1、1.4 The unit impulse and unit step function 1.4.1 The discrete-time unit impulse and unit step sequence1 Unit impulse:0 10 0nnnThroughout the book,we refer interchangeably as the unit impulse or unit sample.n2 Unit step:un0 10 0nnnu3 Relationship between the discrete-time unit impulse and unit step(
2、1)The unit impulse is the first difference of the step:1nunun(2)The unit step is the running sum of the impulse:nmmnuBy changing the variable of summation from m to k=n-m0kknnu0kknnunmmnu0kknnu(a)n0(a)n04 The sampling property of unit impulse:Unit impulse can be used to sample the value of signal at
3、 n=00nxnnxMore generally,if we consider a unit impulse at 0nn0nn 000nnnxnnnxNote:The sampling property of unit impulse is very important.1.4.2 The continuous-time unit step and unit impulse function1 Unit step:0 10 0)(tttu2 Unit impulse:)(tt10 has no duration but unit area,we adopt the graphical not
4、ation for it shown in figure before,where the arrow at t=0 indicates that the area of the pulse is concentrate at t=0 and the height of the arrow and the“1”next to the arrow are used to represent the area of the impulse.)(t)(tktk0More generally,a scaled impulse will have an area k.)(tkThe continuous
5、-time unit step is the running integral of the unit impulse:Unit impulse can be thought of as the first derivative of the continuous-time unit step:dttdut)()(3 relationship between continuous-time unit step and unit impulse:Therefore)()(tkudkt0)()()(ssdtdtutAs with the discrete-time impulse,the cont
6、inuous-time impulse has a very important sampling property.)()0()()(txttxWe also have an analogous expression for an impulse concentrated at an arbitrary point,t0,that is:)()()()(000tttxtttx4 Sampling property 1.5 Continuous-time and discrete-time system Physical system in the broadest sense are an
7、interconnection of components,devices,or subsystem.A system can be viewed as a process in which input signals are transformed by the system or cause the system to respond in some way.outputinputRC circuitsvcvsystemA continuous-time system is a system in which continuous-time input signals are applie
8、d and result in continuous-time output signals.Continuous-time system)(tx)(tyWe often represent the input-output relation by the notion:)()(tytxDiscrete-time system:Discrete-time systemnxnynynxSimple example of system:Rtvtvtics)()()(dttdvCtic)()(1)(2)(1)(1)(tvRCtvRCdttdvsccFinally,we obtain a differ
9、ential equation describing the relationship between the input and output :)(tvs)(tvc1.5.2 Interconnections of system Many real systems are built as interconnections of several subsystems.By viewing system as an interconnection of its components,we can use our understanding of the component systems a
10、nd of how they are interconnected in order to analyze the operation and behavior of the overall system.In addition,by describing a system in terms of an interconnection of simpler subsystem,we may in fact be able to define useful ways in which to synthesize complex systems out of simpler,basic build
11、ing blocks.1 Series interconnection:System 1System 2InputoutputHere,the output of system 1 is the input of system 2.2 Parallel interconnection:System 1System 2InputoutputHere,the same input signals is apply to system 1 and system 2.additionWe can combine both series and parallel interconnections to
12、obtain more complicated interconnections.An example is as follow:System 1System 3InputoutputSystem 2System 43 Feedback interconnection:Here,the output of system 1 is the input to system 2,while the output of system 2 is fed back and added to the external input to produce the actual input to system 1
13、.Ch1.6 Basic system properties 1.System with and without memory 2.Invertibility and inverse system3.Causality 4.Stability 5.Time-Invariance 6.linearity1.6.1 System with and without memory 1.A system is said to be memoryless if its input for each value of the independent variable at a given time is d
14、epended only on the input at that time.For example:22)2(nxnxnymemorylessSimilarly,a resistor is a memoryless system)()(tRxtyVoltage Current One particular simple memoryless system is the identity system,whose output is identical to input.That is)()(txtyOr)()(nxny2.A system with memory if the current
15、 output is dependent on past value or the future value of the input and output.An example of the discrete-time system with is an accumulator or summer:nkkxnyAnd a second example is a delay:1nxnyA capacitor is an example of a continuous-time system with memory:tdxCty)(1)(1.6.2 Invertibility and inver
16、se systemA system is said to be invertible if distinct input lead to distinct output.If an system is invertible,then an inverse system exists.)(2)(txtyThe inverse system is)(21)(tytxAn example of invertibility continuous-time system isAnother example of invertibility system is the accumulator;nkkxny
17、The inverse system is:1nynynwExamples of noninvertibility system are:0ny)()(2txtyThis system produces zero output for any input sequence.This system we cannot determine the sign of input from the knowledge of the outputEx:Moving-Average Systems(滑动平均系统)y(n)=x(n)+x(n-1)+x(n-2)/3 Is this system causal?
