Welcome to the talk, Manipulating Signals, Designing Filters for Signals and Voltages Yourself by Jakob Lerch. This talk is being translated into English for you by Kalluera and Isegrim. Any feedback is appreciated. Please use the hashtag C3Lingo on Twitter or mustered on We hope you enjoyed the talk. Welcome to this topic will be a little more unusual. We are looking into the analog circuit technology. We are looking to annoy a neighbor away from the night neighbor to a filter which reduces these bass signal noises. So there will be a Q&A in the end and if you have questions please ask them at the end and on Twitter or mustered on under the hashtag Rc3Fem or in the IRC or on the rocket chain. Please use the translated audio stream in the web player. Hello, my name is Jakob and now have fun with the talk. Hello, my name is Jakob. I am studying engineer computer science at the Uni Ilmenau and I want to talk about how signals are manipulated and at first what are signals and what is manipulating them. So let's start. Imagine you are having a party and your neighbor is really annoyed because it is at loud and complains and you should turn off the music because otherwise you will call the police and you don't want that but you also don't really want to stop completely. And what would be nice if these deep frequencies which you may be hearing when your neighbors are having a party that is what you can hear especially. So an idea would be if the low frequencies would be filtered out then the neighbors would be less annoyed. It might not be that nice but it's better than turning the music off. So what do we do? We have to put something between the microphone and the speaker so it's a signal. So when we are doing that the neighbor is happy and we are also happy and we can continue our party. That's the aim of this talk that we want to know how the signal looks and how can we manipulate it. So at first I will talk about what is a signal and special audio signal, where the audio signal is a motivation then I will talk about how you manipulate signals and then I will show some circuits which you can use, which you can use in different dimensions with different behaviors or filters and you can set them up and use them. So a small repetition, what is a voltage? It's work that has to be done to move charges from one point to another in our circuit. So I don't want to talk about the physics deeply but we have to know we get a voltage from this voltage source and then it will go down over the rest of the circuit. And some notation, this and you see sometimes this ground symbol on circuits and that is a reference point where the voltage is always zero. If you measure the point from there to there, from that point to that point then it's always zero and the other voltages are also referenced to that point and that's helped with the analogization. If you say the voltage at that point has that and that voltage and then you have it for the difference between these two points and sometimes if it gets more complicated you can leave the blue line out and then everywhere where you have this ground symbol always painted in but there are two different symbols, two different points in the drawing but they are the same point so we can measure the voltage over a component or between a point to ground. And another thing, so you have seen I used before the left one with the line in the middle which goes through, that symbol, I will use the right one here, then the voltage doesn't change but that's easy and I will use the right one and in this talk it's a signal source so the voltage changes often and the voltage of our time is something we can show in a diagram and that's our signal and that's then what we use to transmit information. So a very easy signal is the sign. If you measure the voltage there then it would be changed like this, it would start at zero, go up to one, fall back down to minus one, go back up to one and that's periodical and I prepared something which would look like this. You would have a circuit like this and we see the voltage here in the colors which changes and in the diagram you also see the value from up there to ground. The sinus frequencies have very different rates and it can have a different frequency or it can have a high or low amplitude and you can do some nice things with it and another signal here, it's a square wave so it's like technical informatics so it's for viewers and ones to code them and I don't really mean that here, we don't interpret the signal, we don't interpret the signal and we are just looking how it looks and the other part we are not doing in this context and I could have a triangle wave or I just chose a rectangular wave. So I also drawn this in the simulation here so we see that the voltage changes immediately and so we get the square wave here. Now it's getting interesting, every signal we encounter here typically can be shown as the sum of different sinus waves in technically as a sum of infinite number of sinus waves. I've plotted this here so the blue signal is the, it's stopped after two iterations, two summations, the yellow one after six and the green one after 34 addons and the red one very close to the actual rectangular wave, this is after 258 sine waves, summing up 58 sine waves. So we can basically do this for any kind of wave, sort of waves for triangle waves and so if we look at that we see that we do the plot versus the frequency here which sine waves can be found at which frequency and with this simple example of a sine wave we know that it only consists of one sine wave here so here with 200 hertz and the amplitude of one world. So with a square wave we can create this by summing up lots of different sine waves and this is plotted here. So here we see that we see a very low frequency sine wave with an amplitude of a bit more than 1.2 volts and that's what you see here and you see that the total amplitude is a bit more than 1.2 volts and we can do cool things with that because we can now construct filters that attenuate certain sine waves and amplify other sine waves here. So this diagram here is what we call the spectrum of a signal, this is a different way of showing the same signal but it contains the same information as looking at the time at the time axis here. So that's the basics. So how do we manipulate signals and what do I mean with that? So what we've learned that a signal consists of all these different sine waves here and then I'll play some music that's lots of sine waves being added up and there are low ones among them and especially the low ones are those that travel through walls and so the low ones are the. So in our case we want to filter out the low frequency sine waves from that one. So how do we do this? So as I said we put something in between here but before we want to have a look at how this signal could look like. So I've put a very simple version of that, technically this microphone cable is a lot more complicated. So basically we have a signal source here and some signal receiver and between that we have a simple cable, simple tap wires. And so here in this case we have one wire that's the ground and another wire that contains the signal. So if you measure the voltage between the red one and the blue one you get a voltage signal that contains information for example the sound. So this can have multiple that are shown wet. So for example if you have two you could get a stereo signal through that but right now we are only looking at the simple source. So between so we cut this open and between that we put some electric circuit. So this circuit is what we call an LTE filter. So what's that? An LTE filter is something much more general. It's some electrical circuit taking an input signal here as you in. That's a voltage. We see at these two connections here and outputs a manipulated value out. So I'm just you out here. LTE meaning linear time invariant. I'm going to show what linear means later on but let's show later how this works here. So we putting one signal in and another signal gets out here. So in this case it's an attenuated and you could get other signals that just passes through so factor of one or you could get signals in and which gets amplified here. So for example you could put time waves in that gets amplified here. So which one are and which are not depends on the frequency. So it could be that well it also depends on the system. So there could be systems that so some things we have a high frequency, some have a low frequency. We have high passes which do not filter out the high pass that has low passes. So there are two more words here. So two more ways. This is the homogeneity and the additivity. So the homogeneity meaning if I scale up the input signal then the output signal gets amplified by the same factor. So we have the input signal has the same frequency as on the bottom then above but just a higher amplitude then the output signal is also as the same frequency but different amplitude. The other interesting thing is the sum. So if I add two input signals or I show this here then the output is also the sum of the answers of these two systems. So we can't even just put a sinus waves in but we can also put square waves in or anything else. That's just that you can create by summing up sinus waves. So what we also want to describe how these this kind of LTE systems behave. So there are some low so we know that there are low passes and high passes but there are quite a couple of others that I want to show here. So we can describe this by the transfer function. This is the amplification factor depending on the frequency. And omega is the circular frequency here which is the frequency of the sine wave multiplied by 2 pi. And if the input is a sine wave with one frequency it gets a different amplification than the one as from another frequency. So we can plot this function over the frequency and what we see here is that low frequency gets suppressed and with higher frequency gets amplified or we just push through. So we have one problem. We want for a high pass or low pass filter we want to have to say exactly up until that frequency it let's pass or it amplifies or it attenuates. So this graph doesn't have corners so we don't really have can see a critical frequency here. So what we are doing here we are drawing the asymptotes the straight lines which go closely to the function in the infinity which I show here and where they overlap and then we are drawing a line down where that's our critical frequency which is one for us which I made it easy for myself and our transmission function comes here without a derivation. And it's j omega over 1 over omega j omega plus 1 and we are not going to derivate it here. It's important that it looks like this and later we can vary omega g with in our circuits and later we are just having simple equations where we put our values in and then we get our transmission function and we don't have to think about a transmission function but I want to show you so you know where it comes from. And there are also different filters which have different transmission functions. Here it's a high pass and there are also a band pass which attenuates high and low frequency and just lets the frequency in between pass or amplifies it. And this filters can also be implemented in electric circuits and in the following I want to show you the low pass and the band pass and the very easy filter the amplifier and here you see a low pass filter and it lets low frequencies pass and low frequencies