| 00:13 | Okay, okay. Okay. So to talking about transform, yes, |
|
| 00:31 | way stall the tail end of last and a couple of additional examples quickly |
|
| 00:43 | then all these other things. So you for use here is yes, |
|
| 00:53 | is the best serves up. It's patient enough and go to infinity. |
|
| 01:00 | a while then you can get an representation of the function as long as |
|
| 01:06 | or even just continues on the continues, continues and if it's not |
|
| 01:16 | Then at the jumps it four year here converges to the midpoint 15, |
|
| 01:21 | left and right. So the limit when you approach the and here's what's |
|
| 01:33 | higher compute these coefficients and this is you serious or the coefficients for the |
|
| 01:42 | part of the cosine function and for sign which is kind of papa please |
|
| 01:49 | thanks Frances stick formula. And then went through and said, yeah, |
|
| 01:56 | it's an even function then you don't to worry about the odd part of |
|
| 02:01 | fourier transform because that's from 20 LTD have to work for even less than |
|
| 02:10 | on back. That's the only need the integration point after the interval because |
|
| 02:16 | the symmetry. So from the half and from work to find coefficient and |
|
| 02:23 | , if it's all the same you can only think it's enough to |
|
| 02:26 | more than half interval And only one basis functions to science look at |
|
| 02:35 | And if I want to do the is kind of simple. So depending |
|
| 02:40 | an approximation, is it just for first step. Oh, basis functions |
|
| 02:47 | the cold, serious for your serious of the function. And yeah, |
|
| 02:54 | formula for computing what And it's kind different things of all. And the |
|
| 03:01 | -15 we went through, you socializing can be translation and scaling of |
|
| 03:10 | sport From being symmetric around zero. they shifted and different. So we |
|
| 03:18 | up and then we talked about kind this example actually showing, oh |
|
| 03:28 | you can sort of decided from this and depending on how many terms on |
|
| 03:35 | CS expansion of reviews they got. first one is kind of, it's |
|
| 03:43 | really the first one. The first is the full way became these two |
|
| 03:47 | and then it gets modified by the trick functions. And because around the |
|
| 03:55 | points here it is and odd So that's fine. They won't let |
|
| 04:01 | fine of the basis functions. No sign. It's not kind of odd |
|
| 04:08 | respect to the original thing, but respect to the midpoint. So before |
|
| 04:13 | see this trench and automatically figures it . But it is. And there's |
|
| 04:21 | one that is also an old function in this case about the midpoint, |
|
| 04:26 | can be accountable for first frequency and frequencies and are financially if you add |
|
| 04:33 | all up because closer and closer to to square away. But again, |
|
| 04:41 | I said, it's farther on the , so on this. So there's |
|
| 04:48 | one just together functions. So that just an example, the acoustics are |
|
| 05:05 | little bit far and that's uh so I looked at frequency analysis of signals |
|
| 05:12 | the acoustics or something else, best of the two things I'm interested in |
|
| 05:18 | best the energy that is basically than sum of the squares of the coefficients |
|
| 05:26 | and B. So it's kind of amplitude, sometimes a gift. And |
|
| 05:30 | tells you something about the energy of sun. The difference between that there |
|
| 05:37 | both pros and science kind of a issue um can also have, you |
|
| 05:43 | , check function and argument plus the and I started watching can unravel it |
|
| 05:48 | terms of but this is one way which aptitude for these questions and that |
|
| 06:04 | So any examples of that, I'm to make a little bit more but |
|
| 06:14 | the same time or contact represent. it's common to use whole lot of |
|
| 06:27 | instead of contrition coordinates, but on news, the four star. Um |
|
| 06:36 | so there is that trying to write down in terms of enforcement coordinates, |
|
| 06:45 | imaginary I The Square one. And is kind of related to frequent system |
|
| 06:54 | they come back from the next So it's common to write, introduced |
|
| 07:01 | notion here. Is that yes, with all of P as being the |
|
| 07:07 | period minus 5/2 5/2. And some the formulas last time and two I |
|
| 07:15 | kind of the standard interval for the function from a normal length and then |
|
| 07:21 | of this failing to an arbitrary instead the two points. And so this |
|
| 07:30 | going to show up in a lot sites and that's going to try to |
|
| 07:35 | it and have you have dots, going to be used in the exponent |
|
| 07:42 | peter and of course, um for reasons. And the Israeli ins are |
|
| 07:53 | to Greece and I will be mm . So here it says these are |
|
| 08:04 | coefficients we want to talk about Um the point of that have been |
|
| 08:09 | that in general they helped us a so and come back to that and |
|
| 08:20 | playing around with this notion of the . Okay, I'll pull our coordinates |
|
| 08:28 | appointment, remember that he today are some angle Theta this coast plus I |
|
| 08:37 | minus something. It's just simply that some moments that are So that means |
|
| 08:47 | that's the Express Co Xanterra as well science era in terms of these three |
|
| 08:53 | . Um and that's useful when we back to the for your formula that |
|
| 08:58 | before that. Okay, there was before we get there. So then |
|
| 09:04 | the other thing that is convenient to and we will use it is the |
|
| 09:09 | than and sign is simply just to for them to occupy the part into |
|
| 09:18 | angle. So because it's just go and if I said I would take |
|
| 09:24 | and that's that's what I forgot. that's something happened remarkable then um this |
|
| 09:35 | particular important property but there are other that um I wanna talk about it |
|
| 09:43 | a little bit but um this is normal if you have say that |
|
| 09:50 | what to do uh you invest to around the circle that's a function of |
|
| 09:56 | . I said it was zero and it's one, you know that part |
|
