Re: [time-nuts] AVAR - S_Y conversion
Hello time-nuts community, thanks to your feedback and that of others, I could make additional progress on my journey to understand powerlaw noise :) I would like to reiterate what I have learned so far. Please comment on anything that you think is done wrong. --- 0) Conventions: --- My main goal is to simulate powerlaw noise and analyze it as described in IEEE1139 [1]. I will use the following conventions: f_s sampling frequency of the simulated noise tau_0 time interval between two samples (directly related as 1/f_s) y(t)Fractional frequency deviation at time t S_y(f) one-sided PSD of y, has a shape of as h_alpha * f^alpha alpha has one of the following values: 2 White PM noise 1 Flicker PM noise 0 White FM noise -1 Flicker FM noise -2 Random Walk noise --- 1) Using the formulas in IEEE 1139: --- As I said, I mainly use the formulas given in IEEE1139, especially those in Table B.2, which define the relationship between Allan Variance and the S_y. I have attached a screenshot of those formulas to this mail. I'm not sure what f_h should be in the calculation of the terms D and E. IEEE 1139 defines it as the high-frequency cutoff of an infinitely sharp low-pass filter. I don't sample my noise from a real device, but simulate it. Am I right that I can use f_h = f_s/2 (Nyquist theorem)? I haven't found a publication that explicitly states this, but in my experiments this assumptions works well. --- 2) Noise generation, calculation of h_alpha: --- I generate powerlaw noise according to the publication by Kasdin and Walter [2]. As you might have noticed in my initial mail, I estimated h_alpha from an Allan Variance plot. This is not how it should be done. The better way would be to estimate it from my noise configuration. The reason why I went the other way is that I had trouble to estimate h_alpha from my noise configuration. The approach described in [2] generates white noise, and filters it to get the required PSD shape. The relationship between the standard variance Qd of the white input noise and the scaling factor h_alpha of the powerlaw noise is given in [2] as follows (equ 39): Qd = h_alpha / (2 * (2*pi)^alpha * tau_0^(alpha-1)) However, this definition never worked for me to predict the relationship between h_alpha and Qd. I think the formula should be modified as Qd = h_alpha / (2 * (2*pi)^alpha * tau_0^(alpha+1)) I changed the sign of the 1 in the exponent of tau_0. Using this change, the formula now works for calculating h_alpha and Qd from each other, and the results match if I do a counter-check and estimage h_alpha from the AVAR or the PSD. This change also makes the formula more consistent (e.g. the AVAR is defined so that the standard variance of White FM noise should match the AVAR for tau_0, this holds with the modified formula). --- 3) PSD estimation --- I tried to implement the PSD estimation as a mixture of the information found in [3] and [4]. However, I'm a novice when it comes to PSDs, and my approach had some error (I still don't know exactly what was wrong). As I know now, the 3dB difference that I saw for RW noise in my initial mail is a bug in my clumsy implementation. I tried to learn more about PSDs, and [5] proved to be very useful. I know now that the PSD estimation approach that I tried to use is known as the 'Welch's method' and supported in Matlab as 'pwelch'. -- Using this tool the PSD estimate converges to the expected value for all 5 types of noise! :) As PSD estimation configuration I use non-overlapping segments with a Hann window. There is no deeper reason for this choice (as I said, I'm new to these topics, so I would'nt know any better), it's just what is used in the example in [5] and it provides a PSD estimate as I would expect it. --- 4) All summed up --- With the assumptions and concepts of 1-3 I have finally been able to generate powerlaw noise in a way that the results match what I had configured :) I tried once more to generate the 5 noise types, and compare them with my expectations. I have included the resulting plots in a PDF file, which is available here: https://www.dropbox.com/s/lrdbpxrghkca0y8/Relationship_PSD_AVAR.pdf?dl=0 However, I also found new things that confuse me :P When I try to estimate the PSD with the 'pwelch' function in Matlab, I can select the number of non-overlapping segments my raw data should be divided into. Using a larger number of segments leads to a nicer plot, which converges to clean lines at some point. However, I also see that the lower part
Re: [time-nuts] AVAR - S_Y conversion
Hello Magnus, I'm sorry, but I can't follow you here. I know that time deviation values and fractional frequency values are related via integration, so I think I understand the third line. But I don't know what is meant especially with the first one: What is d(t)? What is D? Is this the frequency drift? - Wolfgang On 03/14/2015 11:12 AM, Magnus Danielson wrote: Wolfgang, Remember to scale the double-integral right: d(t) = D y(t) = integrate(d(t),t) = y_0 + Dt x(t) = integrate(y(t),t) = x_0 + y_0*t + D/2*t^2 Did you miss the 1/2 factor somewhere? That would make sense for the Random Walk phase noise. Cheers, Magnus On 03/09/2015 04:57 PM, Wolfgang Wallner wrote: On 03/06/2015 10:29 PM, Magnus Danielson wrote: I have checked several sources, and they match up with the IEEE 1139 in this regard. I have also evaluated the equation for Allan variance for the random walk noise, and it matches up with the references and what I put here: https://en.wikipedia.org/wiki/Allan_variance#Power-law_noise Thanks a lot for your effort! So, the A formula you have matches up. You will need to find another source of the mismatch. I will take a step back and describe the overall picture of what I'm doing. Maybe someone can help me spot where I do something wrong. (As stated later, the part where I'm quite unsure what I'm doing is the PSD estimation part.) My main goal is to simulate powerlaw noise. I then analyze the generated noise to check if my simulation is reasonable. So the basic workflow would be the following: 1) Generate noise 2) Analyze the noise in the time and frequency domain 3) See that everything agrees and be happy :) Step 1: Noise generation --- I generate powerlaw noise with the method described by Kasdin and Walter in [1]. So basically I generate white noise and apply a filter as described in [1] to get a PSD shape corresponding to the different values of alpha. The part of the PSD that will have the correct shape depends on the filter length and the simulated sampling frequency. Basically: the length of a simulation I would like to carry out specifies a lower bound on the filter length to get correct results. For WPM, WFM and RW noise I can use a shortcut: