{"version":"1.0.0","segments":[{"startTime":0.84,"endTime":5.42,"body":"- Hello, integral transforms is the topic"},{"startTime":5.42,"endTime":7.383,"body":"of today's lecture."},{"startTime":8.7,"endTime":13.64,"body":"And what can I say about"},{"startTime":8.7,"endTime":13.64,"body":"integral transforms,"},{"startTime":13.64,"endTime":16.28,"body":"other than merely murmuring the words"},{"startTime":16.28,"endTime":19.94,"body":"in an undergraduate class,"},{"startTime":16.28,"endTime":19.94,"body":"is usually sufficient"},{"startTime":19.94,"endTime":24.94,"body":"to let rip with the most"},{"startTime":19.94,"endTime":24.94,"body":"appalling screams of anxiety."},{"startTime":25.36,"endTime":27.56,"body":"So, those of you who"},{"startTime":25.36,"endTime":27.56,"body":"are here this evening,"},{"startTime":27.56,"endTime":29.88,"body":"or those of you who're watching online,"},{"startTime":29.88,"endTime":34.77,"body":"I congratulate you for"},{"startTime":29.88,"endTime":34.77,"body":"your enterprise and bravery"},{"startTime":34.77,"endTime":38.422,"body":"in putting up with integral transforms."},{"startTime":38.422,"endTime":39.255,"body":"(smacks lips)"},{"startTime":39.255,"endTime":41.41,"body":"The reason I wanted to talk about them,"},{"startTime":41.41,"endTime":43.27,"body":"is because in this series,"},{"startTime":43.27,"endTime":47.44,"body":"we're talking about inventions"},{"startTime":43.27,"endTime":47.44,"body":"and secret components"},{"startTime":47.44,"endTime":49.17,"body":"of IT systems."},{"startTime":49.17,"endTime":53.34,"body":"And integral transforms are a vital key"},{"startTime":53.34,"endTime":57.28,"body":"to the way modern IT works."},{"startTime":57.28,"endTime":60.35,"body":"But very few people"},{"startTime":57.28,"endTime":60.35,"body":"really know how they work."},{"startTime":60.35,"endTime":62.2,"body":"So, what I want to do,"},{"startTime":62.2,"endTime":67.08,"body":"is to take you through some"},{"startTime":62.2,"endTime":67.08,"body":"of the mathematics and theory"},{"startTime":67.08,"endTime":69.23,"body":"of integral transforms."},{"startTime":69.23,"endTime":72.7,"body":"I'm well aware that some"},{"startTime":69.23,"endTime":72.7,"body":"of us and maybe even me,"},{"startTime":72.7,"endTime":76.03,"body":"will get lost on our journey"},{"startTime":72.7,"endTime":76.03,"body":"through integral transforms."},{"startTime":76.03,"endTime":77.53,"body":"So, if that's the case,"},{"startTime":77.53,"endTime":81.54,"body":"don't worry, don't just"},{"startTime":77.53,"endTime":81.54,"body":"mop your brow gently"},{"startTime":81.54,"endTime":84.72,"body":"and the next slide will come"},{"startTime":81.54,"endTime":84.72,"body":"along and will pick it up."},{"startTime":84.72,"endTime":87.52,"body":"So, when watching a mathematics seminar,"},{"startTime":87.52,"endTime":89.87,"body":"it's not always easy to focus."},{"startTime":89.87,"endTime":92.8,"body":"In fact, I remember speaking"},{"startTime":89.87,"endTime":92.8,"body":"to a professor of maths"},{"startTime":92.8,"endTime":93.633,"body":"and I said,"},{"startTime":93.633,"endTime":96.66,"body":"\"Tim, how much of that"},{"startTime":93.633,"endTime":96.66,"body":"seminar do you understand?\""},{"startTime":96.66,"endTime":99.14,"body":"He said, \"Well, I never"},{"startTime":96.66,"endTime":99.14,"body":"understand maths seminars."},{"startTime":99.14,"endTime":101.09,"body":"I don't really understand"},{"startTime":99.14,"endTime":101.09,"body":"them at all afterwards.\""},{"startTime":101.09,"endTime":103.3,"body":"So, if you don't understand this seminar,"},{"startTime":103.3,"endTime":106.29,"body":"then, there's my excuse."},{"startTime":106.29,"endTime":107.123,"body":"Right."},{"startTime":107.123,"endTime":108.75,"body":"So, for those of you,"},{"startTime":108.75,"endTime":112.73,"body":"who remember your, sort"},{"startTime":108.75,"endTime":112.73,"body":"of, high school physics,"},{"startTime":112.73,"endTime":116.74,"body":"might remember this circuit,"},{"startTime":116.74,"endTime":119.303,"body":"which is called a potential divider."},{"startTime":120.19,"endTime":124.193,"body":"And I wanted a little"},{"startTime":120.19,"endTime":124.193,"body":"example that gets us started."},{"startTime":126.2,"endTime":128.2,"body":"The example that I was going to think of,"},{"startTime":128.2,"endTime":130.283,"body":"was something called impulse testing."},{"startTime":131.35,"endTime":133.74,"body":"There's an old engineering joke which says"},{"startTime":133.74,"endTime":138.23,"body":"that in engineering,"},{"startTime":133.74,"endTime":138.23,"body":"that if something moves"},{"startTime":138.23,"endTime":139.86,"body":"and it shouldn't move,"},{"startTime":139.86,"endTime":141.56,"body":"then, you should use (indistinct)."},{"startTime":142.54,"endTime":145.26,"body":"And if something doesn't"},{"startTime":142.54,"endTime":145.26,"body":"move and it should move,"},{"startTime":145.26,"endTime":146.71,"body":"you should use a hammer."},{"startTime":146.71,"endTime":149.87,"body":"And those are the only two"},{"startTime":146.71,"endTime":149.87,"body":"tools you need for engineering."},{"startTime":149.87,"endTime":153.15,"body":"Well, the hammer is"},{"startTime":149.87,"endTime":153.15,"body":"what we call an impulse."},{"startTime":153.15,"endTime":157.133,"body":"So, it's a blow and is used"},{"startTime":153.15,"endTime":157.133,"body":"as a way of testing things."},{"startTime":158.02,"endTime":163.02,"body":"And this box is known"},{"startTime":158.02,"endTime":163.02,"body":"as a potential divider"},{"startTime":163.05,"endTime":167.01,"body":"and if the boxes contained resistors,"},{"startTime":167.01,"endTime":168.11,"body":"as they do in this case,"},{"startTime":168.11,"endTime":170.67,"body":"then, you might remember"},{"startTime":168.11,"endTime":170.67,"body":"from your school days,"},{"startTime":170.67,"endTime":173.68,"body":"little formula which is the ratio"},{"startTime":173.68,"endTime":175.47,"body":"of the voltage out of this thing"},{"startTime":175.47,"endTime":177.78,"body":"to the voltage into it,"},{"startTime":177.78,"endTime":181.24,"body":"is given by the ratio on"},{"startTime":177.78,"endTime":181.24,"body":"the right-hand side there,"},{"startTime":181.24,"endTime":186.183,"body":"which is the ratio of the two resistances."},{"startTime":189.01,"endTime":191.14,"body":"And a resistor, of course, is a device"},{"startTime":191.14,"endTime":194.25,"body":"which is the voltage"},{"startTime":191.14,"endTime":194.25,"body":"across this is proportional"},{"startTime":195.18,"endTime":196.843,"body":"to the current going through it."},{"startTime":198.04,"endTime":200.58,"body":"Now, if I replaced one of those resistors"},{"startTime":200.58,"endTime":202.49,"body":"with something that looked like this."},{"startTime":202.49,"endTime":205.52,"body":"Then, that funny-looking"},{"startTime":202.49,"endTime":205.52,"body":"thing there is a capacitor."},{"startTime":205.52,"endTime":209.023,"body":"Don't panic if you don't"},{"startTime":205.52,"endTime":209.023,"body":"know what a capacitor is."},{"startTime":209.023,"endTime":212.93,"body":"This is not going to be"},{"startTime":209.023,"endTime":212.93,"body":"a circuit theory lecture."},{"startTime":212.93,"endTime":214.78,"body":"So, just bear with it."},{"startTime":214.78,"endTime":217.4,"body":"And my point really, is that,"},{"startTime":217.4,"endTime":220.06,"body":"in order to describe this circuit,"},{"startTime":220.06,"endTime":222.683,"body":"it's not as easy as just"},{"startTime":220.06,"endTime":222.683,"body":"dividing some resistances."},{"startTime":224.501,"endTime":227.85,"body":"The capacitor has a response"},{"startTime":227.85,"endTime":232.84,"body":"that depends upon the rate of"},{"startTime":227.85,"endTime":232.84,"body":"change of voltage across it."},{"startTime":232.84,"endTime":235.12,"body":"So, the current flowing through it,"},{"startTime":235.12,"endTime":238.3,"body":"depends upon the rate of change of voltage"},{"startTime":238.3,"endTime":239.89,"body":"and we call that \"dv/dt\"."},{"startTime":239.89,"endTime":243.45,"body":"And it's just a bit of"},{"startTime":239.89,"endTime":243.45,"body":"mathematical notation,"},{"startTime":243.45,"endTime":245.88,"body":"multiplied by a number"},{"startTime":243.45,"endTime":245.88,"body":"called \"c\", the capacitor."},{"startTime":245.88,"endTime":249.2,"body":"So, bigger the capacitor,"},{"startTime":245.88,"endTime":249.2,"body":"the higher the current"},{"startTime":249.2,"endTime":250.8,"body":"that flows through it."},{"startTime":250.8,"endTime":252.93,"body":"So, now, we got this rate of change"},{"startTime":252.93,"endTime":254.453,"body":"of something or gradient,"},{"startTime":255.67,"endTime":259.92,"body":"our very simple bit of"},{"startTime":255.67,"endTime":259.92,"body":"circuit analysis has got-"},{"startTime":259.92,"endTime":261.95,"body":"well, it looks likes a bit"},{"startTime":259.92,"endTime":261.95,"body":"more horrifying, really,"},{"startTime":261.95,"endTime":263.7,"body":"'cause it's got a derivative"},{"startTime":263.7,"endTime":265.92,"body":"and it's got this rate of change."},{"startTime":265.92,"endTime":268.3,"body":"And this thing over on"},{"startTime":265.92,"endTime":268.3,"body":"the right-hand side,"},{"startTime":268.3,"endTime":270.4,"body":"is called a differential equation."},{"startTime":270.4,"endTime":272.29,"body":"And I think it's fair to say,"},{"startTime":272.29,"endTime":275.91,"body":"that undergraduates, on the whole,"},{"startTime":275.91,"endTime":278.32,"body":"are not at all keen on such things."},{"startTime":278.32,"endTime":281.12,"body":"And it's probably also fair to say"},{"startTime":281.12,"endTime":283.55,"body":"that professors, however,"},{"startTime":281.12,"endTime":283.55,"body":"are very keen on them."},{"startTime":283.55,"endTime":286.98,"body":"And there's several reasons for that."},{"startTime":286.98,"endTime":287.813,"body":"I mean, partly,"},{"startTime":287.813,"endTime":291.33,"body":"it's because solving differential"},{"startTime":287.813,"endTime":291.33,"body":"equations is something"},{"startTime":291.33,"endTime":293.62,"body":"that can demonstrate your"},{"startTime":291.33,"endTime":293.62,"body":"prowess of mathematics"},{"startTime":293.62,"endTime":296.58,"body":"and professors are very"},{"startTime":293.62,"endTime":296.58,"body":"keen on doing that."},{"startTime":296.58,"endTime":301.26,"body":"And also, each solution of this,"},{"startTime":301.26,"endTime":304.88,"body":"depends upon the type of voltage"},{"startTime":304.88,"endTime":306.34,"body":"that we're going to put into it."},{"startTime":306.34,"endTime":309.07,"body":"So, there's a sort of"},{"startTime":306.34,"endTime":309.07,"body":"infinite number of lectures"},{"startTime":309.07,"endTime":309.99,"body":"that can be given."},{"startTime":309.99,"endTime":312.0,"body":"We just pick the input voltage."},{"startTime":312.0,"endTime":313.48,"body":"So, if we picked a sine wave,"},{"startTime":313.48,"endTime":315.13,"body":"then, that would have one solution"},{"startTime":315.13,"endTime":316.61,"body":"and if picked a cosine wave,"},{"startTime":316.61,"endTime":318.1,"body":"that will be a different solution."},{"startTime":318.1,"endTime":319.187,"body":"If we picked a step function,"},{"startTime":319.187,"endTime":320.56,"body":"that'd be a different solution."},{"startTime":320.56,"endTime":321.82,"body":"If we picked an impulse,"},{"startTime":321.82,"endTime":322.91,"body":"it'd be a different solution."},{"startTime":322.91,"endTime":324.68,"body":"And you can imagine, this is very popular"},{"startTime":324.68,"endTime":326.02,"body":"in undergraduate lecturing course,"},{"startTime":326.02,"endTime":328.83,"body":"'cause we essentially"},{"startTime":326.02,"endTime":328.83,"body":"pick one of these inputs"},{"startTime":328.83,"endTime":332.01,"body":"and then, we spend between"},{"startTime":328.83,"endTime":332.01,"body":"half an hour and 40 minutes"},{"startTime":332.01,"endTime":334.98,"body":"on showing people how to"},{"startTime":332.01,"endTime":334.98,"body":"solve differential equations"},{"startTime":334.98,"endTime":337.04,"body":"for those conditions."},{"startTime":337.04,"endTime":340.0,"body":"And this good if you like"},{"startTime":337.04,"endTime":340.0,"body":"torturing undergraduates,"},{"startTime":340.0,"endTime":344.06,"body":"but it has a slight downside"},{"startTime":344.06,"endTime":345.75,"body":"if we're trying to design things."},{"startTime":345.75,"endTime":347.28,"body":"Because we're going to have to,"},{"startTime":347.28,"endTime":350.49,"body":"sort of, re-solve these"},{"startTime":347.28,"endTime":350.49,"body":"differential equations,"},{"startTime":350.49,"endTime":352.93,"body":"each time we get a different input."},{"startTime":352.93,"endTime":356.39,"body":"So, if, for example, the"},{"startTime":352.93,"endTime":356.39,"body":"input look like this,"},{"startTime":356.39,"endTime":358.35,"body":"this little pink line here,"},{"startTime":358.35,"endTime":361.84,"body":"which is something called a step function."},{"startTime":361.84,"endTime":363.13,"body":"Looks a bit like that."},{"startTime":363.13,"endTime":365.18,"body":"Step, sometimes called the \"Heaviside\"."},{"startTime":365.18,"endTime":368.25,"body":"You need step after Oliver Heaviside,"},{"startTime":368.25,"endTime":370.7,"body":"the Irish mathematician."},{"startTime":370.7,"endTime":374.203,"body":"Then, the output of this circuit"},{"startTime":370.7,"endTime":374.203,"body":"looks like this blue line,"},{"startTime":375.47,"endTime":377.92,"body":"which is exponentially increasing."},{"startTime":377.92,"endTime":379.66,"body":"So, one way of thinking about that"},{"startTime":379.66,"endTime":381.26,"body":"if you're a physics, sort of, person,"},{"startTime":381.26,"endTime":384.02,"body":"is to think about a capacitor charging up."},{"startTime":384.02,"endTime":387.44,"body":"And it depends on the input voltage,"},{"startTime":387.44,"endTime":390.22,"body":"but it also depends on something"},{"startTime":387.44,"endTime":390.22,"body":"called the time constant,"},{"startTime":390.22,"endTime":394.83,"body":"which is how quickly the current"},{"startTime":390.22,"endTime":394.83,"body":"can flow into the capacitor"},{"startTime":394.83,"endTime":397.06,"body":"and that's governed by"},{"startTime":394.83,"endTime":397.06,"body":"how big the capacitor is"},{"startTime":397.06,"endTime":399.17,"body":"and how big that resistor is."},{"startTime":399.17,"endTime":403.368,"body":"So, it's got this \"e\" to"},{"startTime":399.17,"endTime":403.368,"body":"the \"-t\" form in there"},{"startTime":403.368,"endTime":407.203,"body":"with some additive and"},{"startTime":403.368,"endTime":407.203,"body":"multiplicative constants."},{"startTime":408.91,"endTime":411.52,"body":"Now, what to do"},{"startTime":411.52,"endTime":416.32,"body":"if I had a more complicated"},{"startTime":411.52,"endTime":416.32,"body":"waveform in here?"},{"startTime":416.32,"endTime":417.72,"body":"Well, as I indicated,"},{"startTime":417.72,"endTime":419.47,"body":"I'm going to be, sort of, pretty much,"},{"startTime":420.91,"endTime":425.79,"body":"sort of, lumbered doing that."},{"startTime":425.79,"endTime":428.89,"body":"Because I'm going to have"},{"startTime":425.79,"endTime":428.89,"body":"to solve a whole new set"},{"startTime":428.89,"endTime":430.1,"body":"of equations."},{"startTime":430.1,"endTime":431.847,"body":"So, one thing I might do, is I might say,"},{"startTime":431.847,"endTime":435.75,"body":"\"Well, these exponentials"},{"startTime":431.847,"endTime":435.75,"body":"seem to appear a lot"},{"startTime":435.75,"endTime":437.18,"body":"in circuits."},{"startTime":437.18,"endTime":439.11,"body":"So, why don't I just, sort of, solve"},{"startTime":439.11,"endTime":442.535,"body":"for a, sort of, generic exponential?\""},{"startTime":442.535,"endTime":443.368,"body":"(smacks lips)"},{"startTime":443.368,"endTime":444.82,"body":"And this is what I've done here."