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18: Frequency (Fourier and Related Analyses)
615 Plays7 years ago
Duration and proximity are, as demonstrated by Fourier and later Einstein and Heisenberg, very closely related properties. These properties are related by a fundamental concept: frequency. A high frequency describes something which changes many times in a short amount of space or time, and a lower frequency describes something which changes few times in the same time. It is even true that, in a sense, you can ‘rotate’ space into time. So what have we learned from frequencies? How have they been studied? And how do they relate to the rest of mathematics?
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Transcript
Introduction and Equipment Issues
00:00:00
Speaker
I'm Jonathan and I'm Gabriel and today we have on an episode that started out as a minisode but it snowballed into a full episode. Why are we mentioning all this? Because when we record minisodes we still strive for quality because we're giving you something that is ours and we like to be proud of that but we also like to experiment with new equipment and new techniques and that's when we do that.
00:00:22
Speaker
That's right. And recently we just got a new preamp, the thing that connects the mic to the computer and we accidentally over drove it. And you can hear that in this episode. We're also telling you, because this is an episode we're otherwise very proud of. And, uh, we decided it'd be better to let you hear it. Then, uh, not explain why it's like that. That's right. The content is something that we are just so happy with, even if the audio sometimes gets a bit fuzzy.
Podcast Promotion: 'Physical Attraction'
00:00:50
Speaker
So thank you for listening to Breaking Math and we present Episode 18, Frequency. There is a new podcast that I would like to tell you about. It is called Physical Attraction. This is a very fun and clever show. The podcast aims to combine two separate topics, physics and clever picked up lines into one. You heard that correctly. Learn about the universe and tips on chatting up a potential date.
00:01:12
Speaker
You can find the show on Apple Podcasts, or wherever you listen to podcasts, and on Twitter, at PhysicsPod. Check out the show. I think you love it.
Guest Introduction: Tyler Jordan and His Work
00:01:29
Speaker
I'm Jonathan. And I'm Gabriel. And you're listening to the frequency episode of Breaking Math. And today we have on Tyler Jordan. And Tyler, can you tell us a little bit about yourself?
00:01:39
Speaker
I am an electrical engineer. I work in communications engineering, which is basically engineering the components and algorithms that let your cell phone talk to the network or let your computer talk over Wi-Fi to the internet. Those kinds of things.
00:01:56
Speaker
Now you mentioned a few things I have to do a little bit with frequency probably, right? In what capacity does your job have to do with frequency? Almost everything. When you work in signal processing or communications, you think of signals or, you know, these waves in the air. You think of everything in terms of both time and frequency. So when did this occur? And then what frequency was it at? And you can think of those as musical notes kind
Historical Context of Frequency Studies
00:02:22
Speaker
of thing.
00:02:22
Speaker
Awesome, yeah. And before we start talking about frequency, we're going to talk a little bit about the origins of the study of frequency, which have to do a little bit with Aristotle. There's the famous story of Aristotle hearing a bunch of metal workers make beautiful sounds with their hammers by accident. And when he measured the metal, he figured out that there are whole ratios of one another. And that's one of the earliest music theories and one of the earliest, even if it's not completely rigid, theories about frequency itself.
00:02:51
Speaker
Yeah and that is really interesting because you know I'm saying oh we think of these mathematical signals in terms of time and frequency but our ears do that anyway whenever we listen to music. We hear not only that a note occurred at a certain time but we also hear was it low pitch or high pitch and that actually corresponds exactly to frequency you know as you were saying the the length or tension of a string or the metal that the workers are banging on.
Promotion: Brilliant.org and STEM Learning
00:03:17
Speaker
I'm Jonathan. And I'm Gabriel. And we're here once again to talk to you about Brilliant.org. Brilliant.org is where STEM happens. You can learn math, science, and more on their website. This time we're showcasing their CS classes. If you've never done computer science, if you're just starting out, you can take their computer science fundamentals course.
00:03:36
Speaker
Or if you're more experienced, you could challenge yourself with a course like computer memory or neural networks. These can help you in school or even professionally and in life. So to support the show and to maybe support themselves in life, where can they go? They can go to www.brilliant.org slash breakingmath. And yeah, it'll help us out. And now onto the rest of your episode.
00:04:02
Speaker
Yeah, I gotta say, I'm actually really, really excited about this episode. I mean, frequencies are just about everything. There's a quote from Tesla, from Nikolai Tesla, that I didn't write down a specific quote, but he says something to the effect of. His goal is to try to understand everything in the universe in terms of frequencies.
