On 2014-03-28, Emanuel Landeholm wrote:
I agree with the points you raise and I'd like to add that you can also trade bandwidth for bits.
Totally, and you don't even need to go as far as to apply noise shaping. High sampling rates and linear filtering already raises that question. Okay, in audio DSP you'd typically want to do the real, hardcore, noise shaping trick, at least in release formats with insufficient bits like CD, but e.g. in RF work you immediately bump into these kinds of considerations.
One of the nicest examples is something we bumped into a little while ago already, after something Theo said. That's because, as soon as you start doing frequency analyses in the presence of noise, high bandwidth counter-intuitively means that the same precise noise RMS in your signals is spread over a wider bandwidth, so that cutting it out with a filter is per se already an instance of that tradeoff.
That then also means that you can't read spectra at all without invoking the concept of resolution bandwidth. FFT's are slightly easier compared to the analog sweeped ones because thought of as filter banks they're critically sampled by definition, but even they lead to nasty surprises for the uninitiated, because the length of the transform leads the individual bins to be narrower. When that's so, in a longer block the noise is distributed over more bins, but steady state sinusoids -- with their infinitely thin Dirac spectra -- stay within a single bin and hence stick out like a sore thumb. Thus, with a long enough transform, something that in the time domain looks like nothing but noise, in the frequency one suddenly has a spike so high that scaling it to range makes the noise floor round off to invisibility.
Analog spectra are then even nastier, because they're fundamentally overcompete representations where you have two separate things to worry about: the sweep rate which sets the convolutional spreading of peaks due to amplitude modulation, and the resolution bandwidth, which sets integration time, and so both temporal responsiveness and the noise supression of the matched filter.
Funnily enough, eventhough I've been interested in the theoretical aspects of DSP for some two decades now, all such woes of matched filtering and the like are relatively new to me. That leads me to suspect those aspects aren't stressed enough in modern treatments of the subject, and that I might not be the only diginative who has gap in their understanding regarding continuous spectra, matched filtering, statistical detection in the presence of noise, reading the relevant diagrams, and so on. And in fact, while I'm rather critical of Theo's and other audiophile minded folks' claims over things like ultrahigh fidelity formats, I must say their understanding of traditional analog EE and background in tinkering with it probably make them better armed to deal with this side of the field than I'll ever become.
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