},{"startTime":444.82,"endTime":448.97,"body":"So, this on top here, is"},{"startTime":444.82,"endTime":448.97,"body":"my differential equation."},{"startTime":448.97,"endTime":452.94,"body":"And here, I've just assumed"},{"startTime":448.97,"endTime":452.94,"body":"a waveform, a voltage."},{"startTime":452.94,"endTime":455.53,"body":"It is of the form \"e\" to the power \"st\"."},{"startTime":455.53,"endTime":457.19,"body":"So, that's an exponential."},{"startTime":457.19,"endTime":459.69,"body":"And the nice thing about \"e\" to the \"st\"."},{"startTime":459.69,"endTime":461.04,"body":"I'm afraid, I've missed"},{"startTime":459.69,"endTime":461.04,"body":"(indistinct) there,"},{"startTime":461.04,"endTime":463.07,"body":"but I should've put it in there,"},{"startTime":463.07,"endTime":466.725,"body":"it's the derivative of that,"},{"startTime":463.07,"endTime":466.725,"body":"it's just multiplied by \"s\","},{"startTime":466.725,"endTime":469.52,"body":"kind of, a little bit of"},{"startTime":466.725,"endTime":469.52,"body":"algebraic jiggery-pokery."},{"startTime":469.52,"endTime":472.53,"body":"I've converted this into"},{"startTime":469.52,"endTime":472.53,"body":"a nasty-looking thing,"},{"startTime":472.53,"endTime":476.45,"body":"into a nice thing,"},{"startTime":472.53,"endTime":476.45,"body":"which is easily solvable"},{"startTime":476.45,"endTime":479.68,"body":"and, in fact, with a little"},{"startTime":476.45,"endTime":479.68,"body":"bit of algebraic manipulation,"},{"startTime":479.68,"endTime":482.13,"body":"which no need to follow,"},{"startTime":482.13,"endTime":484.75,"body":"I can get it almost into the form I had"},{"startTime":484.75,"endTime":486.73,"body":"for my potential divider."},{"startTime":486.73,"endTime":489.88,"body":"Instead of a resistor, here and here,"},{"startTime":489.88,"endTime":493.25,"body":"as I had in the previous case,"},{"startTime":493.25,"endTime":494.6,"body":"I got something called \"Z\"."},{"startTime":495.65,"endTime":497.7,"body":"And \"Z\" is something called (indistinct),"},{"startTime":499.08,"endTime":503.72,"body":"which is a frequency-varying resistance"},{"startTime":503.72,"endTime":505.67,"body":"and in this case, it's specified in terms"},{"startTime":505.67,"endTime":507.75,"body":"of this variable \"s\"."},{"startTime":507.75,"endTime":509.15,"body":"So, I haven't said what it is,"},{"startTime":509.15,"endTime":513.12,"body":"but \"s\" is that exponent"},{"startTime":509.15,"endTime":513.12,"body":"in the exponential."},{"startTime":513.12,"endTime":515.71,"body":"So, something rather"},{"startTime":513.12,"endTime":515.71,"body":"beautiful has happened."},{"startTime":515.71,"endTime":516.84,"body":"It might not be, sort of, obvious."},{"startTime":516.84,"endTime":518.15,"body":"It might look a big ugly to you."},{"startTime":518.15,"endTime":520.07,"body":"But from my perspective,"},{"startTime":520.07,"endTime":521.81,"body":"it looks rather sort of attractive,"},{"startTime":521.81,"endTime":524.5,"body":"which is I've managed"},{"startTime":521.81,"endTime":524.5,"body":"to make an assumption"},{"startTime":524.5,"endTime":527.17,"body":"about the waveform here."},{"startTime":527.17,"endTime":528.5,"body":"And when I make that assumption,"},{"startTime":528.5,"endTime":531.12,"body":"my nasty differential equation,"},{"startTime":531.12,"endTime":534.15,"body":"becomes a simple, little"},{"startTime":531.12,"endTime":534.15,"body":"polynomial equation,"},{"startTime":534.15,"endTime":537.26,"body":"which I can solve or manipulate"},{"startTime":537.26,"endTime":540.74,"body":"in a, sort of, using my"},{"startTime":537.26,"endTime":540.74,"body":"high school mathematics."},{"startTime":540.74,"endTime":543.7,"body":"So, a very, very attractive idea."},{"startTime":543.7,"endTime":545.33,"body":"Now, you might protest at this point"},{"startTime":545.33,"endTime":546.163,"body":"and you might say,"},{"startTime":546.163,"endTime":546.996,"body":"\"Well, hang on, Richard."},{"startTime":546.996,"endTime":548.827,"body":"I mean, 'e' to the 'st',"},{"startTime":549.97,"endTime":551.25,"body":"this will very well solving for that."},{"startTime":551.25,"endTime":553.0,"body":"But what about the general case"},{"startTime":553.0,"endTime":555.62,"body":"when I got all sorts of other waveforms,"},{"startTime":555.62,"endTime":557.26,"body":"I might want to solve for?\""},{"startTime":557.26,"endTime":558.093,"body":"Ah, right."},{"startTime":558.093,"endTime":560.17,"body":"Well, I've got two things to say."},{"startTime":560.17,"endTime":563.75,"body":"The first one, is there's a good trick"},{"startTime":563.75,"endTime":568.09,"body":"in linear time-independent systems,"},{"startTime":568.09,"endTime":570.62,"body":"which is something called superposition."},{"startTime":570.62,"endTime":572.95,"body":"And superposition is a wonderful trick"},{"startTime":572.95,"endTime":574.713,"body":"that we use a lot."},{"startTime":574.713,"endTime":575.546,"body":"(smacks lips)"},{"startTime":575.546,"endTime":577.41,"body":"And the idea behind superposition,"},{"startTime":577.41,"endTime":580.97,"body":"is to think of the circuit we're analyzing"},{"startTime":580.97,"endTime":582.55,"body":"as a, sort of, black box."},{"startTime":582.55,"endTime":585.55,"body":"So, in this case, I've"},{"startTime":582.55,"endTime":585.55,"body":"got some signal coming in,"},{"startTime":585.55,"endTime":586.7,"body":"which I've called \"Vin\""},{"startTime":588.349,"endTime":591.223,"body":"and I've got some signal leaching out,"},{"startTime":592.19,"endTime":593.78,"body":"which I've called \"Vout\"."},{"startTime":593.78,"endTime":597.75,"body":"And I've said, \"Well, the output"},{"startTime":593.78,"endTime":597.75,"body":"voltage, here, the signal,"},{"startTime":597.75,"endTime":600.35,"body":"is some function of the input voltage.\""},{"startTime":600.35,"endTime":605.35,"body":"So, if my input signal was"},{"startTime":600.35,"endTime":605.35,"body":"\"x(t)\" or something like that,"},{"startTime":611.43,"endTime":615.02,"body":"then, what would come out"},{"startTime":611.43,"endTime":615.02,"body":"is some function of \"x(t)\"."},{"startTime":615.02,"endTime":616.42,"body":"If it was \"y(t)\","},{"startTime":617.32,"endTime":620.18,"body":"then, what would come out"},{"startTime":617.32,"endTime":620.18,"body":"is some function of \"y(t)\"."},{"startTime":621.22,"endTime":622.42,"body":"And superposition says"},{"startTime":622.42,"endTime":624.78,"body":"that if my input signal could be written"},{"startTime":624.78,"endTime":626.4,"body":"as some constant,"},{"startTime":626.4,"endTime":629.04,"body":"multiplied by \"x(t)\" plus"},{"startTime":626.4,"endTime":629.04,"body":"some constant multiplied"},{"startTime":629.04,"endTime":630.29,"body":"by \"y(t)\","},{"startTime":630.29,"endTime":634.38,"body":"then, my output can be written as the sum"},{"startTime":634.38,"endTime":636.573,"body":"of the component outputs."},{"startTime":637.52,"endTime":640.78,"body":"Now, that might not seem terribly useful,"},{"startTime":640.78,"endTime":642.08,"body":"but it's super powerful."},{"startTime":642.08,"endTime":643.15,"body":"Because it means,"},{"startTime":643.15,"endTime":647.15,"body":"we don't have to analyze"},{"startTime":643.15,"endTime":647.15,"body":"big, complicated waveforms"},{"startTime":647.15,"endTime":649.78,"body":"if we can break them down into components."},{"startTime":649.78,"endTime":653.81,"body":"So, if we could break this"},{"startTime":649.78,"endTime":653.81,"body":"complicated waveform input"},{"startTime":653.81,"endTime":656.03,"body":"into additive components,"},{"startTime":656.03,"endTime":657.52,"body":"so, it's a bit of this,"},{"startTime":656.03,"endTime":657.52,"body":"plus a bit of this,"},{"startTime":657.52,"endTime":659.46,"body":"plus a bit of this, plus a bit of this,"},{"startTime":659.46,"endTime":661.82,"body":"then, we only need to find the solutions"},{"startTime":661.82,"endTime":663.25,"body":"for each one of those components"},{"startTime":663.25,"endTime":666.01,"body":"and then, add them back up, the outputs."},{"startTime":666.01,"endTime":669.4,"body":"That's the first little bit of usefulness."},{"startTime":669.4,"endTime":673.35,"body":"So, if I could express"},{"startTime":669.4,"endTime":673.35,"body":"a waveform has some sum"},{"startTime":673.35,"endTime":674.327,"body":"of \"e\" to the \"st\","},{"startTime":675.45,"endTime":677.11,"body":"then, that's great."},{"startTime":677.11,"endTime":680.61,"body":"I've solved that a few slides back."},{"startTime":680.61,"endTime":683.364,"body":"So, all I need to do"},{"startTime":680.61,"endTime":683.364,"body":"is add up the outputs."},{"startTime":683.364,"endTime":684.197,"body":"(smacks lips)"},{"startTime":684.197,"endTime":685.03,"body":"Right."},{"startTime":685.03,"endTime":688.71,"body":"So, how could I convert a waveform"},{"startTime":688.71,"endTime":692.58,"body":"into some sum of \"e\" to the \"st\","},{"startTime":692.58,"endTime":694.96,"body":"or \"e\" to the \"-st\","},{"startTime":692.58,"endTime":694.96,"body":"or something like that?"},{"startTime":694.96,"endTime":696.91,"body":"Ah, well, why don't I measure"},{"startTime":696.91,"endTime":700.99,"body":"how alike my waveform is to that waveform?"},{"startTime":700.99,"endTime":702.667,"body":"So, how can I do that?"},{"startTime":702.667,"endTime":704.42,"body":"(audience coughs)"},{"startTime":704.42,"endTime":708.72,"body":"Well, one way is to multiply a waveform"},{"startTime":708.72,"endTime":710.8,"body":"by the thing I'm interested in measuring,"},{"startTime":710.8,"endTime":712.15,"body":"its alikeness to"},{"startTime":712.15,"endTime":713.76,"body":"and then, average it."},{"startTime":713.76,"endTime":715.47,"body":"So, this is what I could do here."},{"startTime":715.47,"endTime":718.08,"body":"On the top here, that's me, writing that"},{"startTime":718.08,"endTime":720.76,"body":"in, sort of, simple maths."},{"startTime":720.76,"endTime":723.7,"body":"So, this is the waveform"},{"startTime":720.76,"endTime":723.7,"body":"I'm interested in."},{"startTime":723.7,"endTime":726.497,"body":"This is the thing I'm going"},{"startTime":723.7,"endTime":726.497,"body":"to measure it against,"},{"startTime":726.497,"endTime":728.71,"body":"\"e\" to the minus \"st\","},{"startTime":726.497,"endTime":728.71,"body":"I've put a minus in there"},{"startTime":728.71,"endTime":731.94,"body":"to stop this thing shooting"},{"startTime":728.71,"endTime":731.94,"body":"off the slide here."},{"startTime":731.94,"endTime":734.63,"body":"So, this is my \"e\" to"},{"startTime":731.94,"endTime":734.63,"body":"the minus \"st\" waveform"},{"startTime":734.63,"endTime":736.16,"body":"on the top, here."},{"startTime":736.16,"endTime":738.75,"body":"This is the thing I'm going"},{"startTime":736.16,"endTime":738.75,"body":"to measure it against."},{"startTime":738.75,"endTime":741.163,"body":"So, let's just multiply"},{"startTime":738.75,"endTime":741.163,"body":"those two together."},{"startTime":742.35,"endTime":745.25,"body":"And then, what I want to do down here,"},{"startTime":745.25,"endTime":748.47,"body":"is just average all of these."},{"startTime":748.47,"endTime":751.523,"body":"And in mathematics, we"},{"startTime":748.47,"endTime":751.523,"body":"normally write that like that."},{"startTime":751.523,"endTime":755.03,"body":"It's a big elongated \"s\""},{"startTime":751.523,"endTime":755.03,"body":"on the left-hand side,"},{"startTime":755.03,"endTime":756.93,"body":"running from zero to infinity,"},{"startTime":756.93,"endTime":760.556,"body":"stands for a summation or integral."},{"startTime":760.556,"endTime":761.389,"body":"(mumbles)"},{"startTime":761.389,"endTime":762.54,"body":"This \"dt\" here, they always go together"},{"startTime":762.54,"endTime":763.85,"body":"as a sort of brackets."},{"startTime":763.85,"endTime":765.76,"body":"So, I've said \"0.63\","},{"startTime":765.76,"endTime":767.03,"body":"I mean, I should have written clearly."},{"startTime":767.03,"endTime":768.21,"body":"I don't know quite what it is."},{"startTime":768.21,"endTime":772.77,"body":"But the idea here, is"},{"startTime":768.21,"endTime":772.77,"body":"multiply the two together,"},{"startTime":772.77,"endTime":774.49,"body":"average it in time"},{"startTime":774.49,"endTime":779.49,"body":"and that measures how alike"},{"startTime":774.49,"endTime":779.49,"body":"I am to my prototype, here."},{"startTime":780.637,"endTime":782.66,"body":"And this is usually called a kernel"},{"startTime":782.66,"endTime":785.5,"body":"and not a colonel as"},{"startTime":782.66,"endTime":785.5,"body":"in a military colonel,"},{"startTime":785.5,"endTime":788.299,"body":"but a kernel with a \"k\"."},{"startTime":788.299,"endTime":789.38,"body":"(audience coughs)"},{"startTime":789.38,"endTime":790.213,"body":"Right."},{"startTime":790.213,"endTime":792.103,"body":"So, I can put those two"},{"startTime":790.213,"endTime":792.103,"body":"things together now."},{"startTime":793.862,"endTime":796.08,"body":"(clicks)"},{"startTime":796.08,"endTime":800.58,"body":"And that gives me something"},{"startTime":796.08,"endTime":800.58,"body":"that looks pretty horrifying,"},{"startTime":800.58,"endTime":802.17,"body":"but I'm going to break it down for you,"},{"startTime":802.17,"endTime":804.89,"body":"called the Laplace transform."},{"startTime":804.89,"endTime":808.14,"body":"And Laplace transform is merely a measure"},{"startTime":808.14,"endTime":811.65,"body":"of how alike some waveform,"},{"startTime":808.14,"endTime":811.65,"body":"which I've called \"f(t)\","},{"startTime":811.65,"endTime":816.33,"body":"is to a kernel, \"e\" to the \"-st\"."},{"startTime":816.33,"endTime":820.52,"body":"So, that is my time domain"},{"startTime":816.33,"endTime":820.52,"body":"function of the waveform."},{"startTime":820.52,"endTime":822.91,"body":"That's the thing that's probably the input"},{"startTime":822.91,"endTime":825.143,"body":"to the circuit or system."},{"startTime":826.72,"endTime":830.123,"body":"And then, this thing here, is the kernel."},{"startTime":832.02,"endTime":834.04,"body":"And then, these two things here,"},{"startTime":834.04,"endTime":835.09,"body":"that's the averaging."},{"startTime":836.86,"endTime":838.0,"body":"And then, it's also worth thinking"},{"startTime":838.0,"endTime":839.803,"body":"about what I get as the output."},{"startTime":841.96,"endTime":846.08,"body":"And that output, doesn't"},{"startTime":841.96,"endTime":846.08,"body":"depend on time anymore."},{"startTime":846.08,"endTime":849.25,"body":"This waveform, here, depended on time,"},{"startTime":849.25,"endTime":850.083,"body":"this one doesn't."},{"startTime":850.083,"endTime":854.72,"body":"It depends on \"s\" and \"s\" was"},{"startTime":850.083,"endTime":854.72,"body":"a parameter of the kernel."},{"startTime":854.72,"endTime":858.26,"body":"So, at a, sort of, simple algebraic level,"},{"startTime":858.26,"endTime":859.807,"body":"this is a bit of maths."},{"startTime":859.807,"endTime":862.75,"body":"So, it takes something"},{"startTime":859.807,"endTime":862.75,"body":"that depends on time"},{"startTime":862.75,"endTime":864.1,"body":"in one side"},{"startTime":864.1,"endTime":867.01,"body":"and something that depends"},{"startTime":864.1,"endTime":867.01,"body":"on \"s\" on the other side."},{"startTime":867.01,"endTime":868.53,"body":"That's why we call it a transform,"},{"startTime":868.53,"endTime":872.96,"body":"it moves from one domain, as"},{"startTime":868.53,"endTime":872.96,"body":"we call it, the time domain,"},{"startTime":872.96,"endTime":875.893,"body":"into, in this case, the \"s\" domain."},{"startTime":877.164,"endTime":882.12,"body":"And it's called the Laplace"},{"startTime":877.164,"endTime":882.12,"body":"transform after Pierre Laplace."},{"startTime":882.12,"endTime":886.06,"body":"And if you're all, sort of,"},{"startTime":882.12,"endTime":886.06,"body":"freaking out about integrals"},{"startTime":886.06,"endTime":887.66,"body":"and worried about all"},{"startTime":886.06,"endTime":887.66,"body":"these, sort of, things"},{"startTime":887.66,"endTime":888.493,"body":"and well, you might,"},{"startTime":888.493,"endTime":890.52,"body":"'cause these integrals can be a bit tricky"},{"startTime":890.52,"endTime":892.53,"body":"to invert actually."