00:04:17
Speaker
It's just a really cool avenue of mathematics. Also, as you had alluded to, we can talk about frequencies with sound and music and ratios and anything that happens over time, as well as your current job, electrical engineering. I think that this episode is really rich for things like analogies and it's got a lot of
Explaining Frequencies in Engineering
00:04:40
Speaker
applications.
00:04:40
Speaker
I think the goal for this episode, by the way, when we were planning this episode, we wanted to start off with just talking about conceptually what frequencies are, and then also at a very, very basic level, the kinds of engineering that people can do with frequencies, things like AM radio and FM radio and how all that works. Or even civil engineering design with buildings and bridges, and we'll get into that later, right?
00:05:06
Speaker
And the important concept when we talk about frequency is, of course, the sinusoidal wave. I'm gonna give an example of what I came up with, and I think that you have an example that you wanted to give too.
Analogies for Understanding Sinusoidal Waves
00:05:15
Speaker
The one that I wanted to give was, if you're seeing wheels on a train, there's always these two little segments that are connected by a straight line. So as the wheels go around, there's this bar that goes around in a circle. If you only look at the up and down motion of that bar, that's the sinusoidal wave.
00:05:32
Speaker
Yes, so if you could imagine, like we were saying with that kind of wheel analogy, if you see the circle and first you draw a horizontal line through that circle right through its center and then we draw a line from its center to any point on that circle. Now that line is always the same length and it's the radius. So as we rotate that line through that circle we can always draw a triangle.
00:05:59
Speaker
one the long side of the hypotenuse and it goes out to the circle and then comes right down to that horizontal line. So this is a triangle and so some people say a sine sine cosine how does trigonometry have to deal with waves and so we're going to look at that point that line as it rotates around the circle and it always is drawing that triangle
00:06:19
Speaker
And what the sign is, is the ratio of the opposite side from the angle to the hypotenuse. And since the hypotenuse is constant, we can look and we can say, that is growing in a certain rate. It gets positive and then it starts descending and then it gets negative. And so, like he said, as that bar goes up on the rail wheels, we can see it goes up and down, up and down.
Teaching Tools: Visuals and Applets
00:06:46
Speaker
Real quickly, I'd like to say that one of these days, once our Patreon has enough, we'd love to have a live breaking math show where we all have little pads, little sketch pads, are able to show a lot of visuals because I think that would be really helpful in this case.
00:06:59
Speaker
And a visual, if you like one, for this case, you can check out breakingmathpodcast.com slash applets.html and click on the 48 Transform applet. It has a pretty good visual representation of the, not necessarily a triangle that was just mentioned, but of the, the links involved.
00:07:16
Speaker
And actually, I have one other analogy that I wanted to talk about with what a sine wave is. And I think it's really great that we're starting with these fundamentals. Sine waves are oftentimes, there's a term periodic motion. I think of a pendulum or I think of, you know, when we think of hypnosis, you know, somebody swings a pendulum of some sort back and forth and back and forth, there's, you know, one thing that's swinging back and forth. Well, if you imagine a pen or, you know, something to mark on pencil at the end of that pendulum, and it went back and forth,
00:07:46
Speaker
If you move that pendulum forward, or if you move it in one direction as the pen is going back and forth, depending on the rate, you can actually have it draw out a sine wave, sort of. And we call it a sine wave, if you think. I mean, everyone's seen this when ripples in a pond. When I say a wave in water,
00:08:06
Speaker
Don't think of a crashing wave or a surfer wave. Think of a ripple in a pond. And you see these little waves travel through the water. And so when we say a sine wave, we're essentially talking about that perfect ripple that's going through the water there. And so kind of think in terms of that. Almost like an S turned on its side, and then it goes on and on. There. Those are all my analogies for what a sine wave is. I'm assuming that that's quite adequate.
00:08:33
Speaker
Oh yeah, for sure. Now, one thing about sine waves, and we're going to get to other types of waves in a little bit when we talk about different types of transforms, is that sine waves are just a very basic component, and they're used all the time in describing physical waves, EM waves, electromagnetic waves, and the Fourier transform.
Understanding the Fourier Transform
00:08:55
Speaker
Now, the Fourier transform, we touched a little bit on it when I mentioned the applet earlier, but would you like to give an example of when you might want to use the Fourier transform?
00:09:03
Speaker
So the Fourier transform is a really interesting piece of math. And what it does is it lets us take a signal or a function, something that happens in time. And so even my voice being recorded right now by this microphone, basically a few thousand times a second, it measures the amplitude of my voice and that goes through time. Now we can take that
00:09:28
Speaker
And we can send it through a Fourier transform. And the Fourier transform will take this time domain signal and put it into the frequency domain. It is a function that maps the time domain into the frequency domain.
