WEBVTT 00:00:00.000 --> 00:00:05.546 So all these cartoon drawing, hopefully, is intuitive. I want this, this to be 00:00:05.546 --> 00:00:11.019 intuitive to you. This is, critical. The rest is just math, right? The rest is, 00:00:11.019 --> 00:00:16.491 there is a formula, whether you memorize the formula or not doesn't make a 00:00:16.491 --> 00:00:22.999 difference. If you don't know that concept in these cartoons, then you're in trouble, 00:00:22.038 --> 00:00:28.620 okay? Good? Fat window, skinny window, you get to decide and, for the application you 00:00:28.620 --> 00:00:33.208 have, what you get to do is just say, hey, I want to build a Gabor transform. How 00:00:33.208 --> 00:00:37.665 should I, engineer it? Well, you'd better ask what kind of application you're, 00:00:37.665 --> 00:00:42.478 you're looking at. Do I really need good time resolution? Do I need good frequency 00:00:42.478 --> 00:00:47.172 resolution? Which one's more important to me? That would tell you how you should 00:00:47.172 --> 00:00:51.748 dry, design your, your Gabor window and how fat it is, okay? Cuz you get to pick 00:00:51.748 --> 00:00:56.026 it. It's not like there's a formula. Here's what, how you should pick your 00:00:56.026 --> 00:01:03.512 Gabor window to be. There is nothing like that. It's all kind of manipulation that 00:01:03.512 --> 00:01:15.713 you get to do, okay? Questions on that or does everybody feel good about it. Now. 00:01:15.713 --> 00:01:21.799 Yes. I mean, you can like count the pixels cross and then just the Window open. 00:01:21.799 --> 00:01:27.125 Depends on how you use the, the Window. Depends on how you use Windows? Yeah, you 00:01:27.125 --> 00:01:32.388 don't say that constant that window here. You just like change your window every 00:01:32.388 --> 00:01:37.778 time. Okay. Oh, well so what you do is you slide the Window across and by the way if 00:01:37.778 --> 00:01:42.788 you guys, have you guys all gone to the web page? Have you looked at that little 00:01:42.788 --> 00:01:47.492 picture on the right? That's just a spectrogram and you know what the 00:01:47.492 --> 00:01:52.893 difference in the four pictures are? That width so as I run different widths 00:01:52.893 --> 00:01:56.754 through. If you, so if you, we, we're going to do that example in class, but as 00:01:56.754 --> 00:02:01.028 you run the different widths through what you're really doing is saying, Hey, let me 00:02:01.028 --> 00:02:05.147 analyze this, and let me change the width and see kind of what I need to get. You 00:02:05.147 --> 00:02:09.112 don't know ahead of time what you should pick, but since you have freedom, and 00:02:09.112 --> 00:02:13.231 MATLAB was easy, you just do it and see what you get, okay? So, and you can maybe 00:02:13.231 --> 00:02:17.041 use information from both, righ t? Your signal is there and you can process 00:02:17.041 --> 00:02:25.905 it in different ways, okay? a couple comments, first. Just like before you 00:02:25.905 --> 00:02:31.273 transform. When we actually apply this Gabor transform it's not on an infinite 00:02:31.273 --> 00:02:36.504 domain. It's not an integral where you integrate over all time and all space. 00:02:36.504 --> 00:02:42.010 Now, like Fourier transform you're going to say I'm going to discretize my time and 00:02:42.010 --> 00:02:47.379 I'm going to discretize my space. If you have measurements of a signal, presumably 00:02:47.379 --> 00:02:52.885 you've got a clock. You don't have the signal with infinite intestinal precision. 00:02:52.885 --> 00:02:57.909 What you have is okay I'm going to sample my signal every second, for twenty 00:02:57.909 --> 00:03:05.483 minutes. That one second sets your discritization. Right? You always have a 00:03:05.483 --> 00:03:11.856 discritize. So it's always a finite number of frequencies. Finite number of time 00:03:11.856 --> 00:03:18.309 points. Okay. That's comment number one. Number two, I wanted to at least introduce 00:03:18.309 --> 00:03:28.707 you to a couple other kind of important applications of this or not applications 00:03:28.707 --> 00:03:35.470 but a couple other for short fort short, a couple of other short alright ready short 00:03:35.470 --> 00:03:40.489 time Fourier transform methods, okay? That's what I meant to say it took me a 00:03:40.489 --> 00:03:46.133 little while. one of them and by the way the reason I introduced these two in 00:03:46.133 --> 00:03:51.070 