},{"startTime":892.53,"endTime":896.68,"body":"Then, we often use this curly symbol \"l\"."},{"startTime":896.68,"endTime":900.73,"body":"So, take a signal, put it"},{"startTime":896.68,"endTime":900.73,"body":"through the Laplace transform,"},{"startTime":900.73,"endTime":901.88,"body":"out comes \"F(s)\"."},{"startTime":902.74,"endTime":905.38,"body":"And although I haven't proved this,"},{"startTime":905.38,"endTime":906.89,"body":"you can go back."},{"startTime":906.89,"endTime":909.113,"body":"So, we often use this double-headed arrow."},{"startTime":910.8,"endTime":913.78,"body":"Now, something great has"},{"startTime":910.8,"endTime":913.78,"body":"happened at this point."},{"startTime":913.78,"endTime":916.07,"body":"I've now got a magic bit of maths"},{"startTime":916.07,"endTime":920.64,"body":"that can convert signals"},{"startTime":916.07,"endTime":920.64,"body":"into the \"s\" domain."},{"startTime":920.64,"endTime":925.14,"body":"I've, sort of, indicated"},{"startTime":920.64,"endTime":925.14,"body":"that circuit analysis,"},{"startTime":925.14,"endTime":927.12,"body":"by which I mean, electrical circuits"},{"startTime":927.12,"endTime":929.92,"body":"but we could also be talking"},{"startTime":927.12,"endTime":929.92,"body":"about mechanical circuits,"},{"startTime":929.92,"endTime":934.92,"body":"can be analyzed quite easily"},{"startTime":929.92,"endTime":934.92,"body":"in terms of the \"s\" domain."},{"startTime":934.99,"endTime":936.79,"body":"And the output,"},{"startTime":936.79,"endTime":938.76,"body":"well, if we want to get"},{"startTime":936.79,"endTime":938.76,"body":"back to the time domain"},{"startTime":938.76,"endTime":940.35,"body":"and you might think, \"Oh, of"},{"startTime":938.76,"endTime":940.35,"body":"course, we want to get back"},{"startTime":940.35,"endTime":941.27,"body":"to the time domain,\""},{"startTime":941.27,"endTime":943.593,"body":"we could just apply the inverse transform."},{"startTime":944.84,"endTime":947.04,"body":"The inverse transform"},{"startTime":944.84,"endTime":947.04,"body":"can be tricky to compute."},{"startTime":947.04,"endTime":949.7,"body":"But usually, engineers use"},{"startTime":947.04,"endTime":949.7,"body":"a sort of table lookup."},{"startTime":949.7,"endTime":951.71,"body":"So, they say, \"Well, which one of those?"},{"startTime":951.71,"endTime":953.23,"body":"Right, it must be one of those.\""},{"startTime":953.23,"endTime":955.81,"body":"Actually, a lot of"},{"startTime":953.23,"endTime":955.81,"body":"engineering never bothers"},{"startTime":955.81,"endTime":958.24,"body":"to convert it back to the time domain."},{"startTime":958.24,"endTime":962.24,"body":"Engineers, particularly,"},{"startTime":958.24,"endTime":962.24,"body":"electronics and control engineers,"},{"startTime":962.24,"endTime":965.44,"body":"usually do their thinking"},{"startTime":962.24,"endTime":965.44,"body":"in the \"s\" domain"},{"startTime":965.44,"endTime":970.21,"body":"and they usually won't trouble themselves"},{"startTime":970.21,"endTime":973.89,"body":"with explicitly working"},{"startTime":970.21,"endTime":973.89,"body":"out the time domain."},{"startTime":973.89,"endTime":976.82,"body":"There's no need, because"},{"startTime":973.89,"endTime":976.82,"body":"they can visualize"},{"startTime":976.82,"endTime":981.723,"body":"in the \"s\" domain, in"},{"startTime":976.82,"endTime":981.723,"body":"the transform domain."},{"startTime":982.69,"endTime":987.6,"body":"So, that's the essence"},{"startTime":982.69,"endTime":987.6,"body":"of an integral transform."},{"startTime":987.6,"endTime":990.17,"body":"It's a bit of mathematical jiggery-pokery"},{"startTime":990.17,"endTime":992.43,"body":"that makes the analysis easier."},{"startTime":992.43,"endTime":995.64,"body":"And that sense, it's probably,"},{"startTime":995.64,"endTime":998.7,"body":"fairly, sort of, standard affair"},{"startTime":998.7,"endTime":1003.13,"body":"for a lot of engineering"},{"startTime":998.7,"endTime":1003.13,"body":"and physics courses."},{"startTime":1003.13,"endTime":1005.81,"body":"Physics courses, usually, don't really go"},{"startTime":1005.81,"endTime":1007.05,"body":"for the Laplace transform."},{"startTime":1007.05,"endTime":1009.99,"body":"They're usually into a"},{"startTime":1007.05,"endTime":1009.99,"body":"different set of transforms."},{"startTime":1009.99,"endTime":1011.68,"body":"Well, there's a whole book of these"},{"startTime":1011.68,"endTime":1016.26,"body":"and for some time at the end,"},{"startTime":1011.68,"endTime":1016.26,"body":"we can try and think of some."},{"startTime":1016.26,"endTime":1019.37,"body":"But pretty much, any problem"},{"startTime":1016.26,"endTime":1019.37,"body":"that you can think of,"},{"startTime":1019.37,"endTime":1024.14,"body":"will have no appropriate transform."},{"startTime":1024.14,"endTime":1027.45,"body":"Laplace transform is very favored"},{"startTime":1027.45,"endTime":1031.34,"body":"by control engineers and"},{"startTime":1027.45,"endTime":1031.34,"body":"electronics engineers."},{"startTime":1031.34,"endTime":1033.82,"body":"So, you could design specialists."},{"startTime":1033.82,"endTime":1035.32,"body":"Nowadays, of course,"},{"startTime":1035.32,"endTime":1038.79,"body":"we tend not to be designing"},{"startTime":1035.32,"endTime":1038.79,"body":"a lot of analog circuits."},{"startTime":1038.79,"endTime":1041.14,"body":"We're often working in the digital domain."},{"startTime":1041.14,"endTime":1042.67,"body":"And in the digital domain,"},{"startTime":1042.67,"endTime":1046.13,"body":"we tend not to have"},{"startTime":1042.67,"endTime":1046.13,"body":"differential equations,"},{"startTime":1046.13,"endTime":1047.91,"body":"we tend to have difference equations,"},{"startTime":1047.91,"endTime":1049.93,"body":"as it is just differences of things,"},{"startTime":1049.93,"endTime":1051.69,"body":"well, as a transform for that,"},{"startTime":1051.69,"endTime":1053.0,"body":"as a transform for everything"},{"startTime":1053.0,"endTime":1055.29,"body":"and that's called the z-transform."},{"startTime":1055.29,"endTime":1058.48,"body":"Bizarrely, the Laplace"},{"startTime":1055.29,"endTime":1058.48,"body":"transform wasn't invented"},{"startTime":1058.48,"endTime":1059.523,"body":"by Pierre Laplace."},{"startTime":1061.12,"endTime":1065.653,"body":"Laplace transform was probably"},{"startTime":1061.12,"endTime":1065.653,"body":"invented by Niels Abel,"},{"startTime":1067.028,"endTime":1068.88,"body":"I think he is a Norwegian mathematician."},{"startTime":1068.88,"endTime":1073.88,"body":"And Laplace, probably,"},{"startTime":1068.88,"endTime":1073.88,"body":"invented the z-transform"},{"startTime":1074.76,"endTime":1078.923,"body":"in a bizarre, sort of, twist of fate."},{"startTime":1080.055,"endTime":1081.39,"body":"(smacks lips)"},{"startTime":1080.055,"endTime":1081.39,"body":"(inhales)"},{"startTime":1081.39,"endTime":1084.84,"body":"So, I don't want to say too"},{"startTime":1081.39,"endTime":1084.84,"body":"much about the z-transform,"},{"startTime":1084.84,"endTime":1087.79,"body":"but that's the method of choice"},{"startTime":1087.79,"endTime":1089.56,"body":"for designing digital photos"},{"startTime":1089.56,"endTime":1094.33,"body":"and for worrying about"},{"startTime":1089.56,"endTime":1094.33,"body":"the stability of objects."},{"startTime":1096.18,"endTime":1097.47,"body":"Now, I should say at this point,"},{"startTime":1097.47,"endTime":1100.89,"body":"that we're talking about"},{"startTime":1097.47,"endTime":1100.89,"body":"these mathematical tools"},{"startTime":1100.89,"endTime":1102.66,"body":"as the way a mathematician would,"},{"startTime":1102.66,"endTime":1105.307,"body":"as a way of, sort of, either"},{"startTime":1102.66,"endTime":1105.307,"body":"making problem easier to solve"},{"startTime":1105.307,"endTime":1107.24,"body":"and you might not feel"},{"startTime":1105.307,"endTime":1107.24,"body":"they're easier to solve,"},{"startTime":1107.24,"endTime":1109.06,"body":"but once you get the hang of these things,"},{"startTime":1109.06,"endTime":1111.813,"body":"they're fairly, sort of,"},{"startTime":1109.06,"endTime":1111.813,"body":"manageable and manipulable,"},{"startTime":1113.64,"endTime":1115.57,"body":"or a way of understanding,"},{"startTime":1115.57,"endTime":1117.453,"body":"or a way of, sort of, analysis."},{"startTime":1118.34,"endTime":1120.77,"body":"But that's not really-"},{"startTime":1120.77,"endTime":1122.27,"body":"well, computer science has some of that,"},{"startTime":1122.27,"endTime":1124.86,"body":"but I think I would also"},{"startTime":1122.27,"endTime":1124.86,"body":"like to talk about these"},{"startTime":1124.86,"endTime":1127.7,"body":"as, sort of, computational devices."},{"startTime":1127.7,"endTime":1129.66,"body":"And that means switching across"},{"startTime":1129.66,"endTime":1131.95,"body":"to the, sort of, the king of transforms,"},{"startTime":1131.95,"endTime":1135.133,"body":"which is the Fourier transform."},{"startTime":1136.12,"endTime":1139.2,"body":"And I haven't written"},{"startTime":1136.12,"endTime":1139.2,"body":"down the integral form"},{"startTime":1139.2,"endTime":1140.52,"body":"of the Fourier transform,"},{"startTime":1140.52,"endTime":1141.74,"body":"because it's got a complex number in it"},{"startTime":1141.74,"endTime":1145.95,"body":"and that might, I felt, that"},{"startTime":1141.74,"endTime":1145.95,"body":"was probably a step too far"},{"startTime":1145.95,"endTime":1148.85,"body":"amongst civilized company"},{"startTime":1145.95,"endTime":1148.85,"body":"and is not necessary."},{"startTime":1148.85,"endTime":1151.66,"body":"Because once you got the idea, here,"},{"startTime":1151.66,"endTime":1153.74,"body":"instead of \"e\" to the \"st\","},{"startTime":1153.74,"endTime":1157.18,"body":"we have something that depends upon omega,"},{"startTime":1157.18,"endTime":1160.417,"body":"which is the angular frequency"},{"startTime":1157.18,"endTime":1160.417,"body":"or if you prefer, \"f\","},{"startTime":1160.417,"endTime":1163.1,"body":"\"f\" is the non-angular frequency,"},{"startTime":1163.1,"endTime":1164.307,"body":"the frequency we've referred to."},{"startTime":1164.307,"endTime":1165.74,"body":"\"f\" is measured in hertz"},{"startTime":1165.74,"endTime":1168.15,"body":"and omega is measured"},{"startTime":1165.74,"endTime":1168.15,"body":"in radians per second."},{"startTime":1168.15,"endTime":1169.93,"body":"They're coupled by 2 pies"},{"startTime":1169.93,"endTime":1173.707,"body":"and omega equals two times pie times \"f\"."},{"startTime":1174.75,"endTime":1177.623,"body":"Now, the Fourier transform is-"},{"startTime":1180.65,"endTime":1183.08,"body":"did I call it the emperor of transforms?"},{"startTime":1183.08,"endTime":1187.27,"body":"It's the transform that"},{"startTime":1183.08,"endTime":1187.27,"body":"almost everyone knows about,"},{"startTime":1187.27,"endTime":1188.81,"body":"but probably doesn't know about."},{"startTime":1188.81,"endTime":1192.23,"body":"So, they're familiar with the output"},{"startTime":1192.23,"endTime":1194.257,"body":"from the Fourier transform,"},{"startTime":1194.257,"endTime":1198.963,"body":"but they're probably less"},{"startTime":1194.257,"endTime":1198.963,"body":"familiar with its computation."},{"startTime":1199.995,"endTime":1200.828,"body":"(smacks lips)"},{"startTime":1200.828,"endTime":1202.94,"body":"So, let's just have a"},{"startTime":1200.828,"endTime":1202.94,"body":"look at a few of those"},{"startTime":1202.94,"endTime":1207.35,"body":"and try and, sort of, get"},{"startTime":1202.94,"endTime":1207.35,"body":"a handle on how it works."},{"startTime":1207.35,"endTime":1211.72,"body":"So, probably, the simplest"},{"startTime":1207.35,"endTime":1211.72,"body":"thing I could think of,"},{"startTime":1211.72,"endTime":1214.39,"body":"was to refer to this as"},{"startTime":1211.72,"endTime":1214.39,"body":"of start of a lecture,"},{"startTime":1214.39,"endTime":1216.37,"body":"which is thinking about impulse testing."},{"startTime":1216.37,"endTime":1217.77,"body":"So, that's an impulse,"},{"startTime":1217.77,"endTime":1219.85,"body":"is where we hit something with a hammer,"},{"startTime":1219.85,"endTime":1221.76,"body":"or in terms of a voltage,"},{"startTime":1221.76,"endTime":1223.63,"body":"that's where we just, sort of, ping it"},{"startTime":1223.63,"endTime":1227.57,"body":"with a very short pulse."},{"startTime":1227.57,"endTime":1230.78,"body":"Here, I've used a formulation,"},{"startTime":1227.57,"endTime":1230.78,"body":"which is a theoretical one,"},{"startTime":1230.78,"endTime":1232.54,"body":"where instead of drawing a little pulse,"},{"startTime":1232.54,"endTime":1235.31,"body":"I've drawn this little arrow here."},{"startTime":1235.31,"endTime":1239.983,"body":"And this is an example"},{"startTime":1235.31,"endTime":1239.983,"body":"of an idealized pulse."},{"startTime":1241.86,"endTime":1244.88,"body":"It doesn't really exist but it"},{"startTime":1241.86,"endTime":1244.88,"body":"is mathematically convenient."},{"startTime":1244.88,"endTime":1246.82,"body":"So, the little arrow is something"},{"startTime":1246.82,"endTime":1248.21,"body":"that's meant to represent a pulse"},{"startTime":1248.21,"endTime":1251.75,"body":"that's infinitely thin"},{"startTime":1248.21,"endTime":1251.75,"body":"and infinitely high."},{"startTime":1251.75,"endTime":1253.73,"body":"So, it has an area of one,"},{"startTime":1253.73,"endTime":1256.07,"body":"but it's as thin as you can make it"},{"startTime":1256.07,"endTime":1257.75,"body":"and as tall as you can make it."},{"startTime":1257.75,"endTime":1260.37,"body":"They're often called generalized functions"},{"startTime":1260.37,"endTime":1263.677,"body":"and this is an example of"},{"startTime":1260.37,"endTime":1263.677,"body":"a generalized function,"},{"startTime":1263.677,"endTime":1265.64,"body":"or also, called an impulse."},{"startTime":1265.64,"endTime":1268.63,"body":"But also, sometimes called"},{"startTime":1265.64,"endTime":1268.63,"body":"a Dirac delta function,"},{"startTime":1268.63,"endTime":1270.203,"body":"named after Paul Dirac,"},{"startTime":1271.382,"endTime":1275.44,"body":"the Bristolian engineer"},{"startTime":1271.382,"endTime":1275.44,"body":"turned theoretical physicist."},{"startTime":1275.44,"endTime":1277.21,"body":"And over on the right-hand side,"},{"startTime":1277.21,"endTime":1280.563,"body":"I've plotted its Fourier transform."},{"startTime":1281.54,"endTime":1284.24,"body":"Some people would call it the spectrum."},{"startTime":1284.24,"endTime":1286.52,"body":"So, let's just have a look at it."},{"startTime":1286.52,"endTime":1290.88,"body":"So, I've got this highly localized"},{"startTime":1286.52,"endTime":1290.88,"body":"thing in the time domain."},{"startTime":1290.88,"endTime":1292.61,"body":"Beep, click."},{"startTime":1292.61,"endTime":1296.94,"body":"And it contains all"},{"startTime":1292.61,"endTime":1296.94,"body":"frequencies, over here."},{"startTime":1296.94,"endTime":1299.56,"body":"So, that's one of the reasons"},{"startTime":1299.56,"endTime":1302.82,"body":"that impulse testing"},{"startTime":1299.56,"endTime":1302.82,"body":"is often spoken about,"},{"startTime":1302.82,"endTime":1304.29,"body":"particularly, in mechanical engineering,"},{"startTime":1304.29,"endTime":1306.29,"body":"'cause if I tap something with a hammer,"},{"startTime":1306.29,"endTime":1309.61,"body":"it excites all frequencies in the system."},{"startTime":1309.61,"endTime":1314.61,"body":"So, in the old days of steam trains,"},{"startTime":1315.82,"endTime":1317.56,"body":"there was somebody who"},{"startTime":1315.82,"endTime":1317.56,"body":"went around with a hammer,"},{"startTime":1317.56,"endTime":1320.66,"body":"tapping wheels or steam trains,"},{"startTime":1320.66,"endTime":1323.62,"body":"the idea being that, that provided input"},{"startTime":1323.62,"endTime":1325.31,"body":"that contained all frequencies."