00:09:42
Speaker
Yeah. And that is quite a task for an audio podcast. So I'm trying to think of analogies, you know, like how you have the same information, you know, obviously in the case of the FOIA transform, it goes from something represented in time to something represented in frequency. But what, what's, what's some kind of analogy for information, like maybe going from a spoken language to a sign language, you know, that like, I mean, I know that's not a perfect analogy, but I'm trying to convey the basic idea, you know.
00:10:08
Speaker
Well, one way of looking at it, I think, is you could think of when you go to the store, you could think of each moment that you're going to the store as a different thing. Like you're walking to the store, and now you're picking up bread, putting it in the basket. Or you could think about all the frequencies involved. For example, you go to the store once a week, when you put the stuff in your basket, you're swinging your arms, and that has almost a slightly subtle thing.
00:10:33
Speaker
And that's not a perfect analogy either, but it shows kind of the basic philosophy in which a sinusoidal wave can be used to describe any periodic or bounded signal.
00:10:43
Speaker
Yeah, and so what the Fourier transform essentially does, and this is kind of geared towards more mathematical folks who listen to this, but of course it can be understood by anybody who's listening to this, is that if you take a point in a room, you could say, okay, this is that point, it's on my couch, something like that. But what you could also do is you could say, well, let's project it along these three walls. And so you could say that is at this position x, y, and z.
00:11:13
Speaker
in length, width, and height. Yeah, and projecting along something just means if the walls were made out of mirror, and you looked right at the mirror, you would see yourself and the point where it projects it. So it's like the direct shadow. So you're projected onto the globe during high noon, for example.
00:11:32
Speaker
yeah it's like it's like a shadow of the point on each wall and how high it is on the wall how far in from the corner it is and how how far in from the other corner on the floor and so you can think you can think okay i took this point and instead of and instead of just thinking that's on my couch now i can say it's at x y z coordinates four five and six and so what that gives us is
00:11:58
Speaker
Okay, what if it was at 4, 5, 17? We know the dimension in the z direction, you know, is 17 inches off the ground. And so that tells us something more than here's this point in the room. We can actually start looking in these different directions. And much the same way, a foyer transform of a signal
00:12:17
Speaker
Maybe you look at a ripple in a pond, and if there are two frequencies going on there, it's easy to see with one, but with two, you see a ripple on a ripple. It's a ripple on a ripple, and you can hardly see what's going on. It starts to look garbled, but when you take the foyer transform of it, it becomes very clear. You see one frequency, you see one tone at one frequency and one tone at another.
00:12:44
Speaker
Yeah, actually, that's a really great topic. When you talk about one signal riding on top of another signal, I think that really is almost the crux of a lot of signal processing, right? Yes. And so what we're saying, if we put it back into this vectorized thing that I was talking about, and the point in the room is you could say, oh, this is three inches along that wall, five inches along the other wall, and 17 inches up.
00:13:10
Speaker
same thing we could say hey this frequency is you know two parts of a low frequency wave and five parts of a high frequency wave. It's basically decomposing that signal into its frequencies.
Filtering Radio Frequencies
00:13:24
Speaker
So something that I wanted to bring up is that we see this with radio stations okay. If you could see the electromagnetic waves in the air from radio stations it would be garbled and confusing. You wouldn't be able to see
00:13:39
Speaker
anything. But if you take the foyer transformer of that, then you can see it in terms of frequency, and you see these very neat stations. They're all aligned, so you would see along, you know, and we'll talk mathematically, graphically, mathematically along the x-axis. Instead of time, how the signal changes in time, you don't know
00:13:59
Speaker
what happens in terms of time now, your x-axis is now in frequency. And so what's nice is you can say, oh look, at 102.5 MHz, there's a station, and then there's a little gap between that and 1029, which is the hard rock station.
00:14:18
Speaker
And you can see, hey, if I just take everything that isn't my 1025 and I filter that out, then I'm only left with that station. And so you can, you can look at things and this is where it gets into electrical engineering and signal processing is you can filter everything unwanted out by frequency. And then in the time domain, you play through your speakers and you hear wonderful classic rock.
00:14:45
Speaker
Oh, that, that's totally cool. Yeah. So now when you brought up the whole idea of the frequency, you know, like on an old radio and you, and you turn the dial back and forth, that's, that's actually really, really effective. I think that's cool. So I'm wondering, is it a good time to segue into another topic, uh, involving frequencies and that is, uh, wave, uh, modulation, like in other words, how, how is it that in electrical engineering, how do we encode information onto
Modulation in Communication Explained
00:15:09
Speaker
waves? Yes. This is actually my favorite thing to talk about because this is my bread and butter. This is what I work with.
