Re: time-nuts Digest, Vol 184, Issue 35 On Wed, 27 Nov 2019 12:00:02 -0500, time-nuts-request@lists.febo.com wrote: -------------------------------- > > Message: 8 > Date: Wed, 27 Nov 2019 00:29:19 +0100 > From: Jan-Derk Bakker <jdbakker@gmail.com> > To: Discussion of precise time and frequency measurement > <time-nuts@lists.febo.com> > Subject: Re: [time-nuts] A simple sampling DMTD > Message-ID: > <CAEoGJM=DwGUzAuTAAw13948HuAGdZo+YSxymozDnWD07G94Oiw@mail.gmail.com> > Content-Type: text/plain; charset="UTF-8" > > [This thread has started about three months ago; first post with design > considerations is here: > https://www.mail-archive.com/time-nuts@lists.febo.com/msg04265.html ] > > Dear all, > > In the past month I have managed to get the PLL working, and found a > lightweight way to eliminate most if not all of the offset/drift. > > After the discussions with Attila and Bob I have expanded the PLL bandwidth > to 0.5Hz (with a 10Hz PFD frequency I can't go very much higher). The > damping factor is 0.7 for now; I intend to do more tests on how an > increased damping affects the ZCD. To get sufficient ECD drive resolution I > have implemented a 3rd order multibit sigma/delta modulator driving a > 12-bit ADC (with passive filtering and active buffering). The result of the > measured beat note period can be seen at > http://www.lartmaker.nl/time-nuts/PSD%20of%20measured%20ZCD%20period%20with%20PLL%20active.pdf > ; peaking is shown to be limited. > > With an active PLL the beat note spectrum became much narrower (compare > http://www.lartmaker.nl/time-nuts/PSD%20of%20HP10811%20into%20the%20LTC2140%20with%20PLL%20active.pdf > to > http://www.lartmaker.nl/time-nuts/DMTD%20self-noise%20input%20spectrum.pdf > ). This made it possible to use a comb filter to suppress even-order > harmonics (before adding hardware notch filtering at the input). > > The 1001-point FIR band pass filter is a good reference to get an idea of > the best case performance of the system, but it is computationally > infeasible to run on an 8-bit processor. While a cheap comb filter can take > a bite out of the HF noise, canceling offset/drift is harder. Early on I > was looking into ways to average all samples of a single period of the beat > note, but I had trouble finding a closed form solution to the fact that the > beat note period is never an exact integer multiple of the sampling rate. > Through numerical methods I did end up finding a good estimator for the > "leakthrough" caused by the fractional part of the beat note period (as a > function of the measured period and the phase offset), which was fairly > inexpensive to implement. This has yielded a combined "simple" signal > processing path of a differentiator, a double comb filter and the offset > estimator, which is getting very close in performance to the "ideal" band > pass filter (OADEV of 3.77e-13@tau=1s versus 3.25e-13@tau=1s for the BPF; > full plot: > http://www.lartmaker.nl/time-nuts/DMTD%20self-noise%20OADEV%20with%20PLL%20and%20various%20filters.pdf > for this 600000-second recording: > http://www.lartmaker.nl/time-nuts/600ksec%20run%20with%20PLL,%2010811%20through%20splitter.png > . OADEV past ~1000sec is severely compromised by the fact that the > measurement setup is in my home lab which sees temperature swings of up to > 20 degrees C and which does get bumped from time to time. Longer runs in a > more controlled setting forthcoming). In the radar world, the standard solution to the leakthrough problem is to batch the data and apply a windowing function to the data in each batch. Typically, the batches overlap such that every sample appears in two batches. The window functions largely eliminate the splice error due to the FFT, which is fact splices each batch into a circle, causing a discontinuity at the splice. If the window function is very small at the splice, there is little discontinuity power sprayed into innocent FFT bins. In radar, Taylor windows are used, as well as Dolph-Chebyshev if very low sidelobes are needed. .<https://en.wikipedia.org/wiki/Window_function#Dolph%E2%80%93Chebyshev_window> Taylor is mentioned in the above D-C description, and a link is provided. Joe Gwinn

On Wed, Nov 27, 2019 at 6:22 PM Joseph Gwinn <joegwinn@comcast.net> wrote: > in two batches. The window functions largely eliminate the splice > error due to the FFT, which is fact splices each batch into a circle, > causing a discontinuity at the splice. If the window function is very > small at the splice, there is little discontinuity power sprayed into > innocent FFT bins. > > In radar, Taylor windows are used, as well as Dolph-Chebyshev if very > low sidelobes are needed. Related to but distinct from low-sidelobe window functions are _sidelobeless_ windows, which I think may be particularly useful in control-loop applications like PLLs. The gaussian window is obviously sidelobeless, but it has infinite support so it's not useful. But it turns out that there exist finite windows that are completely sidelobeless (a fourier transform of their kernel is monotone). A triangular window convolved with a truncated gaussian window is sidelobeless (with an appropriate choice in parameters), as is a hann-poisson window. While working on the problem of sinusodial modelling ( https://arxiv.org/abs/1602.05900 ), which is essentially the same problem as PLL tracking tones in a noisy signal, I found sidelobeless windows to work much better than even low sidelobe windows like Dolph-Chebyshev ... but I never found a lot of coverage in the literature.

