So, warm welcome to the 28th lecture on the subject of digital signal processing and its applications. We had commenced the discussion of FIR filter design in the previous lecture and we had looked at the simplest approach to FIR filter design taking a cue from how we represented irrational numbers in finite precision. The easy way to do it is simply to truncate up to a certain level of accuracy. So, for example, we agreed that we would simply allow a certain number of impulse response samples to be retained and naturally the choice would be if you were given how many samples you could retain. Then you would choose in some sense the most significant of the samples, the samples that made the most difference and intuitively the samples with maximum magnitude are expected to make the most difference. And typically in most of the responses that we notice the most important samples lie around n equal to 0 and therefore we choose a certain number of samples placed around n equal to 0 depending on what length we are permitted to make the impulse response of the finite impulse response filter. Now, we were trying to understand the consequence of this on the frequency response and we had noted that if we considered an ideal impulse response h ideal n and retain this for n between minus capital N and plus capital N, this is the equivalent to h F r r n or the impulse response of the FR filter given by h ideal n into V n where V n is a window function and of course in the specific case where we are retaining some samples and throwing out the rest, this window function is essentially just a constant in that interval. So, in this case V n is 1 for n between minus n and plus n and 0 else for more n greater than n. Now, if we sketch this sequence which is easy to do, this is the point minus n, this is the point plus n, it is 0 else and if we put around this sequence what we call an envelope like this. An envelope which when sampled at the integers results in the window, then this envelope is rectangular, it is essentially a rectangular pulse and therefore we call this a rectangular window. So far we have not seen any other window but we can of course conceive of other windows by putting other shapes in the same place. So, for example, between minus n and plus n, we could put a triangle if we desire or we could put a cycle of a cosine wave, you know you could put the cos wave displaced by its own amplitude and therefore you could have the cosine function go this way and of course we could conceive of many other windows like this. These are all examples of shapes that one can put between minus n and plus n and then sample this, you see each such shape when it is put between minus n and plus n is then sampled at every integer. So, Vn could be essentially samples of different shapes put between minus n and plus n taken at the integers. Naturally as capital N grows, the number of samples increases. In the previous lecture, although we had introduced the idea of a window, we had not seen why we should write a general sequence Vn, now we know we could have shapes other than rectangular. We had seen last time was the rectangular window, we could have other shapes. In fact, if you look at the triangular shape, it is very easy to see that the triangular shape can be obtained by convolving the rectangular shape with itself. So, if you have a rectangular shape between minus n by 2 and plus n by 2, since this is an envelope, since it is a continuous function, let us say of the variable t, let us call it V continuous t, then V continuous t convolved with itself would give us the triangular window. So, we can see there are relationships between the windows as well and that will also help us obtain the discrete time Fourier transform in case we decide to obtain it for the triangular window appropriately sampled. Anyway, whatever it be, we will agree that in all these windows, once you have sampled that envelope at each of the integers, you get a sequence that sequence is called Vn. So, Vn, let us write down what Vn is, Vn is this continuous window function, window envelope if you like to call it that, sampled, so of course we have called the continuous envelope Vct sampled at t equal to n, we will teach us n and of course Vn has a discrete time Fourier transform. Yes, please, it is a question. So, the question is if the window be other than rectangular, would it distort the original impulse response? Well, indeed it would change the impulse response samples at the points at which they are retained, but that change might as we will see shortly be for the better, right. So, it would of course change the samples, but our ultimate objective is not so much to retain the samples as to get a frequency response which is as close to the ideal as possible. And therefore, what we need to see is what effect does the choice of window have on the frequency response, not so much the raw samples. Anyway, so you see one thing is very clear, Vct needs to be an even function of t. This is because typically impulse responses of ideal filters are typically even or odd in n. So, if we take all the 4 piecewise constant ideal filters, they are all even in n. We shall see some other non-piecewise constant responses which are odd in n, but either even or odd symmetry is or you know