get attenuated. Here you see the asymptones again and here is again the critical frequency and the transmission function is similar as before. It's 1 over 1 over omega g j omega plus 1 and here is the band pass filter. It attenuates low frequencies and attenuates high frequencies and in between it lets the frequencies pass and we are drawing the asymptotes here and then we are seeing the critical frequencies here and we have 2 critical frequencies omega 1 and omega 2 and in reality if we are working with actual audio signals they are actually higher than here but here it's only about the concept and our transmission function is j omega over 1 over omega 1 j omega plus 1 times 1 over omega 2 j omega plus 1 and you can imagine that you put a high pass and a low pass filter just you put them after each other and the high pass attenuates the low frequencies and the low pass attenuates the low frequencies and everything in between gets passed and the amplifier is a really easy filter and it amplifies all the amplitudes independent of the frequency so there is no critical frequency and it has an amplifying factor of k in this time. Here it was 10 and so what did we learn? Well we know we have different ways how to look at voltages which is especially important when I show the images of the circuits where you will use this ground symbol and we have signals and we can show our signals as a sum of assigned frequencies which have a different amplitude for each frequency and then if you add them out if you get that signal and with LTE I filters you can amplify or attenuate these frequencies and the temp depends on the frequency of these so our old goal was to filter a signal and our new goal is to build a circuit which follows this transmission function so we are designing filters and at first I want to show you two components we know the resistance the resistance follows the equation u equals r times i so the voltage over the resistor is the value which is written on it and so if you have a capacity then it's like an ideal capacity and then at first we add a capacity and then we don't just write uc equals c times ic but we write run over j omega c times ic and raise the s dot because we're calculating with imaginary complex numbers and I don't have the time to go into that and that's where the j omega comes from and we will see it later in the transmission function and that's where we get the transmission function from so after the r we have here 1 over omega c and when we have a resistor and a capacity serially we don't have r plus c but r plus 1 over j omega c and another thing is the operational amplifier where we have two inputs one inverting and one non-inverting input which are quite like that we have also a voltage plus it's an example with plus v and minus plus v the minus plus and once we are also connecting it with ground and that's where our signal comes out and then after it depends a little bit how you're doing it and it works a little bit differently but we are just using it once and then it works like we wanted and it helps us filtering signals and I will show it to you in the circuits I'm going to show all the parts which are crossed out here will not be shown but in practice they are there so this makes it a lot easier to actually draw this one here so in reality if you use such an operational amplifier you are going to need these connections so this means that you have a look at a certain number of things but I will mention this later on so in a typical OP that you can order your electronic parts would be an LM358 so if you find this any others have a look at the datasheet and see whether they fit your needs so what we need here to design a filter is the inverting OP circuit which looks at fault so we have the OP here we have the signal source here this signal is delivering the input voltage going into this resistor here in the first branch here which is connected to the inverting input of the OP the non-inverting input is connected to the ground here and the output is connected to R2 with the second resistor which is also connected to the inverting input here so we are picking up the output voltage here which is our output signal here so if we want to actually wire this one so we are putting a wire here one wire here and to ground we put this into our cable and that's our filter this filter here as shown here is just an amplifier so it doesn't filter anything yet but later we are going to show filter circuits that do filter because we place this resistor here with either serial or parallel connections of capacitors so for each of the circuits by now I'll derive the transfer function also I'll derive the amplification factor in the non-anerated area and the cut-off frequency so in depend of what the factor is so the amplifier looks like that so the transfer function is minus R2 divided by R1 which this is because if you analyze this one but I'm not going to do that but let me just sketch this here so you write a minus you write a fraction and on the top you get the resistor coming from the output and on the denominator you get the resistor from the input so let me show this here so this one here is an amplifier we see a 3 kilo ohm here and 1 kilo ohm here so the amplification factor is minus 3 here we see this here, 5 volts input and the output is 50 volts output here so it works so let's have a look at our high pass here so high pass has in the first branch here a serial circuit from a capacitor and a resistor and in this branch here it's just a resistor so this is the transfer function the amplification factor is again R2 divided by R1 and the kind of frequency see the formula here so I will look at this one here so this is at 16 hertz so let me turn up the frequency so we see the input signal is 5 volts and there's almost no attenuation here so almost 5 volts here but this is a high pass so high frequencies are not being attenuated but low frequencies will be attenuated so let's turn down the frequency here we see that the amplitude gets smaller so the amplitude is just about a little less than 1 volt here or even though the input is 1 volt here and that's our high pass here so if you want to build if you want to design a high pass you just take this circuit here you have to commit to one of these components so for example if you have a 1 kilo ohm resistor you set R1 to 1 kilo ohm and then you design which amplification factor you want to have so let's say it's supposed to be 1 so R2 needs to be 1 kilo ohm as well then you write this down and then you do some simple math there and you might be getting some so if you have a capacitor like that you can start with that if you don't and have some others then you maybe have to tweak the other elements as well to get to the same one so how do we get this formula here for the strands of function this one here so with the inverting amplifier we have the feedback branch divided by input branch so the feedback branch here is R2 and the first branch the input branch is a series of goods from R1 plus the 1 divided by j omega c1 so now we do some magic math here multiplying by a 1 so we can move things around here so that's the formula that we get here so we are doing something similar with the low pass as well so we see that we remember up to the kind of frequency we let everything through and above that it's just a tenuous and that's how this looks like here so the feedback branch we have a parallel circuit from a capacitor and a resistor and the input branch we have just a resistor and that's the formula we get from that same amplification factor same again here and that's the formula we get from that same amplification factor same again here and kind of frequency as well so let's have a look at the simulation here so again that's a low pass meaning low frequencies are not getting attenuated so the input signal 5 volt, output signal 4.8 and now we turn up the frequency so if we increase the frequency the amplitude should get smaller and we see that this happens actually happens so our low pass also works so again depending on which frequency you have to modify these three components here so the amplification function can again derive like that so this is the feedback branch and this is the input branch same formula here so the parallel circuit looks like that and then again lots of math a little bit of math and we're getting to the same formula here and that's the transfer function that we get here so now we are having the band pass but we're just a bit lazy we have a low pass and we have a high pass so we plug a band pass, a high pass and a low pass and then we have here these critical frequencies 10 to the power of 0 and 10 to the power of 2 then we are taking a high pass filter with the critical frequency of 10 to the power of 0 and we have lower frequencies stamped and our high pass, our low pass dumps everything above that critical frequency so we just plug our two circuits we had before together we have our amplification factor which is just the both of these amplification factors multiplied for each other and the critical frequencies are those of these thermal circuits and when we're looking at it here we have a very low frequency here and then we're seeing it's 4 hertz and then we have now we are making the frequency a bit higher that was too much so now we make it higher the signal isn't as damped as before and we are in the area where the signal gets passed through and when we increase the frequency again we are seeing again that the amplitude of the signal is damped and that's exactly what we want and that's what succeeded with plugging two different circuits after each other and let's think what was the most important we have a signal which consists of a sum of multiple sinus waves depending on their frequencies they have different amplitudes then we can design circuits which work like LTI systems and which amplify certain frequencies or damp on others and then we have a body diagram which shows that behavior and this filtering behavior we can do in a circuit and there are really many different ones with different critical behaviors it's a really easy design which we have here but I think it's for trying around it's really nice and there are some things we shouldn't not talk about a real operational amplifier it's not ideal we have two things which might be a bit annoying so it starts to damp at a certain frequency and you can see it in its data sheet and it's damps and everything and the voltage which goes out and you see it needs a voltage to work and the output voltage can't be higher than that voltage so if you have an input signal of five volts we want to amplify it by the factor of 10 so we should get 50 volts but if we have a real one which has five volts supply voltage and then our signal doesn't get higher than five volts and we get rectangular waves that might be nice for some overdrive effects but if you want to send some signals it's not really nice that was everything about the op amp and if you want to talk directly to a speaker you probably doesn't work with that one and there are other amplifiers which you can use for that with power amplifiers which you put after the op amp and then the power is amplified and then you can hear it through a speaker and in the last slides I put some interesting links in there and there's also a YouTube video linked of someone who did that and if you're thinking about the health of your signal source because it might break then you should put a non inverted amplifier in there