Thanks for the reference and explanation Robert. Now that you jog my memory I realize you covered this before, some time back. If I can find a few minutes I will try to work out a version including the images from the resampling. The trade-off between resampling and interpolation is not entirely clear to me in this analysis.
So linearly interpolating two adjacent FIR polyphases is equivalent to a single FIR with interpolated coefficients. I.e., we're using an affine approximation of the underlying resampling kernel. And at high resampling ratios this will be a good approximation. IIRC this has been covered on the list before as well. However, it costs twice as much CPU as running a single phase - so isn't the fair comparison to an FIR of double the order (and so, double the resampling ratio)? Ethan D On Wed, Sep 6, 2017 at 9:57 PM, robert bristow-johnson < r...@audioimagination.com> wrote: > > > ---------------------------- Original Message ---------------------------- > Subject: Re: [music-dsp] Sampling theory "best" explanation > From: "Ethan Duni" <ethan.d...@gmail.com> > Date: Wed, September 6, 2017 4:49 pm > To: "robert bristow-johnson" <r...@audioimagination.com> > "A discussion list for music-related DSP" <music-dsp@music.columbia.edu> > -------------------------------------------------------------------------- > > > rbj wrote: > >>what do you mean be "non-ideal"? that it's not an ideal brick wall LPF? > > it's still LTI if it's some other filter **unless** you're meaning that > > the possible aliasing. > > > > Yes, that is exactly what I am talking about. LTI systems cannot produce > > aliasing. > > > > Without an ideal bandlimiting filter, resampling doesn't fulfill either > > definition of time invariance. Not the classic one in terms of sample > > shifts, and not the "common real time" one suggested for multirate cases. > > > > It's easy to demonstrate this by constructing a counterexample. Consider > > downsampling by 2, and an input signal that contains only a single > sinusoid > > with frequency above half the (input) Nyquist rate, and at a frequency > that > > the non-ideal bandlimiting filter fails to completely suppress. To be > LTI, > > shifting the input by one sample should result in a half-sample shift in > > the output (i.e., bandlimited interpolation). But this doesn't happen, > due > > to aliasing. This becomes obvious if you push the frequency of the input > > sinusoid close to the (input) Nyquist frequency - instead of a > half-sample > > shift in the output, you get negation! > > > >>we draw the little arrows with different heights and we draw the impulses > > scaled with samples of negative value as arrows pointing down > > > > But that's just a graph of the discrete time sequence. > > well, even if the *information* necessary is the same, a graph of x[n] > need only be little dots, one per sample. or discrete lines (without > arrowheads). > > but the use of the symbol of an arrow for an impulse is a symbol of > something difficult to graph for a continuous-time function (not to be > confused with a continuous function). if the impulse heights and > directions (up or down) are analog to the sample value magnitude and > polarity, those graphing object suffice to depict these *hypothetical* > impulses in the continuous-time domain. > > > > > >>you could do SRC without linear interpolation (ZOH a.k.a. "drop-sample") > > but you would need a much larger table > >>(if i recall correctly, 1024 times larger, so it would be 512Kx > > oversampling) to get the same S/N. if you use 512x > >>oversampling and ZOH interpolation, you'll only get about 55 dB S/N for > an > > arbitrary conversion ratio. > > > > Interesting stuff, it didn't occur to me that the SNR would be that low. > > How do you estimate SNR for a particular configuration (i.e., target > > resampling ratio, fixed upsampling factor, etc)? Is that for ideal 512x > > resampling, or does it include the effects of a particular filter design > > choice? > > this is what Duane Wise and i ( https://www.researchgate.net/ > publication/266675823_Performance_of_Low-Order_ > Polynomial_Interpolators_in_the_Presence_of_Oversampled_Input ) were > trying to show and Olli Niemitalo (in his pink elephant paper > http://yehar.com/blog/wp-content/uploads/2009/08/deip.pdf ). > > so let's say that you're oversampling by a factor of R. if the sample > rate is 96 kHz and the audio is limited to 20 kHz, the oversampling ratio > is 2.4 . but now imagine it's *highly* oversampled (which we can get from > polyphase FIR resampling) like R=32 or R=512 or R=512K. > > when it's upsampled (like hypothetically stuffing 31 zeros or 511 zeros or > (512K-1) zeros into the stream and brick-wall low-pass filtering) then the > spectrum has