for these types of noise the filter coefficients are basically a discrete derivative, an identity filter and a cumulative sum. This is expected, as it agrees with [2], which states that integration of powerlaw noise decreases alpha by 2 (chapter 3.4 in [2]). Thus for even values of alpha I can even skip the expensive convolution to apply the filter and implement the filters directly. As input white noise I use a Gaussian distribution, mainly because that is what is used in the original paper. (I have also found another implementation [3] that optionally provides a uniform distribution). I'm quite confident that the noise generation part works as expected. However, even if I do something wrong here, it should not influence the analyzing part. Step 2: Analyzing noise --- 2.1 Time domain To analyze powerlaw noise in the time domain, I use a Matlab script called 'allan' [4], which calculates the Allan Deviation. I also found another Matlab tool called 'Stability Analyzer' [5], which can also calculate ADEV values. These two tools are developed by different authors and expect different input formats, but their results agree for any noise example I have tried so far. Thus I would say both of them can be trusted to work as expected. 2.2 Frequency domain IEEE 1139[6] defines S_y as: frequency spectrum Sy(f): One-sided spectral density of the normalized frequency fluctuations, as defined in normalized frequency fluctuations y(t). However, I'm not sure how to calculate this measure for a given noise sample. Anything I describe below is just based on 'I think this might work'. If anyone knows a better way of calculating S_y, or tools that can be used for this task, I would be glad to hear about it :) As already stated in the earlier mail I use the method described in [7] to estimate the one-sided PSD of my noise data in FFD format. These plots are quite noisy, and to improve the graphical presentation I use the averaging method described in [8]. I split the noise vector in parts of equal length, calculate the individual PSDs and average over them. Using this averaging method, the PSD plots converge to lines on a log-log plot with the expected slopes. I have an example figure attached to the mail that shows the effect of the averaging (PSD_Average.png). Step 3: Comparing time and frequency domain results --- At this point I have plots for both the Allan Deviation and the FFD-PSD, and would like to compare them. As first step I estimate h_alpha from the Allan Deviation plot
Re: [time-nuts] AVAR - S_Y conversion
Wolfgang, Remember to scale the double-integral right: d(t) = D y(t) = integrate(d(t),t) = y_0 + Dt x(t) = integrate(y(t),t) = x_0 + y_0*t + D/2*t^2 Did you miss the 1/2 factor somewhere? That would make sense for the Random Walk phase noise. Cheers, Magnus On 03/09/2015 04:57 PM, Wolfgang Wallner wrote: On 03/06/2015 10:29 PM, Magnus Danielson wrote: I have checked several sources, and they match up with the IEEE 1139 in this regard. I have also evaluated the equation for Allan variance for the random walk noise, and it matches up with the references and what I put here: https://en.wikipedia.org/wiki/Allan_variance#Power-law_noise Thanks a lot for your effort! So, the A formula you have matches up. You will need to find another source of the mismatch. I will take a step back and describe the overall picture of what I'm doing. Maybe someone can help me spot where I do something wrong. (As stated later, the part where I'm quite unsure what I'm doing is the PSD estimation part.) My main goal is to simulate powerlaw noise. I then analyze the generated noise to check if my simulation is reasonable. So the basic workflow would be the following: 1) Generate noise 2) Analyze the noise in the time and frequency domain 3) See that everything agrees and be happy :) Step 1: Noise generation --- I generate powerlaw noise with the method described by Kasdin and Walter in [1]. So basically I generate white noise and apply a filter as described in [1] to get a PSD shape corresponding to the different values of alpha. The part of the PSD that will have the correct shape depends on the filter length and the simulated sampling frequency. Basically: the length of a simulation I would like to carry out specifies a lower bound on the filter length to get correct results. For WPM, WFM and RW noise I can use a shortcut: for these types of noise the filter coefficients are basically a discrete derivative, an identity filter and a cumulative sum. This is expected, as it agrees with [2], which states that integration of powerlaw noise decreases alpha by 2 (chapter 3.4 in [2]). Thus for even values of alpha I can even skip the expensive convolution to apply the filter and implement the filters directly. As input white noise I use a Gaussian distribution, mainly because that is what is used in the original paper. (I have also found another implementation [3] that optionally provides a uniform distribution). I'm quite confident that the noise generation part works as expected. However, even if I do something wrong here, it should not influence the analyzing part. Step 2: Analyzing noise --- 2.1 Time domain To analyze powerlaw noise in the time domain, I use a Matlab script called 'allan' [4], which calculates the Allan Deviation. I also found another Matlab tool called 'Stability Analyzer' [5], which can also calculate ADEV values. These two tools are developed by different authors and expect different input formats, but their results agree for any noise example I have tried so far. Thus I would say both of them can be trusted to work as expected. 2.2 Frequency domain IEEE 1139[6] defines S_y as: frequency spectrum Sy(f): One-sided spectral density of the normalized frequency fluctuations, as defined in normalized frequency fluctuations y(t). However, I'm not sure how to calculate this measure for a given noise sample. Anything I describe below is just based on 'I think this might work'. If anyone knows a better way of calculating S_y, or tools that can be used for this task, I would be glad to hear about it :) As already stated in the earlier mail I use the method described in [7] to estimate the one-sided PSD of my noise data in FFD format. These plots are quite noisy, and to improve the graphical presentation I use the averaging method described in [8]. I split the noise vector in parts of equal length, calculate the individual PSDs and average over them. Using this averaging method, the PSD plots converge to lines on a log-log plot with the expected slopes. I have an example figure attached to the mail that shows the effect of the averaging (PSD_Average.png). Step 3: Comparing time and frequency domain results --- At this point I have plots for both the Allan Deviation and the FFD-PSD, and would like to compare them. As first step I estimate h_alpha from the Allan Deviation plot (I'm aware that I need to take care for the Allan Deviation - Allan Variance conversion). Then I try to estimate the expected PSD values and compare them with my actual plot using the formulas from IEEE 1139. However, at this point a see that RW noise behaves unexpected :( Numerical Example: --- Suppose the figure attached as 'Numeric_example.png': At Tau = 0.1s the ADEV