00:15:15
Speaker
Oh, that's so cool. That's so cool. I'm going to let you take the wheel on this one. Well, before we do that, is there any way you could just define what modulation is for us? Yes, exactly. And so you can say modulation is basically how we want to change a physical phenomenon to transmit information.
00:15:35
Speaker
Okay, very good. So you ever do that thing where you get the two cups and you connect them by a string and you can talk over it? What's actually happening is your sound waves from your voice are modulating on that string and so sound waves aren't traveling between your cup and your friend's cup. They're actually traveling on that string and that string is then vibrating your friend's cup and then you can hear it. So that's a form of analog modulation.
00:15:59
Speaker
Awesome. And so everybody knows two different types of modulation just right off the bat, even though they don't know it. AM and FM. So let's go into those. Yes. AM FM. So those stand for AM. That's amplitude modulation. And FM is frequency modulation. So I have a guitar here. I'll kind of display that for you.
00:16:23
Speaker
amplitude modulation is saying okay if I want to send you one signal you know we'll take digital modulation I want to send it one or zero or you know I want to send you two different values I'm going to modulate the amplitude of that signal rather than the frequency so it sound like this a quieter tone and then a louder tone
00:16:45
Speaker
And so using those different volumes, we can transmit information. And that's one very frequent way to transmit information. Another is to use different frequencies. So if I want to say one thing, I will transmit this tone. And another, I'll transmit the higher tone.
00:17:07
Speaker
And there's all sorts of different tones. And this is all different frequencies. And so there are a bunch of ways we can do this. And so if you think we're going to go back to what a frequency actually is, is a cyclical motion. It's you riding your bicycle and your foot going up and down however many times a second it does. And so amplitude in this case would correspond to how large of a circle your feet trace when you pedal your bike.
00:17:34
Speaker
Yeah, so like if you were a little kid with a very small bike, that would correlate to a higher frequency. Not quite a higher frequency, but a lower amplitude. I'm sorry. Correction, yes. Well, it would be a higher frequency if the wheels were smaller, but that's a different topic. We're just going to look at the crankshaft here because, yeah, there's all sorts of gear ratios. But so that would correspond to a larger amplitude is a larger circle that your feet trace.
00:18:02
Speaker
Ooh, I guess I would be right if we were to say, if a bunch of different sized bikes were to go the same distance at the same time, one of them would have to be a higher frequency because you're spinning around. There's more circles involved.
00:18:15
Speaker
Yes, that is true. And then you could say, OK, the frequency, if we're doing the circular motion, the frequency is how many times per second your feet are cranking that bike? And so are you cranking it one time a second or two times per second? That's the frequency. It's just how fast you're going, really.
00:18:36
Speaker
Now as far as I understand it, there's benefits and detriments using AM and FM. And why are those? And what are those? Some of them have to do with the actual frequency these are transmitted at. AM frequencies tend to go further. That's because they're lower frequency. And FM don't go as far.
00:18:57
Speaker
Now, when FM came out it had a much higher fidelity because the noise that is always occurring in the amplifiers or in the air, it doesn't affect angular modulation, such as frequency modulation, as much as it does amplitude modulation.
00:19:14
Speaker
And so it was found that you could get a much higher fidelity recording passed through the air with frequency modulation rather than amplitude modulation, because instead of just looking at what we call the envelope of the signal, which is the loudness of it,
00:19:30
Speaker
The frequency modulation always assumes it's the same loudness, it's only looking at the frequency at which it's transmitted at. So if you do have some noise, it may make your signal sound quieter, but we're not looking at the volume of it. We're actually looking at what frequency we're translating. And so that's kind of a simple explanation why FM is a better modulation, at least for
00:19:58
Speaker
audio transmission. So with respect to AM and FM, there's also a lot of considerations with how they interact with things like clouds. And why is it that AM goes much further than FM? Isn't it based on the size? Is this... I want to make sure I'm not making... The wavelength? That's correct. Yeah, yeah, yeah. Thank you. Based on the wavelength, of course, I believe that AM, it goes right through them, but FM bounces off them, I think.