if the impulse response is even, we call it symmetric around n equal to 0 and if the impulse response is odd, we call it anti-symmetric. So, typically we find that impulse responses are symmetric or anti-symmetric about the center and if we wish to preserve this symmetry or anti-symmetry as it is, so symmetry should be preserved as symmetry and anti-symmetry as anti-symmetry, then the window function needs to be even that is not too difficult to see. Because if it is symmetric of course multiplying by Vn will preserve the symmetry, if it is anti-symmetric it will still preserve this anti-symmetry because corresponding points on the positive and negative side of n will be multiplied by the same number. So anyway this preservation, now why is this preservation of symmetry important? We shall see that it has something very strongly to do with the phase response. In fact, we can see that right away and this also tells us why we are in the first place looking at FR filters, why are we looking at, why are we preserving symmetry or anti-symmetry and the answer is to preserve what is called the linear phase or pseudo linear phase property. The same an FR filter has a symmetric response that means its response is even in n, let FR, let HFRn be even in n extending from n equal to minus n to plus n. The frequency response HFR omega or e raised to power j omega as you please to call it would be summation n going from minus capital N to plus capital N HFRn e raised to power minus j omega n. Now of course we are assuming evenness with the underlying assumption of the impulse response being real as it is for all the ideal filters but if it is not real then we will assume that it is conjugate symmetry. So we will say in general that HFR minus n is HFRn complex conjugated for real HFRn it is simply even. Now let us look at the frequency response from the point of view of its amplitude and phase part.  Now I have expanded showing some typical terms and what we do is to combine corresponding terms now on the negative and positive side. For the moment let us assume that this is real to make matters simple actually we do not really need to. Let us take in general that they are you know conjugate symmetric. So in that case each of the corresponding negative samples is going to be the complex conjugate of the positive sample and therefore we are going to have a combination of capital N terms of the following nature. This can be combined as HFR0 plus summation n going from 1 to capital N HFRn e raised to power minus j omega n plus HFRn complex conjugate e raised to power j omega n minus j omega n also complex conjugate because associated with minus you see HFR minus n is HFRn complex conjugated and the corresponding exponential term associated with it is e raised to power j omega n that can be written as e raised to power minus j omega n complex conjugated. So we have capital N such terms here. Now clearly this is the complex conjugate of this. So this term in its entirety is the complex conjugate of this and therefore the sum is two times the real part of this that is easy to see and that can be written as HFR0 plus summation n going from 1 to capital N two times the real part of HFRn times e raised to power minus j omega n and of course if HFR is real then we have a very simple expression. We have this becomes HFR0 plus summation n going from 1 to capital N two HFRn cos omega n and what is noteworthy anyway whether HFRn is real or not is that this impulse this frequency response is real in either case in any case the DTFT is real this is not worthy. Now when the DTFT is real its phase can either be 0 or pi. So this quantity this expression that we have here or this DTFT itself if you would like to call it that is also called the pseudo magnitude. This DTFT is like a magnitude except that it can be positive or negative. The phase is either 0 or pi but nothing else and therefore it is called the pseudo magnitude. If it happens to be positive all over then it becomes a magnitude the magnitude of the DTFT is itself. Now it should be noted that when you thus truncate so you know when you have taken the ideal impulse response and multiplied it by a window function to retain samples between minus capital N and plus capital N thus the corresponding FHR filter is non-causal. So we have a problem or we think we have a problem there you know if as it stands the FHR filter is non-causal but it is very easy to make it causal all that you need to do is to delay the output by capital N samples. We can easily make it causal by an N sample delay and N sample delay is very easy to understand in the frequency domain. All the N sample delay does is to impose an additional linear phase. The other way of looking at it is you are saying you have the original FHR filter you subject it to an N sample delay that means Y of N is X of N minus capital N and this introduces no change in magnitude. This is a linear shift invariant system very clearly and it is very easy to find its frequency response. The phase response simply becomes minus omega N. In fact you can write down its DTFT the frequency response DTFT of impulse response or frequency response is e raised to power minus j omega N. So it has a magnitude of 1 and a phase of minus omega N and therefore all that this