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What Is the Fourier Transform?

quantamagazine.org

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Re: What Is the Fourier Transform?

#171

Post by goalieca »

astrange wrote: post #64

> A compression algorithm can then remove high-frequency information, which corresponds to small details, without drastically changing how the image looks to the human eye. I slightly object to this. Removing small details = blurring the image, which is actually quite noticeable. For some reason everyone really wants to assume this is true, so for the longest time people would invent new codecs that were prone to thi…

You might be surprised to learn that videos and many images are broken down into brightness + color vectors. Brightness is typically done near the resolution of the image (eg 4K) but color is often 1/4 of that. That’s literally blurring the picture.

Re: What Is the Fourier Transform?

#172

Post by zwnow »

hnuser123456 wrote: post #163
zwnow wrote: post #126

Earlier quoted context omitted.

No worries, as a self proclaimed youngster I didn't manage to understand Fourier in 2 days and never bothered again. Also had no other prior knowledge to algebra so maybe that's why I struggled. Never perceived algebra as useful in anything programming related, will continue to do so as most problems are solvable without it. I'll let the degree havers do all that stuff.

So you're a programmer but you've never assigned a number to a variable or written any math operations? Do you just do string translations or something?

I'm talking algebra you need a degree for. Well algebra u learn while getting one that is.

Re: What Is the Fourier Transform?

#173

Post by slickytail »

trgn wrote: post #170

Can somebody eli5, im an amateur. How does the transform know the frequencies of the output. Do you have to specify a number n, and then it decomposes it into n frequencies. Or do you give it a list of frequencies, and then it decomposes the coefficient or amplitude or something for each? I guess what i want to know, in the examples it always shows like 3 or 4 constituents frequencies as output, but why not hundreds…

In a discrete Fourier transform, the number of frequencies you get out is the number of datapoints you have as input. This is because any frequencies higher than that are impossible to know (ie, they are above the sampling frequency).

But in the continuous Fourier transform, the output you get is a continuous function that's defined on the entire real line.

Re: What Is the Fourier Transform?

#174

Post by astrange »

goalieca wrote: post #171
astrange wrote: post #64

> A compression algorithm can then remove high-frequency information, which corresponds to small details, without drastically changing how the image looks to the human eye. I slightly object to this. Removing small details = blurring the image, which is actually quite noticeable. For some reason everyone really wants to assume this is true, so for the longest time people would invent new codecs that were prone to thi…

You might be surprised to learn that videos and many images are broken down into brightness + color vectors. Brightness is typically done near the resolution of the image (eg 4K) but color is often 1/4 of that. That’s literally blurring the picture.

…Why would you write a comment like this assuming I don't know what 4:2:0 is when I know what a wavelet is?

It doesn't have the same effect because the original high frequency details are correlated and so they're preserved in the Y channel.

Re: What Is the Fourier Transform?

#175

Post by woopsn »

trgn wrote: post #170

Can somebody eli5, im an amateur. How does the transform know the frequencies of the output. Do you have to specify a number n, and then it decomposes it into n frequencies. Or do you give it a list of frequencies, and then it decomposes the coefficient or amplitude or something for each? I guess what i want to know, in the examples it always shows like 3 or 4 constituents frequencies as output, but why not hundreds…

If you take N samples of a real signal you will get N/2+1 bins of information from the DFT, covering 0Hz out to about half the sampling rate.

The bins do not actually measure a specific frequency, more like an average of the power around that frequency.

As you take more and more samples, the bin spacing gets finer and the range of frequencies going into the average becomes tighter (kind of). By collecting enough samples (at an appropriate rate), you can get as precise a measurement as you need around particular frequencies. And by employing other tricks (signal processing).

If you graph the magnitude of the DFT, signals that are a combination of power at just a few frequencies show just a few peaks, around the related bins. Eg a major chord would show 3 fundamental peaks corresponding to the 3 tones (then a bunch of harmonics)

https://en.wikipedia.org/wiki/Fourier_transform#/media/File:...