00:20:24
Speaker
Yeah, so you'll notice if you go on your FM radio, you'll have these stations like 770 or 1080. I don't know, the last time I tuned into an AM radio station. And those are in kilohertz. Those are in thousands of hertz per second. So that's 770,000 times a second this wave is going up and down. And since it's going
00:20:45
Speaker
It has a certain wavelength associated with it, a longer wavelength. That is, as ripples in a pond, you can measure the distance between two peaks in that ripple, and that would be your wavelength. Now, let's say you're splashing your finger in the pond quicker.
00:21:01
Speaker
That frequency, as that frequency goes up, the quicker you do that, that wavelength shortens. And these are all related with the speed of the wave. But anyhow, the wavelength does change. And so with FM radio, they also transmit at higher frequencies. You get 1.025 or 94.1, giving away my musical taste here if you live in Albuquerque.
00:21:26
Speaker
What that is is in megahertz, that's almost a thousand times more frequent that this wave is cycling. And so that wavelength goes down by almost a thousand, it's more of 500 or 200.
Frequency and Wavelength Relationship
00:21:43
Speaker
And so as that wavelength shortens, it actually interacts with physical objects differently. It'll travel through the atmosphere differently.
00:21:51
Speaker
Interesting. One thing that you brought up and I wanted to make sure that's cemented in the minds of the listeners is that frequency is related to wavelength. And that's only because frequency doesn't change the speed in which it propagates, right? Correct. Things for the most part, but yes. Unless you get into advanced stuff, probably. Yes. There are certain materials that, but we won't get into that.
00:22:14
Speaker
All right, wonderful. So yeah, let's talk a little bit more about the Fourier transform. So what was the initial motivation for the Fourier transform? As far as I understand it, it was solving the heat equation on a ring. Now the heat equation, backing up a little bit, is a differential equation, which is just an equation that describes rates of change and how they relate to one another, that describes how heat is transferred.
00:22:42
Speaker
And if you have what's called periodic boundary condition, which you have on a ring, it gets a little bit more complicated and he had to invent this whole type of math for it. Do you have anything to add?
00:22:51
Speaker
Not much to that because I'm not a mechanical engineer or heat transfer expert at all, but it is interesting that something that was invented for one purpose, you know, heat transfer, somebody could apply it to something like audio engineering or electrical engineering, communication systems, signal processing.
00:23:14
Speaker
And I would also like to talk a little bit about the uncertainty principle inherent in the Fourier transform. Do you have anything to say about that? Oh, yes. What's interesting with the Fourier transform, at least when it's implemented practically, see the Fourier transform, it actually takes an infinitely long signal, OK? And then it transforms it into frequency. Now, in real life, we can't record an infinitely long signal.
00:23:40
Speaker
So, when we take a Fourier transform, we only take it over a certain amount of time. Now, what's interesting about that is the longer time we take that Fourier transform over
00:23:53
Speaker
the more information we know about frequency, the higher resolution in frequency we have. And so that's really cool because you can say, oh, that tone is at exactly 440.0015 Hz or something, you know, if you're getting into audio engineering kind of thing. Now, the problem that that gives now is that
00:24:17
Speaker
We know exactly the frequency, but we have no idea within that window when that tone appeared. The more we know about frequency, the less we know about when the signal occurred.
00:24:34
Speaker
And similarly, if you take that when you say, oh, okay, I want to know exactly when that signal occurred, you shorten your window, okay, and then you take this transformer and say, hey, I know where it occurred much more accurately now. Now that you know that, because your transform is over such a smaller window, you don't have the frequency resolution. You know, you'd say, oh, it's at 410 hertz or
00:24:57
Speaker
or 430 Hz, but I'm not sure in there. And so it's interesting, the more you know about one aspect of this occurrence, the less you know about the other.
00:25:10
Speaker
Interesting. And one place where this comes up every day will not necessarily directly in or not in a way that we could actually even see with our big highs every single day is the Heisenberg uncertainty principle. If you take a position of momentum and make them one of them the frequency domain and one of them the space domain, you get the Heisenberg uncertainty principle, which states that the more you know about the momentum of a particle, the less you know about the position and vice versa.
00:25:37
Speaker
I just want to say how happy I am that we're talking about the Heisenberg principle on breaking math. But I did want to, yeah, that's a very interesting application there. But yes, the more we know of a particle's position, the less we know about where it's going. And we can see that exactly in what we're talking about with the time and frequency uncertainty.
00:26:00
Speaker
Now, one topic that we brought up during the planning of this episode was resonance. And an example I wanted to bring up about resonance... Actually, you know how you came up with this one. Why don't you talk about the swinging example?
00:26:13
Speaker
Oh yes, so there's something engineers take into consideration when we make designs, and that's the resonance of something that deals with a reoccurring force. So we can think about this as a swing set. So somebody's on a swing set, and this is essentially a human pendulum.