| 10:00 | then it's data is part of the . Then it's basically coastline is zero |
|
| 10:08 | I is one. So then you to the imaginary one and this is |
|
| 10:14 | to be used in the algorithms from about insurance. I have to compute |
|
| 10:19 | for your transforms but for now it's up it's this thing and the corresponding |
|
| 10:28 | . Oh it's you again the borrowing not in this china is the argument |
|
| 10:35 | me or something. And so um what um so I'm going to go |
|
| 10:47 | and then trying to have you have picture of the unit circle in your |
|
| 10:54 | . I find it very useful to particularly discreet for transform that we talked |
|
| 11:02 | hopefully soon. So um, so is kind of a useful thing. |
|
| 11:10 | ? Right. That's hard times into better and then it's the same as |
|
| 11:14 | and we have to play Just our for one I think on the next |
|
| 11:20 | . Yes. Next to the Is simply listing them. That's that |
|
| 11:29 | remember. And serious serious expansion. was some of things based on co |
|
| 11:36 | and sign of the whole set of . Nothing best to have the property |
|
| 11:42 | there was. Yeah. How about chill out. What do you |
|
| 11:47 | That's right. That's so it goes this. Right. So um ah |
|
| 12:00 | . The end inside that's what's. . Thank you for that. I'm |
|
| 12:12 | . Why don't you come back? here we have this impression for the |
|
| 12:20 | areas of the function over minus pi 2. 5/2. So now you're |
|
| 12:25 | there's kind of an argument um variable the argument of the Cosine and Sine |
|
| 12:31 | that is related to the charming and . So again this was on the |
|
| 12:39 | side and it looked like this. , they tried to use the expression |
|
| 12:45 | the coastline and sine as a function feet to the fire form instead. |
|
| 12:53 | that's the in order to fund things this formula call sign and sign. |
|
| 12:59 | terms of the new professional users the importance. So then uh if I |
|
| 13:08 | these things in here so I can an expression then for the function there |
|
| 13:13 | terms of now using the polar coordinates each or something instead. It's |
|
| 13:22 | Yeah goes in here and there And zero is found a trivial but some |
|
| 13:29 | and papers and that therefore versus sample . And it could plug me in |
|
| 13:37 | . Then this becomes the one. it's clear zero then become centrist the |
|
| 13:44 | of the fixed components concept going around easier to look at this one perhaps |
|
| 13:54 | that if n is positive, one we can do is to occupy this |
|
| 14:02 | and I then they get high on and I squared on the bottom and |
|
| 14:07 | needs to fight. So and then we get the miners are really comes |
|
| 14:12 | this part and you still have to that is common. Hey, they |
|
| 14:17 | pro sign as well as them the market by the a simple everyone can |
|
| 14:26 | then we can simplify the expression to sure I get it. Um And |
|
| 14:32 | kind of look at the negative I just turns out that all of |
|
| 14:38 | All right. But especially when you're these expressions into this formula and then |
|
| 14:55 | So and then it's not writing it , but it thinks using the outputs |
|
| 14:59 | the previous slide, the algorithm needs be gone by during the integration of |
|
| 15:10 | function with respect to. Um So Right inside here and to do that |
|
| 15:22 | one way is tend to multiply first then do the integration. So that |
|
| 15:28 | that's in integration and then we respect . Are you applying the outcasts and |
|
| 15:34 | on the right hand side the And then again expression for F. |
|
| 15:43 | and then you can basically change the of the subversion of the intervention that |
|
| 15:49 | . So then you have this expression and this is kind of the planet |
|
| 15:56 | this argument is zero. The investigator Iran and alpha still not depend on |
|
| 16:08 | . So that means the integration of something and it's basically a whole |
|
| 16:16 | Then it's zero before it becomes there's equal and much more negative and |
|
| 16:23 | . So the only case when this is not zero, it's from these |
|
| 16:28 | guys are equal. So that is you got. So then thanks for |
|
| 16:38 | 10 years, learn best way alpha one M equals m. The integration |
|
| 16:46 | are if this argument zero then it's one. So they will get basically |
|
| 16:53 | . Mm hmm. one over the is um this takes the form and |
|
| 17:06 | and integrated based on the internet. . No later. And that's why |
|
| 17:13 | have a way of death is given you. And this is the scaling |
|
| 17:19 | , the interval given to you. then you've got the Newport christian for |
|
| 17:24 | and to plug in into this So this is kind of representing then |
|
| 17:37 | to transform now using this polar coordinates of calculation coordinates. And uh, |
|
| 17:50 | of discussion of these rules of There is as the same convenient and |
|
| 17:55 | and they don't have this. And so and here is coming back to |
|
| 18:04 | potential formulas that we have a frigid . So so this kind of looks |
|
| 18:11 | ridiculous at first but that's really what want to have. Um So and |
|
| 18:17 | reminds you that any polynomial has a of rules equal to these degrees or |
|
| 18:26 | . And the planes in principle and This equation has 10 rules and I |
|
| 18:34 | want to try to find them by the former representation. And that was |
|
| 18:42 | one. It's just a jewel that order ah Certainly Pretend Equal zero. |
|
| 18:50 | is true. And also you go the circle so we have this part |
|
| 18:59 | from the definitional X. Yeah but for their this is the observation here |
|
| 19:06 | builds the set of explosive power is to one. And if you use |
|
| 19:10 | former representation and use the radio is one and it works so that we |
|
| 19:16 | have that um One equals this They all they also know that the |
|
| 19:29 | that is double period then okay for . We know that means that their |
|
| 19:35 | . So that's true. But it's true for every integer K. Uh |
|
| 19:42 | . Greater number. That's basically to but we're on the unit circle on |
|
| 19:46 | back And it's one. So for case this is also true. So |
|
