Originally posted by: BrownTown
The math with the fourier transform looks right, but im still not entirely sure how you go about reconstructing the signal, just looking at some online programs which let you define signal frequencies and sampling rates it seems (at least visually) like you can get aliasing even above the Nyquist frequency. And then there is the problem that if you sample just at the Nyquist rate you can still get all zeroes which doesn't really tell you anything about the amplitude of the signal...
The math is how we reconstruct the signal. There's a variety of ways to do it specifically, but all it really concerns is doing it in such a way as the math works out. For example, the simplest way to reconstruct the analog signal, and how it is usually explained in advertisements, is a zero-order hold DAC. This is just the classic stairstep waveform that you'll see in advertisements trying to tell you how 96KHz sampling rate is better than 44.1 KHz. But that isn't the end of the DAC stage, you then pass the stairstep waveform through a filter, and the filter is dictated by the math so that the end result is the original waveform. For a zero-order hold, it is an ideal low-pass filter whose amplitude is described by e^x or something like that. But there are other types of DAC and filter combinations. You could do a DAC that outputs a slightly interperlated waveform that would allow for a simpler analog filter after it. The filter following the DAC is your interpolation filter, it takes care of interpolating whatever is sent out of the DAC back to the original waveform. And going from the math, if you know what the output of the DAC is and what the original input was, then you can find the mathematical description of your required filter.
As for aliasing above the Nyquist frequency, not sure what you mean there. One of the stipulations is that you have a bandlimited signal. In real life, when we digitize a waveform, we filter out the higher frequencies that we cannot reproduce. For example, in audio, the audio is sent through a brickwall filter to remove anything above 20 KHz for CD's. The Nyquist frequency for CD's is 44.1KHz, the extra bandwidth of 2.05KHz is there to allow for a little leeway in the interpolating filter without causing aliasing (since we cannot produce an ideal filter we give some extra bandwidth where no signal information is contained so that any aliasing lies outside of the bandwidth of the original signal).