# [time-nuts] Name of integral of timing residual

Christopher Hoover ch at murgatroid.com
Sat May 6 00:22:32 EDT 2017

```+1 to abtime

On Thu, Apr 20, 2017 at 1:35 PM, Tom Van Baak <tvb at leapsecond.com> wrote:

> 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:
>
> https://en.wikipedia.org/wiki/Absement#Higher_integrals
>
> > 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).
>
> https://en.wiktionary.org/wiki/ab-
> http://membean.com/wrotds/ab-away
> https://www.vocabulary.com/lists/135086
>
> 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