 Operations on SequencesA single-input,single-output discrete-time system operates on a sequence,called the input sequence,according to some prescribed rules and develops another sequence,called the output sequence,with more desirable properties.xnynInput s
13、equenceOutput sequenceDiscrete-timesystem2.1 Discrete-Time SignalsBasic OperationsIn signals and systems,we have known that there are three operations:xnynwnyn=xn+wnAdder1.Addition2.1 Discrete-Time SignalsAxnynyn=A.xnMultiplier2.Multiplication3.Unit delaying1zynxnUnit delayyn=xn-12.1 Discrete-Time S
14、ignalsIn additional,there are also some common signal operations:ynxnzUnit advance1.Advance operationyn=xn+12.Time-reversal(folding)operation yn=x-n2.1 Discrete-Time Signals3.Branching operation:Used to provide multiple copies of a sequencexnxnxn3.Product operation:xnynwnyn=xn.wnModulator2.1 Discret
15、e-Time SignalsAn application of the product operation 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.This process is called windowing.2.1 Discrete-Time SignalsWindowing2.1 Discrete-Time SignalsE
16、xample2.1-Consider the following two sequences of length 5 defined for 0 n 4:cn=3.2,41,36,9.5,0 dn=1.7,0.5,0,0.8,1Determine the results of these basic operations between cn and dn.n=02.1 Discrete-Time Signals cn=3.2,41,36,9.5,0 dn=1.7,0.5,0,0.8,1 n=0w1n=cn.dn=5.44,20.5,0,7.6,0w2n=an+bn=4.9,40.5,36,-
17、8.7,1w3n=(7/2)cn=11.2,143.5,126,-33.5,0n=02.1 Discrete-Time SignalsExample2.2 Consider the sequence of length 3 defined for 0 n 2:gn=-21,1.5,3We cannot add the length-3 sequence to any one of the length-5 sequences of Example2.1.For example,cn:cn=3.2,41,36,9.5,0n=0n=02.1 Discrete-Time SignalsThus,li
18、ke we did in Example2.1,we can obtain the results of these operations.n=0gen=-21,1.5,3,0,0n=0We therefore first append gn with 2 zero-valued samples resulting in a length-5 sequence gn=-21,1.5,32.1 Discrete-Time SignalsCombination of Basic OperationsIn most cases,combination of the above basic opera
19、tions are used.Example 2.3yn=1xn+2xn-1+3n-2+4xn-32.1 Discrete-Time SignalsExample2.4 Consider the discrete-time system described by the block diagram:1zb0 xnyn1z1z1zb1b2a1a2yn-a1yn-1-a2yn-2=b0 xn+b1xn-1+b2n-22.1 Discrete-Time Signals2.1.3 Classification of SequencesClassification Based on Symmetry*F
20、or a complex sequence xn,it can be expressed as a sum of its real and imaginary parts:xn=xren+jximnIt has its conjugate counterpart given byx*n=xren-jximn2.1 Discrete-Time SignalsConjugate-symmetric sequence DefinitionA sequence xn is said to be a conjugate-symmetric sequence if xn=x*-n.xn=xren+jxim
21、nx*-n=xre-n-jxim-nWe see that xren must be even and ximn must be odd if xn is a conjugate-symmetric sequence.x*n=xren-jximn2.1 Discrete-Time SignalsConjugate-antisymmetric sequence DefinitionA sequence is said to be a conjugate-antisymmetric sequence if xn=-x*-n.xn=xren+jximn-x*-n=-xre-n+jxim-nWe se
22、e that xren must be odd and ximn must be even if xn is a conjugate-antisymmetric sequence.x*n=xren-jximn2.1 Discrete-Time SignalsAny complex sequence xn can be expressed as a sum of its conjugate-symmetric part xcsn and its conjugate-antisymmetric part xcan.xn=xcsn+xcanwhere the conjugate-symmetric
23、part isxcsn=(1/2)x n+x*-nxcan=(1/2)x n-x*-nand the conjugate-antisymmetric part is2.1 Discrete-Time SignalsIf xn is real valued,and if xn is even,then it must be a conjugate-symmetric sequence.If xn is real valued,and if xn is odd,then it must be a conjugate-antisymmetric sequence.Do you remember th
24、e conclusion:any real valued sequence can be expressed as a sum of its even and odd parts:xn=xen+xon2.1 Discrete-Time SignalsThe computation of the conjugate-symmetric and conjugate-antisymmetric parts of a sequence involves conjugation,time-reversal,addition and multiplication operations.It is impo