},{"startTime":1325.31,"endTime":1327.51,"body":"So, if there was a defect in the wheel,"},{"startTime":1327.51,"endTime":1332.51,"body":"then, there would be enough"},{"startTime":1327.51,"endTime":1332.51,"body":"spectral energy in the impulse"},{"startTime":1332.94,"endTime":1334.46,"body":"to excite that defect"},{"startTime":1334.46,"endTime":1336.08,"body":"and a skilled operator could listen"},{"startTime":1336.08,"endTime":1339.55,"body":"to the ringing associated with the wheel"},{"startTime":1339.55,"endTime":1341.84,"body":"and know that there was a defect."},{"startTime":1341.84,"endTime":1344.623,"body":"Such people are known as wheel-tappers."},{"startTime":1345.618,"endTime":1348.56,"body":"(smacks lips)"},{"startTime":1348.56,"endTime":1350.83,"body":"Let's have a look at something"},{"startTime":1348.56,"endTime":1350.83,"body":"that's a bit more realistic."},{"startTime":1350.83,"endTime":1355.1,"body":"So, here is the Fourier"},{"startTime":1350.83,"endTime":1355.1,"body":"transform of a pulse"},{"startTime":1355.1,"endTime":1358.26,"body":"and this is a rectangular pulse,"},{"startTime":1358.26,"endTime":1360.893,"body":"centered around time zero."},{"startTime":1362.2,"endTime":1365.88,"body":"So, its zero goes up and comes down again"},{"startTime":1365.88,"endTime":1367.91,"body":"and this is its Fourier transform."},{"startTime":1367.91,"endTime":1371.59,"body":"Now, you might be slightly irritated"},{"startTime":1371.59,"endTime":1375.29,"body":"to see frequency, positive and negative."},{"startTime":1375.29,"endTime":1376.14,"body":"This is an artifact"},{"startTime":1376.14,"endTime":1379.0,"body":"of the way most people"},{"startTime":1376.14,"endTime":1379.0,"body":"define the Fourier transform"},{"startTime":1379.0,"endTime":1380.88,"body":"and for all practical purposes,"},{"startTime":1380.88,"endTime":1383.187,"body":"you can just focus on the"},{"startTime":1380.88,"endTime":1383.187,"body":"positive frequency part here"},{"startTime":1383.187,"endTime":1384.42,"body":"and that's what you would see"},{"startTime":1384.42,"endTime":1387.64,"body":"if you had a spectrum"},{"startTime":1384.42,"endTime":1387.64,"body":"analyzer in front of you,"},{"startTime":1387.64,"endTime":1389.123,"body":"that only the positive frequency."},{"startTime":1389.123,"endTime":1391.11,"body":"It's a mathematical artifact having these"},{"startTime":1391.11,"endTime":1392.25,"body":"and, in fact, it's not necessary."},{"startTime":1392.25,"endTime":1394.003,"body":"You could define a Fourier transform"},{"startTime":1394.003,"endTime":1395.03,"body":"that doesn't have them,"},{"startTime":1395.03,"endTime":1397.91,"body":"but convention has it that we have them."},{"startTime":1397.91,"endTime":1399.88,"body":"And you can see here, we've got something."},{"startTime":1399.88,"endTime":1401.293,"body":"This is called a \"sinx over x\" function"},{"startTime":1401.293,"endTime":1403.34,"body":"and \"sinf over f\" function."},{"startTime":1403.34,"endTime":1406.6,"body":"We've got quite a bit of"},{"startTime":1403.34,"endTime":1406.6,"body":"low frequency energy here"},{"startTime":1406.6,"endTime":1410.87,"body":"that's associated with this"},{"startTime":1406.6,"endTime":1410.87,"body":"being lifted off the zero line"},{"startTime":1410.87,"endTime":1415.24,"body":"and then, we've got these"},{"startTime":1410.87,"endTime":1415.24,"body":"sidelobes down here."},{"startTime":1415.24,"endTime":1417.847,"body":"So, high frequency,"},{"startTime":1415.24,"endTime":1417.847,"body":"there's not much energy"},{"startTime":1417.847,"endTime":1419.79,"body":"and the energy there is, is associated"},{"startTime":1419.79,"endTime":1421.823,"body":"with these rapid transitions here."},{"startTime":1423.18,"endTime":1426.56,"body":"If I would've picked a"},{"startTime":1423.18,"endTime":1426.56,"body":"slightly different shaped pulse"},{"startTime":1427.85,"endTime":1429.6,"body":"say, a longer pulse,"},{"startTime":1429.6,"endTime":1432.42,"body":"you might expect that you'd"},{"startTime":1429.6,"endTime":1432.42,"body":"have more energy focused"},{"startTime":1432.42,"endTime":1435.51,"body":"at low frequencies, which it does."},{"startTime":1435.51,"endTime":1438.07,"body":"If I was to pick a shorter pulse,"},{"startTime":1438.07,"endTime":1440.73,"body":"then, you might expect"},{"startTime":1438.07,"endTime":1440.73,"body":"it to have more energy"},{"startTime":1440.73,"endTime":1442.53,"body":"at the higher frequencies."},{"startTime":1442.53,"endTime":1443.4,"body":"So, can you see"},{"startTime":1443.4,"endTime":1447.69,"body":"if I was to make that"},{"startTime":1443.4,"endTime":1447.69,"body":"post thinner and thinner"},{"startTime":1447.69,"endTime":1450.59,"body":"and thinner and taller"},{"startTime":1447.69,"endTime":1450.59,"body":"and taller and taller,"},{"startTime":1450.59,"endTime":1454.82,"body":"this would become wider"},{"startTime":1450.59,"endTime":1454.82,"body":"and wider and wider,"},{"startTime":1454.82,"endTime":1457.87,"body":"until eventually, it would be flat,"},{"startTime":1457.87,"endTime":1459.85,"body":"which was the situation we looked at"},{"startTime":1459.85,"endTime":1461.09,"body":"in the previous slide."},{"startTime":1461.09,"endTime":1465.0,"body":"So, in fact, that is usually the proof"},{"startTime":1465.0,"endTime":1468.28,"body":"of the impulse of zero."},{"startTime":1468.28,"endTime":1473.21,"body":"So, in reality, your"},{"startTime":1468.28,"endTime":1473.21,"body":"wheel-tapper doesn't manage"},{"startTime":1473.21,"endTime":1475.35,"body":"to excite all frequencies,"},{"startTime":1475.35,"endTime":1478.68,"body":"'cause he or she is only"},{"startTime":1475.35,"endTime":1478.68,"body":"able to tap the wheel"},{"startTime":1478.68,"endTime":1482.35,"body":"for a finite amount of time"},{"startTime":1478.68,"endTime":1482.35,"body":"with a finite amount of energy"},{"startTime":1482.35,"endTime":1485.553,"body":"and so, we have the shaping"},{"startTime":1482.35,"endTime":1485.553,"body":"here, to the spectrum."},{"startTime":1487.02,"endTime":1490.343,"body":"Now, to engineers and analysts,"},{"startTime":1491.33,"endTime":1492.81,"body":"you can work in this domain,"},{"startTime":1492.81,"endTime":1494.04,"body":"or you can work in this domain,"},{"startTime":1494.04,"endTime":1497.1,"body":"you can work in the time"},{"startTime":1494.04,"endTime":1497.1,"body":"domain or the frequency domain"},{"startTime":1497.1,"endTime":1498.947,"body":"and everything is (indistinct)."},{"startTime":1500.963,"endTime":1502.24,"body":"(clicks)"},{"startTime":1502.24,"endTime":1505.41,"body":"If I picked something"},{"startTime":1502.24,"endTime":1505.41,"body":"that was well matched"},{"startTime":1505.41,"endTime":1509.48,"body":"to my Fourier transform,"},{"startTime":1505.41,"endTime":1509.48,"body":"like a cosine or sine wave"},{"startTime":1509.48,"endTime":1512.45,"body":"and the Fourier transform can be written"},{"startTime":1512.45,"endTime":1514.94,"body":"in terms of cosines and sine waves,"},{"startTime":1514.94,"endTime":1518.51,"body":"then, I get two generalized functions"},{"startTime":1518.51,"endTime":1519.95,"body":"in the frequency domain."},{"startTime":1519.95,"endTime":1524.95,"body":"So, this is a cosine"},{"startTime":1519.95,"endTime":1524.95,"body":"wave or frequency \"f0\""},{"startTime":1525.6,"endTime":1529.57,"body":"and it has a spike,"},{"startTime":1525.6,"endTime":1529.57,"body":"infinite height, zero width"},{"startTime":1529.57,"endTime":1530.403,"body":"at \"f0\"."},{"startTime":1533.54,"endTime":1538.44,"body":"So, the Fourier transform"},{"startTime":1533.54,"endTime":1538.44,"body":"is particularly well-suited"},{"startTime":1538.44,"endTime":1542.76,"body":"to oscillating waveforms,"},{"startTime":1538.44,"endTime":1542.76,"body":"like cosines and sines."},{"startTime":1542.76,"endTime":1546.41,"body":"And that's one of the"},{"startTime":1542.76,"endTime":1546.41,"body":"reasons it's popular,"},{"startTime":1546.41,"endTime":1548.21,"body":"because a lot of things tend to oscillate"},{"startTime":1548.21,"endTime":1550.603,"body":"and then, it'll go up and down in nature."},{"startTime":1551.89,"endTime":1552.98,"body":"(smacks lips)"},{"startTime":1552.98,"endTime":1555.37,"body":"Right, well, so far, so good."},{"startTime":1555.37,"endTime":1557.34,"body":"I still haven't really got past,"},{"startTime":1557.34,"endTime":1562.34,"body":"the, sort of, using the"},{"startTime":1557.34,"endTime":1562.34,"body":"Fourier transform as a tool"},{"startTime":1562.63,"endTime":1564.73,"body":"for understanding an analysis."},{"startTime":1564.73,"endTime":1566.73,"body":"And in fact, you might be feeling,"},{"startTime":1566.73,"endTime":1568.07,"body":"you're understanding slightly worse"},{"startTime":1568.07,"endTime":1569.53,"body":"than it was at the beginning of this,"},{"startTime":1569.53,"endTime":1572.16,"body":"because of the mathematical"},{"startTime":1569.53,"endTime":1572.16,"body":"complexity associated"},{"startTime":1572.16,"endTime":1572.993,"body":"with transforms."},{"startTime":1574.5,"endTime":1578.44,"body":"I wanted to just do one"},{"startTime":1574.5,"endTime":1578.44,"body":"more little diversion"},{"startTime":1578.44,"endTime":1580.67,"body":"and then, we'll get on"},{"startTime":1578.44,"endTime":1580.67,"body":"to it as an algorithm."},{"startTime":1580.67,"endTime":1583.063,"body":"And that relates to randomness."},{"startTime":1583.93,"endTime":1588.54,"body":"Of course, in reality, signals"},{"startTime":1583.93,"endTime":1588.54,"body":"are most likely to be random"},{"startTime":1588.54,"endTime":1590.88,"body":"and up to this point, I've"},{"startTime":1588.54,"endTime":1590.88,"body":"been talking about signals"},{"startTime":1590.88,"endTime":1594.17,"body":"that are defined by simple equation."},{"startTime":1594.17,"endTime":1597.183,"body":"But the reality isn't like that."},{"startTime":1599.611,"endTime":1600.6,"body":"(clicks)"},{"startTime":1600.6,"endTime":1603.66,"body":"I might have some waveform like this,"},{"startTime":1603.66,"endTime":1606.15,"body":"which is actually me saying"},{"startTime":1603.66,"endTime":1606.15,"body":"the words \"aggression\","},{"startTime":1606.15,"endTime":1608.16,"body":"I think if I've remembered rightly."},{"startTime":1608.16,"endTime":1610.06,"body":"So, how could I handle that?"},{"startTime":1610.06,"endTime":1614.33,"body":"Well, the Fourier transform"},{"startTime":1610.06,"endTime":1614.33,"body":"can certainly be used,"},{"startTime":1614.33,"endTime":1619.33,"body":"but it isn't used directly"},{"startTime":1614.33,"endTime":1619.33,"body":"on the time domain signal."},{"startTime":1619.33,"endTime":1621.17,"body":"It's used on a measure"},{"startTime":1621.17,"endTime":1625.47,"body":"of how alike the time domain"},{"startTime":1621.17,"endTime":1625.47,"body":"signal is with itself."},{"startTime":1625.47,"endTime":1626.9,"body":"We call that autocorrelation"},{"startTime":1627.926,"endTime":1629.45,"body":"and the way we work that out,"},{"startTime":1629.45,"endTime":1633.67,"body":"is to use our trick of"},{"startTime":1629.45,"endTime":1633.67,"body":"multiplying and averaging, again,"},{"startTime":1633.67,"endTime":1636.18,"body":"except we multiply the signal by itself."},{"startTime":1636.18,"endTime":1637.73,"body":"So, we shift the signal"},{"startTime":1637.73,"endTime":1641.367,"body":"by, in this case, an"},{"startTime":1637.73,"endTime":1641.367,"body":"offset called (indistinct)."},{"startTime":1642.854,"endTime":1643.687,"body":"(smacks lips)"},{"startTime":1643.687,"endTime":1645.21,"body":"We multiply the two together"},{"startTime":1645.21,"endTime":1647.81,"body":"and that particular shift here,"},{"startTime":1647.81,"endTime":1649.99,"body":"is associated with one value here"},{"startTime":1649.99,"endTime":1651.44,"body":"at a particular (indistinct)."},{"startTime":1652.37,"endTime":1654.79,"body":"And then, we just shift"},{"startTime":1652.37,"endTime":1654.79,"body":"this thing across here"},{"startTime":1654.79,"endTime":1659.79,"body":"and as (indistinct) varies,"},{"startTime":1654.79,"endTime":1659.79,"body":"we create this thing here,"},{"startTime":1661.26,"endTime":1664.4,"body":"which is not time along here,"},{"startTime":1664.4,"endTime":1668.33,"body":"this is delay, sometimes called lag."},{"startTime":1668.33,"endTime":1673.33,"body":"And if we take the Fourier"},{"startTime":1668.33,"endTime":1673.33,"body":"transform of the autocorrelation,"},{"startTime":1673.9,"endTime":1676.71,"body":"then, we, now, have something"},{"startTime":1673.9,"endTime":1676.71,"body":"that we can work with"},{"startTime":1677.87,"endTime":1681.27,"body":"and that is usually known"},{"startTime":1677.87,"endTime":1681.27,"body":"as the power spectrum."},{"startTime":1681.27,"endTime":1682.79,"body":"So, not quite the same, obviously,"},{"startTime":1682.79,"endTime":1685.58,"body":"spectrum that we've been"},{"startTime":1682.79,"endTime":1685.58,"body":"dealing with as different units"},{"startTime":1685.58,"endTime":1686.48,"body":"for one thing,"},{"startTime":1686.48,"endTime":1690.35,"body":"but it has the same,"},{"startTime":1686.48,"endTime":1690.35,"body":"sort of, intuitive idea,"},{"startTime":1690.35,"endTime":1695.313,"body":"which is these features"},{"startTime":1690.35,"endTime":1695.313,"body":"here, relate to components"},{"startTime":1697.39,"endTime":1699.623,"body":"of interest in the signal."},{"startTime":1700.719,"endTime":1703.483,"body":"And when people are talking about,"},{"startTime":1704.39,"endTime":1707.38,"body":"well, talk is often quite loose"},{"startTime":1707.38,"endTime":1708.88,"body":"on what's meant by these things,"},{"startTime":1708.88,"endTime":1713.55,"body":"but when people are"},{"startTime":1708.88,"endTime":1713.55,"body":"talking about a spectrum,"},{"startTime":1713.55,"endTime":1715.93,"body":"then, that's usually what"},{"startTime":1713.55,"endTime":1715.93,"body":"they're talking about."},{"startTime":1715.93,"endTime":1717.84,"body":"They're talking about something"},{"startTime":1717.84,"endTime":1721.38,"body":"that is technically the Fourier transform"},{"startTime":1721.38,"endTime":1723.59,"body":"of the autocorrelation function."},{"startTime":1723.59,"endTime":1724.883,"body":"Now, in reality,"},{"startTime":1726.15,"endTime":1728.603,"body":"computing the autocorrelation"},{"startTime":1726.15,"endTime":1728.603,"body":"is a bit of a pain."},{"startTime":1731.111,"endTime":1732.25,"body":"There are some technical reasons for that,"},{"startTime":1732.25,"endTime":1733.33,"body":"to do with convergence,"},{"startTime":1733.33,"endTime":1734.83,"body":"but there's also a practical reason"},{"startTime":1734.83,"endTime":1738.24,"body":"which is, it looks like"},{"startTime":1734.83,"endTime":1738.24,"body":"a, sort of, batch process,"},{"startTime":1738.24,"endTime":1739.073,"body":"doesn't it?"},{"startTime":1739.073,"endTime":1741.86,"body":"You have to sit there, measure the signal,"},{"startTime":1741.86,"endTime":1743.72,"body":"shift it around with a copy of itself,"},{"startTime":1743.72,"endTime":1746.29,"body":"multiply it together, do all"},{"startTime":1743.72,"endTime":1746.29,"body":"those integrals and add it up."},{"startTime":1746.29,"endTime":1749.58,"body":"So, in reality, you might"},{"startTime":1746.29,"endTime":1749.58,"body":"not want to do that."},{"startTime":1749.58,"endTime":1752.01,"body":"So, there are various approximations"},{"startTime":1752.01,"endTime":1757.01,"body":"in which you split the"},{"startTime":1752.01,"endTime":1757.01,"body":"waveform into little chunks,"},{"startTime":1757.12,"endTime":1760.29,"body":"you apply the Fourier transform"},{"startTime":1757.12,"endTime":1760.29,"body":"to those little chunks"},{"startTime":1760.29,"endTime":1762.48,"body":"and then, you do some"},{"startTime":1760.29,"endTime":1762.48,"body":"averaging of all those chunks."},{"startTime":1762.48,"endTime":1765.74,"body":"And that can also be shown to"},{"startTime":1762.48,"endTime":1765.74,"body":"produce the power spectrum."