and you can look it up on the internet it's not more complicated than what we've done here and there is some current flowing back if you have something non inverting amplifier then there won't be any current flowing but the voltage is still transmitted and the signal source is protected the images of the circuits were generated with false.com that's nice for small projects and how it looks and there are other tools and if you want to do it more professional there is LT Spice which is an analysis tool which does something similar as false. but it's more efficient and it's not a program on the web and it works really well and the Python library is also with Python control you can plot plots and there are also other libraries but it doesn't really restrict you on your transmission functions and false.com I already said and paySDR is something more about signal manipulation how does it look, how can you control it LearnElectronics is something on YouTube for trying around with this and there are some really nice videos and this video is something how you do it and here is some more literature some more links, some hard tools some calculating helps for high and low pass and I just have to say to you very much fun there will be a round of questions later and you can ask questions and have fun thank you very much Jakob for this really interesting talk so I never learned something learned that much about analog technique so thank you I could learn a lot now we are going to the questions and answers and for those of you who have questions and hadn't posted them somewhere please write them into the IRC hack-end into the kanal AC3-FAM into the AC3 rocket-chip into the channel FAM or on Twitter or in the FIDI-verse the hashtag AC3FAM without the dash let's start with the questions we have a question which was supposed to be what does a Wiener filter has to do with a filter? okay so look at Wikipedia before that one there is a Wiener bridge it's a certain kind of electrical circuit and it can do something with electrical filters there but I don't really know in my university courses it was always in the context of we would like to know something and we use these kind of bridges but you can certainly use this for filters as well is it like a bridge or circuit? is it something for my... I use something for my study IR bridges and then you have for resistors yes exactly that's what this is and you can use this with filters as well or I don't know any examples but we can probably do that well next question the... about the slide 56 what do you recommend against Lippic? so I hope you know the amplitude that's coming from the source so for example you know that's about 10 volts or whatever you have here there and then you need to make sure that the supply voltage for the OOP is... well choose that so you know the input might be 5 volts and we say that we supply it with plus minus 5 volts or maybe a bit more so to have some buffer there some reserve there next question looks... that's not a question that's just a comment here was the wish that you would explain the bandwidth for the bandpass so bandwidth is so the bandpass has two critical frequencies and the upper frequency f2 minus the lower frequency f1 and this is the band with then so the difference between the two critical frequencies so that could be the definition so next question how much would it... how useful would it be to decouple the low noises from the walls and the ground and maybe let it hang from the... from the... I don't know much about that that's not my expertise here so the thing with the microphone and karaoke this is more like motivation for this talk here so we have an audio signal we want to filter that and that's where I jumped right into the discussion but if there's more about... about PA technique then you would have to look into that then there's a question about the images of the circuits and why you have another dot on one of the ground I think it was the slide where there was additional ground in there which just went to a dot which didn't go on further I think you meant the slide where there were several ground cymbals and as I said in the talk so if you have some standard circuit diagram like you know from school without these cymbals here so you could consider one point there to be your reference and that's where you draw the ground cymbal there and you can also draw this on other points as well and if you do that you say that these points are connected meaning the same point there and this makes the circuit... this symbolifies the circuit this representation of the circuit yeah there's another question why don't you do it directly in software why would you do it as a physical circuit? yeah cause it's fun well I thought this talk would make a lot of sense for people who sold a lot who would like to create physical things and that's why I started this with analog filters so if you want to do this with digital filters go ahead can you do that? yeah another question from locally about the high pass filter there we had directly after the input signal there was a capacity and a resistor and the resistor which was in the feedback branch it has the same value as the first one which would lead to an amplification of one can you not just leave the op amp away that doesn't the capacity do everything here now? it can be that you still want to amplify it so for example you could put a variable resistor there which you can then use to increase or lower the amplification factor turn the volume to 11 and there could be some other effects in there but I would have to check here so there are passive high passes or passive low passes which is just combining a capacitor and a resistor but they have a couple of downsides so we would have to check the literature for that there is a really new question now here and do you do it just for fun or was a part of your studies? okay so this was part of my university courses I thought this is a topic that's cool and that's easy to create some slides here but there's not that much media for that so basically the knowledge I presented here is coming from several different courses there and I just wanted to give you the some way for building photos here so it's from my university courses so we have a small comment here in this list somebody wants to recommend the YouTube channel sound file about acoustics in rooms and there was something on the ERC which frequencies do you want to attenuate there was a discussion about that and there was a comment that it depends how the room actually looked like yeah then at last another question in this list two questions what would be necessary to choose the frequency when the filter attenuates a lot you have some knob you can turn around where you can choose so what I meant was so you have the OP filter and the critical frequency of that one this only depends on the actual component if you want to have a higher critical frequency of the OP you need a different one higher cutoff frequency so it means you can't put it very wait let me check the question I got the question wrong it's about critical frequency not cutoff frequency so you have a look at the formula FP is this by that so you take one of these two components either the C or the R and it makes it a variable so for example there are variable resistors but there are also variable capacitors so we can change the capacity there and if you take one of those two as the variable variant then you can change them but you have to be careful of course that you don't change the application factor at the same time so you have to this can become compensated another question which I also asked myself why do I use the inverting input and not the non-inverting input why do we have to invert the signal so two basic circuits on how you can use the OPV maybe a bit more but two are very important so there's the inverting circuit and the non-inverting one the non-inverting looks a bit different so the feedback also goes to the negative one the voltage source is different so the reason why we're using this one here is that the transfer function the formula for that one gets relatively simple so we can realize this function in a rather simple way because you just basically you just divide the feedback loop by the input loop the feedback branch by the input branch for the other so it's just to don't have too hard calculation so with this simple filters it's just three components two resistors, one capacitor and that's it with one of these basic circuits and that's he yeah there is one question do you there are some thoughts about analog computers do we have some? okay so I've heard about analog computers only from some fellow students of mine so they I have never worked with those I haven't done with anything but I know that with OP you can't implement mathematical operations so there's the non-inverting and the inverting circuit but there are addition or subtraction circuits with those so you take the so you have two inputs and you either subtract or add these two and you can use this I think you can use this for analog computers as well I have never really thought about those in detail okay yeah otherwise I see no more questions in the list and the IRC there's also nothing more then I want to say thank you very much for this for the answering the questions and for this really interesting talk yeah and that was it for today that was generally the last talk on the FAM channel thank you all those thanks to all the listeners to the whole audience who are looking and watching so there's only the C3 news show at midnight so we will see tomorrow the other news shows and until then thank you very much for watching thank you for your attention also from the translation booth you have just heard the talk designing filters for voltage signals and it was translated by Isogrim and Kahluera if you have feedback for us please use the hashtag C3lingo on Twitter or on Mastodon


