# [time-nuts] Explanation of how a GPS RX generates time/freqprecision?

Tom Van Baak tvb at LeapSecond.com
Mon Nov 23 10:32:47 EST 2015

```> A 2GHz Intel processor can't measure 10^-12, let alone getting
...
> What are we missing?

Hi Mike,

It sounds like your friend is mixing up time vs. frequency accuracy units. Let's just use time units. Here is an easy 1-2-3 GPS accuracy explanation:

1. The clocks in the GPS satellites are good to one nanosecond, which is 10^-9 second. In fact, if they drift more than half a ns, corrections are applied.

2. Now the bad news. A GPS satellite is far away so the time signal you get at home is late due to the speed of light. Note this is the same problem as any other radio time system (e.g., TV, WWVB, DCF77, Loran-C). In the case of GPS you can assume the satellite is at least 20,000 km away (directly overhead) but it might be 25,000 km away (near the horizon). You don't know. So your time accuracy is only about 10^-2 second. Note this is worse than WWVB.

3. Now the good news. GPS is a PNT (Position, Navigation, and Timing) system, not just a timing system (like WWVB). So the cool thing is that once you receive 4 satellites you can calculate where each of the satellites actually is and where you are. This allows you to back out the previously unknown light travel distance and so your GPS receiver gets time accurate to about 10^-7 second (100 ns). Note this is much better than WWVB because the distance from transmitter to receiver is now precisely known, and also the signal is more direct, there are multiple transmitters, the wavelength is much smaller, etc.

Therefore, almost any GPS receiver with a 1PPS output will give you time accurate to 100 ns. A system (PC or \$1 microcontroller or homebrew TTL circuit) can measure or compare time to 100 ns pretty easy; that's only a 10 MHz clock speed.

Now about frequency, or average frequency accuracy. If you have a homebrew clock that always stays within 100 ns in time, then its average frequency error will stay below 100 ns / N s, where N is the duration of your average. For example, over 100 seconds your clock frequency accuracy will be 10^-9. And over a day it's about 10^-12. Over two weeks it's 10^-13, etc.

Frequency accuracy is measured as the amount of time drift that occurs over a long interval. So that's why GPS frequency accuracy numbers appear so small. Since there are a million seconds in two weeks a frequency accuracy number may look like 10^-13 (0.1 part per trillion) even though the time accuracy number is 10^-7 seconds (100 ns).

/tvb

----- Original Message -----
From: "Mike Baker" <mpb45 at clanbaker.org>
To: <time-nuts at febo.com>
Sent: Sunday, November 22, 2015 6:51 PM
Subject: [time-nuts] Explanation of how a GPS RX generates time/freqprecision?

> Hello, Time-Nutters--
>
> I have a couple of friends who have asked how GPS based
> time & frequency units arrive at their level of precision.
> One sent me some questions and comments (see below)
> which indicate his lack of understanding of how GPS
> disciplined oscillators can deliver the precision they do.
>
> My question is--  Can anyone on the Time-Nuts list point me
> to a source I can give them that will explain how the process
> works?  Seems to me that I have seen some papers describing
> the process but my search for this info has not turned up
> anything suitable that goes into some detail on this.
> Any feedback or suggestions on this is much appreciated!!
>
> Mike Baker
> Micanopy, FL
> ****************
>
> Here is what one of the guys sent me:
>
> A 2GHz Intel processor can't measure 10^-12, let alone getting
> a P4 to do it.  RS232 delay times are measured in mSec, not
> 10^-9.  Yes it can time-stamp PPS at 10^-9 if it had a time source
> that accurate but that has nothing to do with when the pulse
> gets to the receiver or the original time source of the system.
>
> Given that my phone has a 2+ GHz processor / clock and a GPS
> receiver, what is the advantage of a 6 GHz (or 10 MHz) oscillator?
> I could well believe that system is accurate to 100 nSec (10^-7)
> or that a Stratum 0 GPS system good down close to 10^-8
> (40 nSec).  I don't see any way of getting down to 10^-12 with
> that technology or transmitting a time more accurate than 10^-7.
>
> If the original GPS time source is 10^-7 or a little better, how
> can it ever be more accurate than that?
>
> What are we missing?
> ******************************
>
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