| 19:51 | basically have Written one and 2 different this one and this one so that |
|
| 19:59 | can put them equal and said that is true. and then there's the |
|
| 20:07 | whenever theta is equal to support these exponents equal to each other. And |
|
| 20:13 | get for the terror values for which is true. Then that's the route |
|
| 20:20 | this equation. So a couple of is far too long to use |
|
| 20:29 | And these cars will need to make stay at home. So um here |
|
| 20:38 | kind of an an impression or for your top unit. Yes. You |
|
| 20:45 | that square everyone to call it C on this side. But basically There |
|
| 20:50 | two rules and there's one for Um and three that is not commonly |
|
| 20:58 | but use it and best digital 1/3 the way around the unit circle for |
|
| 21:04 | power away. And then they come after having done raised it to the |
|
| 21:10 | for great thanks. And the other is this set of groups. One |
|
| 21:18 | and -1. And either it's To 4th power. Um certainly use it |
|
| 21:24 | verify. One to the fourth car trivial minus four I. To the |
|
| 21:30 | car is for square it's minus one you square it again. Then you |
|
| 21:33 | plus one. The things that you see that these are actually of this |
|
| 21:42 | . So that's one not to talk this. Great. From them. |
|
| 21:49 | kind of pictures is helpful in this me personally and to try to understand |
|
| 21:54 | going on. Yeah. So um so this is the same what they |
|
| 22:08 | step forward. Okay. I guess have an example. So yes. |
|
| 22:12 | yes. So what I want to that often is a common way of |
|
| 22:21 | . Um this is or the series terms of all the coordinates is and |
|
| 22:32 | writing up. So there are two . Okay. You want to buy |
|
| 22:37 | the simple burial rites and omega into car. Okay. And then you |
|
| 22:41 | a subscript of em and the subscript inverse that tells you how many um |
|
| 22:50 | you have. four slices going on unit circle. Oh, so mm |
|
| 22:58 | . All the guys often have tortuous factors from the signal for a bunch |
|
| 23:02 | times but it's important to remember that kind of have to arguments with it |
|
| 23:10 | mm in the case and we will that from the banks of sites |
|
| 23:16 | So, so this is if and um goes between zero and n minus |
|
| 23:28 | . So two pi over N is of the minimum angle or the minimum |
|
| 23:34 | in the unit circle. And then . They just recounts the number of |
|
| 23:42 | . So 01234 exception. And then go back and okay becomes equal to |
|
| 23:49 | then we're back. So that's why goes from zero to environments on um |
|
| 23:59 | , so here was one and equals . Then the only option is K |
|
| 24:03 | zero and that's obviously one that's kind trivial if you have any close to |
|
| 24:10 | I want to these two points and even if okay can number zero or |
|
| 24:15 | can be one and this is the trivial and if it's one what you |
|
| 24:20 | is Into the two parts Times K Divided 1 and two. And this |
|
| 24:28 | actually alright bye. Which is basically moment. I'll have circle 2 -1 |
|
| 24:36 | then you can continue. It gets little bit fussy And the 4th |
|
| 24:44 | there's another four slices for the first is creeping up again. The next |
|
| 24:49 | is 1 and four. So this personal to the right over to and |
|
| 24:56 | best and now you are at this now the point of these things would |
|
| 25:07 | opening clear later when I talked about kids for your transformed because yeah, |
|
| 25:17 | remember the expression for the chanson was sum of our customs in today. |
|
| 25:30 | right. Or something they need to something is here. So it's basically |
|
| 25:36 | of Outkast times on a bus. omega is one, you don't need |
|
| 25:43 | do any work. It was multiplying one. It's kind of done Something |
|
| 25:49 | -1 is not really any work to figure out if we're going to go |
|
| 25:53 | and subtract and yes you have. if there's something that happens to be |
|
| 26:03 | , it doesn't require in the competition just the part of so and the |
|
| 26:13 | called basic for me to ask for transform and hopefully get to talk about |
|
| 26:19 | make use of the fact that some the commenters are simple. It doesn't |
|
| 26:25 | very dramatic operation. So let's say working computer William transforms And this was |
|
| 26:33 | one. Okay. So now I I'm more difficult. So any questions |
|
| 26:38 | that. So try to keep this of unity and the unit circle around |
|
| 26:45 | what you're going to use later you talked about your operating the summer |
|
| 26:51 | first Times omega ah the almond does but tells you how much you want |
|
| 26:58 | wait and then the letter wants. there is no picture. So I |
|
| 27:11 | think they're doing a good job in books to try to figure out continuous |
|
| 27:18 | to be discreet formula. Mm attractive make up for Harley gaps from |
|
| 27:27 | with integral zor continuous function to dealing discrete set of data. And then |
|
| 27:35 | in fits it's a single processing or that will monitor something you're monitored. |
|
| 27:41 | . Single. But I just don't to do for example, the single |
|
| 27:48 | most of this computing today it's So the sample things. And as |
|
| 27:53 | as the digitized today. Seven. for some filtering one that you might |
|
| 27:59 | with that animal form with defects. computers and work on examples. So |
|
| 28:08 | is from the administration of all all the time. Another person in Hong |
|
| 28:15 | . There are some references that show this one. Sorry about the |
|
| 28:18 | All right. So especially it is you know, for simplicity they assume |
|
| 28:26 | sampling rate and the duration of the interval is the capital. So you |
|
| 28:32 | a bunch of values. I should going. So everybody here is that |
|
| 28:39 | actually and a lot of continuous And the sample one is a set |
|
| 28:45 | discrete values that you get overtime for . and then one can index all |
|
| 28:51 | values. Which so and I would to do so one way of all |
|
| 28:59 | them relating the sample data to the continuous functions. There's simply kind of |
|
| 29:09 | is disrupted. That function. There functions is only one when the argument |
