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What Is the Fourier Transform?

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Re: What Is the Fourier Transform?

#141

Post by thyristan »

eric-burel wrote: post #136

What always bothered me when trying to "feel" Fourier transforms is that to compute the oscillations, you need to wait some time. Mathematically, the transformation includes computing integrals. So it's tricky to understand how you compute the Fourier decomposition for a stream. Illustrations always show the whole signal over time but in real life you get the signal progressively. I'd be eager to read more on this.

For a stream, you use a sliding window to compute the FFT. The size of the window of course limits the lowest frequency range that you can 'see', same for the highest frequency through the time quantization that digital data usually has. So there will be an upper and lower frequency limit, beyond those limits the results are meaningless.

And the window of course creates a latency, which is sometimes relevant for realtime audio filtering by FFT.

Re: What Is the Fourier Transform?

#142

Post by dsego »

eric-burel wrote: post #136

What always bothered me when trying to "feel" Fourier transforms is that to compute the oscillations, you need to wait some time. Mathematically, the transformation includes computing integrals. So it's tricky to understand how you compute the Fourier decomposition for a stream. Illustrations always show the whole signal over time but in real life you get the signal progressively. I'd be eager to read more on this.

You do it in short windows, so you get eg. 512 samples and then run a short FFT. Or you can do longer windows that overlap, so e.g. hop by 512 but take 1024, more samples gives you more accurate results.

Re: What Is the Fourier Transform?

#143

Post by jcgrillo »

kragen wrote: post #103
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…

Differential equations aren't limited to describing stable systems, though, and there are chaotic systems that are also in some sense stable. Ordinary differential equations can describe any system with a finite number of state variables that change continuously (as opposed to instantaneously jumping from one state to another without going through states in between) and as a function of the system's current state (as…

One amazing application of spectral theory I always harp on when this topic comes up is Chebfun[1]. Trefethen's Spectral Methods in Matlab is also wonderful.

[1] http://www.chebfun.org/

Re: What Is the Fourier Transform?

#144

Post by AIPedant »

abetusk wrote: post #78

I have a pet theory that the reason why the FT, and other transforms (generating functions, Mellin/Laplace/Legendre/Haar), are so useful is because many real world functions are sparse and lend themselves to compressed sensing. The FT, as are many other transforms, are 1-1, so, in theory, there's no information lost or gained. In many real world conditions, looking at a function in frequency space greatly reduces the…

On the simplest end of that spectrum, Taylor series are useful because many real-world dynamics can be approximated as a "primarily linear behavior" + "nonlinear effects."

(And cases where that isn't true can still be instructive - a Taylor series expansion for air resistance gives a linear term representing the viscosity of the air and a quadratic term representing displacement of volumes of air. For ordinary air the linear component will have a small coefficient compared to the quadratic component.)

Re: What Is the Fourier Transform?

#145

Post by hybrid_study »

oh_fiddlesticks wrote: post #128

This video [1] from a visual effects channel (Captain Disillusion) has an excellent visual illustration of how the Fourier tranform works and how its used in visual effects in his video about blurring and unblurring ("ENHANCE!") images. 1. https://youtu.be/xDLxFGXuPEc?feature=shared

The Carl Sagan bit is also an amusing tribute.

Re: What Is the Fourier Transform?

#146

Post by lock1 »

oh_fiddlesticks wrote: post #128

This video [1] from a visual effects channel (Captain Disillusion) has an excellent visual illustration of how the Fourier tranform works and how its used in visual effects in his video about blurring and unblurring ("ENHANCE!") images. 1. https://youtu.be/xDLxFGXuPEc?feature=shared

While I like CD's works, I would say "CD / Blur" is the least informational of all the CD slash series. I guess it's a fun and more accessible option, but it certainly lacks depth compared to something like 3B1B's video on FT.

Re: What Is the Fourier Transform?

#147

Post by acjohnson55 »

RachelF wrote: post #98
yshklarov wrote: post #41

As everyone in this thread is sharing links, I'm gonna pitch in, too. This lecture by Dennis Freeman from MIT 6.003 "Signals and Systems" gives an intuitive explanation of the connections between the four popular Fourier transforms (the Fourier transform, the discrete Fourier transform, the Fourier series, and the discrete-time Fourier transform): https://ocw.mit.edu/courses/6-003-signals-and-systems-fall-2...

I wonder what happened to Wavelet transforms? The were very popular years ago, and now one never hears about them.

They have specialized applications for sure. I think it's just not as hot an area for new applied math work as 20 years ago.

Re: What Is the Fourier Transform?

#148

Post by jeremyscanvic »

eric-burel wrote: post #136

What always bothered me when trying to "feel" Fourier transforms is that to compute the oscillations, you need to wait some time. Mathematically, the transformation includes computing integrals. So it's tricky to understand how you compute the Fourier decomposition for a stream. Illustrations always show the whole signal over time but in real life you get the signal progressively. I'd be eager to read more on this.

The keyword you're looking for is time-frequency analysis and the main associated tool is the short-time Fourier transform(s). This is the theory underlying spectrograms and all those niceties!

Re: What Is the Fourier Transform?

#149

Post by acjohnson55 »

eric-burel wrote: post #136

What always bothered me when trying to "feel" Fourier transforms is that to compute the oscillations, you need to wait some time. Mathematically, the transformation includes computing integrals. So it's tricky to understand how you compute the Fourier decomposition for a stream. Illustrations always show the whole signal over time but in real life you get the signal progressively. I'd be eager to read more on this.

Check out https://en.m.wikipedia.org/wiki/Short-time_Fourier_transform

This is the "Fourier transform" that is most often used, in practice in computing.

A longer window length gives lower frequency resolution, but longer latency.

Re: What Is the Fourier Transform?

#150

Post by zozbot234 »

segfault99 wrote: post #131
zozbot234 wrote: post #127

Earlier quoted context omitted.

The so-called "Z transform" for discrete sequences is really just a misnomer for the actual method of generating functions (and formal power-series/Laurent-series). You just write a discrete sequence as a power series in z^(-1).

True dat. But you see there's this thing called 'Engineering Maths'. Apparently it's really bad for real mathematicians' blood pressure.

Analytic combinatorics (the rubric where mathematicians would want to place all the region-of-convergence, zeros-poles, etc. analysis of generating functions–formal power/Laurent series–Z transforms that engineering often focuses on) is not exactly easy-going either. Other common methods (relating convolution to multiplication, inverting transforms etc.) would traditionally be comprised under the Operational Calculus of Mikusiński.
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