},{"startTime":1765.74,"endTime":1769.84,"body":"In fact, there are whole"},{"startTime":1765.74,"endTime":1769.84,"body":"thesis, written on methods"},{"startTime":1769.84,"endTime":1773.27,"body":"of estimating this statistic"},{"startTime":1773.27,"endTime":1775.45,"body":"from the original time waveform,"},{"startTime":1775.45,"endTime":1777.61,"body":"without having to go through the tedium"},{"startTime":1777.61,"endTime":1781.273,"body":"of computing the"},{"startTime":1777.61,"endTime":1781.273,"body":"autocorrelation to get that."},{"startTime":1782.745,"endTime":1783.77,"body":"(smacks lips)"},{"startTime":1783.77,"endTime":1784.603,"body":"Right."},{"startTime":1786.11,"endTime":1789.07,"body":"Looks like a bit of a"},{"startTime":1786.11,"endTime":1789.07,"body":"pain to compute this."},{"startTime":1789.07,"endTime":1791.17,"body":"We have to do this"},{"startTime":1789.07,"endTime":1791.17,"body":"thing called an integral"},{"startTime":1791.17,"endTime":1795.56,"body":"and we have to do it for every"},{"startTime":1791.17,"endTime":1795.56,"body":"possible value of frequency."},{"startTime":1795.56,"endTime":1797.15,"body":"So, if you think back to that equation"},{"startTime":1797.15,"endTime":1798.9,"body":"for the Laplace transform,"},{"startTime":1798.9,"endTime":1801.03,"body":"if you imagine writing"},{"startTime":1798.9,"endTime":1801.03,"body":"a program to compute it,"},{"startTime":1801.03,"endTime":1803.84,"body":"you'd think, \"Alright, so, I"},{"startTime":1801.03,"endTime":1803.84,"body":"got to pick a particular 's',"},{"startTime":1803.84,"endTime":1804.673,"body":"right?"},{"startTime":1804.673,"endTime":1806.13,"body":"Get my 'e' to the 'st', right?"},{"startTime":1806.13,"endTime":1808.75,"body":"Draw that out, multiply"},{"startTime":1806.13,"endTime":1808.75,"body":"it by the time waveform,"},{"startTime":1808.75,"endTime":1809.71,"body":"do the integral, average, average,"},{"startTime":1809.71,"endTime":1811.17,"body":"average, average, average, right."},{"startTime":1811.17,"endTime":1812.67,"body":"That's one value, right."},{"startTime":1812.67,"endTime":1814.19,"body":"New 's'."},{"startTime":1814.19,"endTime":1815.244,"body":"Stretch it.\""},{"startTime":1815.244,"endTime":1816.077,"body":"(mumbles)"},{"startTime":1816.077,"endTime":1818.193,"body":"Looks very tiresome, indeed."},{"startTime":1819.55,"endTime":1824.03,"body":"So, there's an approximation that we use"},{"startTime":1824.03,"endTime":1825.6,"body":"to that Fourier transform,"},{"startTime":1825.6,"endTime":1828.99,"body":"which is the Fourier"},{"startTime":1825.6,"endTime":1828.99,"body":"transform applied to signals"},{"startTime":1828.99,"endTime":1831.62,"body":"that are ready, discrete."},{"startTime":1831.62,"endTime":1833.463,"body":"Discrete, here, meaning separate."},{"startTime":1834.53,"endTime":1838.963,"body":"C-R-E-T-E, that sort"},{"startTime":1834.53,"endTime":1838.963,"body":"of discrete, not E-E-T."},{"startTime":1840.77,"endTime":1842.26,"body":"And this is it."},{"startTime":1842.26,"endTime":1845.723,"body":"It's the discrete Fourier transform."},{"startTime":1846.83,"endTime":1848.47,"body":"And you might, sort of, think"},{"startTime":1848.47,"endTime":1850.04,"body":"that, \"Gosh, this doesn't look very much,"},{"startTime":1850.04,"endTime":1851.7,"body":"like, what we had previously.\""},{"startTime":1851.7,"endTime":1855.08,"body":"Actually, it does have"},{"startTime":1851.7,"endTime":1855.08,"body":"almost identical form"},{"startTime":1855.08,"endTime":1857.11,"body":"with a few constants knocking around,"},{"startTime":1857.11,"endTime":1859.01,"body":"different in each case."},{"startTime":1859.01,"endTime":1863.173,"body":"So, the first part of"},{"startTime":1859.01,"endTime":1863.173,"body":"this is the signal itself"},{"startTime":1863.173,"endTime":1865.65,"body":"(clicks)"},{"startTime":1865.65,"endTime":1867.16,"body":"and that's there."},{"startTime":1867.16,"endTime":1868.957,"body":"And I called it a time domain signal"},{"startTime":1868.957,"endTime":1871.3,"body":"and you might say, \"Well, where is time?\""},{"startTime":1871.3,"endTime":1873.99,"body":"Well, because it's a sample,"},{"startTime":1871.3,"endTime":1873.99,"body":"because it's discrete,"},{"startTime":1873.99,"endTime":1877.21,"body":"I'm using \"n\" to index"},{"startTime":1873.99,"endTime":1877.21,"body":"it, rather than \"t\"."},{"startTime":1877.21,"endTime":1878.623,"body":"But it's the same thing."},{"startTime":1879.49,"endTime":1880.8,"body":"So, it's the sample thing."},{"startTime":1880.8,"endTime":1882.44,"body":"And I'm multiplying it by something,"},{"startTime":1882.44,"endTime":1885.51,"body":"that's the thing I want to"},{"startTime":1882.44,"endTime":1885.51,"body":"measure how alike it is to"},{"startTime":1885.51,"endTime":1887.93,"body":"and I'm calling that the kernel,"},{"startTime":1887.93,"endTime":1890.37,"body":"just as I did with the previous thing."},{"startTime":1890.37,"endTime":1892.71,"body":"And because it's a discrete signal now,"},{"startTime":1892.71,"endTime":1894.28,"body":"I'm not going to an integral,"},{"startTime":1894.28,"endTime":1895.89,"body":"I'm just going to do a summation."},{"startTime":1895.89,"endTime":1897.793,"body":"So, that's that bit there."},{"startTime":1899.07,"endTime":1902.92,"body":"And once I've done that average,"},{"startTime":1902.92,"endTime":1903.92,"body":"I called it an average,"},{"startTime":1903.92,"endTime":1907.01,"body":"you might moan that I"},{"startTime":1903.92,"endTime":1907.01,"body":"should be dividing by \"n\""},{"startTime":1907.01,"endTime":1908.27,"body":"for it to be an average."},{"startTime":1908.27,"endTime":1911.817,"body":"Well, let's not worry about"},{"startTime":1908.27,"endTime":1911.817,"body":"\"Divide by n\", \"Divide by t\"."},{"startTime":1914.8,"endTime":1918.27,"body":"It doesn't really matter, I should say."},{"startTime":1918.27,"endTime":1920.373,"body":"And then, over on the left-hand side,"},{"startTime":1921.6,"endTime":1923.29,"body":"I've got my transform."},{"startTime":1923.29,"endTime":1924.36,"body":"And the thing about the transform,"},{"startTime":1924.36,"endTime":1926.397,"body":"is it doesn't depend on \"n\"."},{"startTime":1926.397,"endTime":1929.15,"body":"\"n\" is being summed away, if you like."},{"startTime":1929.15,"endTime":1930.43,"body":"It only depends on \"m\""},{"startTime":1930.43,"endTime":1933.053,"body":"and this is the equivalent"},{"startTime":1930.43,"endTime":1933.053,"body":"of frequency, here."},{"startTime":1933.95,"endTime":1938.75,"body":"Now, this, this is the master work"},{"startTime":1938.75,"endTime":1941.28,"body":"of modern computer science."},{"startTime":1941.28,"endTime":1944.49,"body":"So, this is the discrete"},{"startTime":1941.28,"endTime":1944.49,"body":"Fourier transform,"},{"startTime":1944.49,"endTime":1948.602,"body":"it's the discrete version"},{"startTime":1944.49,"endTime":1948.602,"body":"of the Fourier transform."},{"startTime":1948.602,"endTime":1949.435,"body":"(smacks lips)"},{"startTime":1949.435,"endTime":1953.57,"body":"I hesitated a bit over the \"Divide by n\""},{"startTime":1953.57,"endTime":1955.78,"body":"and all those things."},{"startTime":1955.78,"endTime":1958.09,"body":"It's probably worth"},{"startTime":1955.78,"endTime":1958.09,"body":"saying that in old books,"},{"startTime":1958.09,"endTime":1960.29,"body":"there are quite a few"},{"startTime":1958.09,"endTime":1960.29,"body":"different definitions"},{"startTime":1960.29,"endTime":1962.18,"body":"of Fourier transforms"},{"startTime":1962.18,"endTime":1965.08,"body":"and they differ a little"},{"startTime":1962.18,"endTime":1965.08,"body":"bit in some of the symbols"},{"startTime":1965.08,"endTime":1966.86,"body":"and some of the normalizing constants."},{"startTime":1966.86,"endTime":1970.39,"body":"There are some \"2 pies\""},{"startTime":1966.86,"endTime":1970.39,"body":"that skulk around the place"},{"startTime":1970.39,"endTime":1973.223,"body":"and some people want to"},{"startTime":1970.39,"endTime":1973.223,"body":"divide by \"root 2 pies\","},{"startTime":1973.223,"endTime":1975.095,"body":"so that the inverse transform"},{"startTime":1975.095,"endTime":1976.97,"body":"and the Fourier transform look the same."},{"startTime":1976.97,"endTime":1978.07,"body":"Doesn't really matter."},{"startTime":1978.92,"endTime":1981.11,"body":"Nowadays, there is general agreement"},{"startTime":1981.11,"endTime":1982.53,"body":"on where these constants should sit"},{"startTime":1982.53,"endTime":1983.61,"body":"and that's the definitions"},{"startTime":1983.61,"endTime":1986.37,"body":"as the definitions I'm using here."},{"startTime":1986.37,"endTime":1987.203,"body":"Right."},{"startTime":1987.203,"endTime":1991.53,"body":"So, this thing is the algorithm"},{"startTime":1991.53,"endTime":1995.27,"body":"that dominates modern computing."},{"startTime":1995.27,"endTime":1998.13,"body":"And I'll demonstrate, I'll"},{"startTime":1995.27,"endTime":1998.13,"body":"try to explain why that is"},{"startTime":1998.13,"endTime":2000.9,"body":"in a few moments' time."},{"startTime":2000.9,"endTime":2002.81,"body":"So, let's just spend a short while,"},{"startTime":2002.81,"endTime":2005.8,"body":"just trying to understand what it does."},{"startTime":2005.8,"endTime":2010.8,"body":"So, if I apply the"},{"startTime":2005.8,"endTime":2010.8,"body":"discrete Fourier transform"},{"startTime":2010.97,"endTime":2013.04,"body":"or \"DFT\" as it's almost always called,"},{"startTime":2013.04,"endTime":2016.04,"body":"to a discrete version of the cosine wave,"},{"startTime":2016.04,"endTime":2017.19,"body":"which is what I've drawn here."},{"startTime":2017.19,"endTime":2018.92,"body":"So, this is a cosine wave."},{"startTime":2018.92,"endTime":2020.41,"body":"You can see it's a cosine wave,"},{"startTime":2020.41,"endTime":2024.37,"body":"because it starts up"},{"startTime":2020.41,"endTime":2024.37,"body":"here at its maximum value"},{"startTime":2024.37,"endTime":2025.87,"body":"and then, it goes up and down, up and down"},{"startTime":2025.87,"endTime":2027.35,"body":"and it ends just here."},{"startTime":2027.35,"endTime":2029.25,"body":"And I've chose the cosine wave"},{"startTime":2029.25,"endTime":2031.56,"body":"that fits absolutely perfectly"},{"startTime":2031.56,"endTime":2034.75,"body":"into the number of samples"},{"startTime":2031.56,"endTime":2034.75,"body":"I've in my little graph here,"},{"startTime":2034.75,"endTime":2037.023,"body":"which is, happens to be a 1024."},{"startTime":2038.37,"endTime":2039.81,"body":"And you can see the output,"},{"startTime":2039.81,"endTime":2042.75,"body":"it looks a bit weird, but"},{"startTime":2039.81,"endTime":2042.75,"body":"I'll explain that in a moment."},{"startTime":2042.75,"endTime":2045.62,"body":"Firstly, there is a spike here"},{"startTime":2045.62,"endTime":2050.003,"body":"and that is at the frequency"},{"startTime":2045.62,"endTime":2050.003,"body":"of this cosine wave."},{"startTime":2051.45,"endTime":2054.33,"body":"This spike, here, looks a bit odd."},{"startTime":2054.33,"endTime":2057.36,"body":"The way this thing is formulated,"},{"startTime":2057.36,"endTime":2062.33,"body":"is these are all our positive"},{"startTime":2057.36,"endTime":2062.33,"body":"frequencies up to around 512"},{"startTime":2062.33,"endTime":2064.82,"body":"and then, these have to be"},{"startTime":2062.33,"endTime":2064.82,"body":"wrapped around over here,"},{"startTime":2064.82,"endTime":2066.36,"body":"these are the negative frequencies"},{"startTime":2066.36,"endTime":2069.07,"body":"that we were looking at and"},{"startTime":2066.36,"endTime":2069.07,"body":"I said not to worry about."},{"startTime":2069.07,"endTime":2070.83,"body":"So, just for the time being,"},{"startTime":2070.83,"endTime":2072.75,"body":"just consider this"},{"startTime":2070.83,"endTime":2072.75,"body":"bottom part of the graph."},{"startTime":2072.75,"endTime":2076.18,"body":"So, that looks, well, pretty"},{"startTime":2072.75,"endTime":2076.18,"body":"fab and groovy, really."},{"startTime":2076.18,"endTime":2078.4,"body":"I mean, you've got to compute it,"},{"startTime":2078.4,"endTime":2082.3,"body":"but it gives a reasonable approximation"},{"startTime":2082.3,"endTime":2083.31,"body":"to what we're expecting."},{"startTime":2083.31,"endTime":2086.53,"body":"It's not a Dirac delta,"},{"startTime":2083.31,"endTime":2086.53,"body":"(indistinct) with infinite spike,"},{"startTime":2086.53,"endTime":2088.847,"body":"because we have a finite"},{"startTime":2086.53,"endTime":2088.847,"body":"amount of data here,"},{"startTime":2088.847,"endTime":2093.847,"body":"so, we don't get an infinite"},{"startTime":2088.847,"endTime":2093.847,"body":"spike, we get a finite spike."},{"startTime":2093.95,"endTime":2096.363,"body":"But it looks a bit like what we'd expect."},{"startTime":2097.64,"endTime":2099.9,"body":"But because it's a discrete algorithm,"},{"startTime":2099.9,"endTime":2101.21,"body":"there are some nasties,"},{"startTime":2101.21,"endTime":2103.91,"body":"which we always seem to"},{"startTime":2101.21,"endTime":2103.91,"body":"have to take care of."},{"startTime":2103.91,"endTime":2105.38,"body":"Here's one."},{"startTime":2105.38,"endTime":2107.08,"body":"If I just change the frequency"},{"startTime":2107.08,"endTime":2110.94,"body":"of this waveform a little bit,"},{"startTime":2110.94,"endTime":2112.17,"body":"what you might expect,"},{"startTime":2112.17,"endTime":2116.07,"body":"is that this spike would"},{"startTime":2112.17,"endTime":2116.07,"body":"move this way or this way,"},{"startTime":2116.07,"endTime":2119.01,"body":"left or right, depending on the frequency."},{"startTime":2119.01,"endTime":2120.5,"body":"But in fact, that does happen."},{"startTime":2120.5,"endTime":2124.21,"body":"But what also happens, if I do that,"},{"startTime":2124.21,"endTime":2126.35,"body":"is some artifacts appear."},{"startTime":2126.35,"endTime":2128.01,"body":"I'm not sure if you can"},{"startTime":2126.35,"endTime":2128.01,"body":"see them on the slides,"},{"startTime":2128.01,"endTime":2130.04,"body":"I've just sketched them in here."},{"startTime":2130.04,"endTime":2133.17,"body":"They're called leakage, spectral leakage."},{"startTime":2133.17,"endTime":2136.03,"body":"And the spectral leakage"},{"startTime":2133.17,"endTime":2136.03,"body":"is caused by the fact"},{"startTime":2136.03,"endTime":2140.09,"body":"that this waveform here, at the end,"},{"startTime":2140.09,"endTime":2143.84,"body":"doesn't exactly line up with the start."},{"startTime":2143.84,"endTime":2145.14,"body":"I didn't make this evident,"},{"startTime":2145.14,"endTime":2149.273,"body":"but it's an artifact of the"},{"startTime":2145.14,"endTime":2149.273,"body":"discrete Fourier transform."},{"startTime":2151.26,"endTime":2153.14,"body":"It makes an implicit assumption"},{"startTime":2153.14,"endTime":2156.88,"body":"that the waveform is repeated"},{"startTime":2153.14,"endTime":2156.88,"body":"out of the interval, exactly."},{"startTime":2156.88,"endTime":2158.56,"body":"So, when we choose a frequency"},{"startTime":2158.56,"endTime":2160.83,"body":"that doesn't exactly"},{"startTime":2158.56,"endTime":2160.83,"body":"fit into the interval,"},{"startTime":2160.83,"endTime":2164.03,"body":"we create what look like big jumps."},{"startTime":2164.03,"endTime":2166.18,"body":"These cause spectral energy,"},{"startTime":2166.18,"endTime":2169.883,"body":"which causes imperfections"},{"startTime":2166.18,"endTime":2169.883,"body":"and the spectral leakage."},{"startTime":2171.0,"endTime":2173.63,"body":"Well, there are whole books written"},{"startTime":2173.63,"endTime":2176.12,"body":"on ways to avoid spectral leakage,"},{"startTime":2176.12,"endTime":2180.15,"body":"but the standard way,"},{"startTime":2176.12,"endTime":2180.15,"body":"is to force the issue."},{"startTime":2180.15,"endTime":2182.51,"body":"And the way you force the issue,"},{"startTime":2182.51,"endTime":2185.3,"body":"is by multiplying the input waveform"},{"startTime":2185.3,"endTime":2187.18,"body":"with something called a window, okay?"},{"startTime":2187.18,"endTime":2189.95,"body":"So, this is one that"},{"startTime":2187.18,"endTime":2189.95,"body":"forces the \"n\" to zero."