particular is because they are particularly important for radar sonar 00:03:51.070 --> 00:03:55.466 technologies, okay? So all we're doing here is very broad-brush strokes. This 00:03:55.466 --> 00:04:00.859 stuff was forefront like, research back in the 50s and 60s when radar technology was 00:04:00.859 --> 00:04:05.197 being developed, right? And so, I mean, Gabor, the, the power of what Gabor did 00:04:05.197 --> 00:04:10.063 was that they realized early on that this fourier transform, although cool is so 00:04:10.063 --> 00:04:14.166 limited cause you lose all the time information. Which means you can't 00:04:14.166 --> 00:04:19.014 localize where's this plane coming from, I know there's a plane, right? If you pick 00:04:19.014 --> 00:04:23.888 up a frequency omega not and go hey there's a plane out there somewhere. The 00:04:23.888 --> 00:04:29.410 whole point is to say oh, and its over there about five miles. If you can't make 00:04:29.410 --> 00:04:34.550 that statement, then it's stupid right? sweet, there's a point somewhere, k. So, 00:04:34.550 --> 00:04:39.898 they work on it very early on, and a couple of the more important ones, one of 00:04:39.898 --> 00:04:53.780 them is called the Zac transform also known as the Weil-Brezin, okay? and here 00:04:53.780 --> 00:05:09.780 it is. Someone's got an iPhone. Oh, negative infinity, infinity so here's the 00:05:09.780 --> 00:05:25.868 discreet version of it you take this function, right? And, you see what you do 00:05:25.868 --> 00:05:33.129 is you don't necessarily build a, a filter when instead you do, is you basically 00:05:33.129 --> 00:05:37.872 slide your function across the frequencies, okay? So this is very 00:05:37.872 --> 00:05:43.209 important for periodic signals or quasi-periodic signals. This transform 00:05:43.209 --> 00:05:48.620 here. I don't want to, want to say anything else about it except that I just 00:05:48.620 --> 00:05:55.105 want you to know that transform and the other one I want you to have heard about 00:05:55.105 --> 00:06:04.058 was is this one here the Vigner. Excuse me. Yes. . good question I don't think I 00:06:04.058 --> 00:06:11.221 even specify here what it was. it looks like we won't have time to talk about 00:06:11.221 --> 00:06:15.273 that. I don't know what it is. I, I didn't write it down. But typically what it is, I 00:06:15.273 --> 00:06:19.375 think it's going to be it's a sequence going to be over the domain time where you 00:06:19.375 --> 00:06:23.277 discretize A of N. I think what it does is, this A of N is you take your domain 00:06:23.277 --> 00:06:27.228 from zero to time capital T. You chop it up into a certain number of points and 00:06:27.228 --> 00:06:32.571 then what you do is you recenter this across. What's that? Square root, yeah, 00:06:32.571 --> 00:06:39.720 okay? The other one that's important is the Vigner-Ville. and let me show you this 00:06:39.720 --> 00:06:46.870 one. This is probably the most important in terms of radar applications cuz this is 00:06:46.870 --> 00:06:54.447 kind of what people use a lot and this is and the vigular transform is actually 00:06:54.447 --> 00:06:59.924 probably, one of the more famous ones when you start looking at time frequency 00:06:59.924 --> 00:07:17.875 representations of signals. And here it is. Oh, I just won 25 cents. There's a 00:07:17.875 --> 00:07:23.414 typo in there, you know it's right there. That couch should be up here, not down 00:07:23.414 --> 00:07:28.457 there. I'm going to go give myself 25 cents as soon as I get back to the office. 00:07:28.457 --> 00:07:34.778 And you didn't even catch it did you? He's already got $1.25 from me, but at least I 00:07:34.778 --> 00:07:41.211 saved myself $1.50. So, alright, this is the transform. Again, generically, you can 00:07:41.211 --> 00:07:47.139 pick different Gs But this Vigner-Ville basically, notice what it does. It takes 00:07:47.139 --> 00:07:52.543 this function. Offsets it tell it over two one way. Takes this th ing, tell it over 00:07:52.543 --> 00:07:57.982 two the other way. And then you slide, you slide it this, so in one version of this 00:07:57.982 --> 00:08:02.024 thing, you're doing this, the other version you're doing this, dance moves I 00:08:02.024 --> 00:08:06.502 know, you could go to the Microsoft store, have you ever gone down there, I got kids 00:08:06.502 --> 00:08:09.778 that go down there, I think, I, anyway we won't talk about dancing.