|
| 29:15 | zero and it's true everywhere else. , basically what it says here as |
|
| 29:23 | evolves. And this summation here. one of the terms in the sun |
|
| 29:28 | come to be Non zero. That 20 happens to be for this concept |
|
| 29:36 | K. You got different values the data. So, and then for |
|
| 29:44 | short time will this two then label samples together? Yes. So this |
|
| 29:54 | the way we wanted and representing the between the continuous function and the sample |
|
| 30:05 | . And we want to use that to try to figure out how to |
|
| 30:11 | you think So on the next So it's sort of this this is |
|
| 30:16 | representation of the samples from the continuous . and then two related to work |
|
| 30:23 | and should have been a minus. mean to find the key over to |
|
| 30:27 | an arbitrary capable of them. Um then we had a second rate of |
|
| 30:35 | and the confederates capital and here probably what the number of intervals |
|
| 30:40 | So you know how many um violence have within the period. And so |
|
| 30:49 | means when we take this thing off we don't need to go through um |
|
| 30:55 | because it's periodic function. We only to worry about What happens within one |
|
| 31:01 | from -1 to 2. And with notion of arcade then one can write |
|
| 31:08 | song as coming from minus and over to end of it too. Um |
|
| 31:15 | in this formation and then because again properties of the direct function, practically |
|
| 31:22 | only thing is content with these guys they contribute to this location from the |
|
| 31:27 | life. So now, so the machinery to get the proficient. So |
|
| 31:34 | what we have this most expression for four year series in foreign coordinates. |
|
| 31:41 | what, how we figured out what coefficients. Elf Lands were supposed to |
|
| 31:47 | by integrating this continuous functions multiplied by the function. Yeah. So now |
|
| 32:00 | have our best to find the alphas using all right samples there. So |
|
| 32:09 | , yes, all this expression plug the sample function that the principal looks |
|
| 32:16 | a continuous function with foreign states. continuous function thing. Um but in |
|
| 32:25 | this is not having this expression so is continuous but it's actually only nontrivial |
|
| 32:33 | points dynamic can exchange the integration estimation . They commute and if a co |
|
| 32:41 | data has no dependence on feet so the best we can change the order |
|
| 32:47 | integration and then also and you know expression again for integration over this period |
|
| 32:56 | after this great points when it is zero. Um So if you just |
|
| 33:05 | the tease for um which T. 20 is equal to change the |
|
| 33:14 | K. Then what you get is the only limit is This is non |
|
| 33:23 | , longer reported subsets of T So on the first. Yeah. |
|
| 33:31 | your girl come to the superior because just selecting this Kindergartners for the |
|
| 33:37 | So the song is still there but T just disparate ties into this |
|
| 33:45 | So yes, come on in the the that the out fast in the |
|
| 33:52 | of a sample single is now in some over the sample values within a |
|
| 34:01 | weighted by this or something. So is indeed the discrete fourier transform. |
|
| 34:14 | watching verse three kind of the riots representation of um the samples function from |
|
| 34:25 | continuous function. So I think I this on To have now 40 minutes |
|
| 34:36 | depression for example they are then you the efficient this way and then based |
|
| 34:44 | as I said this translation, this indeed what the discrete fourier transform, |
|
| 34:51 | looks like someone compute a set of , one for each. Ah and |
|
| 35:01 | of them, it is in there's a weighted sum on the samples |
|
| 35:08 | observed within the field and the waiting from the east of the so so |
|
| 35:18 | is that these kids from the transform That the vote 1st, I want |
|
| 35:25 | jump to then see how it relates the continuous thanks give you money for |
|
| 35:34 | dating, continue to transform. So question that rotation that is used from |
|
| 35:49 | uh special soul, okay, on side. So So now that's a |
|
| 36:01 | use extent also the example there as of your input, there was a |
|
| 36:07 | on the previous slides and then there an art book, right, which |
|
| 36:14 | the story of proficient on the previous . So here is a member of |
|
| 36:21 | structure what happened next the input after heat of the audience that before but |
|
| 36:31 | can see that them for each day that was the end on the previous |
|
| 36:35 | , You have a two I subject close. Yeah. And then we |
|
| 36:46 | , as I said, how many that are but this was the ranch |
|
| 36:51 | I was with 102 and oneness oneness was symmetrical. So again, so |
|
| 37:01 | is the typical way things have been and so these no and for twitter |
|
| 37:11 | most monthly on that. Ah so are not depending on the data and |
|
| 37:18 | this is each of the artists civilization on the unit circle and I want |
|
| 37:23 | explain. Um and so typically mm . Can you pre compute these |
|
| 37:38 | Some software they use don't um just of very wasteful someone having happened to |
|
| 37:49 | an effort free routine and depending on you use it, it should be |
|
| 37:55 | and trying see whether it's they're better a set of them ahead of time |
|
| 38:04 | then apply them for all kinds of sequences instead of very confusing them every |
|
| 38:09 | because they do not. They only on the size of the electron. |
|
| 38:16 | that's an important aspect. Remember the . So, another thing that they |
|
| 38:24 | out here, this is basically matrix product text has always been a vector |
|
| 38:33 | each. But today you have one , so to speak in the Matrix |
|
| 38:40 | for EJ. So there is kind the low index and is kind of |
|
| 38:45 | color. So that's the way you the matrix set them off. |
|
| 38:56 | so this is kind of the basic fourier transform computational. You should think |
|
| 39:01 | it as well renting spectrum of profession it's rarely used in this one. |
|