So in my last lecture I discussed about Hilbert matrix and approximating a function using a polynomial function mth order polynomial function and what we saw that we get this Hilbert matrix which for large order polynomial is highly ill conditioned, difficult to invert, well the problem is not different if I just passingly mentioned, yesterday we saw that we had this function ft which belong to set of continuous functions and then maybe we took some specific thing here 0 to 2 pi and then ut which is function ut which is this and then we had this approximation and then we wrote the normal equation and said that particularly when you pick this to be a polynomial that is when you pick this to be alpha 0 plus alpha 1 te plus alpha 2 e square for this particular case I showed that you end up into this Hilbert matrix and this Hilbert matrix is trouble inverting so in principle that you get an equation you cannot you cannot solve it properly because of ill conditioning. Now ill conditioning of a matrix will formally define little later, right now all that I would tell you is that the Eigen values of this matrix or singular values of this matrix to be very precise, singular values or Eigen values of A transpose A, Eigen values or singular values of this matrix the smallest one and the largest one are far apart and that causes problems and computations. Now this is when you are trying to approximate the entire function suppose you know this function instead of the knowing the function knowing the entire function what it is and trying to approximate it using a polynomial suppose you know this function values you do not know the function but that is a more practical problem in engineering you just know function values at different points so t1 this is t2 and so on so tk and in general you know value of this function at several points so these values will be u1, u2, uk, un okay u2, uk, un and then you know at each point you will write this equation that is approximation or ut, uti is equal to alpha 0 plus alpha 1 ti plus up to alpha m ti to power m you get these equations uti is nothing but ui this is nothing but ui so uti, ui okay this plus error i okay and i going from 1 to up to n you have these equations we have done this earlier okay and we get this matrix A matrix so we will get this huge A matrix so this will be 1, 1, 1, t1, t2, tn up to t1 raise to m, tn raise to m and alpha 1 alpha 0 alpha 1 alpha m is equal to u1, u2, un in this case you have this equation you are trying to fit a lower order polynomial say tenth order polynomial and this n here number of data points is let us say 1000 so you have to put a tenth order polynomial in thousand points okay you have lot you know this function value at large number of points simple example cp versus temperature okay i know cp values at many, many temperatures in some range i am trying to fit i am trying to correlate i will get something like this there t will be temperature okay in that case and then u1, u2 will be cp values and of course you have this error coming up here so this error 1, error 2, error n this vector will add here so this is my A matrix this is my A matrix okay so A matrix and we have solved this problem we know that the solution the least square solution the least square solution is given by the least square solution that is alpha least square is A transpose A or i will write it in a little different way so this is A transpose A alpha least square i think we named this as theta right in the earlier this thing we have called as theta so let us keep calling it theta so theta least square theta least square is equal to A transpose u, u is all the values of dependent variable which we know okay we know this now this as i told you is nothing but the Hilbert matrix this is nothing but the Hilbert matrix so this actually what you can show that this matrix this matrix in this normal equation will tend to Hilbert matrix this is not Hilbert matrix in the earlier case it we directly got the Hilbert matrix if you have large number of data points okay what you can show is that this matrix this matrix the proof is just given here i am not going to write it down on the board this matrix will tend to Hilbert matrix so for large values of see A transpose A this tends to this tends to 1, 1 half, 1 third, 1 by m, 1 by 2, 1 by 3, 1 by 4 okay for large number of data points here okay this particular matrix A transpose A okay you can look for the arguments here this matrix will tend to this matrix which means you know basically if you want a good approximation what will you do if you want a good approximation you will take more data points which is logical okay if you take instead of taking you know 100 data points if you take 500 data points you will get a better approximation okay you will get a better approximation but as you take more data points and try to fit the polynomial this matrix A transpose A will tend to the Hilbert matrix and Hilbert matrix is difficult to invert because this is a hill conditioned matrix because this is a hill conditioned matrix we have a problem that we cannot get reliable estimates of theta okay we will not get reliable estimates of theta and because we cannot get reliable estimates of theta approximation that you develop set 10th order or 12th order is not reliable you know is not reliable polynomial approximation a better way here would be to use orthogonal polynomials instead of using orthogonal polynomials we will make this matrix there are slightly different ways of constructing orthogonal polynomials for the discrete data case which I discussed in the notes so I am leaving it for you to read but the idea is that if you use orthogonal polynomials then this difficulty can be avoided there is something that we already have discussed is how do you how do you use the square approximation for fitting certain kind of correlations in chemical engineering we discussed about Sherwood number correlation or we discussed about Prandtl number and you know Nusselt number equal to Reynolds number raise to something and if you if you take a transformation then you can linearize that problem and then solve the problem of estimating you know constants in this correlations using linear v square so that we have already seen so we have already seen one application of linear v square in approximation okay so right now right now when I am doing Cp as a function of temperature I am trying to approximate a continuous function using a polynomial function this is one common application you have some algebraic static map and then you are trying to approximate using some other some other known functions and the problem is transformed to estimating this theta okay and we also saw in any general Hilbert space how you could construct the normal equation I am uploading extra problems today for v square and so you can start looking at the problems where you will see a lot of engineering applications where you can you will be using this v square so one of the things that I promised you is that we will now move on at some point to solving boundary value problems or partial differential equations using this method well that is what I am going to do next that is see how I can apply this method for solving a partial differential equation so now I am going to use the idea of v square for constructing a solution of a boundary value problem okay so right now I am going to restrict here to showing you two methods one is method of v square for a partial for a boundary value problem I will not show explicitly an example of partial differential equations I will also talk about a variant of it called Galerkin's method now these two methods though will be looking at them very briefly right now we are not going to look at the details of this beyond the point because these methods become very complex and what actually you know this is actually we are getting into the method of finite element okay if you have to actually implement this you have to get into method of finite element and finite element methods are fairly complex when it comes to computer implementation so it will go beyond the scope of this particular course you know I have to stop at some point problem discretization and move on into you know solution techniques so I am just going to give you very very brief introduction not really complete introduction but at least the philosophy of this problem discretization that is say boundary value problem how do you discretize this using least squares approach philosophy will be clear we will not go into the details okay so details will be I mean if I have to get into details it will take many lectures and we have to move on and do other things other than just problem discretization okay but this once we do this you know it sort of completes the picture we have three different ways of problem discretization one is problem in this particularly I am referring to boundary value problems or partial differential equations we discretize using Taylor series approximation we discretize using second is interpolation we got collocation methods okay third is this least squares okay we get this finite element method first is finite difference do not confuse finite difference method then we had orthogonal collocations then we also had orthogonal collocations on finite elements okay and this method but that basically is interpolation based method and the third one is this least squares method interpolation we want the polynomial to pass through every point okay that is the difference approximations we are trying to fit a polynomial in some least square sense we do not want the polynomial to pass through every point or the function to pass through every point okay so let us look at this method of least squares for solving so this method is called as minimum residual method and well why can you solve a problem using optimization and just to give you a little bit of a background before I move into least squares okay we will revisit what I am doing right now on the board again but I am just preempting something see we know how to solve a x equal to b right now from your undergraduate how do you solve a x equal to b Gaussian elimination the best method you know is Gaussian elimination you do you know triangularization and then solve in the reverse direction so that is one of the most efficient methods that we know can I use the ideas of optimization to solve this problem okay now let us see whether we can use the ideas of optimization to solve this problem what we wanted in optimization we want to minimize something okay and the minimum the minimum should be the solution okay now let me take this let me take this before I get into this let me let me say that I want to solve a x equal to b I want to solve a x equal to b okay I want to solve this exactly okay I could I could say that well this problem is like saying that you know let me define an error vector let me define an error vector which is a x minus b a x minus b everyone with me on this okay and then I will define an scalar objective function okay I will define a scalar objective function so my scalar objective function is going to be now well what I am going to do now is very very similar to what is happening here so this is my error vector this is a theta and this is u except in this particular case in this particular case we had you know this matrix a was tau matrix it was not a square matrix right now what I am writing there is a square matrix okay square matrix so I want to solve square matrix problem by optimization is it possible so now let me formulate an objective function phi which is e transpose e transpose e I formulate an objective function e transpose e and then what I do is which is what is it e transpose e a x minus b transpose a x minus b okay and now I say that well minimize phi with respect to x how do you get the minimum right so doh phi by doh x is equal to 0 can you solve this what will you get solve this if I use the rules of differentiation of a scalar function phi is phi a scalar function phi is a scalar function right I am differentiating a scalar with respect to a vector we have looked at the rules of differentiation of a scalar with respect to a vector if I use those rules what I will get is this equation a transpose a x is equal to a transpose b I will get a transpose a x minus a transpose b is equal to 0 this is the equation that I will get if I differentiate and set derivative of phi equal to 0 I will get this equation okay is this fine just see this you will get if you expand this you will get a transpose you will get x transpose a transpose a x then you will get a transpose x transpose into b you will get four terms you will get one term which is b transpose b b is not function of x b doh b transpose b by doh x is 0 okay so if you differentiate you will very easily arrive at this particular equation okay I just want to show that this is same as this is same as a transpose a x minus b is equal to 0 okay now if a if matrix a is non-singular if matrix a is non-singular okay when will this be equal to 0 a x minus b equal to 0 okay so actually by solving