I never understood why everything is built from sine waves...until now! (Fourier Transforms) – Transcript

00:00Any shape you can draw with a pen on a paper can be built by adding up a lot of sine waves. Literally any shape at all. This is the idea behind Foryear series and Forier transforms named after Joseph Forier. And without it, you won't be able to watch this video or send a voice message or even take pictures on your phone. Not only does this idea power

00:26Wi-Fi, radar, MRI, you name it, but about a century ago, it helped power the British Empire's global sea trade. And decades later, it decided the date of the largest seabbone invasion in human history. But why is this true in the first place? And how does that help my phone take pictures? And where did it all come from? I never thought I'll get any intuition behind this until now.

00:53Forier says all of this every single aspect of it including all the math can be rediscovered from just one fact that heat flows from hot region to a cold region. That's it. Nothing else.

01:08Wait, what? What? How does this lead to what? For says, Mahesh, here's a hot piece of metal. It's touching a colder one. What happens to its temperature?

01:21Well, heat flows from hot to cold, so its temperature is going to reduce. Obviously, for says wrong. Why? Because it's also touching a hotter piece on the right with the exact same temperature difference. So, the same amount of heat flows back in, making the net heat flow zero. So, our temperature remains unchanged. For you tricked me. What's your point? The point is if the left

01:47piece was only slightly colder then look the amount of heat flowing out would be lesser and now the temperature of our piece will increase. Makes sense. And if this hot piece was even hotter then the temperature is going to increase faster.

02:06Yeah, that also makes sense. So the question is what decides how quickly does the temperature of our piece change? What decides the rate of change of temperature? Well, I I know that it's got something to do with the difference in temperature, but I can't put my finger on it exactly. For says, no worries.

02:29Think geometrically. So, let's go back to our first case and draw the temperature position graph. Remember, the temperature was unchanged over here. What does the graph look like? Well, since the temperature difference is the same, the graph is going to be a straight line. In other words, the slope of that graph is unchanged as I go from here to here. And when the slope is unchanged, the temperature is unchanged.

02:54Ooh, is that is that a coincidence? Let's see. In the second case, this piece was only slightly colder. And so now, as I move from here to here, the the slope increases and the temperature also increases. And in the third case, this piece was even harder, which means the slope increases faster as I go from here to here. And the temperature also increases faster. So there is a nice correlation between this. We see the

03:22rate of change of temperature is directly related to how quickly the slope changes. The rate of change of slope itself. But slope itself is rate of change of temperature. So we get a second derivative. And there we have it, the legendary heat equation.

03:44Okay. But what is this equation telling us? And why is it so legendary? Well, the second derivative means a curvature. So how quickly the temperature changes at a spot depends on the temperature curvature at that spot. For example, if there is no curvature, the temperature is unchanged. If there is a positive upward curvature, look, the temperature increases at that point. If that curvature is stronger, the temperature

04:11increases faster. And of course, if there's a downward curvature, the temperature would reduce. Now, let's put this into action. Suppose we have a rod sitting here at room temperature. We heat up one specific spot. So, the temperature spikes over there. The question is, what happens a little while later? How do we figure that out? Well, just look at the curvature everywhere.

04:35At the peak, the curvature is the strongest. So, that is the point where it'll cool down the fastest. As we go farther away, the curvature reduces and then the curvature goes to zero and then the curvature becomes upward and so these spots actually warm up.

04:54So, a little later our profile is going to look like this. This equation allows us to predict what happens to the temperature profile. It helps us understand how the profile evolves over time. That's awesome. But I still don't get why this is so legendary.

05:18Just like how heat moves from high temperature to low temperature, charges inside a neuron, for example, move from high voltage to low voltage. Which means the exact same physics can be used to model how electricity electricity flows through a neuron. All we need to do is replace temperature with voltage. That's awesome, right? And the exact same equation governs how mutations spread

05:43through a population or how uncertainty spreads through financial markets. Oh my god, why does the single equation work everywhere? Because at the heart of it, the central phenomena is diffusion. So this equation is fundamentally a diffusion equation.

06:02It's one of the most common things that happen in our universe. That's why this equation is legendary. And we derived it from scratch just from the fact that heat moves from hot to cold. That's awesome, right? Okay. But what does this have anything to do with 4year series?

06:18Everything. Because now we have a big challenge in front of us. How do we actually solve this equation? I mean, so far we solved it qualitatively looking at it step by step. But how do we get a mathematical expression for it? U solving differential equations are not easy. So what do we do? Forier uses one of the best problems solving techniques.

06:42If you can't solve a hard problem, see if you can tackle an easier version of the problem. Okay, what's the easier version of the problem here? Well, here if you look at the initial profile, the curvature is all over the place and that's why things look complicated. So, can we start with a profile whose curvature is nice? Yes.

07:06What about a sine wave? How on earth does sign make things easier? Well, Forier says sign has a beautiful property. Its curvature is proportional to the height. What does that mean?

07:21Well, if you look at this point, the height is zero, right? Guess what? The curvature is also zero. This is the point where the height is maximum and the curvature is also maximum. If you consider a point where the height is half the maximum height, the curvature at that point is also half the maximum curvature. See, curvature is proportional to the height. Okay? So this means if the starting profile was a

07:47sign curve then the rate of change of temperature is going to be proportional to the height which means the temperature at every point changes in proportion to its height. And so a little while later the shape looks exactly the same. Sign just shrunk down.

08:09That is cool right? And that continues forever and ever. And the whole point is that is super easy to write down. The whole graph is just shrinking exponentially. So how can we mathematically write down? Well, we can say the temperature profile is basically be sign graph whose amplitude shrinks exponentially.

08:36So this part stays the same because the graph stays the same, the shape stays the same. And this part represents the whole graph shrinking exponentially. So if the initial graph is a sign curve, look, this is the exact equation that represents this evolution. That's awesome. Okay, now here's a question.

08:58What if the initial graph had two sign curves? This also behaves the same way. But the question is, will it shrink at the same rate as the previous one? Will it shrink faster or slower? Why don't you pause the video and think about it?

09:15Well, now we have squeezed two sine waves in that same place as before. And so now the curvature everywhere is much higher, right? We can visually see it. So the temperature change is going to be much quicker. The whole graph is going to shrink much faster than before.

09:30Higher the frequency, the faster it shrinks. Makes sense, right? Okay, this is great. But for what's the point? I mean, this would only work if the initial profile just happens to be a sign curve. Real world profiles are not sign curves. So, what can we do with that? Well, Forier asks, what if we had the initial profile which was sum of two sign curves of different frequencies?

09:57The result will no longer be a sign curve. Its shape will look pretty complicated right now. If you try to solve how it will evolve directly, that would be a nightmare. But we don't have to. We can split it back into its component sign curves. And the whole point is we already know exactly how each one evolves. The high frequency ones die out much faster, right? And so

10:23then what we do is we simply add them back together to get our final profile. Oh my god, that is amazing. So the trick is we never solve a complicated graph directly. Instead, we just break it down into its component sine waves. Let each one evolve over time because we know the exact math for that and then stitch them

10:47back together. That is so ingenious. This works because derivatives are linear operators. Meaning derivative of a + b equals derivative of a plus derivative of b. But again for this only works provided the initial profile just happens to be a sum of two or more sine