| 39:10 | cards, this matrix F that is guns here are various real opportunity. |
|
| 39:21 | this matrix very special property And these special properties is often used to um |
|
| 39:32 | our police are called fast food, is to rely on the top. |
|
| 39:43 | that's what that is. You worried about basically the inverse fourier |
|
| 39:52 | The university ST fourier transform is based using kind of it inverse of the |
|
| 39:58 | several factors. And if it works out to see that to do the |
|
| 40:09 | fourier transform, you don't really need new set of federal factors. So |
|
| 40:14 | nice recognize what's it to the minus instrumented for each of them. Thank |
|
| 40:22 | for your practice for whereas the But they said the other thing to |
|
| 40:31 | on this side is the way they oh will be transformed and it's important |
|
| 40:41 | coordinate. I am not normalized from I mentioned last time. So I'm |
|
| 40:50 | on you views things um, you root and scaling for the forward and |
|
| 40:56 | backward. You don't need an extra . But if you didn't scale the |
|
| 41:01 | person need to scale it for them . They came first. So if |
|
| 41:08 | has to be, but making sure you get something the same. So |
|
| 41:14 | curious in the former transform, why is here, wow, This |
|
| 41:24 | that they need to divided by end order to get the original. Mm |
|
| 41:37 | . Oh thank you. And the so far as welcome continues to be |
|
| 41:48 | , wow. Now, first of of the books slides voice make some |
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| 42:02 | about that. Mhm Yeah. Mm . So remember again how the F |
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| 42:12 | the collection of several factors. Has role for each of the wives or |
|
| 42:19 | outcomes of the proficient and the fourier . Uh it's uh rescuing the size |
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| 42:29 | the circle in two halves and there's then the subsequent is the number of |
|
| 42:36 | um in this case here, Is going to each roll of this |
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| 42:45 | of the matrix. So for each put substantive buying the portal transform into |
|
| 42:53 | input then in the expo owner and is okay. And then you have |
|
| 43:04 | hmm. Finding the why is this to the J. K. So |
|
| 43:09 | product of J and K is what up in this matrix here. |
|
| 43:15 | they the worst sub script is a of slices. So in this case |
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| 43:21 | four, let's get started for and there's cape times day showing up on |
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| 43:27 | here as the power of each of two parties divided by that. So |
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| 43:35 | time J zero. So the broadcast zero everywhere. Um similar in the |
|
| 43:41 | column, K is zero. Also the product of J&K is zero |
|
| 43:48 | And then you go to the 2nd . Now jay is fine. And |
|
| 43:54 | you see here is One times K saying No 1, 2, |
|
| 43:58 | Next time is jay is to some this two times 02 times 12 times |
|
| 44:05 | times three for them. 1, , 3 being the case. Did |
|
| 44:12 | get this set of exponents and then a perfect now probably this each of |
|
| 44:18 | empire Now divide by four and Then they just have each of the |
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| 44:27 | parts or final. The 22 parts the horses powers of That's the |
|
| 44:34 | Mm hmm. In order to ah power is zeros and then it's all |
|
| 44:41 | and it's all one in the E. I. Over two is |
|
| 44:47 | imaginary hiked all the way around healing . So that means ah this is |
|
| 45:00 | um we are I think I did pay attention to. So there is |
|
| 45:10 | of a sudden inconsistency and contradiction whether use plus or minus here. The |
|
| 45:18 | thing that matters is whether ah when do the inverse transform that they use |
|
| 45:27 | opposite sign from the forward. so look at this thing now that |
|
| 45:37 | best for you. If there was over to each of the pi over |
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| 45:41 | times one, that's plus I not . And that's why I discovered that |
|
| 45:47 | . Sure that in your mind this and then the next time is basically |
|
| 45:53 | squared with his mind as well. so they did. There's a difference |
|
| 46:04 | the sign. While I said it matter for the correct outcome. So |
|
| 46:10 | you have plus. All right, it for her picture on here in |
|
| 46:21 | stuff. So if it's plus in experiment, you go around this |
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| 46:28 | if you have minus in the experiment go around this. So it's under |
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| 46:33 | direction but it doesn't change the collection values. So, um so if |
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| 46:48 | was a minus here, you will this matrix. And I said, |
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| 46:54 | this essentially when you do that's four here. So this one man. |
|
| 47:05 | if the core here says this is native quantum occupy again and what this |
|
| 47:11 | and there's really no or maybe feeding let's see that At the time of |
|
| 47:16 | month, like three or -1 is subtraction is going to happen instead of |
|
| 47:22 | . And when you do create more modification and I definitely need start to |
|
| 47:29 | the real components, an imaginary components resources. So North is needed To |
|
| 47:36 | compute at four times or no twiddle them up. So here is on |
|
| 47:49 | same side. They were working it at four And doing destiny there four |
|
| 47:55 | 6 outcome in terms of that they for the transform and this is again |
|
| 48:03 | pain versus from now. And this again what the head of the previous |
|
| 48:13 | and front therefore we got the respondents to remember what it is, the |
|
| 48:21 | kind tricks. There's a reason for of petitioning this thing that will come |
|
| 48:26 | for the next slide and so the of this, next slide is really |
|
| 48:38 | . Go on to make this observation the structural properties of the FBI fast |
