You can also use the continuous Fourier transforms and not sample so there won’t be a nyquist rate. I also think neural networks have some really interesting properties when viewed from a signal processing perspective. In fact you can view the correct outputs of a neural network as a signal and the incorrect outputs as noise, and use conventional SP techniques to narrow down what parts of a network are informative.
Practically speaking, how would you use a continuous Fourier transform? Do you mean passing it through a "Fourier transform analog circuit" (does that even exist?), and then sample the output?
Build a bank of bandpass filters with different center frequencies, and you have an analog Fourier transform. (Or at least the power spectrum equivalent; it's trickier to get the phase info.) Analog vocoders are one example of this and they were invented long before discrete FTs.
See my other answer - I meant applying it analytically. Your idea is interesting though, and I do know that it should be possible to build a Fourier transform analog circuit. Once you sample though you will be subject to having a nyquist frequency and introducing noise to your signal and the advantage of performing it with an analog circuit will be lost.
Yea, actually I thought a bit more about it as well, and any real (analog) circuit will have a frequency response that tapers of to zero as the frequency goes to infinity. So that basically determines a Nyquist frequency for any analog circuit, even if that frequency is so high that it has no practical implications.
I think what he means is that in signal processing you can define the signal analytically and perform the Fourier transform analytically, without sampling. The Nyquist rate only comes in when you talk about digital signal processing.