Re: [time-nuts] AVAR - S_Y conversion
Hello Tom, hello time-nuts, Have a look at the 20 plots in: http://leapsecond.com/pages/allan/Exploring_Allan_Deviation_v2.pdf Thanks for sharing the data and the PDF! It's good to have reference values. I also had a look at the stable32 user manual [1] to see how it calculates the PSD graphs and compare it to my approach. On page 195 there is a paragraph about PSD Averaging, and it describes basically the same averaging procedure that I use, so I guess this approach is justified. The same page also has a paragraph about PSD Windowing, which I currently do not apply. I will have to read further on that topic. What I could not find in the manual is how the green lines in your PSD plots are calculated. The manual also states that stable32 uses the noise generation as described by Todd Walter, so it's the same approach that I apply. See if your results agree. I have tried to analyze your data with my methods. To compare our results I made screenshots of your graphs and used them as background images in my plots. The PDF with my results is available at [2]. It is not explicitly stated in your PDF file, but from the graph axis I assume a sampling frequency of 1Hz. The images in the left column of my PDF show the estimated ADEV, and i mostly agrees with your plots (at least below Tau = 10^2s). If I use an averaging factor of 2 in the PSD estimation, the PSD plots match yours very closely. These are the images in the center column in my PDF. What was the averaging configuration for your plots? The images on the right show the estimated PSD if I use an averaging factor of 200. In these plots my estimates converge to single lines, but they do not match the green lines of your graphs for any of the noise types. It looks like they would be too large by a factor of 2 for any noise type. I have used both the frequency and phase data you provided for these plots, and results agree (so at least my FFD/TD conversion scripts work :) ) I would interpret the comparison graphs as follows (please correct me if you think otherwise): *) The tool I use for ADEV calculation should be fine. *) The script I use for PSD estimations returns usable results for an averaging number of 2. However, I'm still not sure how I this helps me to apply the domain conversion formulas found in IEEE 1139 formulas to relate these plots together. Questions about the PSD plot generation: *) How is the green line in your plots calculated? Or more generally: From a given set of noise data, how does one estimate h_alpha? *) Is averaging the PSD until it converges to a line the wrong way to go? General question about the IEEE1139 formulas: *) What values do I need to use for f_h and Tau_0 when calculating the terms D and E for WPM and FPM noise. If I have noise sampled at f_s = 1kHz, the smallest Tau value would be 1ms. Does that mean that Tau_0 = 1ms? And would f_h be f_s/2? Thanks for your help so far, Wolfgang [1] http://www.wriley.com/Manual150.pdf [2] http://leapsecond.com/tmp/graphs.pdf ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR - S_Y conversion
On 03/06/2015 10:29 PM, Magnus Danielson wrote: I have checked several sources, and they match up with the IEEE 1139 in this regard. I have also evaluated the equation for Allan variance for the random walk noise, and it matches up with the references and what I put here: https://en.wikipedia.org/wiki/Allan_variance#Power-law_noise Thanks a lot for your effort! So, the A formula you have matches up. You will need to find another source of the mismatch. I will take a step back and describe the overall picture of what I'm doing. Maybe someone can help me spot where I do something wrong. (As stated later, the part where I'm quite unsure what I'm doing is the PSD estimation part.) My main goal is to simulate powerlaw noise. I then analyze the generated noise to check if my simulation is reasonable. So the basic workflow would be the following: 1) Generate noise 2) Analyze the noise in the time and frequency domain 3) See that everything agrees and be happy :) Step 1: Noise generation --- I generate powerlaw noise with the method described by Kasdin and Walter in [1]. So basically I generate white noise and apply a filter as described in [1] to get a PSD shape corresponding to the different values of alpha. The part of the PSD that will have the correct shape depends on the filter length and the simulated sampling frequency. Basically: the length of a simulation I would like to carry out specifies a lower bound on the filter length to get correct results. For WPM, WFM and RW noise I can use a shortcut: for these types of noise the filter coefficients are basically a discrete derivative, an identity filter and a cumulative sum. This is expected, as it agrees with [2], which states that integration of powerlaw noise decreases alpha by 2 (chapter 3.4 in [2]). Thus for even values of alpha I can even skip the expensive convolution to apply the filter and implement the filters directly. As input white noise I use a Gaussian distribution, mainly because that is what is used in the original paper. (I have also found another implementation [3] that optionally provides a uniform distribution). I'm quite confident that the noise generation part works as expected. However, even if I do something wrong here, it should not influence the analyzing part. Step 2: Analyzing noise --- 2.1 Time domain To analyze powerlaw noise in the time domain, I use a Matlab script called 'allan' [4], which calculates the Allan Deviation. I also found another Matlab tool called 'Stability Analyzer' [5], which can also calculate ADEV values. These two tools are developed by different authors and expect different input formats, but their results agree for any noise example I have tried so far. Thus I would say both of them can be trusted to work as expected. 