energy at the baseband (from -Nyquist to +Nyquist of the > original sample rate, Fs) and is empty for 31 (or 511 or (512K-1)) image > widths of Nyquist, and the first non-zero image is at 32 or 512 or 512K x > Fs. > > now if you're drop-sample or ZOH interpolating it's convolving the train > of weighted impulses with a rect() pulse function and in the frequency > domain you are multiplying by a sinc() function with zeros through every > integer times R x Fs except for the one at 0 x Fs (the baseband, where the > sinc multiplies by virtually 1) those reduce your image by a known amount. > multiplying the magnitude by sinc() is the same as multiplying the power > spectrum by sinc^2(). > > with linear interpolation, you're convolving with a triangular pulse, > multiplying the sample values by the triangular pulse function and in the > frequency domain you're multiplying by a sinc^2() function and in the power > spectrum you're multiplying by a sinc^4() function. > > now that sinc^2 or sinc^4 functions really puts a hole in those images, > reducing the area in those power spectrum images greatly. > > now when we resample at an arbitrary rate, we can expect in worse case > that *all* of those images get folded back into the baseband. > > with 3rd-order B-spline (which i don't recommend) it's sinc^4 in the > frequency domain and sinc^8 in the power spectrum. > > so you have to determine the area of those images (compared to the area of > the baseband image) and add up all of the areas of those non-baseband > images in the power spectrum and that is the noise power (N) in this > conversion process. and the area of the baseband image is the signal power > (S). S/N is your signal-to-noise ratio and 10 log10( S/N ) is your S/R in > dB. > > it **assumes** that your filter design choice is perfect and **all** of > those images in between the baseband and (R Fs) are zero. as a *separate* > exercise, you can try to get a handle on the images between baseband and (R > Fs) that your optimally-designed polyphase FIR filter didn't completely > kill and that noise power should add to the other noise power, N, above. > > the point is that, with optimal upsampling, those images at non-zero > integer multiples of (R Fs) get whacked pretty good with the sinc^4 > function, which is what the linear interpolation between subsamples gets > you. > > take a look at the "Drop Sample Interpolation" section and the previous > section to get how we get a handle on the theoretical approximations to S/N. > > ____ > > hay, about mixing it up here on music-dsp, i'll take blame for some of it. > i really do sorta pick fights with people who don't approve of my naked > dirac delta functions. here is why i think i can get away with it: > > if you have a nascent impulse (that is a limiting approximation that has > constant area and shrinking width, it could be a rect() or a triangular or > Gaussian pulse), we can't hear the difference between these two pulses if > the area stays constant and the width is reduced from 2 microseconds to 1 > microsecond. as long as no voltage limits are exceeded and everything > stays LTI. the inherent LPF in our ears and head will make those two thin > pulses (but one is twice as thin as the other) sound the same. > > so i don't give a rat's ass about the difference between the nascent delta > functions and the ideal impulse that they are approximating (which is "not > a function", but is a "distribution"). but the "sampling" or "sifting" > property of a single dirac impulse function loses *all* of the data of the > (hopefully bandlimited) input signal that is sampled, except for the for > the value at precisely the impulse time. it doesn't matter what your ADC > does, we can take the numbers that come outa the ADC and attach them to > hypothetical impulses that are uniformly spaced in time, and pass that > result through a brick-wall low-pass filter and get our original > bandlimited continuous-time function back. that's the sampling theorem. > > so, in interpolation, whether it's for resampling for either pitch > shifting or for sample rate conversion or for precision delay, we > mathematically emulate, with a hypothetically zero-stuff discrete-time > input, a brick-wall filter (and, for an FIR, we need not include the > hypothetical zero-stuffed samples in the FIR summation), and resampling at > a different continuous time potentially between adjacent discrete sample > instances in the dirac impulse train that isturned into a windowed-sinc() > train by the practical brick wall filter. > > -- > > r b-j r...@audioimagination.com > > "Imagination is more important than knowledge." > > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp >
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