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This article was interesting but didn't really tell me anything I don't already know. Does anyone know where I can find a good article that actually explains the mathematics of performing a Fourier transformation? I thought that is what this article was going to be about.
Earlier quoted context omitted.
Thanks for your feedback. Sure, any digital signal is by definition finite in its resolution (the sample rate or bits). I was trying to address the distinction between wave files of the type stored on audio CDs, and MP3s - both digital signals. I agree that the Fourier transform is in principle lossless, but it's particularly useful to use it in a lossy way, i.e. to throw out the least important (to us) components of…
Well, if you want to be 100% accurate, I think the section talking about how the high notes aren't important could be clarified. The really high "notes" have already been lost when you recoded the wav file digitally. The lossy step of mp3 encoding is not a result of the transform, but what you do with that information and is more complex than just discarding high frequency components. Also, the the word "note" is con…
In any case, if discarding high frequency information is all you needed to do to compress, you could simply low-pass filter the time-domain signal. A better description of what goes into MP3 compression is that it omits frequency components in sound that we can't hear because they are shadowed by nearby (in time and/or frequency) components that are louder.
> You could just tell them a handful of numbers—the sizes of the different circles in the picture above. Maybe I'm crazy and just missing something, but this feels a little too good to be true. This would put the set of smooth curves in 1-1 correspondence with the set of finite sets (since each curve is being specified completely by a finite set of numbers). But the set of finite sets is countably infinite since it's…
D=continuous domain, d=discrete domain, E=infinite extent, e=finite extent
time domain frequency domain
Fourier Transform: DEDE
Fourier Series: DedE
Discrete Time Fourier Transform: dEDe
Discrete Fourier Transform: dede
The cardinality mismatch shows up for FS and DTFT. It is resolved through a different notion of equality than you may be used to. The "distance" between two functions can be measured using several norms, and when this distance is zero we claim two functions are equal. L1, L2, and Linf are the common ones. Linf corresponds to pointwise equality, which is what you assumed (not unreasonably) that they meant. L1 and L2 do not.
L1 norm: d(f,g)=integrate(abs(f(x)-g(x)),a,b)
L2 norm: d(f,g)^2=integrate(abs(f(x)-g(x))^2,a,b)
Linf norm: d(f,g)=max(f(x)-g(x)) over the interval (a,b)
Fourier Series only guarantee reproduction (take the transform and then take the inverse to get the reproduced curve) up to the L2 norm. This makes physicists and engineers happy, since L2 norms correspond to energy measurements, and if the difference between two physical quantities has no energy then it's effectively irrelevant to physical processes.
The remaining hurdles for mathematicians are to show that
1) The reproduction coefficients found by the transform are the best possible coefficients (the curve they generate is closer to the original than for any other set of coefficients of the same size). This is a trivial proof using inner products that usually goes by a name like "best approximation theorem" or "generalized pythagorean theorem."
2) Reproducible curves are dense in the space of actual curves. That is, there is always a reproducible curve that has zero L2 difference from an arbitrary input curve. If this is true, then by #1 we know that the transforms will find it. Unfortunately, this proof is quite involved. A "quick and dirty" method uses the Stone-Wierstrass theorem (polynomials can reproduce continuous functions with arbitrary L1 or L2 fidelity + the FS can reproduce polynomials with arbitrary L2 fidelity). A better method arises in the context of Sturm-Liouville theory that generalizes "density" to solutions of a large class of differential equations. This is important for physicists and engineers because it justifies normal mode expansions even when the normal modes aren't perfect sinusoids.
If you want to know more, any linear algebra book that discusses inner products should hit on #1. For #2 you want an Analysis textbook. Analysis is a tough subject so it's far more important to be sure that the book starts at your level and has an understandable proof of Stone-Wierstrass than it is to be sure that it covers this exact topic. If it doesn't, supplement it with the first few chapters from "Completeness and Basis Properties of Sets of Special Functions" by Higgins and you'll be set.
Earlier quoted context omitted.
Yes, any digital signal has a limited bandwidth However, MP3s throw more high frequencies than the digital signal at that given sampling rate allows.
Well, sort of. If you have a wav file and an mp3 file, both of which have a sample rate of 44kHz, they will both be able to represent the same maximal frequency of 22kHz. The mp3 wouldn't necessarily discard the high frequency information, but it may do so when it is deemed that the sound wouldn't be perceivable.
Are there any good books/papers/web articles on this topic that are accessible? I often find myself reading papers where some of the math goes over my head.
Something with examples/code (code makes me understand math so much easier!) would be fantastic!
I would love to know more about FT's, along with FFT's and how they help with for example signal processing or finding a signal when looking at a sample or multiple samples of a SDR. Are there any good books/papers/web articles on this topic that are accessible? I often find myself reading papers where some of the math goes over my head. Something with examples/code (code makes me understand math so much easier!) wou…
I would love to know more about FT's, along with FFT's and how they help with for example signal processing or finding a signal when looking at a sample or multiple samples of a SDR. Are there any good books/papers/web articles on this topic that are accessible? I often find myself reading papers where some of the math goes over my head. Something with examples/code (code makes me understand math so much easier!) wou…
I'm sorry, but what's an SDR?