|
| 48:46 | transform. They want us to call tune transform. This is coming out |
|
| 48:54 | the book. Try to explain it a different motion autopsy. Yes. |
|
| 49:02 | the past four years transformed. Mhm. This matrix now is not |
|
| 49:12 | there is the force that we have . Ah and here was the ability |
|
| 49:18 | Matrix 16 x two matrix. Now turns out that if you do so |
|
| 49:25 | effect for the permutations of this Um You can in fact get the |
|
| 49:33 | subjects like this. So this is commuting rows and columns and just make |
|
| 49:41 | now after you've done this permutation. this. Uh huh. The original |
|
| 49:48 | nature that is comfortable. Um Now got this picture form of the resulting |
|
| 49:58 | nuclear in nature. Now I'll be that um this is in fact the |
|
| 50:07 | . Two and this is also the nature. So that explains this notation |
|
| 50:16 | this column. Now This is two and this is a two. |
|
| 50:21 | This is kind of almost f. except for their I. And the |
|
| 50:29 | and the same thing is also almost except the signs and things have become |
|
| 50:37 | not supplied by the high. So this is also kind of related to |
|
| 50:43 | two but it's multiplied by the diagonal in the sense of The first row |
|
| 50:49 | the same as in f. The next show has been modified by |
|
| 50:53 | design. So that's kind of multiplying a diagonal matrix tax ft Um this |
|
| 51:01 | is essential listing with the minus sign negate all the things on top this |
|
| 51:09 | here. So in the end somewhere , is that showing what diagonal matrix |
|
| 51:17 | plugs in order to that's always take diagonal Party supply. This F two |
|
| 51:27 | yes, 13 months of Matrix. what's the point of this? The |
|
| 51:34 | is, yeah, If you have at two again at four doesn't require |
|
| 51:44 | more work than this failing Of F and the summation. So you're welcome |
|
| 51:56 | that. So there are many things happens in this process and use the |
|
| 52:01 | of what happens, you know? it turns this permitting. So now |
|
| 52:06 | need to fill it up. So we introduced try to do this by |
|
| 52:12 | the structure of this thing. They need to Correent the consequence of the |
|
| 52:19 | on that. So you can do competition in one part and then you |
|
| 52:26 | to work on there were x also inverse fermentation nature. So if you |
|
| 52:33 | that, you'll get a good structure looks like this that doing in fact |
|
| 52:42 | of sort in this case the input to even index mm hmm input values |
|
| 52:50 | our index into the fire and then do and that will transform half size |
|
| 53:01 | XC and into another half size on on guard. So the question |
|
| 53:10 | what was the point of this? yes. So instead of doing 14 |
|
| 53:16 | four brand that are giving to So it doesn't seem to be much |
|
| 53:23 | a game. But the point is can do this very personally. |
|
| 53:31 | that's what I'm trying to show Yes. So that's why instead of |
|
| 53:37 | something the have more interest plantations ah your book. So I'll just common |
|
| 53:48 | . So matrix vector multiplication in terms arithmetic operations we know is n squared |
|
| 53:55 | in the end violent matrix and takes squared operations some of protection but exploiting |
|
| 54:02 | properties of the S matrix of these . You in fact and um there |
|
| 54:10 | animals changing the arithmetic complexity 10 times . They constantly personally apply this game |
|
| 54:18 | moving forward and forward. Thanks to guys twice as many half sites at |
|
| 54:25 | time we do it. But the repeated a number of steps. So |
|
| 54:31 | the work can accept this kind of and a bunch of so I will |
|
| 54:46 | to talk about. Okay, what you doing this? I think. |
|
| 54:52 | . Because if farmers aspects important computation but this is a personal way of |
|
| 55:01 | it is known as the school to associated with it. But they were |
|
| 55:09 | the first to discover it. But even if it was discovered that it |
|
| 55:16 | the first ah and stuff. All actually I don't know what the person |
|
| 55:23 | , but it was just one scientists their own industry industry that ended up |
|
| 55:31 | for the transformers? That seems a . I don't like everybody else. |
|
| 55:45 | . So the one thing that is too as that. It's your |
|
| 56:02 | They are originally in there. So nicely from zero to N. In |
|
| 56:09 | of your samples now you kind of them so you could even once before |
|
| 56:15 | other ones. And now if you this recursive lee to go from there |
|
| 56:23 | or big size too small to buy that is the smallest too, then |
|
| 56:31 | hands up the order this time and . And then what happens is what's |
|
| 56:42 | thing here. So it's known as controversial and honor show that on some |
|
| 56:52 | from that faction that works. But . So the one way of doing |
|
| 56:57 | competitions is you can write down barn coat what saved the day is here |
|
| 57:06 | then you can just reverse the order the bits in that fourth step. |
|
| 57:12 | I'm trying to show that what the means. So you have zero and |
|
| 57:19 | from left to right and then you to write right to left instead of |
|
| 57:24 | that can change zero. This If you start to write it down |
|
| 57:29 | the opposite order. This one from goes to Now, then eight or |
|
| 57:34 | highest numbers of one is not first of that. So this is just |
|
| 57:39 | example of taking this and then but think things done and will suffer officer |
|
| 57:44 | the order of debates. So this that's where the picture ourselves impact |
|
| 57:54 | So this is important. Um, was implicit In the past four means |
|
| 58:02 | it is causing this the organization because the sorting into order and even the |
|
| 58:11 | the documents to have accepted input variables apply the platform and transformed to it |
|
| 58:20 | the output is not necessarily in the party expected it to be the boss |
|
| 58:26 | to find it by the controversial by the actual suffers who's that and I'll |
|
| 58:33 | to show if anyone is interested in due north Carolina sympathies in particular on |
|
