Earlier quoted context omitted.
You got confused before we even started so maybe you are not the target audience.
If you can't communicate clearly there's no reason to try to take you seriously.
Is there a reason to take you seriously?
What Is the Fourier Transform?
191–200 of 214 posts
Re: What Is the Fourier Transform?
#192Earlier quoted context omitted.
You are being confused with #samples needed for perfect reconstruction, i.e. Nyquist sampling frequency. Fourier series/transforms work regardless of the bandwidth of the signal, as long as the integral exists, i.e. it must vanish at infinity. Essentially it's just projection in infinite-dimensional vector spaces.
That's what is commonly understood by reconstruction: perfect reconstruction. And for that you need a band-limited signal. Otherwise he would have said approximate- or lossy reconstruction.
> And for that you need a band-limited signal.
Luckily, we live in a physical universe, where such mathematical oddities, like infinite bandwidth signals, cannot exist, so this isn't an actual issue. Any signal that that contains infinite bandwidths only exists because it has sampling artifacts. You would, necessarily, be attempting to reconstruct errors. There are many "tricks" around dealing with such flawed signals. But yes, you can't fully reconstruct impossible signals with FFT.
Re: What Is the Fourier Transform?
#193If you like Fourier, you're going to love Laplace (or its discrete counterpart, the z transform). This took me down a very fascinating and intricate rabbit hole years ago, and is still one of my favorite hobbies. Application of Fourier, Laplace, and z transforms is (famously) useful in an incredibly wide variety of fields. I mostly use it for signal processing and analog electronics.
When I first learned Laplace transform in university, it was my goto for differential equations of any kind. I was even naive enough to believe well this is a solved problem now. Eventually found out this wasnt the case after studying PDEs. Its still my favourite transform. Immensely useful not to mention the whole method of moments in random variables is basically laplace transform.
I don't like Fourier transform but for petty reasons. In the engineering exams, I messed up a Fourier Transform calculation and ended up just a few points short of a perfect score. Hate it ever since :)
Re: What Is the Fourier Transform?
#194> Fourier argued that the distribution of heat through the rod could be written as a sum of simple waves. How do you even begin to think of such things? Some people are wired differently.
I don't know/remember the historical derivation, but the story you might get in a class goes something like:
Energy can't be created or destroyed, so it follows a continuity equation: du/dt + dq/dx = 0. Roughly, the only way for energy to change in time is by coming from somewhere in space. There are no magic sources/sinks (a source or sink would be a nonzero term on the right).
Then you have Fourier's law/Newton's law of cooling: heat flows proportional to temperature difference, from high to low: q = -du/dx.
Combining these, you get the heat equation: du/dt = d^2 u/dx^2.
Now if you're very fancy, you can find deeper reasons for this, but otherwise if you're in engineering analysis class, just guess that u(t,x)=T(t)X(x). i.e. it cleanly factors along time/space.
But then T'(t)X(x)=X''(x)T(t), so T'(t)/T(t) = X''(x)/X(x). But the left and right are functions of different independent variables, so they must be constant. So you get X''= λX for some lambda. But then from calc1, X is sin/cos.
Likewise T' = λ T so T is e^-λt from calc 1.
Then since it's a linear differential equation, the most general solution (assuming it splits the way we guessed) is a weighted sum of any allowable T(t)X(x), so you get a sum of exponentially decaying (in time) waves (in space).
Re: What Is the Fourier Transform?
#195Earlier 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.)
Was this professionally or in school? I still did this in an EE program 15 years ago and I can't imagine things have changed since then. I think kids still have to do lots of ugly math in EE classes.
Undergrad. Mid-late 1980s.
I wasn't making point about mathematics qua mathematics. Was thinking that if I were doing EE undergrad today, I'd use SageMath or Mathematica to crunch the mechanical algebraic manipulations involved in doing a z-transform.
Re: What Is the Fourier Transform?
#196Earlier 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?
Plot twist: He's a Haskell guru juggling hylomorphisms blindfolded.
Re: What Is the Fourier Transform?
#197> 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…
Well, there are use cases for lossy compression as well as non-lossy, and nobody is saying they are the same. If you really need to heavily compress to reduce file size or transmission bandwidth then you'll likely need to use a lossy CODEC, so the question then becomes how can you minimize the reduction in perceived quality of whatever you are compressing (photos, video, audio), which comes down to how these various…
> large scale detail (corresponding to low frequency FFT components)
This isn't true in practice - images are not bandlimited like audio so there aren't really visual elements of images corresponding to low frequency cosine waves. That's why the lowest frequency DCT coefficient in a JPEG image is 16x16 pixels, which is hardly large scale.
But you do quantize all components of the DCT transform, not just the highest ones.
Actually in the default JPEG quantization matrix it's the coefficient to the upper-left of the last one that gets the most quantization: https://en.wikipedia.org/wiki/Quantization_(image_processing...
Re: What Is the Fourier Transform?
#198Can 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…
Oki thx, it's starting to click :)
Re: What Is the Fourier Transform?
#199I found the 3Blue1Brown explanation to be more intuitive to understand, but this article has more practical examples/details.
Here's the 3Blue1Brown explanation:
Re: What Is the Fourier Transform?
#200If you like Fourier, you're going to love Laplace (or its discrete counterpart, the z transform). This took me down a very fascinating and intricate rabbit hole years ago, and is still one of my favorite hobbies. Application of Fourier, Laplace, and z transforms is (famously) useful in an incredibly wide variety of fields. I mostly use it for signal processing and analog electronics.
When I first learned Laplace transform in university, it was my goto for differential equations of any kind. I was even naive enough to believe well this is a solved problem now. Eventually found out this wasnt the case after studying PDEs. Its still my favourite transform. Immensely useful not to mention the whole method of moments in random variables is basically laplace transform. I don't like Fourier transform bu…
You know that if you have the Laplace transform, you can just insert s = iω and then you have the Fourier transform, right? :-P
(Or jω, if you prefer that notation)