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The Fourier Transform, explained in one sentence (2014)

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Re: The Fourier Transform, explained in one sentence (2014)

#61

Post by nuancebydefault »

thrdbndndn wrote: post #19
munchler wrote: post #17

I’m a layman, but I always think of the Fourier Transform as “an algorithm that converts amplitude over time into frequency intensities”. I guess that’s more of a What than a How, but it still seems good enough for a single sentence.

It's close enough. But then, how is it different from Laplace transform? (I have to admit, I actually learned about both during my uni time. But now I totally forgot them all.)

Laplace is a generalisation of Fourier. Fourier can in principle only represent periodic signals. To circumvent this limitation, they invented the windowing of consecuive parts of signals and do the transform on each window. Laplace adds the transient part, that's what makes it more complicated.

Re: The Fourier Transform, explained in one sentence (2014)

#62

Post by quercusa »

dec0dedab0de wrote: post #48
jn5 wrote: post #36

Earlier quoted context omitted.

If you lose something, you always find it in the last place you search... because why would you keep searching after you found it

It took me an embarrassingly long time to realize that it was a joke when people said "It's always the last place you look." Like well into my teens. But ever since I figured it out, I always look at least one more place after finding something.

I think that there are many people who don't recognize it as a joke and pass it on as great wisdom. I was also pretty old when I realized the tautology.

Re: The Fourier Transform, explained in one sentence (2014)

#63

Post by sam_lowry_ »

tel wrote: post #16

Honestly, it's not so bad. It's easy to pick any such attempt apart. This is close to my favorite pithy way of explaining it, too, which is to break it down component-wise using the idea of filter banks. It's not a single sentence, but here's what I tend to say: Any signal—like sounds or electrical signals, or even images—can be thought of as having a certain amount of 'energy' at any choice of frequency. This makes…

I like your explanation in that it gives reason to the rotation.

Re: The Fourier Transform, explained in one sentence (2014)

#64

Post by selimthegrim »

dec0dedab0de wrote: post #48
jn5 wrote: post #36

Earlier quoted context omitted.

If you lose something, you always find it in the last place you search... because why would you keep searching after you found it

It took me an embarrassingly long time to realize that it was a joke when people said "It's always the last place you look." Like well into my teens. But ever since I figured it out, I always look at least one more place after finding something.

Last is semantically ambiguous here, you’re not wrong, could also mean “last place (most unobvious) you would think to look” as well

Re: The Fourier Transform, explained in one sentence (2014)

#65

Post by nuancebydefault »

rectang wrote: post #53
fortran77 wrote: post #47

Earlier quoted context omitted.

Or even more simply, if you know that e^x on the complex plane rotates you around the origin.

People told me that over and over but it didn’t help — because it didn’t make sense why repeated multiplication would cause rotation! Later in that video, we see a visualization of the rotation. I was able to grasp how the exp function could yield rotation where I’d never been able to understand why e*e*e*e… did. https://www.youtube.com/watch?v=ZxYOEwM6Wbk&t=2178s

I think you need to study the Euler equation to understand the relationship between goniometric functions and exponential functions when calculating with complex numbers. One easy to remember formula that connects those functions.

Re: The Fourier Transform, explained in one sentence (2014)

#66

Post by teldau »

voidhorse wrote: post #29

I think the best semi-intuitive, non rigorous explanation I've seen of the Fourier transform is still one that first explained signal correlation in the time domain and then described the transform as basically performing correlation on the signal for all the possible sines at different frequencies. Essentially you're just testing for the presence of individual sine waves (of different frequencies) within the signal.…

I agree, FFT is simply a correlation of a bunch of sine waveforms.

The extra tidbit - the sine waveforms are orthogonal to each other (their cross correlation is zero), so the transform can be inverted. (in math terms: they form a basis set. I'm mostly referring the STFT here, which is the 'common' FFT use. not infinite FFT).

Re: The Fourier Transform, explained in one sentence (2014)

#67

Post by Misdicorl »

IME explanations of the Fourier transform focus far too much on the specific mechanics of it without motivating the reason. I find people often don't even realize the time space and frequency space functions are the same function! I dive in like so

"There are many different ways to write down the number 5. Tally marks, the sigil 5, 4+1, 2.5 times 2, 10/2. Each of them is more or less useful in different circumstances. The same is true of functions! There are many different ways to write the same function. The mechanics of taking a function in one form and writing it in a new form is called a transformation."

Above needs to be said a hundred times. Then you talk about why a frequency space function can be useful. Then you talk about how to transform a time space function to the frequency space representation.

Re: The Fourier Transform, explained in one sentence (2014)

#68

Post by rectang »

nuancebydefault wrote: post #65
rectang wrote: post #53

Earlier quoted context omitted.

People told me that over and over but it didn’t help — because it didn’t make sense why repeated multiplication would cause rotation! Later in that video, we see a visualization of the rotation. I was able to grasp how the exp function could yield rotation where I’d never been able to understand why e*e*e*e… did. https://www.youtube.com/watch?v=ZxYOEwM6Wbk&t=2178s

I think you need to study the Euler equation to understand the relationship between goniometric functions and exponential functions when calculating with complex numbers. One easy to remember formula that connects those functions.

Indeed. The title of that 3Blue1Brown video is, “What is Euler’s Formula actually saying?”

That “easy to remember” formula had been presented to me countless times. It only made sense once I stopped thinking about it in terms of repeated multiplication.

Re: The Fourier Transform, explained in one sentence (2014)

#69

Post by prof-dr-ir »

Does "blog post learning" ever really work?

I have taught these kind of undergraduate subjects and, in the context of a course, the Fourier transform never struck me as something very complicated. First, it feels completely natural to write down a Fourier decomposition for periodic functions; the inverse transform to determine the coefficient is just (real or complex) calculus; and all that is left is to convince the audience of completeness which is best done through a few examples as well as the time-honoured method of "proof by intimidation". (Note that the blog post does not do a better job here.)

By contrast, these posts seem to fulfil a demand where people look for an isolated explanation of concept X, like Fourier transforms, matrix determinants, monoids, etc. But they are necessarily too isolated to really convey the subject. For example, one does not normally teach (complex) Fourier transforms without first dedicating significant time to topics like complex exponentials and integrals of trig functions, and likewise one normally teaches monoids only after introducing functors and applicatives.

In other words, without having the necessary background at your fingertips it is hard to really grasp the core concepts. That is why, IMHO, reading these blog posts may feel good but will rarely make things really stick.

Re: The Fourier Transform, explained in one sentence (2014)

#70

Post by narag »

selimthegrim wrote: post #64
dec0dedab0de wrote: post #48

Earlier quoted context omitted.

It took me an embarrassingly long time to realize that it was a joke when people said "It's always the last place you look." Like well into my teens. But ever since I figured it out, I always look at least one more place after finding something.

Last is semantically ambiguous here, you’re not wrong, could also mean “last place (most unobvious) you would think to look” as well

Because it it were obvious, you had already found it.
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