this optimization problem by solving this optimization problem you have solved you have reached the solution of a x equal to b and later at some point we are going to see a method for solving a x equal to b using optimization and iterative procedure which is faster in arriving at a solution many times then is this the optimum why is this the optimum why is this minimum if you take second derivative what will you get a transpose a transpose is always a symmetric matrix whatever is a a transpose is symmetric positive definite matrix very nice matrix okay so we have reached the minimum okay so this is so I can actually formulate a problem of solving a x equal to b as an optimization problem yeah this will be exact solution okay earlier we saw approximate solution so in the situation where in the situation where you know a is a tall matrix okay which means a is a non-square matrix it is it has more number of rows than the columns okay then you will get the least square solution when you when a is a you know square matrix invertible square matrix this will give you exact solution so look at look here this will give you exact solution if a is invertible when can you solve that problem a x equal to b when a is invertible if a is not invertible you cannot solve that problem right so now we are starting with the assumption that a is invertible okay and we want to solve the problem so this obviously gives me the solution if I if I if I do this okay so optimization could be a root to arrive at a solution of a particular problem okay what I am going to do for what I am going to do for the case of boundary value problem is more similar is similar to the case where a is tall okay so a is non-square what I am going to do for boundary value problem is qualitatively similar to a being non-square that is a has more number of rows than the columns okay it is possible to do at least formulate the problem to get an exact solution of a boundary value problem using optimization and that discussion is included here in the notes under the it is very very nice you should read this section though I am not going to do it on the board it takes lot of time and this actually forms the foundation of finite element method and finite element methods are very very useful in solving partial differential equations so I would say there are these two competitors finite difference is very easy to understand but finite difference is not so efficient because you need large number of grid points okay two methods that are competing good methods you know which balance between computational efficiency and good solutions really a good balance of these two things are orthogonal collocation collocation interpolation based methods or this finite element methods so now let us move on to boundary value problem okay now I want to consider a specific problem here boundary value problem so there is some operator L which is operating on UZ okay I suppose now you are familiar with these kind of things because you are most of you are attending the other course on mathematics mathematical methods and you have looked at these kind of problems right L operating on Z U of Z is equal to this is equal to some FZ and boundary condition one this corresponds to U0 is equal to 0 and BC2 is equal to U1 is equal to 0 so this is a classical boundary value problem that you get when you are solving a partial differential equation this will typically yield if you solve it exactly if you know the solution exactly it will yield the eigen function, eigen function expansions are you doing eigen function expansions yeah so this will yield the eigen function expansions right now my mandate is not to solve it exactly my mandate is to solve it approximately okay I am going to solve it now here I am assuming that L is a differential operator which is operating on U conceptually we are not doing anything different than ax equal to b and we have talked about it right this is a this is equivalent to operator ax equal to b we are trying to find out inverse problem classic inverse problem so right now I am not interested in the exact solution I am interested in some approximate solution to this problem and I am going to hypothesize a solution approximate solution U cap Z is equal to alpha 1 okay I am going to hypothesize a solution now what are these U1 U2 U3 these are some known functions these could be simple polynomial if you want it that way but you know polynomials is not a good choice okay this could be this could be Lichander polynomials shifter Lichander polynomials it could be sin and cos it depends upon how you choose these functions it is up to you and you have to choose these functions such that the boundary conditions are met of course so I have these functions which I have chosen here let us assume that so let us assume that we have chosen these functions for the time being okay that such that at the two ends they satisfy the boundary conditions okay we have chosen the functions such that at two boundary points we satisfy the boundary conditions okay so you have to choose the choose these functions in this particular case carefully now what I am going to do is I am going to define this residual I am going to define this residual okay the terms used are little different but the meaning is same in the square parameter estimation we call this error vector okay here we call residual vector okay residual is between left hand side and the right hand side okay so residual is between L uz minus fz this is my error vector in the case of finite dimensional least squares what did we do how did we find the minimum sum of the square of errors was minimum what was actually sum of the square of errors 2 norm square what is 2 norm square in a product of the vector with itself right in a product of vector with itself so for the finite dimensional case for the finite dimensional case we had this just keep this in the background in the finite dimensional case we wanted to obtain theta so theta least square was minimum with respect to theta you know in a product of error and error right sum of the square of errors least square estimation error vector was where error was defined as a theta u minus a theta right let's keep this in the background okay I just want to do the exactly same thing here for this particular problem now I have this I have this functions I have this functions u1 u2 u3 so what I am going to do is I am going to pose this problem okay I am going to pose this problem as minimize I am skipping the algebra which can be a little bit tedious I will write the final expressions but whatever we have done till now from that you can very easily derive what I am doing if I have to do the same thing here I will have to write this as summation of these functions and then start working with those summations and the expression will become complex but that is just an algebra the concept is exactly identical so you should not so what are the unknown here what is the theta here what is theta here alpha values alpha 1 alpha 2 alpha m right alpha 1 alpha 2 alpha m okay now what is r u zi what is r u zi well you will have to substitute one by one each that is not ready to go about okay so we want here r zi other what is r zi what is r zi r zi is l operating on summation i going from 1 to m alpha i ui ui of zi ui of zi let me let me do a small modification here let me let us call this j let us call this j and let us call this j this is this is residual at this is the residual at z equal to this is the residual at z equal to this is the residual at z equal to zi what do I need the points here I am not going to go same way as the grid points I am going to go in the I am going to go in the parameter space directly so what I am going to do here is instead of forcing the residual to be 0 at each grid point well what I am going to do here is to pose my problem as minimize with respect to theta phi theta which is equal to inner product of r r z well this is this is something different from okay you should realize this is something different from I am not taking any grid points right now I am not taking any grid points okay so what is what is this what is the inner product what is the inner product defined as inner product integral say 0 to 1 now how do I get the equations from this how do I get equations I have to differentiate this I have to differentiate this with respect to also with respect to theta right so so now what I have to do is to get the solution to get the solution I should see how many equation how many unknowns are there there are m unknowns I need m equations okay so if my phi theta is this then what is how do you get the optimum doh phi doh theta is equal to 0 okay so which actually means doh by doh alpha i of r is equal to 0 for i going from 1 to up to is this okay doh phi by doh theta equal to 0 which is same as this is my objective function error right this is my error square some of the square of errors but it is not some it is integral of error it is integral of error not at one point not at few points but over the entire domain okay there is a difference you can see here between what I am getting the way I am constructing the solution and finite difference method or orthogonal collocation method here we are setting this with respect to 0 okay if I if I solve for this if I solve for this okay what do I expect to get see what is rz here so you have to you have to actually solve this problem of doh by doh alpha i okay into what is this this is fz minus summation i going from 1 to m alpha i u cap i right what is rz rz is fz minus this right this equal to 0 okay and this equation you have to solve for i going from 1 to up to m oh yeah l of this right right operator l operating on this operator l operating on it yeah so thanks for this correction so l operating on this and l operating on this and I have to take I have to take derivative and set it equal to 0 but if l is a linear operator if we make an assumption l is a linear operator okay then this alpha i will come out and then you can differentiate and solve the problem very very easily so if l is a linear operator then what I can do is if I actually write those equations and collect the terms together okay I will get this I will get this equation which looks very very similar to the normal equation you can work out the algebra I am just skipping it it is just matter of doing it meticulously you will get this equation u1 okay you get this equation if you rearrange if you actually differentiate do a little bit of take a trouble of rearranging all the equations and then put them together you will get this equation which looks very very similar to the normal equation okay except that operator l is operating on each one of those basis functions which you have chosen okay and then if I solve this if I solve this then I will get alpha 1 alpha 2 alpha 3 alpha m I am getting an approximate solution in the least square sense okay there is one difference there is one difference here between the previous methods of discretization and this method of discretization what is that what is the difference here here these integrals are over the entire domain not at few points okay I am not evaluating this integral at few points I am evaluating this integral over the entire domain okay so the finite dimensional approximation comes because of the finite number of basis vectors we have taken the finite dimension business here comes because of not because of the grid points okay not because of the grid points here you have constructed a solution