25、rtant to point out that these operations can not change the range that the sequence is defined.2.1 Discrete-Time SignalsThus,these operations are possible if the parent sequence is of odd length defined for a symmetric interval,-MnM.After the class,you take some time to do Example 2.5 yourself.2.1 D
26、iscrete-Time SignalsFor a length-N sequence xn defined for 0nN-1,as the following sequence:xcsn=(1/2)x n+x*-nxcan=(1/2)x n-x*-n-are not applicable.2.1 Discrete-Time SignalsHere(-n)N is called modulo operation.xpcsn=(1/2)x n+x*(-n)N =(1/2)x n+x*N-nxpcan=(1/2)x n-x*(-n)N =(1/2)x n-x*N-nIn stead by(-6)
27、8=2(10)8=2(17)8=1(-17)8=7called periodic conjugate-symmetric part.called periodic conjugate-antisymmetric part.and2.1 Discrete-Time SignalsConstruct a periodic sequence from the N-point sequence:x(-5)6=x1x(-4)6=x2x(-3)6=x3x(-2)6=x4x(-1)6=x5x(-6)6=x02.1 Discrete-Time SignalsDetermine its conjugate-sy
28、mmetric and conjugate-antisymmetric parts.un=1+j4,-2+j3,4-j2,-5-j6n=0Example 2.6 Consider the finite-length sequence of length 4 defined for 0n3:N=4Problem 2-6.2.1 Discrete-Time SignalsPeriodic and Aperiodic SignalsA sequence satisfying the periodicity condition is called an periodic sequencexn=xn+k
29、Nperiodicity condition2.1 Discrete-Time SignalsOther types of Classification Bounded signalA sequence xn is said to be bounded ifxBnxExample-The sequence xn=cos(0.3 n)is a bounded sequence as13.0cosnnx2.1 Discrete-Time Signals Absolutely summableA sequence xn is said to be absolutely summable ifnnxE
30、xample-The sequence00030nnnyn,.is an absolutely summable sequence as4285713011300.nn2.1 Discrete-Time Signals Square-summable signalA sequence xn is said to be square-summable ifnnx2Example-The sequencennnh4.0sinis square-summable.2.2 Typical Sequences and Sequence RepresentationTypical Sequences:1.
31、Unit impulse sequence n and unit step sequence n;2.Real sinusoidal sequence;3.Exponential sequence;Goback2.2 Typical Sequences and Sequence RepresentationUnit sample sequence-0,00,1nnn143210123456n0,00,1nnn432101234561n Unit step sequence-Goback2.2 Typical Sequences and Sequence RepresentationReal s
32、inusoidal sequence-xn=Acos(0n+)where A is the amplitude,0 is the angular frequency,and is the phase.Example-Goback2.2 Typical Sequences and Sequence RepresentationExponential sequence-,nAnxnA general form of a complex exponential sequence iswhere A and are real or complex numbers.Different choices o
33、f the parameters A and lead to different characteristics.2.2 Typical Sequences and Sequence Representation,nAnxn,)(ooje,jeAA,)(nxjnxeeAnximrenjjoothen we can express xn asLet),cos(neAnxonreo)sin(neAnxonimowhere2.2 Typical Sequences and Sequence RepresentationReal exponential sequence-xn=A n,-n 2.2 T
34、ypical Sequences and Sequence RepresentationLet 1=0.02 and 2=0.2 Goback2.2 Typical Sequences and Sequence Representation2.2.2 Sequence Generation Using MATLABTwo MATLAB programs in Page 59.2.2 Typical Sequences and Sequence Representation%Program 2_1%Generation of complex exponential sequencea=input
35、(Type in real exponent=);b=input(Type in imaginary exponent=);c=a+b*i;%Form a complex exponentK=input(Type in the gain constant=);N=input(Type in length of sequence=);n=1:N;x=K*exp(c*n);stem(n,real(x);xlabel(Time index n);ylabel(Amplitude);title(Real part);2.2 Typical Sequences and Sequence Represen
36、tationdisp(PRESS RETURN for imaginary part);pausestem(n,imag(x);xlabel(Time index n);title(Imaginary part);2.2 Typical Sequences and Sequence Representation%Program 2_2%Generation of real exponential sequencea=input(Type in argument =);K=input(Type in the gain constant=);N=input(Type in length of se
37、quence=);n=0:N;x=K*a.n;stem(n,real(x);xlabel(Time index n);ylabel(Amplitude);title(alpha=,num2str(a);2.2 Typical Sequences and Sequence RepresentationGoback2.2 Typical Sequences and Sequence Representation2.2.3 Representation of an Arbitrary SequenceAn arbitrary sequence can be represented as a weig