18、1.6.3 Causality A system is causal if the output at any time depends only on value of the input at the present time and in the past.All memoryless system are causal.Why?Series RC circuit driven from an ideal voltage source v1(t),producing output voltage v2(t).Ex:Consider the RC circuit.Is this syste
19、m causal or non-causal?Stable system：Bounded Input-Bounded Output.yxMtyMtx)()(yxMnyMnx)()(Unstable system：Bounded input-Unbounded output 1.6.4 Stability Continuous-time system)(tx)(tyDiscrete-time systemnxny Ex 1:Moving-Average Systems(滑动平均系统).Show that the System is BIBO stable:y(n)=x(n)+x(n-1)+x(n
20、-2)/3.Solution：y(n)=x(n)+x(n-1)+x(n-2)/3 （Mx+Mx+Mx）/3=Mx y(n)Bounded Solution：if x(n)Mx1，rn，y(n)diverge.Ex 2:Unstable System.,r1nxrnynDramatic photographs showing the collapse of the Tacoma Narrows suspension bridge on November 7,1940.(a)Photograph showing the twisting motion of the bridges center s
21、pan just before failure.(b)A few minutes after the first piece of concrete fell,this second photograph shows a 600-ft section of the bridge breaking out of the suspension span and turning upside down as it crashed in Puget Sound,Washington.Note the car in the top right-hand corner of the photograph.
22、An Unstable System1.6.5 Time Invariance A system is time invariant if the behavior and characteristic of the system is fixed over time.For example,the RC circuit is time invariant if the resistance and capacitance value R and C are constant over time.On the other hand,if the resistance and capacitan
23、ce value R and C are changed or fluctuate over time,then it is not time invariant.Because we would expect the result of our experiment to depend on the time at which we run it.Ex:Are these systems time-invariant?time-invarianttime-variantA system is time-invariant if the input (1)y(t)=3x(t)(2)y(t)=t
24、f(t)()(0ttxtxThen the output)()(0ttyty0nnxnx0nnyny)()(11tytxif)()(11taytaxthen)()(),()(2211tytxtytxif A linearity system is a system that possesses the important property of superposition.)()()()(2121tytytxtxthen1.6.6 Linearity)()()()(2121tybtyatxbtxa2121nybnyanxbnxaThe two properties defining a lin
25、ear system can be combined into a single statement:Continuous-time:Discrete-time:a and b are complex constants.4)(3)()1(txty)()()2(2txttyttxtyd)(d)()3(Analysis method:)()()()(2121tybtyatxbtxaCheck 4)(3)()(222txtytx4)(3)()1(txty4)(3)()(111txtytxThe system is nonlinear.4)(3)()1(txty8)(3)(3)()(2121tytxtyty)()(121txtty)()(222txtty)(3)(2)(3)(221221txtxttyty)()()2(2txttyThe system is linear。The system is linear。ttxtxtxtxd)(3)(2d4)(3)(22121ttxtyd)(d4)()3(ttxtxd)(d4)(22ttxtxd)(d4)(11ttxttxd)(d12d)(d821