| 58:46 | . So they were distributed to transform is a good one to best in |
|
| 58:55 | of network trying to help. It one of the politicians that Children so |
|
| 59:05 | I want to know and probably um are algorithms, yeah, some of |
|
| 59:16 | there are so called sorting algorithms that of built in the implementation to undo |
|
| 59:27 | effects of the implicit controversial that transform so for somebody to use libraries I |
|
| 59:38 | need to be aware of where there's self supporting version or The most informal |
|
| 59:47 | . Okay, now the things I to highlight, I'll talk about in |
|
| 59:52 | book. So sorry, two Basic and I'll talk about what happened Most |
|
| 60:01 | us share of books and seen they about this one and they are no |
|
| 60:09 | in computational complexity. So that the in terms of how they need with |
|
| 60:18 | rotational factors and the trivial factors and different and how you get memory |
|
| 60:25 | So they have very different behaviors and a particular report distributed cluster. Um |
|
| 60:36 | I'm trying to feel it and then . Trying to cover this. Thank |
|
| 60:49 | . So for computation efficiency in particular gas And computational efficiency worry about two |
|
| 61:02 | . one is the number of our which is important sometimes, but overall |
|
| 61:11 | the number of memory references you which is an ordinance to discuss access |
|
| 61:16 | need. It's also more important than account of operation. And that's why |
|
| 61:22 | things come from. Mm hmm. now let's start with this upon the |
|
| 61:30 | and frequency wants to do. So the gun. Now, the general |
|
| 61:38 | of these little factor matrix. They mentioned some of the previous ones. |
|
| 61:45 | one thing to notice them because of property. Uh the field factors began |
|
| 61:49 | to go around the during a circle . Uh that's the other piece is |
|
| 61:57 | as one. So even though this looks different, they are now. |
|
| 62:03 | basically we should go down halfway down to be over too. The rest |
|
| 62:07 | the role here is basically the same the first part. Except there's a |
|
| 62:14 | . So what is the one way uh huh being one of their projects |
|
| 62:20 | trying to do that here in expression terms of then. So project um |
|
| 62:29 | there was this a matrix vector You have reported factors Now the key |
|
| 62:37 | Transformers, the three points and then compute and the other version of the |
|
| 62:45 | . And then there's this case today the summation index going through a role |
|
| 62:51 | I always found them take the first of the role and the next second |
|
| 62:55 | of the roll, then we can get their proper both input factor and |
|
| 63:03 | factor about this. Today parents sent rain entity over to if you do |
|
| 63:11 | then you can uh yeah, there some current theater. Right, so |
|
| 63:19 | have this benefactor times P over two that was effectively -1 to the power |
|
| 63:26 | my God, a subscript p to power of two is -1. And |
|
| 63:30 | there is tell here that uh Kind expansion exponential -1 here. And then |
|
| 63:39 | have so basically you get the same factor multiplied by either a sum or |
|
| 63:48 | or to improve. So here's the and that was that Okay, so |
|
| 63:57 | we can do one more thing. this was so this just kind of |
|
| 64:03 | size transform if you like the summation half you have two elements here. |
|
| 64:11 | there is an additional kind of trickery by looking at even advertising. So |
|
| 64:21 | such as so if they look at components then you basically have the place |
|
| 64:31 | how to do so quit crying then now that he wants From there or |
|
| 64:38 | two more cartoons. So so you plug it in. It's nothing |
|
| 64:45 | It's not now but So this one basically from 0 to T -1 And |
|
| 64:52 | not trying to value from 0 to . So if I get in here |
|
| 64:59 | once and for getting the odd ones once when you have a two and |
|
| 65:05 | experiment is the same as having P there too in the orphanage. So |
|
| 65:12 | all that happened here I think. now for the off the shadows here |
|
| 65:20 | still having to circle it down for little bit here. So this gets |
|
| 65:25 | no, the first part is this and then we have a one |
|
| 65:31 | That is this additional diagnostic. So this is, it's kind of one |
|
| 65:41 | sized transport. The fact is Fruity number two. The range of |
|
| 65:49 | compute author is has a range of have the number of values. As |
|
| 65:58 | as the number of values. I that's the same here. The difference |
|
| 66:04 | the first one is that this is . So there's a minus here and |
|
| 66:08 | there's they stood the factor that Yeah, the results of subtracting so |
|
| 66:15 | can help. So this is the . Um you put that into the |
|
| 66:24 | this is also have transformed over the . So the bottom of life is |
|
| 66:30 | have such things as these. But these properties so you have the competition |
|
| 66:36 | is to some two values or subtract to values and multiply what the |
|
| 66:42 | So one year this was popularly known butterflies as for your transform literature to |
|
| 66:49 | about my butterfly computation. And this says we have these two input values |
|
| 66:57 | either add or subtract them. Oh okay and then the one that got |
|
| 67:04 | just multiplied by the factor. So kind of the competition that is done |
|
| 67:10 | each step. Oh this ah fast to transport. So it looks like |
|
| 67:16 | is what I did. So basically take an input value and you combine |
|
| 67:21 | where the other riders that was half distance away. That's the P over |
|
| 67:27 | . And then you do to have transports. Now what they also did |
|
| 67:33 | that case was yeah we have these so the given ones and then we |
|
| 67:41 | the old ones. So in this what we get is um the ones |
|
| 67:47 | did just the addition that corresponded to even ones for the ones that actually |
|
| 67:53 | by the tribal factor after subtraction. correspondent to the audience and then you |
|
| 68:00 | this for something and then you get like this. So that's good. |
|