which is not see earlier case when you get the solution you must have done it now for the boundary value problem which we are looking at tram problem it is some points right finite after all you get only few points actually you know that the solution is a continuous function by this approach we are getting the solution which is a continuous function I will just take a simple example so that the ideas will become clear okay is this algebra clear is there any doubt about this see here l this residual is this residual is fz-l operating on this function okay fz-l operating on this function we are taking l to be a linear operator okay linear differential equation and then we are solving this problem of you know estimating alpha 1 alpha 2 alpha m. Now let us look at a specific problem well this particular method will work by this way because when the operator is linear if the operator is not linear what do I do we will come to that a little later but right now right now we are just looking at the basis of the method which is least squares method and the operator is linear so we get this nice so let us look at a specific problem okay that will give you better insights into well I want to solve this problem of my example is l uz this is same as dou 2u or d2u by dz square minus u equal to 1 so this is my fz I want to solve this problem I want to solve this problem this is equal to 1 okay over the entire domain this should be equal to 1 and my boundary condition 1 is u0 is equal to 0 my boundary condition 2 is u1 equal to 0 well u is not equal to 1 please note that u is not equal to 1 we are saying that this operator operating on u should be equal to 1 okay do not confuse okay I am going to take a very very simple solution to this problem this particular problem you know you must have seen in the other course that this particular problem can be solved analytically you will get analytical solution right now the motivation is to convey some concept okay so that is more important right now so let us take this approximate solution u cap z okay so my approximate solution is alpha 1 sin pi z plus alpha 2 sin 2 pi z does this satisfy the boundary conditions satisfies the boundary conditions at 0 and at 1 boundary condition is satisfied now I want to see if I get difference between finite difference method okay and this method is that if I get alpha 1 alpha 2 I can find out value or approximate value of the solution at every point z not at finite number of points you get a difference if I can get least square estimates of alpha 1 alpha 2 I get okay so now what is what is the what is the normal equation the normal equation normal equation so what is my u1 cap what is my u1 cap z is sin pi z and what is my u2 cap z sin 2 pi z right sin pi z and sin 2 pi z these are my u1 cap and u2 cap right okay what is the normal equation what is the normal what is the equation that you get well it normal like equation let me not call it normal equation because not exactly the normal equation so what is the equation that you will get here how do you get alpha 1 alpha 2 so this is inner product of l operating on u1 l operating on u1 cap inner product of l operating on u1 l operating on u2 cap l operating on u2 cap l operating on u1 cap u2 cap l operating on u2 cap this matrix okay this matrix times into alpha 1 alpha 2 this matrix time alpha 1 alpha 2 will be equal to to get alpha 1 alpha 2 what should this be equal to inner product of u1 cap 1 inner product of u2 cap 1 so what is inner product of l operating on u1 what is l so let's go back here so what I do what is the first inner product l u1 l u1 this is inner product 0 to 1 what is l u1 d by dc square of what is u1 sin pi z minus d2 by dc square d2 by dc square of sin pi z minus sin pi z this so this integral I have this integral I can do I can just it is not so difficult to do this integral you have to be little bit patient and then do these integrals okay if you work out these okay well I will write down what you get here actually just if you take this what you can show is that l u1 this is equal to minus pi square plus 1 sin pi z you can show this very very easily you can also show what is l u2 okay and then you just have to find those integrals if you find those integrals what you what do you see here is you will get pi square plus 1 whole square by 2 0 0 and 4 pi square plus 1 whole square by 2 right hand side will turn out to be minus 2 pi square plus 1 by pi and 0 if I do patiently all those four integrals okay then I get this I get this equation I have chosen orthogonal basis functions here sin pi z and sin 2 pi z are orthogonal so that helps me in this particular case that helps me in getting these to be 0 this comes out to be diagonal matrix and what happens here is that I get the final solution so what is the final solution if I if I compute alpha 1 and alpha 2 I get the final solution to be my solution comes out to be minus 4 by pi pi square plus 1 sin pi z so which means alpha 2 comes out to be 0 alpha 1 comes out to be this and this solution is pretty close to the in this particular case the true solution can be found out the true solution is the true solution is uz equal to this is the true solution this is the approximate solution to solution can be found by some different means not by this this approach yeah yeah and then force the residual to 0 so now I choose some of these functions from a standard basis no I can choose them from a standard basis like sin cos I could choose them from the general polynomials shifted general polynomials I could I could have worked out this problem using shifted general polynomials if I have those shifted general polynomials which satisfy the two boundary conditions in this in this particular case sin and sin functions are convenient because they satisfy the two boundary conditions they are not be orthogonal but in this particular case orthogonality always helps you always work with orthogonal functions okay we try to work with orthogonal functions as far as possible orthogonality is but this solution there is one difference between this other solution other solution is exact at known points exact means it is not exact it exactly satisfies the differential equation at known points here this solution is obeying the differential equation in the least square sense okay at every point it is not equal to 0 but if you take the least square sense it is 0 okay integral of the square of error is 0 error is not equal to 0 okay integral of the residual square is 0 okay even in the other case we are forcing residual to 0 only at finite point what happens to the residual at the point which are in between they will not be 0 right so we are only forcing at a finite number of points even here you know you do not know which points exactly the residual will be 0 but the solution will be crossing you know 0 at many points so this is both approximate solution this is one way of getting approximate solution and you have other way of getting approximate solution which is what we have seen earlier okay now the problem is that in every everywhere where you are solving this your L will not be in general you know it will not be linear there could be a nonlinear differential equation we are solving this Tram problem right it is not a linear operator okay so what do I do what will I get what will I what how will I solve that problem so in that case you know I just use the concept of orthogonality between the subspace and error the concept that we had earlier and we will get the method of you know Galerkin method so I will discuss it briefly in my next class but the next class after doing that half an hour of Galerkin method something about a Galerkin method I want to move on to solving problems because we are only two classes away from the end same and then complete same and we will solve some problems of v square so that you get better insight into what is happening but the key thing here is that right now I have not used polynomial approximation have you general functional approximation the solution is obtained by posing the problem as a v square problem okay and then you are getting a solution approximate solution of the boundary value problem using v square support okay it is possible to extend this to partial differential equations we will have to compute double integrals in x and y and so on okay it is possible to do that in algebra will become complex the concept is not different


Thank you for coming this early morning. So I'm going to start with the aggregate relationship between aid and growth. And go start off with a basic question. Thinking about aid and growth, why are some countries poor? Many of them. And actually, there is a pretty simple answer. It's that they just don't produce very much. I started doing work with household surveys in Mozambique. And we did a household survey, and we did a counterfactual of a perfect transfer program. So every household has the same income. And in that case, in 1997, every household in Mozambique would have been poor. So it's the lack of goods that makes up the problem. Why do they produce so little? Well, they have rudimentary technology. They don't have much human capital. They don't have much physical capital. And even in cases where sometimes they add some physical capital or some human capital, they often lack the institutional structures to make this function well. Well, why is it that they lack this wherewithal? And the reason is they've failed to accumulate. There is this long process of accumulation. We stand here in this building, in this university, that's been built, buildings built probably before we were born. That's part of the accumulation. Many of you have children who are going to school this morning. Those children won't contribute to GDP for another 10, 15, 20 years. This is a long process of accumulation. So the role of aid is to facilitate this long run and often fragile process of accumulation. And that's what we're looking at today, have we managed to facilitate this long run process of accumulation? So and looking at the recent cross-country literature, there's been a piece published in 2008 that has been quite influential. And they look at lots of countries through time. And they look to see whether aid has contributed to growth across this set of countries. And they conclude that no, that there is no detectable impact of aid in this sample of countries through various time periods and using various methods. And so in essence, this is what's been enabled the micro-macro paradox. And it's been revived. Why is it called the micro-macro paradox? And this is fairly well accepted. By and large, through the aid literature, if you look at project evaluations, not every project succeeds. It's true. There are failures, as Finn was pointing out. But by and large, projects evaluated ex post receive positive evaluations. On average, we have positive project evaluations. We have positive what we call impact evaluations. Development economics has been at the forefront of what has been called the credibility revolution in econometrics. So they look in a very rigorous way at whether certain interventions with the project and without the project or with and without the intervention have actually had impact. And again, not every impact evaluation conducted comes to a positive conclusion. But most do. And then we often get positive sector evaluations. Aid does seem to have an impact on education, enrollments at least, on health. And yet, we find no detectable impact on growth. So there