does is to add a linear phase. Now of course this is not very difficult to understand intuitively. If you delay the output obviously the nature of the output is unchanged it is just going to come later and that coming later is reflected in the phase. So now we also see what happens when you have a phase response. The role played by a phase response is essentially shifting in time. If all components are shifted by the same amount in time then the phase response is transparent. All that you will see is that your output comes later. But if some components come earlier than others then you have a problem and that is what you mean by non-linear phase response. If the phase response is non-linear then some frequency components come before the others and then we have a problem then the fundamental nature of the signal changes. So obviously the desirable situation is to have as linear as possible a phase response and in fact even after thus making the FR filter causal the phase response of the FR filter is almost exactly linear. The only little bit of non-linearity is caused by the fact that the original DTFTO the original frequency response could be either positive or negative. So it is a pseudo-magnitude multiplied by a linear phase and that pseudo-magnitude is a magnitude then you have exact linear phase. If the pseudo-magnitude becomes negative at places then you have a problem that the phase becomes 0 plus a linear phase at some places and pi plus a linear phase at other places. That is much better than having non-linear phase. So this is clearly the reason why FIR filters have a great advantage over IIR filters. Unfortunately it is impossible to design infinite impulse response filters with linear phase. In fact I put this as a challenge before you. Prove that I know your rational filters if causal filters cannot have linear phase. That precludes a good phase response and we saw that in the Butterworth and the Chebyshev filters. There was a trade-off. When you went from the Butterworth to Chebyshev you would not do any worse in terms of resource as far as meeting magnitude specifications were concerned. But when it comes to meeting phase response which we have not specified at all the Chebyshev filter cannot do better than the Butterworth filter. So although the Chebyshev filter does not do worse than the Butterworth filter or might even do better in terms of magnitude specifications that means it conserves on resources to meet magnitude specifications. It unfortunately degrades the phase response and in fact as you go from Chebyshev to elliptic that is even more true though we have not talked about elliptic filter design. So this whole problem manifests itself in IR filters and in FIR filters we obliterate it entirely. We do not have a problem of phase response at all. We can get exactly linear phase and of course I told you right in the beginning when we started the engineering part of this course nothing comes for free in engineering systems. What you lose in terms of I mean what you gain in terms of phase response here you lose then in terms of resource. Ultimately a lot of engineering systems come down to the simple maximum invest more gain more. I can scarcely think they are very I mean it requires a lot of ingenuity it requires a lot of deep understanding to violate this principle into some extent. Of course we do to an extent at places but by and large it is always a game of investing more and gaining more or investing less and then also losing something. Of course the game also pertains to what you can lose. So you know sometimes you do not mind losing something and then you can gain something else and that is also game then what are you willing to lose. Anyway here if you wish to gain in terms of phase response you have to lose in terms of resource. So what typically happens is that to meet the same magnitude specifications you would have to keep a much larger length for the FIR filter and in the design exercise that has been administered to all students in this course you are of course expected to observe that. So you would be designing an IRR filter with given specifications and you would also be designing an FIR filter with the same magnitude specifications and you would compare the requirement of resource when you design it with the IRR approach and when you design it with the finite impulse response approach to meet the same magnitude specifications. But anyway we now need to complete our discussion on how you can meet a set of magnitude specifications.


The Dependency Scanning feature can automatically find security vulnerabilities in your dependencies while you're developing and testing your applications. For example, Dependency Scanning lets you know if your application uses an internal library that is known to be vulnerable. Here we see a merge request. When expanding the Security Scanning section, we can scroll down and see that Dependency Scanning has detected 10 potential vulnerabilities. They're sorted by severity. When clicking on a vulnerability, we can see detailed information on that vulnerability as well as how to resolve it. These vulnerabilities are also actionable. You can dismiss it and provide a comment to the security team. You can also create a confidential issue to collaborate on it later.