2.2 Frequency domain IEEE 1139[6] defines S_y as: frequency spectrum Sy(f): One-sided spectral density of the normalized frequency fluctuations, as defined in normalized frequency fluctuations y(t). However, I'm not sure how to calculate this measure for a given noise sample. Anything I describe below is just based on 'I think this might work'. If anyone knows a better way of calculating S_y, or tools that can be used for this task, I would be glad to hear about it :) As already stated in the earlier mail I use the method described in [7] to estimate the one-sided PSD of my noise data in FFD format. These plots are quite noisy, and to improve the graphical presentation I use the averaging method described in [8]. I split the noise vector in parts of equal length, calculate the individual PSDs and average over them. Using this averaging method, the PSD plots converge to lines on a log-log plot with the expected slopes. I have an example figure attached to the mail that shows the effect of the averaging (PSD_Average.png). Step 3: Comparing time and frequency domain results --- At this point I have plots for both the Allan Deviation and the FFD-PSD, and would like to compare them. As first step I estimate h_alpha from the Allan Deviation plot (I'm aware that I need to take care for the Allan Deviation - Allan Variance conversion). Then I try to estimate the expected PSD values and compare them with my actual plot using the formulas from IEEE 1139. However, at this point a see that RW noise behaves unexpected :( Numerical Example: --- Suppose the figure attached as 'Numeric_example.png': At Tau = 0.1s the ADEV plot has a value of 0.005849, so de AVAR would be 3.4211e-05 at this point. The constant A is 2 * pi^2/3 = 6.5797. Thus the value of h_-2 could be roughly estimated as AVAR / (Tau * A) = ~5.2e-05. This would lead to an expected S_y value at a frequency f = 10Hz of h_-2 * f = 5.2000e-07, or -62.84dB
Re: [time-nuts] AVAR - S_Y conversion
Hi Wolfgang, Have a look at the 20 plots in: http://leapsecond.com/pages/allan/Exploring_Allan_Deviation_v2.pdf This shows phase/frequency/ADEV+MDEV and PSD for 5 noise types. Zoom the PDF 400x if necessary. This was generated with Stable32 and should be 100% correct. See if your results agree. The raw data is at: http://leapsecond.com/pages/allan/ /tvb - Original Message - From: Wolfgang Wallner wolfgang-wall...@gmx.at To: time-nuts@febo.com Sent: Monday, March 09, 2015 8:57 AM Subject: Re: [time-nuts] AVAR - S_Y conversion On 03/06/2015 10:29 PM, Magnus Danielson wrote: I have checked several sources, and they match up with the IEEE 1139 in this regard. I have also evaluated the equation for Allan variance for the random walk noise, and it matches up with the references and what I put here: https://en.wikipedia.org/wiki/Allan_variance#Power-law_noise Thanks a lot for your effort! ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR - S_Y conversion
Wolfgang, I have checked several sources, and they match up with the IEEE 1139 in this regard. I have also evaluated the equation for Allan variance for the random walk noise, and it matches up with the references and what I put here: https://en.wikipedia.org/wiki/Allan_variance#Power-law_noise So, the A formula you have matches up. You will need to find another source of the mismatch. Cheers, Magnus On 03/06/2015 11:04 AM, Wolfgang Wallner wrote: On 03/05/2015 07:23 PM, Attila Kinali wrote: Servus! Servus :) On Thu, 05 Mar 2015 14:35:51 +0100 Wolfgang Wallner wolfgang-wall...@gmx.at wrote: For the random walk noise the expected line is off by a factor of exactly 2 from the calculated plot, and I don't know how to explain this behavior. I'm probably the wrong one to answer, as I have never done any noise simulation or even read up the relevant papers, but... A factor of 2 sounds like the difference you would get between one sided and two sided noise PSD's. I calculate the one-sided PSD of the FFD data as described in [1] (first paragraph), so the code looks like this: xdft = fft(x); xdft = xdft(1:N/2+1); psdx = (1/(Fs*N)) * abs(xdft).^2; psdx(2:end-1) = 2*psdx(2:end-1); Remark: Before calculating the PSD, I split the data into parts of equal size, calculate the PSD for each one, and average over the set of PSDs. This improves the graphical visualization a lot. As the result matches my expectation exactly for 4 different kinds of noise, I would have assumed that this PSD calculation approach is quite reasonable. As I see the unexpected behavior only with random walk noise, and the main difference in the calculation is the term A, I would suspect that it has something to do with it. However, I'm a novice in this field, so any hint is very appreciated. regards, Wolfgang [1] http://de.mathworks.com/help/signal/ug/psd-estimate-using-fft.html ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there. ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR - S_Y conversion
On 03/05/2015 07:23 PM, Attila Kinali wrote: Servus! Servus :) On Thu, 05 Mar 2015 14:35:51 +0100 Wolfgang Wallner wolfgang-wall...@gmx.at wrote: For the random walk noise the expected line is off by a factor of exactly 2 from the calculated plot, and I don't know how to explain this behavior. I'm probably the wrong one to answer, as I have never done any noise simulation or even read up the relevant papers, but... A factor of 2 sounds like the difference you would get between one sided and two sided noise PSD's. I calculate the one-sided PSD of the FFD data as described in [1] (first paragraph), so the code looks like this: xdft = fft(x); xdft = xdft(1:N/2+1); psdx = (1/(Fs*N)) * abs(xdft).^2; psdx(2:end-1) = 2*psdx(2:end-1); Remark: Before calculating the PSD, I split the data into parts of equal size, calculate the PSD for each one, and average over the set of PSDs. This improves the graphical visualization a lot. As the result matches my expectation exactly for 4 different kinds of noise, I would have assumed that this PSD calculation approach is quite reasonable. As I see the unexpected behavior only with random walk noise, and the main difference in the calculation is the term A, I would suspect that it has something to do with it. However, I'm a novice in this field, so any hint is very appreciated. regards, Wolfgang [1] http://de.mathworks.com/help/signal/ug/psd-estimate-using-fft.html ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR - S_Y conversion
Servus! On Thu, 05 Mar 2015 14:35:51 +0100 Wolfgang Wallner wolfgang-wall...@gmx.at wrote: For the random walk noise the expected line is off by a factor of exactly 2 from the calculated plot, and I don't know how to explain this behavior. I'm probably the wrong one to answer, as I have never done any noise simulation or even read up the relevant papers, but... A factor of 2 sounds like the difference you would get between one sided and two sided noise PSD's. Attila Kinali -- _av500_ phd is easy _av500_ getting dsl is hard ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
[time-nuts] AVAR - S_Y conversion