is some incoherence in this. And the macro literature is the outlier. There are a series of positive contributions of RS08. Rajan was the chief economist of the International Monetary Fund. Subramanian is not there as a charity case. They're smart guys. And there's a series of things that help really to advance the literature. And one is they establish a clear prior. So they look at modern growth theory. And they say, well, how much growth should we get if aid is being invested according to modern growth theory? And they come up that if aid is about 1% of GDP, then the increment to GDP growth should be about 0.1 percentage points. So if a country without aid was growing at 1% per capita per year with aid at 1% of GDP, it would grow at 1.1% per year. Second, they take a long-run perspective. Not throughout their entire article, but their principal conclusions are based on long-run cross sections, looking from 1960 to 2000 and 1970 to 2000. We agree with that because of this idea that growth is a long-run cumulative process. Finally, they set the standard for addressing the endogeneity issue. And this is what Finn was mentioning as the attribution problem. As Finn was mentioning, we are now, Denita is in the process of leaving Vietnam. All the monies that are being allocated to Vietnam will go somewhere else in a few years. Maybe that money will go to, hopefully that money, will be able to go to Somalia. They need it, but that money is going to Somalia, if it goes there, because they're poor. This is the attribution. And if we're not careful statistically, we can be concluding, we're in danger of concluding that something like mud causes rain rather than the other way around. So what have we done? We begin, and again, Rajan and Subramani kindly shared their data and their codes. So we're able to begin from exactly where they start. And this is the standard in this literature. So we want to know what exactly the differences are. So if we begin from exactly where they begin, then we know what the differences are. So we are able to reproduce their results exactly, and then we look at what's happening. And we make three sets of improvements. I mentioned the credibility revolution in econometrics program evaluation. This is the new perspective. And we develop what's called an estimator, an approach for looking at this that is based on this new approach in econometrics. This approach has implications for what we call the specification. These are our controls, what we're controlling for. So we make some changes for the way we control for other factors that are influencing growth aside from aid. This turns out to be one of the least having a smaller effect. Finally, we strengthen the instrument. And the instrument is what we're using to try to deal with this endogeneity or attribution issue. And here, as I said, Rajan and Subramani have set the standard. They do a smart thing, which is instead of trying to look at aid from the perspective of the recipient, they look at aid from the perspective of the donor. How much aid is being supplied rather than how much aid is coming in. And so there is a database at the OECD DAC that will tell you this kind of information. So you can go in there and you can hunt and see if how much aid did France give to Guinea-Bissau in 1978? The principle should be there. But in a lot of cases, it's not.  And what does it mean? Well, potentially two things. One, they didn't give any. Often, we don't report zeros. Or France gave something, but OECD doesn't know what it is. It's missing. And we worked on trying to figure out which one it is. Finally, I won't tell you how long it took us to come to the conclusion that perhaps we should ask somebody at the OECD DAC what we should do. And they came back very quickly and said, if it's not there, it's zero. Or it should be almost all of the time. And Rajan and Subramanian had treated it as missing, which creates quite a difference. There's an error in the instrument part. We do some other things to make the instrument stronger. But this turns out to be one of the more important things that we actually do. So what does it do at the end of the day? These are the sets of results that we have according to the various permutations. And the first thing to notice that their result, the result that they actually obtain, is exactly their pride. Point one, that's what we pointed out. It's exactly what they got. It's just not statistically significant. It's not estimated precisely enough. That's the first thing to point out from this slide. The second is, when we shift over to more this credibility revolution program evaluation perspective, the estimator, then in all cases, regardless of what else is done, the estimate is positive, significant, and in the range implied by growth theory. Alternatively, when we use an instrument that is missing the data error and has certain other additions, then in every case, regardless of if we use their specification, our estimator, their estimator, we get positive and significant implications for growth in the long run. Finally, the more of these improvements that we add, the more precisely the parameter is estimated. We end up with significance at the 99% level. So a quite precisely estimated coefficient, quite close to what we think is a reasonable prior. So what we get is for every percent of GDP and aid, you get 0.13 increment in per capita growth. So if it was without aid, 1%, with 1% of GDP, you would get 1.13% growth rate. What do we conclude? We conclude that on average and over time, aid contributes positively growth at levels predicted by growth theory. This is over periods of three decades or more. We also conclude that there is no micro-macro paradox, that in fact there's consistency across the project evaluations, the program evaluations, the sector level, and the macro level when taken over a period of time. And this we've published in a new journal edited by Joseph Stiglitz in their first issue. And this has been a very, very popular paper. One would think that an article by Joseph Stiglitz in his new journal on the financial crisis would be a popular paper. And it is a popular paper. But in fact, this paper has been downloaded more often than Joseph Stiglitz's paper since the papers have been published. It's generated a lot of interest. And many people think that it sets the standard for this particular set of analysis. So that's my contribution. And Sam is going to come up and talk about it, a different form of consistency.


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Like when I look over and I see you instead of the other thing. I think you've said that before not in those words though But it's true Hey, welcome back to our stupid or EXO Corbin You can follow us on Instagram Twitter for more juicy content Thank you to everybody's forts on patreon follow to the count subscribe like button and welcome back also This is my wife Hi for all of those new people who are confused who always ask who's that girl in that are talking about andrani That's not my wife. That is Rick's wife. This is my beautiful wife. I Just there's like every single time they feel like there's a new person on it's always who's that if Lexi's here or Vashly's here Who's that? This is my beautiful wife. Hello Today we're doing a watch along Thanks for stopping by and you do see Stephanie. So you're like, is this a show Rakan movie? No But we will be getting to those very soon. So But this is a another animated film because if you watched Stephanie's last watch along we watched Ramayana, right? That's how you pronounce them the 1993 anime film about Ramayana. I don't have to explain that to you guys But we really really enjoyed that one. Yeah, it made me go on a deep dive and I read Comic book adaptation. Yeah of it. I'm aware. It's not a comic book. Yes, but the adaptation they did in comic book form Yes, that was fun learned more stuff. Yeah So this is actually in the comments of that watch long if you haven't go watch that, please go watch that Everybody was like watch the 2005 animated film Hanuman And so it's not anime, but it's animated so 2005 animation though, so I'll be prepared for that But this is more Hanuman centric. So I think you'll still have some of the similar story But his story and other like his side stories and that kind of stuff Which is great because it was with my favorite character. I liked Hanuman Even Adi Pradesh or whatever that movie was called. I liked Hanuman as well He seems like a fun character, but if you're watching this on YouTube You're gonna see a very cut-up version two to three parts probably Depending but for copyright reasons we can only use very cut-up footage on YouTube about three to five minutes If you'd like to see the full-on cut versions of this and other things that Rick and I have reacted to Steph and I have reacted to you can have her to patreon. I'll put that in the pinned comment and the description below so you can go check that out Let's just get into it if we get up if we get up at all or pause this It's cuz we have babies that are hopefully just be asleep the entire time, but they're babies We got tea this time instead of alcohol Oh Ravaged the land and brought with them Jesus in the basket This was in great peril and had no way to combat these demonic forces I will be born again as round The king of Ayutthaya to destroy all evil this task. I shall carry out with Shiva No, but for in my 11th form I shall choose to remain celibate An unusual one and I shall be called Hanuman It's me and Jimmy. I'm The Wingard Lord Shiva has sent as for the divine plan. He has blessed you and soon You will have a baby boy and he shall be called I Hanuman was born Wait, I thought he was a reincarnation of I'm confused. I don't know anything Oh, yeah, he swallowed the Sun.  Sugri will help you but you must Someone will help him recall all his divine King Dashrath wishes, what is it sir? To have your monkey for Prince Ram. Pick me up promise You must always give him a place in your heart Hanuman was wise beyond this year How could I tell where I do was blessed with divine wisdom remember? I became very fond of Ram and his three brothers As Lord Surya had predicted I met Sugri Greetings my friend. I am Sugri younger brother Bali has my brother Bali been killed. Oh, no He blocked the entrance of the cave to make sure General go out in every direction and Tracy does well about You are a wise and brave warrior No one can match you in spirit and valor Will you leave this mission? Oh, divine lady? We're looking for Lady Sita. Please guide us. I shall transport you to the seashore from where your task will be easier He transported him up. There was no road ahead. That wasn't in the other one Oh Sorry somebody we thought you knew why if you wish mighty mountain I Cannot even think of rest Certainly not before I complete my task. I saw her a woman of unsurpassed grace and ethereal beauty the lady Sita Have you abandoned me Lord? Will you rescue me Ram? Oh And that at the end of our part one reaction, how was it great beautiful? If you'd like to see the full uncut unadulterated version of our reaction you can head over to patreon We already watched obviously the entire thing and so it's already on patreon so you can go check that out I'll put the link description below and the pinned comment and Let's just get into it Part two let's get into part two It's confusing