The people, the people are great. Well, it's really nice It's it's it's a new company. It's a startup. So it has a lot of flexibility I used to work for a bank so I can really come make it the comparison when you have a really like fixed environment highly regulated and and on the other hand you have the startup which you have time Flexibility, so it's great the company you have there in this this interesting topic So you have the the the opportunity to work on a little nice people study nice things So it's great for me. It's great


Hey, hey Sniffers. Okay, remember when I rescued Cody? This is another one of those situations. Okay, so I may post this in real time. Yeah, old school YouTube stuff because I want to get this information out to you as fast as possible. Sorry, I'm kind of in a hurry. I needed somebody to help me and luckily my friend happened to come over and volunteer. Yeah, I ignore the mess in my room. I've been working really hard to try to get a video for you, but here's what happened, okay? Somebody called my phone number and said, I'm sorry you lost your bird. I was wondering if you're interested in mine. Okay, well, you guys know that like that's not how it works. I don't just want to replace that bird at all. However, the reason I called this person back is because my thought process is you're willing to just give away your bird. The bird might need a better home. So I called her. I just want to see do you need help? Is everything okay? Like I want to know about every situation for every bird that I come across.  Now I have to go down there. She lives close by and I kind of told George about this, but I would need access to his van and I don't have it. So I have to go with my trustworthy Mini Cooper and the roof down and I wanted to go with somebody so I won't be alone and my friend just happened to come over who I haven't seen in two months by the way. Yeah, you Tracy. So that's what's about to happen. She actually doesn't. She actually has no idea what we're doing, but she kind of took a sort of guess. So I'm going to get blankets for the car. All right. So I'm getting some blankets, like old blankets to put in the car, just something to like cover the car and make sure I don't know how dirty the cage is or whatever. And I'm also going to be bringing, I'm also going to bring a carry-on cage so that, you know, obviously my roof's going to be down. The bird can't be in the cage. Okay. So Tracy doesn't know where we're going. She doesn't know what I'm doing at all either, do you Tracy? I'm suspicious because of the blank of that way that I'm picking up a bird. Only time will tell. I have to get something else to no, I think it's something else for our little trip. So it might be a clue for you. Okay. I need this or a squirrel. Why would I pick up? I mean, I would pick up a squirrel. Here's the other thing. I actually probably need two of those. I don't know if that's a word, but that's a good one. If I've ever heard a blonde make up a word, I like that one. Okay. Push these two down, push these in and move them over. Oh, I did that. Yeah. And then move the other one over. Okay. Now open it. Let's throw whatever's out. It's not like this. It's just simple. Yeah, I know, but it's so hard to get it out. Okay. Tracy's not going to be happy. The reason I'm going to do this in real time, probably without editing is because it's going to be the easiest way to get information faster out to you guys. That doesn't go in there. That's for the car, but you could put it in there for now so that you could see everything in real time. And the reason being is if I have to edit it, I'm going to be late on everything. And I don't want to be late on everything for you guys. I really want to keep up and let you guys see in real time exactly what happens. If you have some knowledge in birds and birds come your way to help, I honestly think it becomes a little bit your duty. So I'm not going to keep these birds, but I do kind of have a home in mind, but I just got to assess the situation, see how friendly they are, see what we're dealing with. I have no idea what we're dealing with. All I know is that she's really sick and can't give time to the birds and the birds might need to be separated. So let's see what happens. Why did I go to the passenger seat? Like I'm so not used to driving. Like what? I don't get out much anymore guys. Okay guys, we're on our way. I don't know if we're going to be able to film there. If it feels weird for any reason, like I'm invading their privacy, like we're not going to film, but I'll get as much as I can like after that. So