麻省理工学院公开课：信号与系统：模拟与数字信号处理 第15集英中字幕.doc

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1、目 录第1集 引言2第2集 正弦信号和指数信号13第3集 信号和系统：第二部分31第4集 卷积53第5集 模拟与数字信号处理73第1集 引言 E01 Im Alan Oppenheimand Id like to welcome you to this video tape courseon signals and systemsSignals, at least as an informal definitionare functions of one or more independent variablesthat typically carry some type of informa

2、tionSystems in our settingwould typically be used to process signalsOne very common example of a signal, might belets say, a speech signaland you might think of the air pressure as a function of timeor perhaps the electrical signal after itgoes through the microphone transducer as a function of time

3、As representing the speech signaland we might see a typical speech signallooking something like Ive indicated hereits a function of time in this particular caseand the independent variable being time is in fact continuousand so a signal like thiswe will typically be referring to as a continuous time

4、 signalNow, it also, for this particular exampleis a function of one independent variableand that will be referred to as a one dimensional signalcorresponding to the fact thattheres only one independent variableinstead of several independent variablesSo the speech signal is an example ofa continuous

5、 time, one dimensional signalNow, signals can of course, be multidimensionaland they may not have as there independent variablestime variablesOne very common exampleare the examples represented by imagesImages as signalswe might think of as representing brightnessas it varies in a horizontal and ver

6、tical directionand so the brightness as a function of these two spatial variablesis then a two dimensional signalAnd it, the independent variables would typically be continuousBut of course, theyre not time variablesand incidentally, its worth just commenting thatvery often, simply for conveniencewe

7、ll have a tendency to refer to the independent variableswhen we talk about signals as time variableswhether or not they really do represent timeWell, let me illustrate one example of an imageThis is a picture of J. B. J. Fourierwho perhaps more than anyone else is responsiblefor the elegance and bea

8、uty of a lot of the conceptsthat well be talking about throughout this courseAnd when you look at thisin addition to seeing Fourier himselfyou should recognize what youre looking at is basically a signalwhich is brightness as the function of the horizontaland vertical spatial variablesAs another exa

9、mple of an image as a signallets look at an aerial photographthis is an aerial photograph taken over a set of roadswhich you can more or less recognize in the pictureAnd one of the difficulties with this signal is thatthe road system is somewhat obscured by cloud coverand what I wanna show later as

10、an example ofwhat a system might do to such a signal in terms of processing itis an attempt to at least compensate somewhatfor the cloud cover thats represented in the photographAlthough, in terms of the detailed analysisthat we go through during the courseour focus of attention is pretty muchrestri

11、cted to one dimensional signalsIn fact, well be using two dimensional signalsmore specifically images very oftento illustrate a variety of conceptsNow, speech and images are examples of whatwe refer to as continuous time signals inthat they are functions of continuous variablesAn equally important c

12、lass of signals thatwell be concentrating on in the course aresignals that are discrete time signalswhere by discrete-time, what we mean is thatthe signal is a function of an integer variableand so specifically only takes on valuesat integer values of the argumentSo here is a graphical illustration

13、of a discrete time signalAnd discrete time signals arise in a variety of waysone very common example that youve seen fairly often isthe discrete-time signals in the context of economic time seriesand for example stock market analysisSo what I show hereis one very commonly occurring example of a disc

14、rete time signalit represents the weekly stock market indexThe independent variable in this case is the week numberand we see what the stock market isdoing over this particular periodas a function of the number of the weekand of course, along the vertical axis is the weekly indexIncidentally, this p

15、articular period was not chosen at randomit in fact captures a very interesting aspect of stock market historynamely the Stock Market Crash in 1929which in fact is represented by the behaviorof this discrete time signal or sequence in this particular areaSo this dramatic dip in fact is the Stock Mar

16、ket Crash of 1929Well, the Dow Jones weekly averageis an example of a one dimensional discrete time signalAnd just as with continuous timewe had not just one dimensional but multidimensional signalsLikewisewe have multidimensional signals in the discrete time casewhere, in that case thenthe discrete

17、 time signal that we were talking about are sequenceis a function of 2 integer variablesAnd as one example this mightlets say, represent a spatial and time raywhere this is array number in, lets say, the horizontal directionand this is array number in the vertical directionBoth classes of signals, c

18、ontinuous time and discrete timeas Ive indicated, are very importantAnd it should be emphasize thatthe importance of discrete time signalsin an associated processingcontinuous to grow in large partbecause of the current and emerging technologies thatpermit basically the processing of continuous time

19、 signalsby first converting them to discrete time signalsand processing them with discrete time systemsand that in fact, is a topic that we will discussin a fair amount of detail later on in the courseLets now turn our attention to systemsand as I indicateda system basically processes signalsand the

20、y have of course, inputs and outputsand depending on whether were talking aboutcontinuous time or discrete timethe system maybe a continuous time systemor a discrete time systemSo in the continuous time caseI indicate here an input x(t), and an output y(t)If we were talking about a discrete time sys

21、temI would represent the input in terms of a discrete time variableand of course, the output in terms of a discrete time variable alsoNow, in very general terms, systems are hard to deal withbecause they are defined very broadly and very generallyAnd in dealing with systems and analyzing themwhat we

22、 will do is attempt toexploit some very specific and as well seevery useful system propertiesTo indicate what I mean, and how things might be split upwe could talk about systemsand well talk about systems that are linearand we could divide systems basically intosystems that are either linear or non-

23、linearand we willor also divide systems into systems that arewhat we referred to as time-invariant or time-varying systemsand these arent terms that weve defined yet of coursebut well be defining in the course very shortlyAnd while in some sense this division represents all systemsand this does toot