What's up everybody, I'm EvilRabbit, we're here on Drift21, it was a big update with some forced feedback updates as well as a new car, so we're gonna hit it in Drift21 today, make sure you follow me on Instagram and Twitter, all that you found in the description box below. Let's get to it! Alright, so if you guys have been following the channel, you guys know we were working on the 15 which is over here, we did put some wide spaces and everything on it to get somewhat of a fitment, it does have a thousand horsepower and they did say they adjusted the driving feel, so we're gonna go take a look at that but we're also going to be taking a look at the new car. So we have the 15 fully built, we never finished the BRZ but we now have the Mazda RX8 which you know we're gonna bring out because well why not, so we're gonna buy the RX8 in today's episode which is gonna pretty much drain our credits, so we're gonna start getting some credits and stuff like that to start building the car up. So first we're gonna go drive the thousand horsepower 15 and see how this does, so we do have North, we do have school and we have Miname, so I feel like we should just go to Miname and Free Mode, we're just gonna go send this and see what we can come up with and see how the driving feels differently. So it's not as twitchy now, oh I'm noticing it's not as twitchy at all, feels a little bit better, well as I say that we kinda, okay so that's a little bit better, so the wheel still has a bit of a different wheel, you know feel to it, it's still, I don't believe it's perfect, there's probably some things that could be a little bit better, it doesn't seem to rotate that easy still and it still gets that little bit of slide, so there was somebody who told me to change a setting in the game to eliminate that, so we're gonna go take a look and see what we can determine, oh they have new bases, we're gonna go with the Fanatec, Fanatec 2.5 base so it's going to give us a force feedback for that, okay, alright that's definitely pretty cool, so I'm trying to figure out what setting somebody was telling me about, it's probably another setting, something about automatically press clutch and handbrake, steering limiting engine stall, that's probably the engine stall setting, so now we need ignition, so ignition on we're gonna go here, ignition off we're gonna go here, okay so now let's see what this, oh yep now we've stalled the car, can we start the car, oh do I not have an ignition, do I not have a starter button, oops wrong button, did I not put a starter button now, no ignition on off, there's no starter button, so unsure how that works, so we'll just go back to, so now it's set for the Fanatec so let's see how it feels, it does feel a little bit better, I think I'm still just roasting these tires to be quite honest with you, yeah these tires are just getting smoked, so the feeling of the wheel is a little bit better, it does feel not as bad, we could be slipping the clutch because of, yeah clutch slip, I think that is our problem, we do have clutch slip in the car, so apparently our clutch is not good enough, because you know some force gear I get no power, so we definitely need to work on the clutch and stuff in the 15, but we're here to take a look at that RX8, let's go back to the garage, take a look at the RX8, like I said it's going to plead our funds, but we do have the BRZ to do stuff as well as the S15, we're gonna have to tear the entire S15 apart though in order to mess with that clutch, we have to pull the engine, pull everything apart, so we will save that for another time, we have parts everywhere, I feel like we should just clean up our garage, yep alright, so now let us go and buy a new car and deplete our funds completely, and then we will have to get our funds back, I want to buy that yes, so now my question does it come with a rotary, because there's still no rotary available in the garage, so I can't open anything, you can't open anything unless you put it to drift mode, that's unfortunate, we can't even see what's in there, your car receives basic modifications, no alright well I guess we're gonna go drive this car first before we send it into oblivion of drifting, probably have to pull a motor out of another car, but let me say we're gonna lose the rotary, rotary getting wrapped, oh I'm gonna be kind of really sad that we're gonna lose the rotary, because from what I know, we don't have a handbrake really, from what I know when you do the drift mode you are gonna basically, it's gonna pull the motor and everything out, is there a time trial we can do with this to get some money in stock mode, so we're gonna go see what it does when we actually hit that button for drift mode, I'm gonna be super sad if we lose that rotary, because we cannot buy a rotary, because if you go into buying a warehouse there is no rotaries, you only have inline four and inline six motors, no rotary motor, oh that's sad, please don't get rid of the motor, please don't get rid of the motor, oh they don't let you keep the rotary in it, oh that's super sad, put a radium in there already though, they don't let you keep the rotary motor in the car, so unfortunately we are going to have to build a motor or pull a motor out, so if we pull the motor out of there we could put it in the, you know, the RX-8, but unfortunately that's gonna, it's gonna definitely make for an unfortunate time, I'm gonna take off that and see if we can pull this motor off into the stand, move to RX-8, okay, well then move to engine stand and then take the pipes off, move to RX-8, so we can only put the forebanger in the RX-8, that is unfortunate to me and well that's kind of sad, I was hoping we'd keep the the rotary in there, nope I don't want to drop but unfortunately they don't have the rotary in there, so that's a little bit unfortunate, let's go put this on the lift, we have to put this car in the air and put all the missing bits in it, like big intercooler, and we gotta do the intercooler piping, do we have enough money for all this, a lot of the stuff we already bought, so luckily a lot of the things other than these new parts are still good, and let's do a short gear ratio, oh no I don't want to move as a dyno rip, we gotta continue and put a bunch of things in this thing so we can get it on the ground and driving, I'm very curious as to how it reacts with the turbo inline 4, hopefully we have enough money to buy all these parts for this, we do so we're good, oh we're gonna have to put the front pipe in there, so now don't drop that on the ground, drop a lot of parts here in drift 21, if you guys have not noticed, there's no way to get the front pipe in there, then I believe we need to put the interior bits in, maybe put steering wheel in, I think we're just gonna put our normal universal wheel in there, our big universal handbrake, we need to just put factory seats in for right now, oh no I don't want that, we're just gonna do all the basics on it and we will get more into it later on, but today I kind of want to drive this thing, so we drove it with the rotary, but now we don't have a rotary anymore, so now we have a turbo 4 and we got a lot of clutch slip, so so unfortunately we lost the rotary that was in the car, still sounds like one though, the physics and the wheel do feel a little bit better now, it's not as uh let's say loose, but we're gonna have to be doing some challenges and stuff to give money, so before we end this episode off let's go take a look at if we can do one or two challenges maybe and get some cash to reimburse us because we're down on cash now, so select mode, solo run, no wait we want to go solo, select, that's just, what is this just a time, you can get a decent amount of money for that, all right we'll try it, apparently you can get 1800 bucks for getting gold in this one, oh don't miss checkpoints, what are we completed with, bronze, I mean all right, so it is quite easy to actually get uh get some money in here, we'll try and get this faster, oh as I said that we lose it, so if you guys are doing challenges I really gotta fix that clutch slip, we definitely gotta fix the clutch slip issues, we got a gold trophy, earn credits 2700, quick way easy way to earn money just by doing something like that, yeah so did we just gain a lot of money, so that's a quick and simple way to earn some cash in this game if we actually did, we did, so good news is we can earn cash easily, so now we got well a couple cars in our garage here, I think before we leave we should definitely and try and make a basket because I did find the basketball hoop outside, so make sure you guys follow me on instagram and twitter all which are found in the description box below, I'll make this, please hit that subscribe button, oh well I mean you could probably still hit that subscribe button if you would like, make sure you guys follow me on all social media, apparently I'm really bad at basketball in drift 21, so as always I thank you guys for watching, I'm even rabbit, oh I think that went in but it glitched, I'll see you guys on the track


Hey guys, welcome back to my YouTube channel, this is Daniel Rosel here. I'm currently down with a bit of a head cold, so this is always my excuse to indulge in a bit of hibiscus tea in Arabic Shai al-Kurqadiyah. Kurqadiyah is very popular in Egypt, Jamaica, and the Caribbean. Some places serve it ice, some places drink it hot like a tea as I'm going to do, and it's been proven to be moderately useful in lowering blood pressure, so it's quite healthy.  You can find these in health food stores, and if you have a Middle Eastern store in your locality they might also stock it because it's very popular as I mentioned in Egypt. This is the strainer that I use, I bought it on the internet on eBay I think, and it's great, it just has a little thing that sits over the lip of the cup you're drinking it from, and these are the dry hibiscus flowers. Now something that you want to do is after putting them in the strainer I always rinse them under tap water just for a few seconds, because these can be stored for quite a amount of time, and there might be dust and other sort of contaminants sitting around there. So I use three tablespoons of hibiscus flowers for a cup, but you can experiment with different strength. It is very strong and tart, but adding a bit of sugar which is something I only recently started doing, and that's very traditional in Egypt does really help to kind of cut across the tartness. So what you want to do is as soon as your water is off the boil put your strainer thing into your cup, and just pour hot water over the hibiscus flowers until you sort of cover them at the top, and then if you do want to add your sugar this is the time to do it, just pour it straight over the strainer. I use something like a tablespoon, the recipes you'll see from Egypt use crazy quantities of sugar. I find that this quantity works perfectly, and then also just give it a little bit of a stir so that the sugar mixes in perfectly and doesn't settle at the bottom of the cup. Now you want to leave it sitting for about three minutes, that's enough to let it infuse properly, and after that just pull it out and drip off any remaining hibiscus. It looks kind of like red wine, a hot red wine. Of course it is non-alcoholic and it's caffeine free, but it does stain quite easily, so just make sure that you've something to catch your stain or so it doesn't get all over your countertop, and that's pretty much it. It's ready to drink, drink it straight away while it's hot. Some people add a bit of lemon or even mint just to add a bit more flavors. Enjoy!