let's see what happens. I've been out the driveway for a long time, but she's not answering the phone. And I can't get in the gate. So Tracy's going to see if there's like a buzzer somewhere. I've called her twice. I hope this isn't a fail because she said like, I have to do this before 7 p.m. Okay. This is Monty. Monty is in a wonderful home, but he's going to get re-homed due to some unfortunate circumstances. He's a cutie. He's like, I'm not leaving my toys. Okay, so that's Monty. And now in here, we're going to put Nelly. So the reason I'm not putting them in there just so you understand is because I don't want their first interaction with me to be about me trapping them or taking them away from anything. And sometimes birds don't like to get in carriers. So that cannot be my first interaction with them. I want them to feel very happy and comfortable with me when we get home because everything that happens right now is new. And since they're so well taken care of and happy, it's not going to be the easiest thing for them to adjust to being without their family. But it should be a good transition because it'll be okay. So who is it that likes to go upside down? Was that Nelly? Monty. Monty. Oh, Tracy going for it. Yeah. Good job, Tracy. I guess it's okay. I know. I know. It's scary noise. Go ahead and know. You can put on my arm. There we go. Okay. All right. Make sure this is closed. Snap tight. Make sure all the sections are closed, guys. And we're going to we're getting the car ready first. Can't see but the roof is going down. So we're trying to get this in there. So this is happening. So her son helped me put this in the car. Here's the update. She's really sick. She doesn't know what she has. So I'm definitely quarantining the birds. I would have done that anyway, but I'm going to have to be very, very careful. Here he comes. Tracy's got the two birds. We're not going far, but my seat is stuck. Put my seatbelt on for us really quick. Yeah, do that. See, you'll also have to hold this.  The window? Okay. Okay. Thanks. You just want to move the car if you can. Okay, we can't back out like this. So he's going to move the car. Okay. So here's the situation. Not only are the birds going to be quarantined, we're going to be quarantined. It just means that, like, when you go home, if you can before you touch Bella's shower, there's no like, she obviously has something viral within her, but I don't want to take any chances. But anyway, she's a really good person. She's really suffering right now. It's hard to be sick and take care of birds, but they were out and happy and, and well taken care of. Yeah, she's sad about it. So this happens all the time, guys. Sometimes it is better to get your birds to a place that will emotionally. Why is the car stopped? What's wrong with the car? Do you have to put your foot on the brake? Oh no. The car's not starting. Is it a car? Is it because your seat's not locked? I don't know. Oh no. Is it? It might be because the car knows your seat's not locked safe. Oh my god, it says that. It does? Oh yeah, like, it won't let me drive. Mini Cooper, you're pretty cool. Shoot, hold on. Okay, we're going, turning off the camera, being safe. Here we go. It's going to be a very slow drive. Guys, I'm driving up to my driveway and I see a car in the driveway, which means it's George and the lights are still on. I don't know if he sees me. He probably spots me.   He's looked, but I don't know if he's processed what's happening. Like, did he not process these? He must be in a serious conversation. He's not even, like, he's deep in this conversation. Like, he doesn't know I have a situation here. Like, even if I put the camera on, you won't even see it, but he's like, hi. Turn your light on. Just turn it on so my fans can see you. Are we on live? Do you see the situation here? Yeah. Come out of the car. Okay. What's their names? Nellie. Nellie? Nellie. My grandmother. Aw. And how did that happen? What did she say? Morty or Monty? Monty. Monty and Monty. What are they? Irish? No, I can't see you. Come into my camera, please. Hi guys. We're not live, but I thought it was live. No, but I, so here's the thing though. He's looking at me. Who's that? Monty. The one on the bottom is Monty? This is Monty. That's Nellie. Okay. Here's the thing. She has a viral infection. She doesn't know what she has. She thinks she has, like, West Nile or something. So these birds, this cage, then we need to make sure that they're, this cage, I think needs to kind of maybe just go in the backyard until we could pressure wash it. I can pressure wash it. Yeah. West