So you detect these peaks to find out what frequency components are present. (though peak detection itself is complicated)

Re: What Is the Fourier Transform?

#176

Post by IAmBroom »

trgn wrote: post #170

Can somebody eli5, im an amateur. How does the transform know the frequencies of the output. Do you have to specify a number n, and then it decomposes it into n frequencies. Or do you give it a list of frequencies, and then it decomposes the coefficient or amplitude or something for each? I guess what i want to know, in the examples it always shows like 3 or 4 constituents frequencies as output, but why not hundreds…

I'll try.

Suppose you blinde, and are on one side of a black picket fence. Let's say the fence uprights are 1" wide, and 1" apart.

If you aim a light meter at the fence, it will sum up all the light of the visible slices of the background. (This meter reads its values out loud!)

Suppose the background is all white - the meter sees white slices, and get a 50% reading (because the fence stripes are black). It's as high as you can get.

Now move the picket fence half an inch to the left - still all white, still a 50% reading. Now you know the background is 100% white, no other colors.

But if when you move it, black stripes are exposed, then the meter reads 0%, and you know that the background is 50/50 white and black, in stripes.

Congratulations! You've just done a Fourier transform that can detect the difference between 0-frequency and a 1"-striped frequency pattern. In reality, you're going to continuously move that fence to the left, watching all the time, but this is simple.

But instead, imagine you got readings of 40% and 10%. What kind of pattern is that? Gray stripes? White stripes that aren't as wide as the fence posts? You'll have to build another fence, with another spacing - say 1/2" fence posts 1/2" apart. Repeat.

Re: What Is the Fourier Transform?

#177

Post by IAmBroom »

zwnow wrote: post #126
segfault99 wrote: post #69

Earlier quoted context omitted.

When I did EE, didn't have access to any kind of computer algebra system. Have 'fond' memories of taking Laplace transform transfer functions and converting to z-transform form. Expand and then re-group and factor. Used a lot of pencil, eraser and line printer fanfold paper for doing the very basic but very tedious algebra. Youngsters today don't know how lucky.. (ties onion to belt, etc., etc.)

No worries, as a self proclaimed youngster I didn't manage to understand Fourier in 2 days and never bothered again. Also had no other prior knowledge to algebra so maybe that's why I struggled. Never perceived algebra as useful in anything programming related, will continue to do so as most problems are solvable without it. I'll let the degree havers do all that stuff.

"Book learnin' didn't do me no good no how!"

Re: What Is the Fourier Transform?

#178

Post by IAmBroom »

schlauerfox wrote: post #169

All these comments and no Steinmetz mention. The original papers he wrote applying the transforms to electrical engineering work are so genius and simplifying for the craft, and he's such a nicer person for one of the founders of the field of EE. Also the origin of that old story about $10 for hitting the machine with a hammer $9990 knowing where to hit it, from what I can gather.

Steinmetz is an unsung hero of tech, except to well-educated EE's.

Was he a dwarf or a hunchback? I seem to recall a physical setback.

Re: What Is the Fourier Transform?

#180

Post by abdullahkhalids »

im3w1l wrote: post #105
abdullahkhalids wrote: post #88

Earlier quoted context omitted.

The question is why "so many real-world systems are governed by differential equations" and "so many real-world systems involve periodic motion". Well, stable systems are can either be stationary or oscillatory. If the world didn't contain so many stable systems, or equivalently if the laws of physics didn't allow so, then likely life would not have existed. All life is complex chemical structures, and they require s…

> Well, stable systems are can either be stationary or oscillatory. In practice this is probably true, but I can see another possibility. The system could follow a trajectory that bounces around endlessly in some box without ever repeating or escaping the box.

You can treat that, and scientist often do treat it, as a stationary system with some error bounds.

For example, the concept of homeostasis in biology is like this. Lots of things are happening inside the living body, but it's still effectively at a functional equilibrium.

Similarly, lots of dynamic things are happening inside the Sun (or any star), but from the perspective of Earth, it is more or less stationary, because the behavior of the sun won't escape some bounds for billions of years.

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