| 68:05 | here is that the first stage that halfway across, That's the way you |
|
| 68:10 | input values and then one just multiply there and fiddle factors and then you |
|
| 68:19 | to have size and then you repeat same thing. So here's 1/2 size |
|
| 68:24 | has one half size and public dependent now they turn the factors is if |
|
| 68:30 | use the same p bass, it's the 024 and six 412, 3.5 |
|
| 68:38 | the team. So the best and is the one that repeats itself. |
|
| 68:44 | as I said, the first thing started with them as what's known as |
|
| 68:47 | order input And we're taking within this from zero and 1 words and what |
|
| 68:54 | gaps on the output and um resorting odds and even in terms of the |
|
| 69:02 | , this is in fact where the values are so to invest in do |
|
| 69:09 | things and right things back where they from in memory. Then as I |
|
| 69:13 | that a result appears Uh in the location on the themed books some of |
|
| 69:21 | one. So this is kind of # 14 or whatever it is. |
|
| 69:28 | location instead of this location comparative. Sorry, this is occasion or it |
|
| 69:37 | but it's It's the intervals from the 16 input. This so these investor |
|
| 69:44 | traverse studies we found on the Yeah, so this is probably what |
|
| 69:51 | controversy. This submission. Made So this is normal and that was |
|
| 69:59 | traverse and so no mm mm And practices usually 50 thing in particular. |
|
| 70:09 | it's distributed these so it's important to out if the former assume that you |
|
| 70:16 | your data sets you know some sub other processors question between the factories and |
|
| 70:24 | not going to be too much into time but this what I'm trying to |
|
| 70:29 | here is that together so and this going to be a difference compared to |
|
| 70:36 | destination of time. That is often in articles or books and protected |
|
| 70:43 | So in this kind of estimation frequency it's called. The view is kind |
|
| 70:48 | the maximum set with benefactors in the branch and then it is the subset |
|
| 70:53 | other one And every 4th 1 etc . These critical factors is the substance |
|
| 71:00 | what to use in the first stage this okay and there was the civic |
|
| 71:12 | of factors based on the stage. I'm sorry so you do things and |
|
| 71:21 | the other destination in time that tends be the one that's most distressed it |
|
| 71:26 | with out instead of trying to do in terms of output they do um |
|
| 71:34 | even respected the same gang loca half the input data points and the other |
|
| 71:44 | of the data points so still half the inner product this one on evening |
|
| 71:54 | potato plug it in and simplify the . Yeah again through an experiment that |
|
| 72:03 | replace the doctor too and the subscript thing for the other trump card |
|
| 72:11 | So now we have something that looks this. Now the range of |
|
| 72:15 | Is still the old ranch where it the summation range is half of them |
|
| 72:21 | we want to try to find we also use half the range for |
|
| 72:26 | outputs and expertise. So now you instead of mm hmm with L and |
|
| 72:35 | even else now you have not even ah as opposed to halfway the input |
|
| 72:44 | on the evening the input halfway and output and the buckling to the formulas |
|
| 72:52 | you think that happens here that if few of the two here arrange ah |
|
| 73:03 | we have this guy here To be the two And it's the one place |
|
| 73:13 | The over two, it's the same the base for them that stays in |
|
| 73:18 | so we don't have never bought that and then we have Plus B over |
|
| 73:22 | and this one too Plus B over and this one and that's where so |
|
| 73:31 | Iran that means that becomes a negative . Yes sir. So this is |
|
| 73:39 | now have the range in terms of after that and from that and there's |
|
| 73:45 | place for teams ah you have successfully the differences kind of looks like this |
|
| 73:51 | that case one of the here is of the inputs is occupied by this |
|
| 73:59 | for cases and then there's at the . So in this case the twitter |
|
| 74:04 | is the input and then it's the artist so it's a difference where the |
|
| 74:10 | effect that happens to be on the around the output from this and so |
|
| 74:17 | what this thing is kind of reverses there, I think input there then |
|
| 74:26 | goes backwards but it's that so long , halfway across in terms of ah |
|
| 74:34 | data and then twitter was the So you can gamble but then you |
|
| 74:45 | required to process them have to go come The last stage towards the first |
|
| 74:51 | when you're adopted tested 2x2s and there's past minus because is on the unit |
|
| 74:59 | and if you do this thing so know, normally when you discover that |
|
| 75:04 | the result order made the input and disorder in order to. Ah yes |
|
| 75:10 | course for the normal order. Uh . The other thing is also the |
|
| 75:18 | thing in terms of the swivels, use them All the twitters in the |
|
| 75:23 | stage of his F50 and suppose the stage and previous ones are always the |
|
| 75:29 | . Um Now so this was I want to get to restore order and |
|
| 75:37 | one of the factors once again said one last point questions well I'll do |
|
| 75:50 | and I will repeat that next The point is there are a few |
|
| 75:55 | about these things. So we'll talk and more so all some of its |
|
| 76:03 | as well. For the desperation interpersonal violence is not true. What |
|
| 76:13 | true is that affected the traverse is order. So you can also use |
|
| 76:20 | destination time for normal order in from chooses, but then your similarly you |
|
| 76:27 | use destination time, destination frequency, to hearst inputs if that's what you |
|
| 76:35 | . And I'll talk about that next . I'm supposed to also have a |
|
| 76:39 | forward and inverse because many applications they a piece to come with certain computations |
|
| 76:47 | they said for your space so they back and forth between forward and |
|
| 76:52 | So I will talk about thank You're welcome. Mm hmm, mm |
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| 5999:59 | |
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