00:26:34
Speaker
and we know we've seen people get very high on swing sets but they never got that high with one push and you'd have to be pretty strong to do that and you'd have to put a lot of force wouldn't be very comfortable but what we do is you push them a little bit and then they swing back and we push them again they get a little higher swing back push them again they get a little higher and what this is doing is we're applying a periodic force
00:26:59
Speaker
to the person in the swing set. This is a frequency. This is frequency math. If we were to take this, you're saying, oh, you know, I'm just pushing every one to two seconds.
00:27:09
Speaker
Now, we could take the Fourier transform of that and say, oh, so that's about 0.8 to 1 Hertz. And so I just give that as an example that, hey, we can take anything and look at it in the frequency domain. But you could have never put that much energy into that person in one push. But over seven or eight pushes, you can get them at the top of that swing set. And they have a lot of energy going there. Now, how is that different from using something like a jack on a car?
00:27:40
Speaker
where you do the same motion over and over again and you get an accumulated effect.
00:27:45
Speaker
Yeah, so the jack on the car, that is kind of an interesting example. It's not dynamic, it's static, and so it has a little latch on it. First of all, you use a lever, which multiplies your torque or your force, and you just go up a little bit at a time. And so you are still applying the same amount of force needed, but you're just latching it up a little bit at a time. And it is kind of a similar example, but we have to do this dynamically, and at the right time, apply the force.
Resonance Explained with Analogies
00:28:14
Speaker
So it would be true resonance if the car bounced up and down. Yes. We see these guys on Central Saturday nights do that. It's really interesting.
00:28:23
Speaker
And of course there's the famous, we'd probably be amiss if we didn't talk about it, the possibly apocryphal example of Nikola Tesla vibrating his building in New York. He had a machine, it was a very, very well designed machine. It took into account the difference between air pressure densities and things like that to create mechanical vibrations. And he was trying to tune it to the vibration of his building and apparently he heard a cracking sound or something.
00:28:51
Speaker
Yeah, so this is kind of, you know, it's a myth, kind of, I don't know, MythBusters did a little episode on it and basically what it is is a little machine and had a little oscillating weight in it and it worked the same way as a swing set. Imagine somebody very smalling an ant pushing on a swing set. Theoretically, if there's not so much damping that is friction, they could just keep pushing just a little bit at the right time
00:29:18
Speaker
And that energy would accumulate. And so that's what he was trying to do with this, a little weight just moving back and forth. And it would find the natural frequency of a building. And so when we strike a piece of metal like, what did you say, Archimedes? Oh yeah, I think it was Archimedes. Archimedes or a guitar string.
00:29:39
Speaker
It vibrates at its particular resonant frequency. And so what this thing was trying to do was trying to find that resonance and push it just at the right times. And does resonance have anything to do with the Dirac delta function? The Dirac delta function for those unacquainted being like you clap in a room and you hear the echoes of the clap so you know what the shape of the room is.
00:30:04
Speaker
Yeah, a little bit, yes. That would actually give us the impulse response of that room, and we talk about that in terms of filtering, that the Dirac delta is essentially a clap. And that clap is this impulse, and that impulse contains all frequencies possible, but in theoretically zero amount of time, or infinitely small amount of time, rather.
00:30:29
Speaker
And of course if you take a if you clap in the frequency domain you get a pure tone in the time domain. Yes. So if you know if you take like we said like that clap is just an impulse gets really loud really quick and quiet right back down. And so when we take that into the.
00:30:48
Speaker
a Fourier transform of that, it's a straight line through all frequency. It contains all frequency. Now what this does is you're saying, look at the shape of the room. It's letting that sound go out to the shape of the room. The tones that don't resonate just get quieted down.
00:31:05
Speaker
The tones that do, they linger a little bit, and then they come back to your microphone. And your microphone picks up only those tones that linger. And so the room is actually acting like a filter, kind of like we were saying with the radio stations. You're putting out all frequencies all at once in this clap, and then you're listening back for the frequencies that remain and aren't damped.
00:31:30
Speaker
Wonderful.
00:31:47
Speaker
The bridge had been shaking and during one day they went out and filmed it when it was going out of control. And just this wind blowing over this bridge, these aerodynamic forces, it found the resonant frequency of this bridge. And you can look this up online, please
Case Study: Tacoma Narrows Bridge Collapse
00:32:04
Speaker
do. If you're listening to this podcast, the Tacoma Narrows Bridge, it's an amazing piece of footage. And this bridge is swinging like a rope.