Hello time-nuts community, I hope this is the right place for the following question :) I'm dealing with the simulation of powerlaw noise, and I stumbled upon something I cannot explain when I tried out some formulas of IEEE 1139 [1]: According to Table B.2 in [1] the one-sided power spectral density of fractional frequency data and the Allan variance of this data can be related as follows: PSD of FFD: S_y(f) = h_alpha * f1^alpha AVAR:Sigma_y(Tau) = K * h_alpha * f ^ x where K and x depend on the type of noise (alpha). For your convenience I made a screenshot of the formulas I'm referring to: http://postimg.org/image/6qcx3ggu9/ I have generated data sets with simulated powerlaw noise for different values of alpha, and did the following for each of these noise vectors: *) Calculated and plotted the Allan Variance *) Used to formulas of [1] to calculate h_alpha *) Calculated and plotted the FFD-PSD (the PSD plot is averaged the get better visual results) *) Added colored lines to both plots according to the calculated h_alpha values I would have expected that the colored lines would match each plot. However, this is only the case for White PM, Flicker PM, White FM and Flicker FM noise. To my surprise the calculated line for random walk noise does not match the PSD plot. For the random walk noise the expected line is off by a factor of exactly 2 from the calculated plot, and I don't know how to explain this behavior. I supposed that maybe the factor A is twice as large as it should be, and thus I searched in other powerlaw noise publications for different formulas. However, as far as I can see they agree with the definition given in [1]. I could only find one paper with another definition: [2] In that other paper A is defined as 2 pi^2/6 instead of 2 pi^2/3 (equation 24). Using this definition would result in a plot that matches what I would have expected. These are the graphs I'm referring to: White PM: http://postimg.org/image/fk059s243/full/ Flicker PM: http://postimg.org/image/6q71l03cp/full/ White FM: http://postimg.org/image/mxhxeszqx/full/ Flicker FM: http://postimg.org/image/3vzpj7jmf/full/ Random Walk: http://postimg.org/image/hxad6okwv/full/ -- the bad guy Does anyone know the reason for the behavior I see? best regards, Wolfgang Wallner PS: I tried to keep this mail short. If I have left out any information that would be useful feel free to ask, please :) [1] 1139-2008 - IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology [2] Gaderer, et al - Achieving a Realistic Notion of Time in Discrete Event Simulation ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR Femtoseconds
Hi The use of femtoseconds come from the AVAR it's self. It was originally defined by time domain people. It's delineated by a Tau dimensioned in seconds. The time domain noise that's 1x10^-12 or 1x10^-15 down at one second does indeed have units of 1x10^-12 or 1x10^-15 seconds. As with any real world system one has to be very careful about the difference between resolution and accuracy. Resolution generally is easy, accuracy is more difficult. Any of the commonly used measurement techniques used to drive the AVAR are capable of enormous resolution. The problem is that past a certain point the added digits are simply internal noise and do not represent the DUT or the reference. One very simple example: A heterodyne system beats two 10 MHz oscillators down to a 1 Hz note. That gives you a 1x10^7 expansion. Drive the note directly into any of the common 11 digit per second counter (no limiters, no amps, straight in). You now have a LSD that has dimensions of 1x10^-18. You could claim that you have a system with a resolution of 1 atto-seocnd. A quick look at the output of the counter would show you that a lot of those digits were simply random numbers. You could do equally well by taking a 6 digit / second counter and a simple PIC program to make up another 5 digits of data. Bob On Jun 20, 2010, at 10:46 AM, Robert Benward wrote: Steve, I am a professional engineer, but in this arena I am an amateur. That is why I'm asking the questions, not to put down, but to understand some of the claims made. And as I said in one of my previous emails, I've seen amateurs run circles around the professionals, and those professional admitting utter astonishment at those amateur accomplishments (this is in the area of amateur astrophotography). What I have heard throughout this thread is a lot of bashing of those asking the questions, surfacing as derogatory and berating comments on other's understanding. I have also heard much claims to a certain procedure without one iota of numerical mumbo-jumbo to back it up. The issue here is an inability to describe a simple claim. Pete has attempted to put things in simple numbers, and I see where he is going, and I concur with some of his calculations. If one can not describe what appears to be a simple procedure, then I must question the basic understanding behind the explanation. If you make a wild claim, and then you can't even get the bullet on the paper, then I must question the shooter's understanding. I guess I am not comfortable with the use of femtoseconds to describe frequency accuracy. Technically, a locked PLL is at the exact frequency as the reference, as measured in the long term. The phase between the two may not be at zero, that depends on the type of phase detector and the DC offsets in the system. On the short term, phase noise of the reference will cause the loop to generate error terms which will change the phase of the DUT. Oscillators are also specified using phase noise, e.g. 135dB down @ 100Hz. That specifies how much energy is not in the bandwidth of the carrier. It also implies the phase is constantly changing! If the phase is changing, the error term is changing, and so forth and so on.Your measurement can only be as good as your reference oscillator. A DVM can only average this error, it can't give you the instantaneous value of the peak deviation of the error signal, which is what you would need to claim fs cycle to cycle timing. Fs units are appropriate for cycle to cycle variation, not long term or multicycle assements. Even the best HP DVM is only good to 3ppm on the 100mV scale and the shortest reading is 167us. That's 10 orders of magnitude greater that the deviation you are trying to measure. If you average the mixer output, you can no longer claim fs timing. What you can claim is a long term frequency stability in ppm. This is my simple understanding of phase detectors and mixers. You might get there by dividing down a bunch of numbers but I don't think the method supports the claim (of fs timing). Bob - Original Message - From: Steve Rooke To: Discussion of precise time and frequency measurement Sent: Sunday, June 20, 2010 2:00 AM Subject: Re: [time-nuts] Advantages Disadvantages of the TPLL Method Bob, Can I answer this one. On 20 June 2010 04:36, Robert Benward rbenw...@verizon.net wrote: Warren, I was responding to ke5fx comment using a 12-bit, 480-Hz serial DAQ in place of the voltage-to-frequency converter in the diagram above. A DAQ is a multifaceted data acquisition system, where as in your annotated diagram you showed an ADC. The DAQ that Warren is referring to to has a 12bit ADC input capable of performing up to 480 samples per second. I understand it's analog, but you said: Say you have a nice logic gate with 1 ns delay . So back to the analog