38、hted sum of some basic sequence(as an impulse sequence)and its delayed versions.2.2 Typical Sequences and Sequence RepresentationMore general,kknkxnx2 15.125.0nnnnx675.04nnGoback2.3 The Sampling ProcessWe have known,some discrete-time signals are inherently discrete,but most of discrete-time sequenc
39、es are generated by sampling continuous-time signals,as illustrated in the following figure.2.3 The Sampling ProcessAssume that the sampling interval is denoted by T,thenxn=xa(t)|t=nT=xa(nT),n=,-2,-1,0,1,2,nTt TFnTn2where FT is the sampling frequencyand T is the sampling angular frequency2.3 The Sam
40、pling ProcessThe corresponding discrete-time signal is)cos()(0tAtxIf the continuous-time signal is)cos(0nTAnx)2cos(0nAT)cos(0nFAT)cos(0nATTFT00002where-the normalized digital angular frequency2.3 The Sampling ProcessExample 2.11 Consider the three sequences generated by uniformly the three cosine fu
41、nctions of frequencies 3Hz,7Hz,and 13Hz,respectively:)6cos()(1ttg)14cos()(2ttg)26cos()(3ttgwith a sampling rate of 10Hz.The derived sequences are:)6.0cos(1nng)4.1cos(2nng)6.2cos(3nng)6.0cos(n)6.0cos(n2.3 The Sampling ProcessThe curves of the signals in Example 2.11)6cos()(1ttg)14cos()(2ttg)26cos()(3
42、ttg2.3 The Sampling ProcessThe above phenomenon of a continuous-time sinusoidal signal of higher frequency acquiring the identity of a sinusoidal sequence of lower frequency after sampling is called aliasing.Sampling theorem:Goback2.4 Discrete-Time SystemsThe function of a discrete-time system is to
43、 process a given input sequence xn to generate an output sequence yn with more desirable properties.2.4 Discrete-Time SystemsIn our textbook,we only focus our attention on the single-input,single-output systems(SISO systems):xnynInput sequenceOutput sequenceDiscrete-TimeSystem2.4 Discrete-Time Syste
44、msExample 2.13 A very simple example of a discrete-time system is the accumulator defined by the input-output relationship:nllxny1nxlxnlThe input-output relationship can also be written in the form:nlllxlxny01,10nllxy0n 1nxny2.4 Discrete-Time SystemsExample 2.14 Another simple example of a discrete-
45、time system is the M-point moving-average filter defined by101MkknxMnySuch a system is often used to smoothing random variations in data.Consider a signal sn corrupted by a noise dn for n0,xn=sn+dn2.4 Discrete-Time SystemsProgram 2_3R=50;M=5;d=rand(R,1)-0.5;m=0:1:R-1;s=2*m.*(0.9.m);x=s+d;r=zeros(1,M
46、);y=zeros(1,R);%filteringfor i=1:R-1 for j=1:M-1 r(M-j+1)=r(M-j);end r(1)=x(i);y(i)=sum(r)/M;endsubplot(221)stem(m,s,.)title(Uncorrupted sequence)subplot(222)stem(m,d,.)Title(Noise)subplot(223)stem(m,x,.)title(Smoothed sequence)2.4 Discrete-Time SystemsGoback2.4 Discrete-Time Systems2.4.1 Classifica
47、tion of Discrete-Time Systems Linear System Shift-Invariant System Causal System Stable System Passive and Lossless Systems Impulse and Step ResponseGobackLinearityLinearityA system is said to be a linear system if it satisfies both the additivity and the homogeneity properties.system1nx1nysystem2nx
48、2ny21nxnxnx21nynynysystemnxnyLinearityHomogeneitysystem1nx1nysystem1nax1nayCombining the two properties:system1nx1nysystem2nx2nysystemnxny21nbxnaxnx21nbynaynyGobackTime-InvarianceTime-InvarianceA system is said to be time-invariant if it satisfies the time-invariance property.system1nx1nysystem 11nx
49、 11nyLTI Discrete-Time SystemLinear Time-Invariant(LTI)System A system satisfying both the linearity and the time-invariance properties is said to be Linear Time-Invariant System.LTI systems are mathematically easy to analyze and characterize,and consequently,easy to design.GobackCausalityCausalityA
50、 system is said to be causal if its n0th output sample yn0 depends only on input samples xn for nn0.This means the output signal can not be produced before the input signal is applied to the system.GobackStabilityStabilityA system is said to be stable if a bounded input leads to a bounded output.Thi