On 11/27/19 9:48 AM, Joseph Gwinn wrote: > Re: time-nuts Digest, Vol 184, Issue 35 > On Wed, 27 Nov 2019 12:00:02 -0500, time-nuts-request@lists.febo.com > wrote: > -------------------------------- >> >> Message: 8 >> Date: Wed, 27 Nov 2019 00:29:19 +0100 >> From: Jan-Derk Bakker <jdbakker@gmail.com> >> To: Discussion of precise time and frequency measurement >> <time-nuts@lists.febo.com> >> Subject: Re: [time-nuts] A simple sampling DMTD >> Message-ID: >> <CAEoGJM=DwGUzAuTAAw13948HuAGdZo+YSxymozDnWD07G94Oiw@mail.gmail.com> >> Content-Type: text/plain; charset="UTF-8" >> >> [This thread has started about three months ago; first post with design >> considerations is here: >> https://www.mail-archive.com/time-nuts@lists.febo.com/msg04265.html ] >> >> Dear all, >> >> In the past month I have managed to get the PLL working, and found a >> lightweight way to eliminate most if not all of the offset/drift. >> >> After the discussions with Attila and Bob I have expanded the PLL bandwidth >> to 0.5Hz (with a 10Hz PFD frequency I can't go very much higher). The >> damping factor is 0.7 for now; I intend to do more tests on how an >> increased damping affects the ZCD. To get sufficient ECD drive resolution I >> have implemented a 3rd order multibit sigma/delta modulator driving a >> 12-bit ADC (with passive filtering and active buffering). The result of the >> measured beat note period can be seen at >> > http://www.lartmaker.nl/time-nuts/PSD%20of%20measured%20ZCD%20period%20with%20PLL%20active.pdf >> ; peaking is shown to be limited. >> >> With an active PLL the beat note spectrum became much narrower (compare >> > http://www.lartmaker.nl/time-nuts/PSD%20of%20HP10811%20into%20the%20LTC2140%20with%20PLL%20active.pdf >> to >> http://www.lartmaker.nl/time-nuts/DMTD%20self-noise%20input%20spectrum.pdf >> ). This made it possible to use a comb filter to suppress even-order >> harmonics (before adding hardware notch filtering at the input). >> >> The 1001-point FIR band pass filter is a good reference to get an idea of >> the best case performance of the system, but it is computationally >> infeasible to run on an 8-bit processor. While a cheap comb filter can take >> a bite out of the HF noise, canceling offset/drift is harder. Early on I >> was looking into ways to average all samples of a single period of the beat >> note, but I had trouble finding a closed form solution to the fact that the >> beat note period is never an exact integer multiple of the sampling rate. >> Through numerical methods I did end up finding a good estimator for the >> "leakthrough" caused by the fractional part of the beat note period (as a >> function of the measured period and the phase offset), which was fairly >> inexpensive to implement. This has yielded a combined "simple" signal >> processing path of a differentiator, a double comb filter and the offset >> estimator, which is getting very close in performance to the "ideal" band >> pass filter (OADEV of 3.77e-13@tau=1s versus 3.25e-13@tau=1s for the BPF; >> full plot: >> > http://www.lartmaker.nl/time-nuts/DMTD%20self-noise%20OADEV%20with%20PLL%20and%20various%20filters.pdf >> for this 600000-second recording: >> > http://www.lartmaker.nl/time-nuts/600ksec%20run%20with%20PLL,%2010811%20through%20splitter.png >> . OADEV past ~1000sec is severely compromised by the fact that the >> measurement setup is in my home lab which sees temperature swings of up to >> 20 degrees C and which does get bumped from time to time. Longer runs in a >> more controlled setting forthcoming). > > In the radar world, the standard solution to the leakthrough problem is > to batch the data and apply a windowing function to the data in each > batch. Typically, the batches overlap such that every sample appears > in two batches. The window functions largely eliminate the splice > error due to the FFT, which is fact splices each batch into a circle, > causing a discontinuity at the splice. If the window function is very > small at the splice, there is little discontinuity power sprayed into > innocent FFT bins. > > In radar, Taylor windows are used, as well as Dolph-Chebyshev if very > low sidelobes are needed. > > .<https://en.wikipedia.org/wiki/Window_function#Dolph%E2%80%93Chebyshev_window> > > Taylor is mentioned in the above D-C description, and a link is > provided. > There are also some very nice low sidelobe, rectangular top, windows like the Blackman-Harris, Nuttall, Solomon, etc. Albrecht, 2001 has a collection of windows with max sidelobe levels more than 200dB down. fred harris has a great 32 page summary paper in the Jan 1978 Proceedings of IEEE which collects a bunch of the windows and their properties. "On the use of windows for harmonic analysis with discrete fourier transform" v66, #1, pp51-83