00:32:14
Speaker
this is it has cars on it people are trying to run off this bridge and it's uh it's swinging back and forth because that wind has excited that bridge at its resonant frequency and when it resonated there it got out of control and the bridge just tears apart and absolutely destroys itself.
00:32:33
Speaker
Yeah, definitely. With Resonance, you can get all kinds of bizarre effects. I mean, the one that I think a lot of people are familiar with too, and this might be one of the last examples we give, is the opera singer who sings and then breaks glass. Apparently they are not able to replicate that on MythBusters, I think, but I might be wrong.
00:32:55
Speaker
You know, actually, I have heard of one other example. There was an earthquake in Mexico and certain buildings that had a concrete that was the same resonant frequency as the earthquake. There was some tragic collapsing of buildings. And I think that learning from that, buildings are no longer made the same way. In fact, there's a whole field into how to construct buildings that are earthquake proof, in fact.
00:33:18
Speaker
Yes, there are. So with the Tacoma Narrows Bridge, it's just a perfect example of, hey, civil engineers, let's definitely do some frequency analysis on all the civil engineering, all these buildings, so that we don't hit a resonant frequency. And so the way they avoid this is essentially using filters. Part of the building is made out of something with a certain resonant frequency, but they know that.
00:33:46
Speaker
but it does not resonate at other frequencies. Then they use another part of the building is made out of material that will resonate at another frequency, but not at the first material's frequency. And so they actually filter out each other's frequencies, which is really interesting. The other thing that they do is you can look this up too. It's incredible. They have these giant weights that are in these skyscrapers in San Francisco.
00:34:11
Speaker
So when the building shakes, this weight kind of stays in the same place and it acts as a damping force. It acts as something it can anchor itself to rather than the earth. Wow. The math that is used to analyze these resonant frequencies, it's very similar to the foyer transform and it's called a Laplace transform. It's actually a generalization of the foyer transform. And before we talk about that, let's do a quick refresher on the complex plane.
00:34:42
Speaker
The complex plane is like the complex number line, but instead of having one dimension, you have two dimensions. So, um, we're one dimension, you have things multiplied by the square root of negative one. If you'd like to know more about that, you could check out our episode, wreaking chaos. It's episode two. But we should do an imagine. So yeah, there, there is an imaginary episode of breaking math about imaginary numbers. Perhaps it'll become real. Only if we multiply it by its complex conjugate.
00:35:09
Speaker
It's going to be quite a complex episode. I have to follow along. So if the foyer transform is taking a signal and mapping it into frequencies, the Laplace transform takes a signal or a measurement through time and maps it into the complex and real exponentials. And now this is like kind of mind blowing if you're listening because you go, wow, like,
00:35:38
Speaker
complex exponentials but we'll get into that. If you think of an exponential that's something that either increases at an increasing rate or decreases at a decreasing rate if it's negative. And so what the Laplace transform lets you do is it lets you take a response of a material or building and our control system or a filter and it lets you look at it in terms of these exponentials. Now a complex exponential
00:36:07
Speaker
Let me just say this, it's going to take some proving to do, but a complex exponential is a sine wave. It's a periodic wave, and it never decays. When we look at the Laplace response of something such as how a building moves during an earthquake, or how a control system moves an arm or something, or a robot,
00:36:32
Speaker
We can see if we map this to These exponentials if if anything maps to a positive exponential we can say oh That response that is the shaking the vibration whatever it will increase and an increasing rate That is not good. That is resonance, and that is what makes the Tacoma narrow bridge blow up interesting and
00:36:59
Speaker
You said that during the planning, there's a transform related to the Laplace transform, but for discrete time signals. Do you want to talk at all about that? Yes, and so maybe we'll have some reference sheet on this that you can look at later. But it's essentially the same thing, but instead of applying to exponentials,
00:37:20
Speaker
that is e to the something times something we actually just map it to a complex number z and that can be anywhere in the plane and what that does is if the purely imaginary axis of the Laplace transform okay we're in the plane just the imaginary axis which conventionally is the vertical axis
00:37:42
Speaker
That is the Fourier transform. Okay, that is a purely imaginary exponential, which is a frequency. So you could take Laplace transform and only look along that axis, get the Fourier transform. With the Z transform, it actually maps the Fourier transform onto the complex plane, but actually in a circle.
00:38:04
Speaker
It's the unit circle about the origin. This is important because when we get into discrete signals, we talk about sampling. And that sampling is actually cyclic. And so if we follow along that Fourier transform, which is that unit circle, eventually we'll end up back where we started. And that's something that brings up, I think, our next topic, which is sampling, Nyquist sampling.