Re: [time-nuts] AVAR Femtoseconds
Bob Boy, you guys are really making me read a lot. I'm digesting Wiki right now. I see tau, but does identifying a tau of 1E-14 allow you to say you are locked to 10fs? The smallest tau I've seen in my E1938 collection is 1E-1. Bob - Original Message - From: Bob Camp To: Discussion of precise time and frequency measurement Sent: Sunday, June 20, 2010 11:32 AM Subject: Re: [time-nuts] AVAR Femtoseconds Hi The use of femtoseconds come from the AVAR it's self. It was originally defined by time domain people. It's delineated by a Tau dimensioned in seconds. The time domain noise that's 1x10^-12 or 1x10^-15 down at one second does indeed have units of 1x10^-12 or 1x10^-15 seconds. As with any real world system one has to be very careful about the difference between resolution and accuracy. Resolution generally is easy, accuracy is more difficult. Any of the commonly used measurement techniques used to drive the AVAR are capable of enormous resolution. The problem is that past a certain point the added digits are simply internal noise and do not represent the DUT or the reference. One very simple example: A heterodyne system beats two 10 MHz oscillators down to a 1 Hz note. That gives you a 1x10^7 expansion. Drive the note directly into any of the common 11 digit per second counter (no limiters, no amps, straight in). You now have a LSD that has dimensions of 1x10^-18. You could claim that you have a system with a resolution of 1 atto-seocnd. A quick look at the output of the counter would show you that a lot of those digits were simply random numbers. You could do equally well by taking a 6 digit / second counter and a simple PIC program to make up another 5 digits of data. Bob On Jun 20, 2010, at 10:46 AM, Robert Benward wrote: Steve, I am a professional engineer, but in this arena I am an amateur. That is why I'm asking the questions, not to put down, but to understand some of the claims made. And as I said in one of my previous emails, I've seen amateurs run circles around the professionals, and those professional admitting utter astonishment at those amateur accomplishments (this is in the area of amateur astrophotography). What I have heard throughout this thread is a lot of bashing of those asking the questions, surfacing as derogatory and berating comments on other's understanding. I have also heard much claims to a certain procedure without one iota of numerical mumbo-jumbo to back it up. The issue here is an inability to describe a simple claim. Pete has attempted to put things in simple numbers, and I see where he is going, and I concur with some of his calculations. If one can not describe what appears to be a simple procedure, then I must question the basic understanding behind the explanation. If you make a wild claim, and then you can't even get the bullet on the paper, then I must question the shooter's understanding. I guess I am not comfortable with the use of femtoseconds to describe frequency accuracy. Technically, a locked PLL is at the exact frequency as the reference, as measured in the long term. The phase between the two may not be at zero, that depends on the type of phase detector and the DC offsets in the system. On the short term, phase noise of the reference will cause the loop to generate error terms which will change the phase of the DUT. Oscillators are also specified using phase noise, e.g. 135dB down @ 100Hz. That specifies how much energy is not in the bandwidth of the carrier. It also implies the phase is constantly changing! If the phase is changing, the error term is changing, and so forth and so on.Your measurement can only be as good as your reference oscillator. A DVM can only average this error, it can't give you the instantaneous value of the peak deviation of the error signal, which is what you would need to claim fs cycle to cycle timing. Fs units are appropriate for cycle to cycle variation, not long term or multicycle assements. Even the best HP DVM is only good to 3ppm on the 100mV scale and the shortest reading is 167us. That's 10 orders of magnitude greater that the deviation you are trying to measure. If you average the mixer output, you can no longer claim fs timing. What you can claim is a long term frequency stability in ppm. This is my simple understanding of phase detectors and mixers. You might get there by dividing down a bunch of numbers but I don't think the method supports the claim (of fs timing). Bob - Original Message - From: Steve Rooke To: Discussion of precise time and frequency measurement Sent: Sunday, June 20, 2010 2:00 AM Subject: Re: [time-nuts] Advantages Disadvantages of the TPLL Method Bob, Can I answer this one. On 20 June 2010 04
[time-nuts] AVAR for
Hi, I'm trying to compare a various methods of time and frequency comparision: GPS common-view, GPS carrier-phase and GPS TWSTT. Does anyone know what is the floor level for these methods in AVAR? Filip. ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR Femtoseconds
Robert Benward wrote: Bob Boy, you guys are really making me read a lot. I'm digesting Wiki right now. I see tau, but does identifying a tau of 1E-14 allow you to say you are locked to 10fs? The smallest tau I've seen in my E1938 collection is 1E-1. Bob tau is the time over which the measurement is made, typically 1 second or greater. loosely speaking, the 1e-14 is the average fractional deviation of frequency over that time period. ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR Femtoseconds