},{"startTime":2189.95,"endTime":2191.8,"body":"If you force the \"n\" to zero,"},{"startTime":2191.8,"endTime":2194.36,"body":"obviously this thing lines up"},{"startTime":2194.36,"endTime":2196.543,"body":"and the spectral leakage disappears."},{"startTime":2198.09,"endTime":2199.35,"body":"It does have a consequence,"},{"startTime":2199.35,"endTime":2203.96,"body":"which is, as we squeeze"},{"startTime":2199.35,"endTime":2203.96,"body":"those \"n\" down to zero,"},{"startTime":2203.96,"endTime":2208.03,"body":"we broaden the peaks."},{"startTime":2208.03,"endTime":2208.863,"body":"Perhaps, I should've said this"},{"startTime":2208.863,"endTime":2212.4,"body":"when I was talking about"},{"startTime":2208.863,"endTime":2212.4,"body":"that rectangular pulse."},{"startTime":2212.4,"endTime":2213.41,"body":"I don't know if you noticed,"},{"startTime":2213.41,"endTime":2217.973,"body":"but as the rectangular"},{"startTime":2213.41,"endTime":2217.973,"body":"pulse changed width,"},{"startTime":2219.53,"endTime":2223.83,"body":"the width of the \"sinx over"},{"startTime":2219.53,"endTime":2223.83,"body":"x\" function also changed"},{"startTime":2223.83,"endTime":2226.543,"body":"in exactly the opposite ratio."},{"startTime":2227.77,"endTime":2232.77,"body":"So, that's something called"},{"startTime":2227.77,"endTime":2232.77,"body":"the time bandwidth product,"},{"startTime":2233.34,"endTime":2234.95,"body":"which is always constant."},{"startTime":2234.95,"endTime":2236.63,"body":"And if you are a physicist,"},{"startTime":2236.63,"endTime":2237.86,"body":"you would say, \"Ah, yes."},{"startTime":2237.86,"endTime":2242.05,"body":"Well, I know that position and momentum"},{"startTime":2242.05,"endTime":2243.92,"body":"of Fourier transform pairs"},{"startTime":2243.92,"endTime":2246.92,"body":"and I know, due to the"},{"startTime":2243.92,"endTime":2246.92,"body":"Heisenberg uncertainty principle,"},{"startTime":2246.92,"endTime":2250.08,"body":"that position times momentum"},{"startTime":2246.92,"endTime":2250.08,"body":"is always a constant.\""},{"startTime":2250.08,"endTime":2254.54,"body":"So, in other words, you can"},{"startTime":2250.08,"endTime":2254.54,"body":"know precisely where you are,"},{"startTime":2254.54,"endTime":2256.31,"body":"but not know how quickly you're going."},{"startTime":2256.31,"endTime":2257.95,"body":"We can know precisely"},{"startTime":2256.31,"endTime":2257.95,"body":"how quickly you're going,"},{"startTime":2257.95,"endTime":2261.705,"body":"but not how and precisely where you are."},{"startTime":2261.705,"endTime":2262.538,"body":"Always a good excuse"},{"startTime":2262.538,"endTime":2264.51,"body":"if you're stopped by a"},{"startTime":2262.538,"endTime":2264.51,"body":"policeman on the motorway"},{"startTime":2264.51,"endTime":2265.66,"body":"and you're a physicist."},{"startTime":2267.642,"endTime":2270.53,"body":"And the same thing"},{"startTime":2267.642,"endTime":2270.53,"body":"applies in everyday life."},{"startTime":2270.53,"endTime":2275.53,"body":"So, you can either know"},{"startTime":2270.53,"endTime":2275.53,"body":"precisely where you are in time,"},{"startTime":2276.43,"endTime":2278.28,"body":"but not know the frequency very precisely,"},{"startTime":2278.28,"endTime":2280.38,"body":"or you can know the frequency precisely,"},{"startTime":2280.38,"endTime":2281.79,"body":"but not know where you are."},{"startTime":2281.79,"endTime":2285.43,"body":"And that's a consequence of"},{"startTime":2281.79,"endTime":2285.43,"body":"the finite window of analysis,"},{"startTime":2285.43,"endTime":2287.27,"body":"which was seen in this case."},{"startTime":2287.27,"endTime":2292.27,"body":"So, there's a whole, sort"},{"startTime":2287.27,"endTime":2292.27,"body":"of, sub-genre associated"},{"startTime":2292.74,"endTime":2297.74,"body":"with using the discrete Fourier transform"},{"startTime":2298.83,"endTime":2301.003,"body":"in an elegant way."},{"startTime":2302.29,"endTime":2305.13,"body":"Now, I should say, that you probably heard"},{"startTime":2305.13,"endTime":2307.71,"body":"that the term \"DFT\","},{"startTime":2305.13,"endTime":2307.71,"body":"discrete Fourier transform,"},{"startTime":2307.71,"endTime":2310.1,"body":"isn't actually used that much nowadays."},{"startTime":2310.1,"endTime":2313.31,"body":"Almost everyone would use the term \"FFT\"."},{"startTime":2313.31,"endTime":2318.31,"body":"And the \"FFT\" stands for"},{"startTime":2313.31,"endTime":2318.31,"body":"fast Fourier transform."},{"startTime":2321.26,"endTime":2324.2,"body":"It always used to stand for one algorithm,"},{"startTime":2324.2,"endTime":2327.64,"body":"known as the \"Cooley-Tukey"},{"startTime":2324.2,"endTime":2327.64,"body":"radix-2 algorithm\"."},{"startTime":2327.64,"endTime":2329.58,"body":"But there is now a whole library"},{"startTime":2329.58,"endTime":2333.67,"body":"of fast Fourier transform algorithms."},{"startTime":2333.67,"endTime":2335.79,"body":"So, when people say the \"FFT\","},{"startTime":2335.79,"endTime":2340.273,"body":"they could mean any algorithm"},{"startTime":2335.79,"endTime":2340.273,"body":"for computing it quickly."},{"startTime":2341.28,"endTime":2345.41,"body":"And I may have used them"},{"startTime":2341.28,"endTime":2345.41,"body":"interchangeably in this lecture"},{"startTime":2345.41,"endTime":2349.15,"body":"and if I have, I think"},{"startTime":2345.41,"endTime":2349.15,"body":"that's now acceptable."},{"startTime":2349.15,"endTime":2350.89,"body":"If we were in an undergraduate lecture,"},{"startTime":2350.89,"endTime":2352.97,"body":"they'd be saying, \"No,"},{"startTime":2350.89,"endTime":2352.97,"body":"they're completely different.\""},{"startTime":2352.97,"endTime":2355.67,"body":"But I think in common parlance,"},{"startTime":2355.67,"endTime":2358.53,"body":"nobody computes a \"DFT\" the slow way."},{"startTime":2358.53,"endTime":2359.61,"body":"Everybody computes it,"},{"startTime":2359.61,"endTime":2362.96,"body":"using one of the numerous"},{"startTime":2359.61,"endTime":2362.96,"body":"fast algorithms there are,"},{"startTime":2362.96,"endTime":2364.2,"body":"out there."},{"startTime":2364.2,"endTime":2365.99,"body":"The fast algorithms arise"},{"startTime":2365.99,"endTime":2367.91,"body":"because the kernel function,"},{"startTime":2367.91,"endTime":2371.27,"body":"in a Fourier transform, rotates."},{"startTime":2371.27,"endTime":2372.38,"body":"And because it rotates,"},{"startTime":2372.38,"endTime":2377.31,"body":"we can find symmetry in the algorithm."},{"startTime":2377.31,"endTime":2380.06,"body":"So, we can split the multiplications,"},{"startTime":2380.06,"endTime":2381.87,"body":"associated with the kernel function"},{"startTime":2381.87,"endTime":2384.247,"body":"into multiple separate multiplications"},{"startTime":2384.247,"endTime":2386.423,"body":"and that gives us"},{"startTime":2384.247,"endTime":2386.423,"body":"computational efficiency."},{"startTime":2387.636,"endTime":2389.22,"body":"(clicks)"},{"startTime":2389.22,"endTime":2390.213,"body":"Right."},{"startTime":2391.188,"endTime":2393.39,"body":"Let's look at some examples"},{"startTime":2391.188,"endTime":2393.39,"body":"of how it's applied."},{"startTime":2393.39,"endTime":2395.92,"body":"Because I want to, sort of, dwell a bit"},{"startTime":2395.92,"endTime":2400.92,"body":"on how the \"FFT\", the \"DFT\" is used."},{"startTime":2401.97,"endTime":2405.44,"body":"And the first example of this I've picked,"},{"startTime":2405.44,"endTime":2407.25,"body":"is something I've been"},{"startTime":2405.44,"endTime":2407.25,"body":"longing to talk about"},{"startTime":2407.25,"endTime":2408.23,"body":"for quite a few lectures"},{"startTime":2408.23,"endTime":2410.41,"body":"and I realize, \"I"},{"startTime":2408.23,"endTime":2410.41,"body":"should've talked about it"},{"startTime":2410.41,"endTime":2412.25,"body":"in my compression lecture,\""},{"startTime":2412.25,"endTime":2413.59,"body":"but it just wasn't enough time."},{"startTime":2413.59,"endTime":2415.68,"body":"So, I've, sort of, snuck"},{"startTime":2413.59,"endTime":2415.68,"body":"a bit of compression"},{"startTime":2415.68,"endTime":2418.943,"body":"into this lecture, partly to fill a hole"},{"startTime":2420.7,"endTime":2422.223,"body":"that I felt need to be filled."},{"startTime":2424.26,"endTime":2429.26,"body":"This graph, which is"},{"startTime":2424.26,"endTime":2429.26,"body":"quite an old graph now,"},{"startTime":2430.7,"endTime":2433.77,"body":"talks about the response of the human ear"},{"startTime":2433.77,"endTime":2438.77,"body":"and we're describing an"},{"startTime":2433.77,"endTime":2438.77,"body":"effect here known as masking."},{"startTime":2439.07,"endTime":2443.37,"body":"And masking is a property that arises"},{"startTime":2443.37,"endTime":2445.363,"body":"from the way the human cochlea,"},{"startTime":2446.33,"endTime":2450.1,"body":"which is the human Fourier"},{"startTime":2446.33,"endTime":2450.1,"body":"transformer, that's in your ear,"},{"startTime":2450.1,"endTime":2451.1,"body":"that's how it works."},{"startTime":2452.31,"endTime":2457.07,"body":"And the human ear is incapable"},{"startTime":2452.31,"endTime":2457.07,"body":"of hearing certain things"},{"startTime":2457.07,"endTime":2460.79,"body":"when certain signals are present."},{"startTime":2460.79,"endTime":2462.633,"body":"So, for example,"},{"startTime":2464.45,"endTime":2469.45,"body":"if a signal falls below"},{"startTime":2464.45,"endTime":2469.45,"body":"this smooth threshold here,"},{"startTime":2472.66,"endTime":2475.29,"body":"labeled \"threshold in quiet\","},{"startTime":2475.29,"endTime":2477.24,"body":"then, we probably won't hear it at all."},{"startTime":2478.35,"endTime":2480.91,"body":"If you can't read this on the bottom here,"},{"startTime":2480.91,"endTime":2484.35,"body":"we have a frequency axis,"},{"startTime":2480.91,"endTime":2484.35,"body":"measured in kilohertz"},{"startTime":2484.35,"endTime":2487.12,"body":"and up here, we have sound pressure level,"},{"startTime":2487.12,"endTime":2490.33,"body":"measured with respect to some reference."},{"startTime":2490.33,"endTime":2492.28,"body":"I can't remember what the reference is."},{"startTime":2493.62,"endTime":2495.69,"body":"There's also, I don't"},{"startTime":2493.62,"endTime":2495.69,"body":"know if you can see this,"},{"startTime":2495.69,"endTime":2500.01,"body":"but there's a second graph here"},{"startTime":2500.01,"endTime":2502.28,"body":"and this threshold arises"},{"startTime":2502.28,"endTime":2505.81,"body":"from the presence of a tone."},{"startTime":2505.81,"endTime":2508.41,"body":"So, if we had a tone,"},{"startTime":2505.81,"endTime":2508.41,"body":"which I've sketched here"},{"startTime":2508.41,"endTime":2510.36,"body":"as a Dirac delta symbol."},{"startTime":2510.36,"endTime":2515.36,"body":"If we had a tone, here, of this intensity,"},{"startTime":2516.94,"endTime":2520.78,"body":"then, any signal that falls within this,"},{"startTime":2520.78,"endTime":2524.84,"body":"is not going to be"},{"startTime":2520.78,"endTime":2524.84,"body":"distinguishable by human ear."},{"startTime":2528.09,"endTime":2530.7,"body":"So, if you think about the signals"},{"startTime":2530.7,"endTime":2532.96,"body":"that you're used to listening to,"},{"startTime":2532.96,"endTime":2536.37,"body":"if there is a substantial tonal component"},{"startTime":2536.37,"endTime":2537.597,"body":"of, say, a music signal,"},{"startTime":2537.597,"endTime":2540.68,"body":"let's say you're listening to"},{"startTime":2537.597,"endTime":2540.68,"body":"someone playing the violin,"},{"startTime":2540.68,"endTime":2542.45,"body":"so, maybe the fundamental"},{"startTime":2542.45,"endTime":2545.39,"body":"or the second harmonic"},{"startTime":2542.45,"endTime":2545.39,"body":"of the violin dominates"},{"startTime":2545.39,"endTime":2546.81,"body":"what you're listening to,"},{"startTime":2546.81,"endTime":2549.11,"body":"there are other bits of the musical signal"},{"startTime":2549.11,"endTime":2550.82,"body":"that you can't hear."},{"startTime":2550.82,"endTime":2553.43,"body":"So, there's clearly some potential there"},{"startTime":2553.43,"endTime":2557.04,"body":"for not sending those across the airwaves"},{"startTime":2557.04,"endTime":2559.57,"body":"or onto the disk at all."},{"startTime":2559.57,"endTime":2562.05,"body":"And that's the basis of what's"},{"startTime":2559.57,"endTime":2562.05,"body":"called lossy compression."},{"startTime":2562.05,"endTime":2563.79,"body":"Lossy compression is when we're prepared"},{"startTime":2563.79,"endTime":2567.07,"body":"to completely eliminate"},{"startTime":2563.79,"endTime":2567.07,"body":"parts of the signal,"},{"startTime":2567.07,"endTime":2570.09,"body":"because we think the"},{"startTime":2567.07,"endTime":2570.09,"body":"human at the other end,"},{"startTime":2570.09,"endTime":2571.8,"body":"won't receive it."},{"startTime":2571.8,"endTime":2573.66,"body":"So, several important points there."},{"startTime":2573.66,"endTime":2575.1,"body":"It's the human on the other end."},{"startTime":2575.1,"endTime":2578.47,"body":"So, if your dog is listening,"},{"startTime":2578.47,"endTime":2579.51,"body":"I'm not so confident,"},{"startTime":2579.51,"endTime":2583.46,"body":"because we don't know how"},{"startTime":2579.51,"endTime":2583.46,"body":"dogs and cats and non-humans,"},{"startTime":2583.46,"endTime":2585.943,"body":"well, this is a human-only system."},{"startTime":2587.06,"endTime":2590.31,"body":"So, if you are listening to,"},{"startTime":2590.31,"endTime":2595.31,"body":"or sending compressed digital audio"},{"startTime":2596.41,"endTime":2599.73,"body":"and you almost certainly are doing that,"},{"startTime":2599.73,"endTime":2604.73,"body":"well, certainly during the"},{"startTime":2599.73,"endTime":2604.73,"body":"duration of the pandemic,"},{"startTime":2604.93,"endTime":2608.81,"body":"if you're on \"Zoom\", \"Teams\","},{"startTime":2608.81,"endTime":2612.01,"body":"most of us were virtually"},{"startTime":2608.81,"endTime":2612.01,"body":"every hour of the waking day"},{"startTime":2612.01,"endTime":2614.0,"body":"and several hours of a non-waking day"},{"startTime":2614.0,"endTime":2616.01,"body":"if you're working in other countries,"},{"startTime":2616.01,"endTime":2620.477,"body":"you were doing this every single moment."},{"startTime":2620.477,"endTime":2621.31,"body":"(clicks)"},{"startTime":2621.31,"endTime":2622.6,"body":"And what you were doing,"},{"startTime":2622.6,"endTime":2626.613,"body":"was following something like"},{"startTime":2622.6,"endTime":2626.613,"body":"the MP3 coding standard."},{"startTime":2627.65,"endTime":2629.55,"body":"And this is a diagram of it,"},{"startTime":2629.55,"endTime":2631.55,"body":"but there are different"},{"startTime":2629.55,"endTime":2631.55,"body":"acoustic standards."},{"startTime":2631.55,"endTime":2634.64,"body":"But the MP3 one is, sort of, canonical"},{"startTime":2634.64,"endTime":2639.64,"body":"and the basic idea is"},{"startTime":2634.64,"endTime":2639.64,"body":"repeated again and again."},{"startTime":2639.96,"endTime":2642.44,"body":"And what I'd like to draw"},{"startTime":2639.96,"endTime":2642.44,"body":"your attention to here,"},{"startTime":2642.44,"endTime":2645.13,"body":"is we've got this digital"},{"startTime":2642.44,"endTime":2645.13,"body":"audio coming in here,"},{"startTime":2645.13,"endTime":2647.96,"body":"the first thing that's"},{"startTime":2645.13,"endTime":2647.96,"body":"happening over here,"},{"startTime":2647.96,"endTime":2650.65,"body":"is a fast Fourier transform algorithm."},{"startTime":2650.65,"endTime":2652.65,"body":"Well, we've just talked about that."},{"startTime":2652.65,"endTime":2654.04,"body":"Then, we're running this model."},{"startTime":2654.04,"endTime":2656.02,"body":"So, what the model is doing,"},{"startTime":2656.02,"endTime":2658.1,"body":"is it's looking for these spikes"},{"startTime":2658.1,"endTime":2659.42,"body":"and it's trying to work out"},{"startTime":2659.42,"endTime":2661.4,"body":"what the masking threshold is."