[time-nuts] Name of integral of timing residual

Tom Van Baak tvb at LeapSecond.com
Thu Apr 20 16:35:06 EDT 2017

Jim Palfreyman writes:

> Consider a plot of a timing residual vs time. Say a watch against a maser,
> residual=watch-maser.

We usually don't use the word residual for this. When you compare a watch with a maser, or any DUT time against REF time, you get a quantity like: phase difference, or sometimes just "phase", or time difference, time error, time interval, time interval error, etc.

What residual usually refers to is if you post-process the raw time or frequency data in some way to better expose underlying structure. For example, if you remove a linear or quadratic fit from your phase data the resulting data set can be called phase residuals. This is done with free-running clocks because both frequency, and especially phase, diverge badly over time. So plotting residuals removes large systematic effects and exposes small effects of interest.

> Now if I now plot the cumulative sum (think integral) of the residual,
> that's going to give me an overall view of how the clock is performing over time.

A traditional phase plot of residuals is itself "an overall view of how the clock is performing over time". That's why even before we make ADEV plots we want to see the phase (actually, phase difference) plot and maybe also the frequency (usually, normalized frequency) plot. Both give an overall view of how the clock is performing, not to mention the ADEV plot which even further summarizes clock performance.

A cumulative sum, an integral, of the timing residuals is a bit odd, but not wrong. This is the "area under the curve" of any residual phase plot. A traditional phase plot gives you a series of points on a line -- these tell you your clock error as a function of elapsed time. But plots are 2D, so your eye also senses the amount of area under the line -- this tells you not only how far off your clock is, but how long your clock has been how far off. The plot shows, and the eye recognizes both the line (how far) and the area (how far x how long).

> (If it helps, think of PID controllers and how they work in the "I" part.)

Yes, exactly. And the reason this is explicit in PID (or PIID) is that there is no human eye and no 2D plot. Therefore the PID algorithm has to manually compute the "area under the curve"; it has to calculate the cumulative sum as a scaler value. And it sounds like this single scaler value, as opposed to a rendered plot image, is what you're after.

> Now if you look at *motion* of an object over time, and you integrate its
> acceleration you get velocity, integrate again you get displacement.
> Integrate again and you get "absement" and again you get "abcity" (I only
> recently discovered these terms).

Ok, thanks for that word of the day! Full list here:


> Does the integral of a timing residual have a name, and does the integral
> of *that* have a name as well?

Nope. But let's make one up in honor of your time spent doing Pulsar work. Some sources suggest absement is a portmanteau of absent and displacement. Ok, could be, but just as likely ab- is a fine Latin prefix on its own, meaning away, depart. Think of abnormal, abhor, absent, abdicate, aberrant. Or the German abfahren, to depart from. (Ah, I finally got to put my Latin and German to use; or is that abuse).


Anyway, in the world of space / distance:

-4 abserk
-3 abseleration
-2 absity
-1 absement
 0 displacement
+1 velocity
+2 acceleration
+3 jerk

So how about for the world of time, we call integrated phase error: abtimer, or just abtime:

-1 abtime (integrated phase error, cumulative sum of time error, etc.) units: s^2
 0 time (phase, time error, phase difference, etc.) units: s
+1 frequency (rate of phase change, etc.) units: /s, Hz
+2 drift (linear frequency change) units: /s^2, Hz/s

I can imagine cases where abtime would be useful, especially for closed loops. Units are seconds^2, or second*days, etc. For example, it may come in handy when I post plots of the new WWVB receiver, or characterizing a sloppy GPSDO timing receiver.


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