On 06/20/2010 11:53 PM, jimlux wrote: Robert Benward wrote: Bob Boy, you guys are really making me read a lot. I'm digesting Wiki right now. I see tau, but does identifying a tau of 1E-14 allow you to say you are locked to 10fs? The smallest tau I've seen in my E1938 collection is 1E-1. Bob tau is the time over which the measurement is made, typically 1 second or greater. loosely speaking, the 1e-14 is the average fractional deviation of frequency over that time period. It is a RMS type (much like statisticians standard deviation) of frequency stability over the observation interval of tau (little greek letter looking similar but not quite like a little t, which is the real reason for using it). Since it is a RMS type of measure, it indicates the effective power of noise, but not what the actual deviation in frequency will be, it's just a statistical measure. You may form a confidence interval such as that for 99,7 % or something which forms a scale-factor, quite similar to the use of the error function for the Gaussian distribution. An Allan deviation measure of 1E-14 is however not quite the same as 10 fs. Besides the units being wrong (Allan deviation is a relative and unit-less measure, essentially Hz/Hz) the Allan variance (and hence the Allan deviation) is a frequency stability measure, indicating the stability of normalized frequency rather than stability of normalized phase. The time deviation represents the stability of phase over some observation time. Assuming the nominal frequency and linear effects removed, then this would indicate the time error noise of the phase, here use of seconds could be used, but it would be to stretch things a bit. The time and frequency world has it's own qualities of noise... Cheers, Magnus ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR Femtoseconds
Hi While the units are more properly femto seconds/ second you do indeed see 1s AVAR plots labeled in fS. Bob On Jun 20, 2010, at 6:25 PM, Magnus Danielson wrote: On 06/20/2010 11:53 PM, jimlux wrote: Robert Benward wrote: Bob Boy, you guys are really making me read a lot. I'm digesting Wiki right now. I see tau, but does identifying a tau of 1E-14 allow you to say you are locked to 10fs? The smallest tau I've seen in my E1938 collection is 1E-1. Bob tau is the time over which the measurement is made, typically 1 second or greater. loosely speaking, the 1e-14 is the average fractional deviation of frequency over that time period. It is a RMS type (much like statisticians standard deviation) of frequency stability over the observation interval of tau (little greek letter looking similar but not quite like a little t, which is the real reason for using it). Since it is a RMS type of measure, it indicates the effective power of noise, but not what the actual deviation in frequency will be, it's just a statistical measure. You may form a confidence interval such as that for 99,7 % or something which forms a scale-factor, quite similar to the use of the error function for the Gaussian distribution. An Allan deviation measure of 1E-14 is however not quite the same as 10 fs. Besides the units being wrong (Allan deviation is a relative and unit-less measure, essentially Hz/Hz) the Allan variance (and hence the Allan deviation) is a frequency stability measure, indicating the stability of normalized frequency rather than stability of normalized phase. The time deviation represents the stability of phase over some observation time. Assuming the nominal frequency and linear effects removed, then this would indicate the time error noise of the phase, here use of seconds could be used, but it would be to stretch things a bit. The time and frequency world has it's own qualities of noise... Cheers, Magnus ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there. ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR calculation
Second thought, look at the tau=1 to 10 s and you see that it first rises before the usual slope. This is the effect of averaging in the counter. I would suspect that a HP53132A is being used. Magnus, You're correct about averaging effects; for example, see: http://www.leapsecond.com/pages/adev-avg/ But, I think in this case it's not the measurement system that's causing it because I saw the same PRS10 hump: http://www.leapsecond.com/museum/prs10/1sigma2.gif http://www.leapsecond.com/museum/prs10/log16084v.gif http://www.leapsecond.com/museum/prs10/ What we see then would appear to be a feature of the PRS10 itself and not the measurement system. Since the PRS10 is based on a quality SC-cut OCXO, this might be an artifact of their blending the OCXO and Rb cell; i.e., it looks just like a GPSDO PLL hump, only moved two decades to the left. /tvb ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR calculation
Tom Van Baak pisze: Hi Filip, See attached. Let me know how your results differ from this. It looks like 1.75 hours of frequency data of a 250 MHz DUT. You first convert your raw frequency measurement data into normalized frequency error data (i.e., subtract and divide by f0 = ~250 009 770 Hz). You can run the y- form (frequency) of an adev calculation directly, or integrate the frequency error series into a time error series and run the x- form (phase) of adev calculation. I can explain more if you wish. JohnM and UlrichB can double check this plot with their tools. Thank you all for your charts. I forgot to substract and divide by center frequency ;-) That data represents a beat note between two optical frequencies: Nd:YAG laser stabilized to hyperfine transitions of molecular iodine and optical frequency comb that is phase locked to the SRS FS-725 rubidium standard. And it seems that AVAR of this measurement is lower than values for FS-725; AVAR for Nd:YAG laser can be found here: http://www.innolight.de/pdf/laser_accessories.pdf . Does anybody know what is the floor level of AVAR just by counting frequency (for example using 53132 Stable32 or SRS620)? Filip. ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR calculation
That data represents a beat note between two optical frequencies: Nd:YAG laser stabilized to hyperfine transitions of molecular iodine and optical frequency comb that is phase locked to the SRS FS-725 rubidium standard. And it seems that AVAR of this measurement is lower than values for FS-725; Which values of the FS-725 are you referring to? I'm pretty sure the FS-725 has a PRS10 Rb in it. If so, have a quick look at these PRS10 performance plots: http://www.leapsecond.com/museum/prs10/ In particular, the ADEV of your data (pomiar61-frep.txt) which I attached in a previous posting (pomiar61-adev.gif) looks almost exactly like the ADEV of the PRS10 that I measured: http://www.leapsecond.com/museum/prs10/1sigma2.gif /tvb ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
[time-nuts] AVAR calculation
I'm trying to calculate AVAR from collected data, but I would expect a different behavior of my DUT. Does anybody can calculate the AVAR for me? Gate time was 1s, without any deadtime between measurements, this is a file http://www.fuw.edu.pl/~fozimek/pomiar61-frep.txt with collected data. Thanks, Filip Ozimek ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR calculation