},{"startTime":2661.4,"endTime":2664.45,"body":"And having worked out what"},{"startTime":2661.4,"endTime":2664.45,"body":"the masking threshold is,"},{"startTime":2664.45,"endTime":2668.66,"body":"it's then, going to use more or fewer bits"},{"startTime":2668.66,"endTime":2671.84,"body":"to code bits of that signal."},{"startTime":2671.84,"endTime":2675.377,"body":"To do that, it has to do"},{"startTime":2671.84,"endTime":2675.377,"body":"it in the frequency domain"},{"startTime":2675.377,"endTime":2677.31,"body":"and not in the time domain."},{"startTime":2677.31,"endTime":2682.31,"body":"So, this part of the"},{"startTime":2677.31,"endTime":2682.31,"body":"algorithm, up here, top-left,"},{"startTime":2683.7,"endTime":2688.213,"body":"splits the input signal"},{"startTime":2683.7,"endTime":2688.213,"body":"into sub-bands, alright?"},{"startTime":2689.15,"endTime":2690.77,"body":"Then, transforms them,"},{"startTime":2690.77,"endTime":2695.77,"body":"usually using a modified"},{"startTime":2690.77,"endTime":2695.77,"body":"discrete cosine transform,"},{"startTime":2695.97,"endTime":2697.82,"body":"which is a type of Fourier transform."},{"startTime":2698.72,"endTime":2701.29,"body":"It then, applies a"},{"startTime":2698.72,"endTime":2701.29,"body":"different number of bits"},{"startTime":2701.29,"endTime":2703.15,"body":"to each one of these bands."},{"startTime":2703.15,"endTime":2706.01,"body":"It then, does some standard"},{"startTime":2703.15,"endTime":2706.01,"body":"run-length encoding"},{"startTime":2706.01,"endTime":2707.02,"body":"and compression"},{"startTime":2707.02,"endTime":2711.66,"body":"and slams it down the audio stream."},{"startTime":2711.66,"endTime":2714.37,"body":"So, if you're listening"},{"startTime":2711.66,"endTime":2714.37,"body":"to a digital television,"},{"startTime":2714.37,"endTime":2716.053,"body":"or a digital radio,"},{"startTime":2716.894,"endTime":2720.44,"body":"or you're broadcasting"},{"startTime":2716.894,"endTime":2720.44,"body":"via \"Zoom\" or \"Teams\","},{"startTime":2720.44,"endTime":2722.25,"body":"or whatever you're using,"},{"startTime":2722.25,"endTime":2724.67,"body":"you're doing something like this."},{"startTime":2724.67,"endTime":2727.603,"body":"And the rate that you're"},{"startTime":2724.67,"endTime":2727.603,"body":"doing this is fairly amazing,"},{"startTime":2728.54,"endTime":2732.94,"body":"doing this, probably, well,"},{"startTime":2728.54,"endTime":2732.94,"body":"if it's high-quality audio,"},{"startTime":2732.94,"endTime":2737.1,"body":"44.1 or 48 kilo samples per second."},{"startTime":2737.1,"endTime":2739.913,"body":"So, you're doing a lot of it."},{"startTime":2740.76,"endTime":2745.24,"body":"Now, this is one of the algorithms"},{"startTime":2745.24,"endTime":2748.62,"body":"that fall, I think, in the"},{"startTime":2745.24,"endTime":2748.62,"body":"discrete Fourier transform"},{"startTime":2748.62,"endTime":2751.37,"body":"or \"FFT\" hall of fame."},{"startTime":2751.37,"endTime":2754.63,"body":"And I'll talk a little bit"},{"startTime":2751.37,"endTime":2754.63,"body":"about some of these now."},{"startTime":2754.63,"endTime":2755.59,"body":"But before I do that,"},{"startTime":2755.59,"endTime":2757.44,"body":"I thought I'd draw your"},{"startTime":2755.59,"endTime":2757.44,"body":"attention to this quote"},{"startTime":2757.44,"endTime":2758.47,"body":"from Gilbert Strang."},{"startTime":2758.47,"endTime":2761.16,"body":"Strang is bit of a hero"},{"startTime":2761.16,"endTime":2764.34,"body":"of undergraduates for his explanations"},{"startTime":2764.34,"endTime":2768.87,"body":"of matrix, maths and linear algebra."},{"startTime":2768.87,"endTime":2773.42,"body":"And he has described this as"},{"startTime":2768.87,"endTime":2773.42,"body":"the most important algorithm"},{"startTime":2773.42,"endTime":2775.163,"body":"of our lifetime."},{"startTime":2776.31,"endTime":2780.23,"body":"And let's just talk about"},{"startTime":2776.31,"endTime":2780.23,"body":"some of these briefly."},{"startTime":2780.23,"endTime":2781.93,"body":"So, audio coding we've talked about."},{"startTime":2781.93,"endTime":2785.13,"body":"The idea there is the"},{"startTime":2781.93,"endTime":2785.13,"body":"\"FFT\" is certainly used"},{"startTime":2785.13,"endTime":2788.61,"body":"for computing the psychoacoustic masking."},{"startTime":2788.61,"endTime":2790.29,"body":"It may, in some versions,"},{"startTime":2790.29,"endTime":2793.9,"body":"also be used for the sub-band filtering,"},{"startTime":2793.9,"endTime":2795.74,"body":"filtering the bands and applying them."},{"startTime":2795.74,"endTime":2796.573,"body":"Subsequently,"},{"startTime":2796.573,"endTime":2798.89,"body":"to that, you're going to apply"},{"startTime":2796.573,"endTime":2798.89,"body":"different amount of bits."},{"startTime":2798.89,"endTime":2800.36,"body":"Not all standards use that."},{"startTime":2800.36,"endTime":2803.07,"body":"Some of them use filters,"},{"startTime":2800.36,"endTime":2803.07,"body":"some use the \"DFT\"."},{"startTime":2803.07,"endTime":2805.93,"body":"Video coding, I've talked"},{"startTime":2803.07,"endTime":2805.93,"body":"about in previous lectures."},{"startTime":2805.93,"endTime":2809.093,"body":"But if you're watching this online,"},{"startTime":2812.56,"endTime":2815.57,"body":"the video that you're"},{"startTime":2812.56,"endTime":2815.57,"body":"watching has been split"},{"startTime":2815.57,"endTime":2818.97,"body":"into eight-by-eight pixel blocks."},{"startTime":2818.97,"endTime":2820.81,"body":"Those blocks have been transformed,"},{"startTime":2820.81,"endTime":2823.71,"body":"using a version of the"},{"startTime":2820.81,"endTime":2823.71,"body":"discrete Fourier transform,"},{"startTime":2823.71,"endTime":2825.54,"body":"or the discrete cosine transform,"},{"startTime":2825.54,"endTime":2828.18,"body":"which is tuned to all positive values"},{"startTime":2828.18,"endTime":2830.48,"body":"of these all that you're getting, images."},{"startTime":2830.48,"endTime":2834.5,"body":"Then, some of those coefficients are sent"},{"startTime":2834.5,"endTime":2835.58,"body":"but others are discarded,"},{"startTime":2835.58,"endTime":2838.26,"body":"because we don't think that"},{"startTime":2835.58,"endTime":2838.26,"body":"human eye can look at them."},{"startTime":2838.26,"endTime":2839.48,"body":"And in previous lectures,"},{"startTime":2839.48,"endTime":2842.58,"body":"I've certainly looked"},{"startTime":2839.48,"endTime":2842.58,"body":"and the \"DCT\" does that."},{"startTime":2842.58,"endTime":2844.77,"body":"So, not only was your \"Zoom\" call,"},{"startTime":2844.77,"endTime":2848.06,"body":"it was the audio coded using the \"DFT\"."},{"startTime":2848.06,"endTime":2850.859,"body":"The video was coded using the \"DCT\","},{"startTime":2850.859,"endTime":2855.6,"body":"which is a type of Fourier transform."},{"startTime":2855.6,"endTime":2856.92,"body":"Speech recognition."},{"startTime":2856.92,"endTime":2858.81,"body":"There are various ways of"},{"startTime":2856.92,"endTime":2858.81,"body":"doing speech recognition,"},{"startTime":2858.81,"endTime":2860.38,"body":"but one of the standard ways"},{"startTime":2860.38,"endTime":2865.38,"body":"and I think I discussed this"},{"startTime":2860.38,"endTime":2865.38,"body":"in a lecture a few years ago,"},{"startTime":2867.6,"endTime":2871.547,"body":"is (indistinct) use \"Mel-frequency"},{"startTime":2867.6,"endTime":2871.547,"body":"cepstral coefficients\"."},{"startTime":2871.547,"endTime":2874.18,"body":"\"MFCC\" is, as they're"},{"startTime":2871.547,"endTime":2874.18,"body":"called, are a measure"},{"startTime":2874.18,"endTime":2878.57,"body":"of how the spectrum of the audio changes"},{"startTime":2878.57,"endTime":2880.883,"body":"or what the shape of the spectrum is."},{"startTime":2881.78,"endTime":2884.98,"body":"And the way they measure that"},{"startTime":2881.78,"endTime":2884.98,"body":"shape, is rather interesting."},{"startTime":2884.98,"endTime":2888.06,"body":"They use, first of all, the"},{"startTime":2884.98,"endTime":2888.06,"body":"discrete Fourier transform"},{"startTime":2888.06,"endTime":2891.59,"body":"to compute estimates of"},{"startTime":2888.06,"endTime":2891.59,"body":"the spectrum in blocks."},{"startTime":2891.59,"endTime":2896.59,"body":"And then, they measure the"},{"startTime":2891.59,"endTime":2896.59,"body":"spectrum of the spectrum."},{"startTime":2897.35,"endTime":2900.85,"body":"So, they apply the Fourier"},{"startTime":2897.35,"endTime":2900.85,"body":"transform to the spectrum"},{"startTime":2900.85,"endTime":2901.96,"body":"to work out that shape"},{"startTime":2901.96,"endTime":2904.02,"body":"and it's the spectrum of the spectrum,"},{"startTime":2904.02,"endTime":2905.98,"body":"or cepstrum as it is often called."},{"startTime":2905.98,"endTime":2907.26,"body":"Cepstrum is a little joke,"},{"startTime":2907.26,"endTime":2910.08,"body":"where you interchange some of the letters"},{"startTime":2910.08,"endTime":2913.02,"body":"to indicate that its slightly"},{"startTime":2910.08,"endTime":2913.02,"body":"different from its spectrum,"},{"startTime":2913.02,"endTime":2914.27,"body":"spectrum of the spectrum,"},{"startTime":2915.897,"endTime":2919.81,"body":"to characterize the voice."},{"startTime":2919.81,"endTime":2921.83,"body":"One interesting things about"},{"startTime":2919.81,"endTime":2921.83,"body":"the speech recognition,"},{"startTime":2921.83,"endTime":2924.903,"body":"of course, is that it has"},{"startTime":2921.83,"endTime":2924.903,"body":"to happen continuously."},{"startTime":2926.571,"endTime":2929.69,"body":"It's no good to, sort of,"},{"startTime":2926.571,"endTime":2929.69,"body":"waiting a goodly amount of time,"},{"startTime":2929.69,"endTime":2930.523,"body":"computing everything."},{"startTime":2930.523,"endTime":2932.07,"body":"We have to do it now,"},{"startTime":2932.07,"endTime":2934.17,"body":"otherwise, there's too"},{"startTime":2932.07,"endTime":2934.17,"body":"much delay in the system."},{"startTime":2934.17,"endTime":2937.3,"body":"So, the speech engineers divide the speech"},{"startTime":2937.3,"endTime":2940.02,"body":"into little blocks,"},{"startTime":2937.3,"endTime":2940.02,"body":"usually, 20 milliseconds."},{"startTime":2940.02,"endTime":2944.52,"body":"They compute the \"DFT\" in a"},{"startTime":2940.02,"endTime":2944.52,"body":"block and then, they overlap."},{"startTime":2944.52,"endTime":2945.88,"body":"So, they wait 10 milliseconds"},{"startTime":2945.88,"endTime":2948.02,"body":"before computing another"},{"startTime":2945.88,"endTime":2948.02,"body":"block and another block"},{"startTime":2948.02,"endTime":2949.16,"body":"and another block and another block."},{"startTime":2949.16,"endTime":2950.59,"body":"So, there's always this overlap."},{"startTime":2950.59,"endTime":2953.32,"body":"They use windows that exactly overlap."},{"startTime":2953.32,"endTime":2956.4,"body":"So, in principle, you could"},{"startTime":2953.32,"endTime":2956.4,"body":"add them back together"},{"startTime":2956.4,"endTime":2959.02,"body":"and get the original waveform."},{"startTime":2959.02,"endTime":2961.28,"body":"So, they use this overlap idea"},{"startTime":2962.32,"endTime":2965.41,"body":"and that allows them"},{"startTime":2965.41,"endTime":2970.37,"body":"to either do recognition or speech coding."},{"startTime":2970.37,"endTime":2973.04,"body":"The overlapped transform,"},{"startTime":2973.04,"endTime":2976.47,"body":"that is particularly tuned to being able"},{"startTime":2976.47,"endTime":2978.64,"body":"to add them back together in an easy way,"},{"startTime":2978.64,"endTime":2982.167,"body":"is known as the modified \"DCT\"."},{"startTime":2983.155,"endTime":2985.71,"body":"So, health system, they"},{"startTime":2983.155,"endTime":2985.71,"body":"use for that or \"MDCT\","},{"startTime":2985.71,"endTime":2990.233,"body":"which again, is a form of"},{"startTime":2985.71,"endTime":2990.233,"body":"discrete Fourier transform."},{"startTime":2991.25,"endTime":2992.32,"body":"Digital TV."},{"startTime":2992.32,"endTime":2995.33,"body":"Well, we've talked about"},{"startTime":2992.32,"endTime":2995.33,"body":"digital TV in the coding,"},{"startTime":2995.33,"endTime":2998.93,"body":"but the \"FFT\" is also"},{"startTime":2995.33,"endTime":2998.93,"body":"used in the transmission."},{"startTime":2998.93,"endTime":3000.07,"body":"Because they use,"},{"startTime":3000.07,"endTime":3003.56,"body":"as does cellular telephone"},{"startTime":3000.07,"endTime":3003.56,"body":"transmission nowadays,"},{"startTime":3003.56,"endTime":3006.107,"body":"5G uses something called \"OFDM\","},{"startTime":3006.107,"endTime":3010.623,"body":"\"Orthogonal Frequency Domain"},{"startTime":3006.107,"endTime":3010.623,"body":"Multiplexing\" modulation,"},{"startTime":3012.93,"endTime":3014.933,"body":"yeah, modulation, I think."},{"startTime":3016.35,"endTime":3018.74,"body":"And the way that works,"},{"startTime":3018.74,"endTime":3021.21,"body":"is they say, \"Well,"},{"startTime":3018.74,"endTime":3021.21,"body":"it'd be very convenient"},{"startTime":3021.21,"endTime":3024.293,"body":"if the band, frequency"},{"startTime":3021.21,"endTime":3024.293,"body":"band we had available,"},{"startTime":3025.9,"endTime":3028.2,"body":"wasn't completely used by one person."},{"startTime":3028.2,"endTime":3032.51,"body":"So, we want to split it into"},{"startTime":3028.2,"endTime":3032.51,"body":"very precise little sub-bands"},{"startTime":3032.51,"endTime":3033.343,"body":"and we want to,"},{"startTime":3033.343,"endTime":3036.08,"body":"say, (indistinct) phone to"},{"startTime":3033.343,"endTime":3036.08,"body":"go down this little band"},{"startTime":3036.08,"endTime":3038.73,"body":"and Keith's phone to go down here"},{"startTime":3038.73,"endTime":3040.96,"body":"and Vierra's phone to go down this one."},{"startTime":3040.96,"endTime":3044.01,"body":"Right, but we want to be"},{"startTime":3040.96,"endTime":3044.01,"body":"able to do this adaptively."},{"startTime":3044.01,"endTime":3045.74,"body":"Because sometimes, we're very busy"},{"startTime":3045.74,"endTime":3047.22,"body":"and we want to restrict people's access"},{"startTime":3047.22,"endTime":3048.41,"body":"and sometimes, it's not so busy"},{"startTime":3048.41,"endTime":3049.74,"body":"and we don't mind having that."},{"startTime":3049.74,"endTime":3051.94,"body":"Is there a way of doing this?\""},{"startTime":3051.94,"endTime":3052.907,"body":"Well, what they do, is they say,"},{"startTime":3052.907,"endTime":3055.02,"body":"\"Right, well, we'd like"},{"startTime":3055.02,"endTime":3058.47,"body":"to put them onto a"},{"startTime":3055.02,"endTime":3058.47,"body":"little carrier waveforms"},{"startTime":3058.47,"endTime":3061.15,"body":"and we'll multiply those carrier waveforms"},{"startTime":3061.15,"endTime":3063.143,"body":"by the signal from, say, (indistinct).\""},{"startTime":3064.82,"endTime":3065.653,"body":"How can I do that"},{"startTime":3065.653,"endTime":3067.33,"body":"when I'm not quite sure"},{"startTime":3065.653,"endTime":3067.33,"body":"what the carriers are"},{"startTime":3067.33,"endTime":3068.73,"body":"and they're going to have to vary,"},{"startTime":3068.73,"endTime":3070.86,"body":"when I'm overseas, it's different."},{"startTime":3070.86,"endTime":3075.15,"body":"I know what I'll do, is"},{"startTime":3070.86,"endTime":3075.15,"body":"I'll use an inverse \"DFT\""},{"startTime":3075.15,"endTime":3078.16,"body":"to create all of these."},{"startTime":3078.16,"endTime":3081.31,"body":"So, I'll multiply the"},{"startTime":3078.16,"endTime":3081.31,"body":"signal by a sine wave"},{"startTime":3081.31,"endTime":3085.58,"body":"and then, I'll put them"},{"startTime":3081.31,"endTime":3085.58,"body":"into an inverse \"DFT\""},{"startTime":3085.58,"endTime":3087.42,"body":"and create a time waveform,"},{"startTime":3087.42,"endTime":3090.9,"body":"that is essentially the sum"},{"startTime":3087.42,"endTime":3090.9,"body":"of all these little channels."},{"startTime":3090.9,"endTime":3093.28,"body":"So, your mobile telephone,"},{"startTime":3093.28,"endTime":3097.26,"body":"is doing a discrete Fourier transform"},{"startTime":3097.26,"endTime":3098.85,"body":"as often as it can."