24、he focus of the course is really going to be principally onlinear time-invariant systemsSo its basically these systems that we will be focusing onand well be referring to those systemsas linear time-invariant systemsWell, as a the brief glimpse saidsome of the kinds of things that systems can dolet

25、me illustrate first in a one dimensional continuous time contextand then later with a discrete time exampleone example of some processing of signalswith an appropriate systemthe particular example that I wanna illustraterelates to the restoration of old recordingsand this is some work that was done

26、by professor Tom Stockhamwho is at the University of Utahand work that he had done a number of years agorelating to the fact thatin old recordingsfor example, in Crusade recordingsthe recording was done through a mechanical hornand the characteristics of the horn tended to vary from day to dayand be

27、cause of the characteristics of the hornthe recording tended to have a muffled qualitysomething like thissort of the sense you would getif you were speaking through a megaphoneWhat professor Stockham did was develop a system to processthese old recordings in such a way thata lot of the characteristi

28、csand distortion due to that recording system was removedSo Id like to illustrate that as one example ofsome signal processing withan appropriate continuous time systemand what youll hear is a two-track recordingon the first track is the original, un-restored Caruso recordingand on the second track

29、is the result of the restorationand so as I switch back and forthfrom channel one to channel twowell be switching from the original to the restoredWell begin the tape by playing the originaland then as it proceeds, well switchSo we begin on channel oneThats the original recordingand switch now to th

30、e processedNow lets switch backback to the originalback to the restorationand once again, back to the originalAnd presumably and hopefully what you heard was thatin the restorationin fact a lot of the muffled characteristics of the original recordingwere compensated for or removedNow, one of the int

31、eresting things that happened in factin the work did, professor Stockham didis that in the process of the restorationand perhaps youve heard thisin the process of the restorationin fact some of the background noiseon the recording was emphasizedand so he processed the signal furtherin an attempt to

32、remove that background noiseand with that particular processingthe processing was very highly non-linearA very interesting thing happenedwhich was thatnot only in that processing was the background noise removedbut somewhat surprisinglyalso the orchestra was removedand let me just play that now as a

33、n example ofsome very sophisticated processingwith an non-linear systemwhat youll hear on channel oneis the restoration as we had just playedwhen I switch to channel twoit will be after the processingwith an attempt to remove the orchestra and the background noiseChannel one nowand now with the nois

34、e and orchestra removedBack to channel oneAnd finally, once again, with the orchestra removedOkay, so thats an example ofprocessing of a continuous time signalwith a corresponding continuous time systemNow Id like to illustrate an example ofsome processing on a discrete time signaland Id like to do

35、that in the context of the examplethat I showed before of a discrete time signalwhich was the Dow Jones industrial weekly stock market indexI had showed it before as Ive showed it here againOver a period of slightly more than a yearwhere this is the number of weeksand this is the weekly indexAnd to

36、illustrate some of the processingwhat Id like to do is to show the stock market indexthe weekly index over a much longer time periodin particular, the weekly index over a ten-year periodand thats what I show hereSo what this covers is roughly 1927 to 1937and in this case, although this is still a di

37、screte time signaljust simply for the purposes of displaywhat weve done is to essentially to connect the dotsand draw a continuous curve through the pointsso that this picture isnt filled up with a lot of vertical linesSo this is the discrete time sequence thatrepresents the weekly Dow Jones index o

38、ver a ten-year periodAnd here by the wayagain is the Crash of 1929Its interesting to note by the way thatactually the disaster in the stock marketwasnt so much the 1929 Crashbut the long downward trend that followed thatand you can see that here by kind of filtering through by eyethe rapid variation

39、s in the indexand what you see is this smooth downward trendfollowed eventually by an upward trendNow, this issue of looking at something like thislooking at a sequenceand following the smoother parts of it, namely the long term trendsis in fact, something that is done quite typicallywith economic t

40、ime series like thisand in particular whats done is to smooth itor average over some time periodto emphasize the slow variationsand deemphasize the rapid variationsand that in factis processing that is done with a discrete time systemSo when you hear referred tolets say, in stock market reportsthe 5

41、1-day moving averagethat in fact is processing this stock market indexwith a particular discrete time systemThe result of doing that on this particular examplegenerates a smooth version of the curvewhich I overlay hereand the overlay then is really attempting totrack the smoother variationsand deemp

42、hasize the more rapid variationsLet me just slightly offset thatso that the difference stands out a little moreand so here you see what is the original weekly indexand this is the result of processing that sequencewith an appropriate system to apply smoothingand in fact what it is, is a moving avera

43、geand so here again, you can see in the smoother curvethis general downward trendup until this time period followed by eventually a recoveryRight, well, weve seen an example with a continuous time signalthe Caruso recordingand an example of the discrete time signalthis stock market indexAnd what Id

44、also like to show is a third examplewhich is the result of some processing on an imagein particular, the image that we talked about beforewhich was the aerial photographthat had the problem of some cloud coverSo once againwhat we see here is the original aerial photographwith the cloud coverand some

45、 processing was applied to this using a systemwhich in fact was both non-linear and co-time-varying orin the case of this independent variablesyou would refer to it as spatially varyingAnd the result of applying that processing is shownin the adjoining pictureand what we see there ishopefully a reas

46、onable attempt to compensatefor the cloud coverAnd this by the way was some workthat was done by professor Lim at MITand has been a very successful type ofprocessing for aerial photographsI should say also, that this particular example is onewhere although the original signalwas a signal that is con

47、tinuous timethat is the independent variables are continuousas they are in a spatial aerial photographIn fact for the processingthat picture was first converted to a sequencethrough a process called samplingwhich well be talking about laterand then the processingin fact was done on a digital computerOkay, well, these are some examples of the use of some systemsto process some signals both in continuous time and discrete ti