Hi, I'm going to have a go at answering, but there will be points where I'm not sure exactly what you're saying. I'll be as clear as possible and request clarification.
On Tue, Nov 12, 2013 at 3:56 PM, Theo Verelst <theo...@theover.org> wrote: > Suppose we take a simple 1st order low-pass (or high-pass) filter in the > digital domain, and compare it with the corresponding electronic > implementation, preferably as a network theory example, ignoring > electro-magnetics, Johnson noise, and physical delay of currents through > wires, which are reasonable assumptions for audio filters for instance. > > input signal -->o------ > | > - > | | R > | | > - > | > |------o--> output signal > | > ----- C > ----- > | > | > |-->o------------o-->| > | | > --- --- > > > We know that the practical, passive real world electronics circuit is > highly linear, highly accurate one precise pole that never changes, noise > isn't much for large signals, and of course this circuit doesn't work with > samples. So it is possible to see this as an idealized filter of simplest > form, for which there are simple network theoretical analysis schemes, for > DC and AC behavior, replacement values for the 2 network elements in the > complex frequency domain, and of course the impulse response is an > exponential, and the cutoff-frequency of the filter 1/( 2*PI*R*C ). > > So, we're considering H(s)=s/(1+s). Standard 1st order HPF. > So lets take the "equivalent" of this simple filter in the digital domain > (e.g. as function of a z^-1 network, and see what happens. > > Well, there's obviously no direct "equivalent". It comes down to the choice of method of discretisation how good an approximation we get. Obvious case in point: H(s) at nyquist is strictly less than 1. As a general rule, any conformal mapping will underperform at this task. This said with some good heuristics, one can do very well. > Or we could take some trivial or less trivial practical implementation > form, like a single delay, moderate sampling frequency one-to-one related > implementation with some number of bits accuracy, or we could take a > infinite length formal convolution integral of a formally bandwidth limited > input signal with the formal impulse response of this filter, state the > output of that as a sampled signal (with limited frequency range), and > sinc-reconstruct the analog signal coming from that, and see if this result > is theoretically correct (which I predict from hard theory, it is, given > that you factor the required frequency limitation in). > > I'm afraid I don't follow you here. Are you suggesting discretisation methods that are available to use? Is "like a single delay, moderate sampling frequency one-to-one related implementation with some number of bits accuracy" suggesting some xfer function of the form H(z)=Az+b ? With: "we could take a infinite length formal convolution integral of a formally bandwidth limited input signal with the formal impulse response of this filter, state the output of that as a sampled signal (with limited frequency range), and sinc-reconstruct the analog signal coming from that, and see if this result is theoretically correct" are you describing the case whereby an extremely long FIR is used given a bandlimited evaluation of the impulse response? That, of course, will produce the correct output. You appear, and I apologise if I misunderstand, to be explicitly stating the bandwidth limitation requirements. When processing a digital signal, we tacitly assume that the signal we have been presented with is the one that the user intends us to process. This is one of the unwritten, though generally understood axioms of audio processing - we are given bandlimited data, and we generate bandlimited data. Such is the nature of the space. It would be not only unfair, but irrelevant to measure a discretization of a system with signals entirely outside of it's bandwidth. The objective of any discretization is to generate the correctly bandlimited output of the analogue processor under measurement. Because failure to do so is aliasing. I feel like I might have missed your point here, so please do set me straight. Anyone want to explain to me what their most or least favorite > implementation of thus simple, single order filter without feedback in the > digital domain will do when we feed it with an impulse, a shifted step > (occurring between samples), a cascading of its own output after applying a > perfect phase shift c.q. delay, in terms of the exact, one-to-one > correspondence of the digital and the analog domain ? > > Again, I'm not sure why you specify "without feedback". Surely we would use feedback? As for an impulse, of course we're feeding it with a sinc function. A bandlimited shifted step is interesting - presumably that's the integral of a sinc function over a short range. Viz: shstep(t)=\int_a^b sin(t+x))/(t+x) dx with some normalisation to taste? Looks like it has a fairly smooth falloff in energy by frequency, the slope being determined by the distance between a and b, but that's just by eyeballing the equation. So, this is a function that probably has more practical utility in the analogue domain, but it's a rare bandlimited signal. A cascading of it's own output after applying a perfect phase shift…So you're saying we're discussing system A, and you want to know what A.S.A where S is a perfect phase shift would look like? Clearly it's A^2 S, but what's the input? Is this particularly interesting? Again, I think you're being too subtle with the point you're making for me to grasp what you're saying. The phrasing of the paragraph has an unusual lack of precision - your writing usually makes very explicit bandlimiting requirements, but here you skip that. Is that intentional? So, again, my apologies for not following your meaning here. > I'll help you a bit, you can take a single sine wave as an input, with > 'infinite" (say 10 seconds) duration, which makes the impulse response > easier to compute, Ok, here I think you're referring to the trivial application of |H(z)| and Arg(H(z)) for z=e^if for some f to phase-offset and scale the sinewave… though without taking the fourier transform across all frequencies that doesn't give you the ir. Sounds as if you're hinting at something here, but it's really not clear to me what. > and the bandwidth limitation easy to deal with, Again, I'm not succeeding with this. > and you want to be aware of the following property of the correct sampling > and sampling reconstruction theory (oh, and don't try to prove network > theory transforms or the s-transform (or the comparable Fourier transform) > itself, that's way to involved, of course presume they're a given): > > "the s-transform (or the comparable Fourier transform)" What? I don't know in what terms you understand this maths, but it's different to mine. That or you're being very loose with your phraseology. The Fourier transform (of an algebra partnered with a normed space) is a consequence of the Gelfand-Naimark representation theorem, which builds a secondary, related space from a basis of the characters of the algebra underlying the space in question. > * the shift invariance theorem > > Link? > (that states that the output of a filter remains shift the same if you > sub-sample shift the input of a sampled system. > > Can you formalise the parenthesized section for me? Because what I read here is: 1. Let S be an operator which sub-sample shift a given signal. 2. Let F be a filter, and X an input signal. 3. The Shift Invariance Theorem states that F(X)=F(SX). Which is trivially nonsense, since SF(X)=F(SX) by linearity. But this is childsplay for any linear system. I can't possibly see how it could be relevant here. Again, please correct me. Dave. > T.V. > > > -- > dupswapdrop -- the music-dsp mailing list and website: > subscription info, FAQ, source code archive, list archive, book reviews, > dsp links > http://music.columbia.edu/cmc/music-dsp > http://music.columbia.edu/mailman/listinfo/music-dsp > -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