},{"startTime":3098.85,"endTime":3102.43,"body":"In fact, one of the reasons"},{"startTime":3098.85,"endTime":3102.43,"body":"your mobile phone gets hot"},{"startTime":3102.43,"endTime":3103.53,"body":"and then, one of the reasons"},{"startTime":3103.53,"endTime":3105.78,"body":"why your digital TV gets hot,"},{"startTime":3105.78,"endTime":3107.98,"body":"is because there's a chip in there,"},{"startTime":3107.98,"endTime":3111.74,"body":"just hammering out the"},{"startTime":3107.98,"endTime":3111.74,"body":"discrete Fourier transform,"},{"startTime":3111.74,"endTime":3112.923,"body":"without halt."},{"startTime":3114.18,"endTime":3116.94,"body":"Okay, I haven't really got time"},{"startTime":3116.94,"endTime":3118.57,"body":"to talk about these other applications,"},{"startTime":3118.57,"endTime":3119.85,"body":"but there's an awful lot of them."},{"startTime":3119.85,"endTime":3121.66,"body":"But I just wanted to mention one"},{"startTime":3121.66,"endTime":3124.27,"body":"that well, once you get to"},{"startTime":3121.66,"endTime":3124.27,"body":"a certain venerable age,"},{"startTime":3124.27,"endTime":3127.23,"body":"you start to worry about various things,"},{"startTime":3127.23,"endTime":3128.78,"body":"including your health."},{"startTime":3128.78,"endTime":3130.95,"body":"And health is always popular,"},{"startTime":3130.95,"endTime":3134.3,"body":"so, I thought I'd quickly"},{"startTime":3130.95,"endTime":3134.3,"body":"talk about the connection"},{"startTime":3134.3,"endTime":3135.84,"body":"with one of these things."},{"startTime":3135.84,"endTime":3137.273,"body":"So, this is,"},{"startTime":3138.13,"endTime":3139.98,"body":"actually, I think I was"},{"startTime":3138.13,"endTime":3139.98,"body":"going to say it's a CT scan."},{"startTime":3139.98,"endTime":3142.24,"body":"I don't think it might be an \"MRI\"."},{"startTime":3142.24,"endTime":3143.34,"body":"Doesn't really matter."},{"startTime":3144.6,"endTime":3146.62,"body":"The idea behind these things,"},{"startTime":3146.62,"endTime":3151.06,"body":"is that on one side of"},{"startTime":3146.62,"endTime":3151.06,"body":"your head or your body,"},{"startTime":3151.06,"endTime":3153.16,"body":"is something that transmits something,"},{"startTime":3153.16,"endTime":3157.14,"body":"maybe some ionizing radiation"},{"startTime":3153.16,"endTime":3157.14,"body":"or non-ionizing radiation."},{"startTime":3157.14,"endTime":3159.68,"body":"And on the other side, we have a receiver"},{"startTime":3159.68,"endTime":3161.4,"body":"and they rotate around."},{"startTime":3161.4,"endTime":3164.65,"body":"You can hear them rotating"},{"startTime":3161.4,"endTime":3164.65,"body":"as you sit in these things."},{"startTime":3164.65,"endTime":3167.06,"body":"So, over on the right-hand side,"},{"startTime":3167.06,"endTime":3168.76,"body":"here, is the basic idea."},{"startTime":3168.76,"endTime":3170.86,"body":"We've got something that rotating"},{"startTime":3170.86,"endTime":3175.25,"body":"and what we measure is"},{"startTime":3170.86,"endTime":3175.25,"body":"the total transmission"},{"startTime":3175.25,"endTime":3176.89,"body":"through the body."},{"startTime":3176.89,"endTime":3179.75,"body":"So, we're not quite measuring,"},{"startTime":3179.75,"endTime":3181.73,"body":"you can't really do very much with this."},{"startTime":3181.73,"endTime":3183.73,"body":"This thing is usually called a sinogram."},{"startTime":3184.69,"endTime":3188.91,"body":"And this thing on the"},{"startTime":3184.69,"endTime":3188.91,"body":"right-hand side of the sinogram,"},{"startTime":3188.91,"endTime":3191.887,"body":"is actually another form"},{"startTime":3188.91,"endTime":3191.887,"body":"of integral transform"},{"startTime":3191.887,"endTime":3194.023,"body":"and it's called a radon transform."},{"startTime":3196.434,"endTime":3198.41,"body":"Of course, you've got this"},{"startTime":3196.434,"endTime":3198.41,"body":"thing on the right-hand side"},{"startTime":3198.41,"endTime":3201.67,"body":"and you would like this"},{"startTime":3198.41,"endTime":3201.67,"body":"lovely on the right,"},{"startTime":3201.67,"endTime":3205.06,"body":"you'd like to see into your"},{"startTime":3201.67,"endTime":3205.06,"body":"brain or see into your pancreas"},{"startTime":3205.06,"endTime":3207.27,"body":"or whatever it is you're imaging."},{"startTime":3207.27,"endTime":3212.05,"body":"And so, you have to solve this"},{"startTime":3207.27,"endTime":3212.05,"body":"inverse transform problem."},{"startTime":3212.05,"endTime":3214.92,"body":"I haven't talked much"},{"startTime":3212.05,"endTime":3214.92,"body":"about inverse transforms"},{"startTime":3214.92,"endTime":3215.753,"body":"and that's because,"},{"startTime":3215.753,"endTime":3217.53,"body":"the mathematics is a bit more complicated."},{"startTime":3217.53,"endTime":3222.41,"body":"But in order to solve the"},{"startTime":3217.53,"endTime":3222.41,"body":"inverse radon transform problem,"},{"startTime":3222.41,"endTime":3226.76,"body":"you usually use the"},{"startTime":3222.41,"endTime":3226.76,"body":"discrete Fourier transform."},{"startTime":3226.76,"endTime":3228.99,"body":"Yes, it even appears in medicine."},{"startTime":3228.99,"endTime":3231.087,"body":"So, you take the Fourier"},{"startTime":3228.99,"endTime":3231.087,"body":"transform of this sinogram"},{"startTime":3231.087,"endTime":3233.67,"body":"and then, you multiply it by wavenumber,"},{"startTime":3233.67,"endTime":3237.14,"body":"which is the spatial"},{"startTime":3233.67,"endTime":3237.14,"body":"equivalent of frequency,"},{"startTime":3237.14,"endTime":3238.85,"body":"take the inverse, \"FFT\""},{"startTime":3238.85,"endTime":3241.33,"body":"and that's when you get"},{"startTime":3238.85,"endTime":3241.33,"body":"a picture of your brain."},{"startTime":3241.33,"endTime":3244.82,"body":"So, the next time you're"},{"startTime":3241.33,"endTime":3244.82,"body":"lying in one of these things,"},{"startTime":3244.82,"endTime":3246.67,"body":"feeling a bit scared for your life"},{"startTime":3246.67,"endTime":3248.58,"body":"and wondering what the hell is going on,"},{"startTime":3248.58,"endTime":3253.03,"body":"you can engage the radiologist"},{"startTime":3248.58,"endTime":3253.03,"body":"in an entertaining repartee"},{"startTime":3253.03,"endTime":3256.4,"body":"about which Fourier transform"},{"startTime":3253.03,"endTime":3256.4,"body":"algorithm they're using"},{"startTime":3256.4,"endTime":3257.95,"body":"and the window that they used"},{"startTime":3257.95,"endTime":3259.89,"body":"in order to get it looking good or not"},{"startTime":3259.89,"endTime":3261.053,"body":"as the case maybe."},{"startTime":3262.75,"endTime":3266.093,"body":"Now, I can see time is pressing upon us."},{"startTime":3267.36,"endTime":3269.12,"body":"I do want to say one thing though,"},{"startTime":3269.12,"endTime":3274.12,"body":"which is the Fourier transform"},{"startTime":3269.12,"endTime":3274.12,"body":"is a rather interesting thing"},{"startTime":3274.51,"endTime":3277.68,"body":"and it's well understood"},{"startTime":3274.51,"endTime":3277.68,"body":"by physicists and engineers"},{"startTime":3277.68,"endTime":3279.3,"body":"on his part of the syllabus."},{"startTime":3279.3,"endTime":3282.15,"body":"It isn't part of the syllabus"},{"startTime":3279.3,"endTime":3282.15,"body":"for computer scientists"},{"startTime":3282.15,"endTime":3284.32,"body":"and programmers and software engineers."},{"startTime":3284.32,"endTime":3286.95,"body":"They may know it, they"},{"startTime":3284.32,"endTime":3286.95,"body":"may know something of it."},{"startTime":3286.95,"endTime":3288.44,"body":"But for a lot of people,"},{"startTime":3288.44,"endTime":3291.55,"body":"the understanding is"},{"startTime":3288.44,"endTime":3291.55,"body":"probably a little bit less"},{"startTime":3291.55,"endTime":3292.6,"body":"than a black box."},{"startTime":3292.6,"endTime":3294.57,"body":"It's a thing you put numbers into"},{"startTime":3294.57,"endTime":3296.58,"body":"and out comes some other numbers."},{"startTime":3296.58,"endTime":3300.48,"body":"So, that's a bit of a concern to me."},{"startTime":3300.48,"endTime":3302.46,"body":"Because it's far more than that"},{"startTime":3302.46,"endTime":3305.18,"body":"and, in fact, there are"},{"startTime":3302.46,"endTime":3305.18,"body":"simple tests you can do"},{"startTime":3305.18,"endTime":3306.22,"body":"to see that it works"},{"startTime":3306.22,"endTime":3307.66,"body":"and we've talked about some of them here."},{"startTime":3307.66,"endTime":3311.66,"body":"If we put in, say, a spike"},{"startTime":3311.66,"endTime":3312.53,"body":"then, we should expect"},{"startTime":3312.53,"endTime":3314.22,"body":"to see all the numbers come out equal."},{"startTime":3314.22,"endTime":3315.51,"body":"If we put in a cosine wave,"},{"startTime":3315.51,"endTime":3317.837,"body":"we should expect to see"},{"startTime":3315.51,"endTime":3317.837,"body":"spikes come out and so on."},{"startTime":3317.837,"endTime":3320.1,"body":"And it is quite easy to be intuitive"},{"startTime":3320.1,"endTime":3322.34,"body":"about what these things are working"},{"startTime":3322.34,"endTime":3324.35,"body":"and whether they're working or not."},{"startTime":3324.35,"endTime":3326.7,"body":"And that's an increasing"},{"startTime":3324.35,"endTime":3326.7,"body":"issue for modern computers"},{"startTime":3326.7,"endTime":3328.54,"body":"that bothers me a bit"},{"startTime":3328.54,"endTime":3331.22,"body":"that there's a sort of black-box approach"},{"startTime":3331.22,"endTime":3334.21,"body":"to design and understanding,"},{"startTime":3334.21,"endTime":3336.97,"body":"which means it's difficult to be confident"},{"startTime":3336.97,"endTime":3338.86,"body":"that the system you're using,"},{"startTime":3338.86,"endTime":3341.227,"body":"is, in fact, doing what"},{"startTime":3338.86,"endTime":3341.227,"body":"it's meant to be doing."},{"startTime":3341.227,"endTime":3344.58,"body":"And Fourier transform is"},{"startTime":3341.227,"endTime":3344.58,"body":"perhaps the most low-level box,"},{"startTime":3344.58,"endTime":3345.87,"body":"I could think of."},{"startTime":3345.87,"endTime":3348.2,"body":"It's probably the lowest-level box"},{"startTime":3348.2,"endTime":3350.09,"body":"that isn't understood by most people"},{"startTime":3350.09,"endTime":3351.93,"body":"that I could think of."},{"startTime":3351.93,"endTime":3353.41,"body":"But there are others."},{"startTime":3353.41,"endTime":3357.23,"body":"And the others, you"},{"startTime":3353.41,"endTime":3357.23,"body":"could, sort of, wrap up"},{"startTime":3357.23,"endTime":3360.92,"body":"into that interface between the real world"},{"startTime":3360.92,"endTime":3364.97,"body":"and the thing that you're"},{"startTime":3360.92,"endTime":3364.97,"body":"doing on the real world."},{"startTime":3364.97,"endTime":3367.57,"body":"And that interface with the real world,"},{"startTime":3367.57,"endTime":3369.02,"body":"in a computer system,"},{"startTime":3369.02,"endTime":3372.35,"body":"is usually called an operating system."},{"startTime":3372.35,"endTime":3376.96,"body":"And so, that is the"},{"startTime":3372.35,"endTime":3376.96,"body":"topic of the next lecture"},{"startTime":3376.96,"endTime":3380.92,"body":"and in fact, will be the"},{"startTime":3376.96,"endTime":3380.92,"body":"topic of my final lecture."},{"startTime":3380.92,"endTime":3381.753,"body":"Thank you."},{"startTime":3382.937,"endTime":3385.937,"body":"(audience applauds)"},{"startTime":3388.79,"endTime":3390.23,"body":"- Thanks so much, Professor Harvey."},{"startTime":3390.23,"endTime":3393.69,"body":"We've got a couple of questions"},{"startTime":3390.23,"endTime":3393.69,"body":"from the online audience,"},{"startTime":3393.69,"endTime":3395.0,"body":"which I was going to start with"},{"startTime":3395.0,"endTime":3398.47,"body":"and then, we can move on to"},{"startTime":3395.0,"endTime":3398.47,"body":"the audience in the room."},{"startTime":3398.47,"endTime":3400.107,"body":"The first question is,"},{"startTime":3400.107,"endTime":3403.677,"body":"\"Why is the length of the"},{"startTime":3400.107,"endTime":3403.677,"body":"FFT always a power of two?\""},{"startTime":3405.46,"endTime":3406.789,"body":"- It's not."},{"startTime":3406.789,"endTime":3407.622,"body":"(thuds)"},{"startTime":3407.622,"endTime":3410.36,"body":"So, it always used to be,"},{"startTime":3410.36,"endTime":3411.79,"body":"because there was only one algorithm"},{"startTime":3411.79,"endTime":3413.59,"body":"for computing the \"FFT\" quickly"},{"startTime":3413.59,"endTime":3415.27,"body":"and it was called the"},{"startTime":3413.59,"endTime":3415.27,"body":"Cooley-Tukey algorithm"},{"startTime":3415.27,"endTime":3417.18,"body":"and it worked in powers of two,"},{"startTime":3417.18,"endTime":3419.15,"body":"worked on lengths of"},{"startTime":3417.18,"endTime":3419.15,"body":"signals of powers of two."},{"startTime":3419.15,"endTime":3420.76,"body":"Nowadays, there are"},{"startTime":3419.15,"endTime":3420.76,"body":"plenty of other choices,"},{"startTime":3420.76,"endTime":3422.65,"body":"prime number transforms and so on."},{"startTime":3422.65,"endTime":3424.13,"body":"And there's a whole"},{"startTime":3422.65,"endTime":3424.13,"body":"books of these choices."},{"startTime":3424.13,"endTime":3425.89,"body":"So, it doesn't need to be power of two."},{"startTime":3425.89,"endTime":3428.85,"body":"So, it's pure mystique"},{"startTime":3425.89,"endTime":3428.85,"body":"that it is a power of two."},{"startTime":3428.85,"endTime":3431.327,"body":"And perhaps, what the"},{"startTime":3428.85,"endTime":3431.327,"body":"question really means is,"},{"startTime":3431.327,"endTime":3434.18,"body":"\"Why the hell did you pick a power of two"},{"startTime":3434.18,"endTime":3435.17,"body":"to demonstrate it?\""},{"startTime":3435.17,"endTime":3436.3,"body":"And I don't know the answer to that."},{"startTime":3436.3,"endTime":3438.045,"body":"I should have picked something"},{"startTime":3436.3,"endTime":3438.045,"body":"that wasn't a power of two."},{"startTime":3438.045,"endTime":3438.878,"body":"(claps)"},{"startTime":3438.878,"endTime":3439.711,"body":"- Second question from online,"},{"startTime":3439.711,"endTime":3441.56,"body":"\"What is a spectrogram?\""},{"startTime":3441.56,"endTime":3442.393,"body":"- Ah, spectrogram."},{"startTime":3442.393,"endTime":3444.58,"body":"So, the (indistinct)"},{"startTime":3442.393,"endTime":3444.58,"body":"illustrated this lecture"},{"startTime":3444.58,"endTime":3446.343,"body":"with a spectrogram, actually."},{"startTime":3447.234,"endTime":3451.07,"body":"And a spectrogram is a set of \"DFT\","},{"startTime":3451.07,"endTime":3453.05,"body":"taken one after the another"},{"startTime":3453.05,"endTime":3455.19,"body":"and displayed using pseudo colors."},{"startTime":3455.19,"endTime":3457.54,"body":"So, they're very popular"},{"startTime":3455.19,"endTime":3457.54,"body":"with speech engineers."},{"startTime":3462.09,"endTime":3463.76,"body":"A friend of mine can"},{"startTime":3462.09,"endTime":3463.76,"body":"look at the spectrogram"},{"startTime":3463.76,"endTime":3464.98,"body":"and know what was said,"},{"startTime":3464.98,"endTime":3466.473,"body":"which is a wonderful trick."},{"startTime":3467.888,"endTime":3469.76,"body":"If you'd like to know"},{"startTime":3467.888,"endTime":3469.76,"body":"more about spectrograms,"},{"startTime":3469.76,"endTime":3471.21,"body":"then, there's an online lecture from me"},{"startTime":3471.21,"endTime":3475.2,"body":"on speech recognition called"},{"startTime":3471.21,"endTime":3475.2,"body":"\"How to recognize speech?\""},{"startTime":3475.2,"endTime":3477.05,"body":"- Thanks so much, Professor Harvey"},{"startTime":3477.05,"endTime":3479.38,"body":"and please do come to the next lecture"},{"startTime":3479.38,"endTime":3482.63,"body":"on the 31st of May, 6"},{"startTime":3479.38,"endTime":3482.63,"body":"o'clock on operating systems."},{"startTime":3482.63,"endTime":3483.84,"body":"Thank you so much."},{"startTime":3483.84,"endTime":3484.848,"body":"- Thanks."},{"startTime":3484.848,"endTime":3487.848,"body":"(audience applauds)"}]}