Filip Ozimek wrote: I'm trying to calculate AVAR from collected data, but I would expect a different behavior of my DUT. Does anybody can calculate the AVAR for me? Gate time was 1s, without any deadtime between measurements, this is a file http://www.fuw.edu.pl/~fozimek/pomiar61-frep.txt with collected data. Thanks, Filip Ozimek Filip If you have a windows machine download Ulrich's plotter program and use it to analyse the data. http://ulrich-bangert.de/html/downloads.html Otherwise you should be able to run it either in a virtual machine or under wine with Linux, FreeBSD etc. If the input data is frequency one will get a plot something like that attached. Bruce attachment: AVAR_1.gif___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR calculation
Hi Filip, See attached. Let me know how your results differ from this. It looks like 1.75 hours of frequency data of a 250 MHz DUT. You first convert your raw frequency measurement data into normalized frequency error data (i.e., subtract and divide by f0 = ~250 009 770 Hz). You can run the y- form (frequency) of an adev calculation directly, or integrate the frequency error series into a time error series and run the x- form (phase) of adev calculation. I can explain more if you wish. JohnM and UlrichB can double check this plot with their tools. /tvb - Original Message - From: Filip Ozimek me_su...@o2.pl To: time-nuts@febo.com Sent: Sunday, January 10, 2010 1:20 PM Subject: [time-nuts] AVAR calculation I'm trying to calculate AVAR from collected data, but I would expect a different behavior of my DUT. Does anybody can calculate the AVAR for me? Gate time was 1s, without any deadtime between measurements, this is a file http://www.fuw.edu.pl/~fozimek/pomiar61-frep.txt with collected data. Thanks, Filip Ozimek attachment: pomiar61-adev.gif___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR calculation
Filip Ozimek wrote: I'm trying to calculate AVAR from collected data, but I would expect a different behavior of my DUT. Does anybody can calculate the AVAR for me? Gate time was 1s, without any deadtime between measurements, this is a file http://www.fuw.edu.pl/~fozimek/pomiar61-frep.txt with collected data. Thanks, Filip Ozimek Comparing the overlapped Hadamard deviation and overlapped ADEV plots indicates that linear frequency drift may be significant for tau 100sec or so. Bruce ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR calculation
John, You can convert frequency to phase (time interval) using fr2ti under www.leapsecond.com/tools/ /tvb - Original Message - From: John Miles jmi...@pop.net To: Discussion of precise time and frequency measurement time-nuts@febo.com Sent: Sunday, January 10, 2010 3:04 PM Subject: Re: [time-nuts] AVAR calculation Ulrich's PLOTTER application is a good tool for that; mine doesn't support frequency input yet. If you haven't tried PLOTTER, it's the second download on the page at http://ulrich-bangert.de/html/downloads.html . Be sure to select Data is frequency from the Time Stability Statistics menu. -- john, KE5FX -Original Message- From: time-nuts-boun...@febo.com [mailto:time-nuts-boun...@febo.com]on Behalf Of Tom Van Baak Sent: Sunday, January 10, 2010 2:01 PM To: Discussion of precise time and frequency measurement Subject: Re: [time-nuts] AVAR calculation Hi Filip, See attached. Let me know how your results differ from this. It looks like 1.75 hours of frequency data of a 250 MHz DUT. You first convert your raw frequency measurement data into normalized frequency error data (i.e., subtract and divide by f0 = ~250 009 770 Hz). You can run the y- form (frequency) of an adev calculation directly, or integrate the frequency error series into a time error series and run the x- form (phase) of adev calculation. I can explain more if you wish. JohnM and UlrichB can double check this plot with their tools. /tvb - Original Message - From: Filip Ozimek me_su...@o2.pl To: time-nuts@febo.com Sent: Sunday, January 10, 2010 1:20 PM Subject: [time-nuts] AVAR calculation I'm trying to calculate AVAR from collected data, but I would expect a different behavior of my DUT. Does anybody can calculate the AVAR for me? Gate time was 1s, without any deadtime between measurements, this is a file http://www.fuw.edu.pl/~fozimek/pomiar61-frep.txt with collected data. Thanks, Filip Ozimek ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there. ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
Re: [time-nuts] AVAR calculation
Yep, it's just a matter of laziness on my part. Frequency input is one of several dozen items that were on my to-do list when I got sidetracked with the ADCs, and which are almost all still there... -- john, KE5FX -Original Message- From: time-nuts-boun...@febo.com [mailto:time-nuts-boun...@febo.com]on Behalf Of Tom Van Baak Sent: Sunday, January 10, 2010 3:09 PM To: Discussion of precise time and frequency measurement Subject: Re: [time-nuts] AVAR calculation John, You can convert frequency to phase (time interval) using fr2ti under www.leapsecond.com/tools/ /tvb - Original Message - From: John Miles jmi...@pop.net To: Discussion of precise time and frequency measurement time-nuts@febo.com Sent: Sunday, January 10, 2010 3:04 PM Subject: Re: [time-nuts] AVAR calculation Ulrich's PLOTTER application is a good tool for that; mine doesn't support frequency input yet. If you haven't tried PLOTTER, it's the second download on the page at http://ulrich-bangert.de/html/downloads.html . Be sure to select Data is frequency from the Time Stability Statistics menu. -- john, KE5FX -Original Message- From: time-nuts-boun...@febo.com [mailto:time-nuts-boun...@febo.com]on Behalf Of Tom Van Baak Sent: Sunday, January 10, 2010 2:01 PM To: Discussion of precise time and frequency measurement Subject: Re: [time-nuts] AVAR calculation Hi Filip, See attached. Let me know how your results differ from this. It looks like 1.75 hours of frequency data of a 250 MHz DUT. You first convert your raw frequency measurement data into normalized frequency error data (i.e., subtract and divide by f0 = ~250 009 770 Hz). You can run the y- form (frequency) of an adev calculation directly, or integrate the frequency error series into a time error series and run the x- form (phase) of adev calculation. I can explain more if you wish. JohnM and UlrichB can double check this plot with their tools. /tvb - Original Message - From: Filip Ozimek me_su...@o2.pl To: time-nuts@febo.com Sent: Sunday, January 10, 2010 1:20 PM Subject: [time-nuts] AVAR calculation I'm trying to calculate AVAR from collected data, but I would expect a different behavior of my DUT. Does anybody can calculate the AVAR for me? Gate time was 1s, without any deadtime between measurements, this is a file http://www.fuw.edu.pl/~fozimek/pomiar61-frep.txt with collected data. Thanks, Filip Ozimek ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there. ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there. ___ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