Nile, is she okay? I don't know if that's what she has, but let's consider the worst, okay? Now her son doesn't have anything and he's been with her taking care of her. So there's no way that is. So like, I don't know if it's contagious. And have the birds okay? However, these birds, yeah, these birds have to be quarantined from our birds, which you know. Yeah. Except we might have to quarantine ourselves. So like, we have to continue to wash our hands, shower. Like, we can't handle these birds and then accidentally go handle ours. Right, of course. You want to talk to maybe Lou about it? Yeah, we'll talk to Lou about it. We'll go to Lou, we'll like investigate. Hi, Tracy. Hi. Tracy got roped up into this. She just wanted to come over and show me Bella's costume. And Bella? She didn't care about me. She only cares about me seeing Bella. She needed help. So I sent her on. But she did come and help me out. Thank Tracy very much. And the only thing that any of them says is Nelly says Nelly neglected. No way. Yeah. Does she really say that? Yeah. And she, it's from the home that they were in before. So it made her really sad. And she's a good bird owner. She really, the birds were out. They were happy. They had the plastic under. You could tell the birds are out all the time. Everything is good. Like she's great. She, whatever. Are we going to keep them in a certain room instead? We have to keep them somewhere else. They can be in my room. Can they? Yeah. That's very generous of you, George. Thank you very much. You're very welcome. I know you'd appreciate it. I do appreciate that.  We'll see. My gecko and my chameleon.


I was 18 when I looked through a telescope for the first time and the first thing I saw was Saturn, and I remember thinking, it really has a ring. To me, that was the coolest thing I had ever seen. And it made me wonder, what else is out there? Since Galileo pointed a telescope to the sky for the first time, scientists have been looking into the cosmos for over hundreds of years. Electromagnetic waves have shown us that there are billions of galaxies, trillions of stars, and an inconceivable number of planets and asteroids. That accounts for only 5% of the universe. The rest, we just couldn't see, let alone hear. What had changed on the night of September 14, 2015, I was just a young physicist freshly graduated, operating one of the two gravitational wave detectors in the US. At midnight, I pushed the observe button, and after 50 years of effort, humans were able to catch a gravitational wave for the first time. The world heard a sound of the two black holes colliding, 1.4 billion light-years away from us. This work won a Nobel Prize in 2017. Gravitational waves are all around us, they pass through us undetected. It is happening to you, me, and my three-minute thesis slide right now. Our existence is consistently being distorted, but the distortion is so tiny that even Einstein himself who predicted the existence of these waves doubted they could ever be detected. But we did. Scientists used light to catch gravitational waves, and light is made of tiny little photons. And if you were to zoom in all the way into the quantum realm, you will find that these photons act randomly. Everything is fuzzy in a quantum scale. It's like a sea of noise down there, and this noise is drowning out the already minuscule gravitational wave information. Scientists want a clearer signal. Last year I researched, built, and commissioned a photon-ricking machine called a quantum squeezer. Using entangled pair of photons that share quantum information by telling one what to do, I was able to manipulate the other. And by doing so, I organized this naturally chaotic sea of photons and enhanced gravitational wave signals. And with my work, scientists were able to detect 56 gravitational waves in just under one year, five times more than the previous total. And one of these waves revealed an astronomical object that scientists didn't even think could exist. Maybe now, I am closer to pushing beyond the 5% of the universe known to us. Not only because I will always be listening, but also I will be helping us listen better. Thank you.


It's a stilling process where you stop and pause. We can't always pause. I have complete empathy for why you've got to put it away just to go be an adult. But to get it back, I think it's the practice of noticing whatever is authentic to you, like when you feel that little tickle on your heart. You feel that little pull. Huh, what does that mean? Or you read a headline there that just gives you that eerie feeling of wanting to know more.