00:38:28
Speaker
Indeed yeah Nyquist sampling is Fascinating we did a lot of planning and our guests did a lot of planning for this episode you had a great example of aliasing and aliasing for those who are unacquainted is Like actually why don't you take it away?
00:38:44
Speaker
So when we get into sampling, what we're saying is, okay, if we want to record something, let's say, you know, let's say you're doing a scientific measurement, you want to record temperature every every second. And so you open your eyes, you look at the gauge, and then you write down the temperature and you do that every second. Okay, you could do the same thing in terms of any measurement you take. And so this is really important when you're doing these scientific studies and you want to measure these things. However, there's some things you got to be careful of. When we're looking at these frequencies,
00:39:14
Speaker
There's some assumptions that, you know, if you make that could be fatal to your experiment, to your analysis. So let's take the crankshaft of the bicycle into account again, okay? We're going to look at the bike from the left side of the bike, if you're standing on the bike, and someone's riding forward. Maybe it's a stationary bike, that's easier.
00:39:37
Speaker
and they're going to start with their left foot and it's going to be on the back at 90 degrees and we'll call that zero degrees actually it's going to be you know the back the left pedal will be near the back the right pedal near the front okay now let's say somebody starts riding forward and you're going to look up once a second and you're going to record the position of that person's foot and then you can say okay well i know the position
00:40:02
Speaker
and I know the time, now I can take the frequency analysis and know the frequency of how fast this guy is pedaling.
00:40:10
Speaker
Now that works up to a point. So if you take a measurement every second and this person is turning the crankshaft a third of a rotation every second, you're going to open your eyes and see they're at that starting point. Then you're going to look down, write your number down, you're going to look up and you open your eyes and they're a third of the way around the crank. You write that down. The next time they're two-thirds. The next time they're backward they started.
00:40:35
Speaker
So now you go, okay, so this is good. I know that they're going at one-third Hertz. Every second that crank travels one-third of the way around the circle. And you'd be right in this case. However, you start running into problems
00:40:53
Speaker
when that frequency gets faster than half your sampling rate, and let me explain this. Let's say instead of a third of a rotation per second, they did an entire rotation per second. Now every time you look up and you open your eyes, okay, where they started, you look down, you write down the measurement,
00:41:14
Speaker
then they've traveled all the way around that circle and they're back where they started. You look up and you say, oh, they haven't moved an inch. And you record zero degrees again. You look down, look up, zero degrees. Now, is that an accurate measurement? No. That is not an accurate measurement because you made the assumption that they are moving less than half your sample rate per second.
00:41:37
Speaker
And that's a fatal assumption, and this is something we call aliasing. So that is aliasing a one hertz rotation, or frequency, into zero.
00:41:48
Speaker
And if you want to ever look at an example of 2D aliasing, if you've ever played a video game that was made maybe in the 90s where there's a checkerboard pattern and the checkerboard looks wonky, or even what's called a moiré pattern, where you take two grids and put them in front of one another, you'll notice that there's like a weird sort of third pattern that crops up, and that's aliasing right there.
00:42:14
Speaker
Yeah, so we see that there's this third pattern and it's a low frequency sinusoid. And so for an example of that, let's say instead of one third rotation a second, they did two thirds. So you look, they're at their starting point, you record, you open your eyes one second later, they're two thirds through that circle and you record that position. And you keep doing this, now they go another two thirds, another one third of the entire circle.
00:42:43
Speaker
you keep recording this and you look back at your data
00:42:46
Speaker
If you look at that, it looks like your foot, the person's foot started in one place and went backwards by one third of the circle. It looks like instead of this person going forward at two thirds rotations a second, they're going backwards at one third rotation and that is aliasing. Now that can be very fatal to your measurement because now you're saying, you know, something's moving forward and you're assuming that it's moving backwards.
00:43:17
Speaker
I want to say this real quick. I know that we are a bit limited on time.
Conclusion and Future Episode Teasers
00:43:20
Speaker
I want to just offer a proposal. What if we were to do frequency part two someday? Then we could do all the fun stuff that you'd like to talk about. We could do that. We could do it with information theory, bandwidth. We could do a communication systems engineering episode, if you like. I am so down for that. I am so down for that. Maybe even an episode that's catered to those who are perhaps considering that degree path or just anyone in particular who wants to know more about that topic.
00:43:46
Speaker
Yeah. Well, thank you guys so much for having me. I love talking about this stuff. Absolutely. You've been a very, very well-informed guest. Yes. Let's do this again. Adios amigos.