Dear Joe, On Wed, Nov 27, 2019 at 7:22 PM Joseph Gwinn <joegwinn@comcast.net> wrote: > > [snip] > > The 1001-point FIR band pass filter is a good reference to get an idea of > > the best case performance of the system, but it is computationally > > infeasible to run on an 8-bit processor. While a cheap comb filter can > take > > a bite out of the HF noise, canceling offset/drift is harder. Early on I > > was looking into ways to average all samples of a single period of the > beat > > note, but I had trouble finding a closed form solution to the fact that > the > > beat note period is never an exact integer multiple of the sampling rate. > > Through numerical methods I did end up finding a good estimator for the > > "leakthrough" caused by the fractional part of the beat note period (as a > > function of the measured period and the phase offset), which was fairly > > inexpensive to implement. [snip] > > In the radar world, the standard solution to the leakthrough problem is > to batch the data and apply a windowing function to the data in each > batch. Typically, the batches overlap such that every sample appears > in two batches. The window functions largely eliminate the splice > error due to the FFT, which is fact splices each batch into a circle, > causing a discontinuity at the splice. If the window function is very > small at the splice, there is little discontinuity power sprayed into > innocent FFT bins. > This is an interesting approach, and one which hadn't occurred to me while I was working on the offset/drift cancellation. After some consideration I am afraid it will not help much in this particular case. What I want is to filter out as much DC/LF energy before running the zero crossing detector (ZCD) on the rising edge of the sampled sine wave. To do this right now I also run the ZCD on falling edges, and I remember the (sub)sample position of the current and previous falling edges. I then average a full nominal period of the sine wave between these two falling edges(200 samples at 10Hz beat frequency and 2ksps after the first sinc^2 filter/decimator), to determine the offset at the rising edge in between. This is indeed mathematically equivalent to calculating bin 0 of a forward Fourier transform with a rectangular window. Because the actual period is not an exact integer number of samples, in practice this bin is contaminated by spectral leakage. Windowing would help here, if it weren't for the fact that all of my wanted signal energy is in bin 1. As any windowing function other than the rectangular window trades side-lobe suppression for main lobe width, the cure would likely be worse than the disease. This could be ameliorated by averaging over multiple periods (increasing the transform length), but that would necessarily spread the LF noise energy over multiple bins, again making it all but impossible to get optimal rejection. What I've done instead is make an estimator for the leakage. For a period of 200 +/- 1 sample, the following has an estimator error of better than -40dB: alpha = 0.5*perOffset^2 + (1.5 - phase)*perOffset - phase correction = (1-alpha) * pointN + alpha * pointN1 where correction is the estimator (in sample values) perOffset is the difference between the measured period (199...201 samples) and the nominal period (200 samples) phase is the estimated starting phase of the averaging interval, in fractional samples (range: [0,1>) pointN is the value of the sample immediately after the averaging interval (ie sample[200], when the averaging interval is sum(sample[0]...sample[199]) pointN1 is the value of the second sample after the averaging interval (ie sample[201]). This works very well, both on simulated and actual data. I am embarrassed to admit that I cannot explain *why* it must be this particular formula (and I'm very open to get hints here). Sincerely, JD 'if you can't do the math, run simulations and do curve fits until it works' B.