# [time-nuts] Measuring phase shift between 1 Hz DMTD signals by I+Q processing

Joe Gwinn joegwinn at comcast.net
Sun Jul 26 00:23:07 UTC 2009

```Magnus,

At 4:01 PM +0000 7/25/09, time-nuts-request at febo.com wrote:
>
>Message: 5
>Date: Sat, 25 Jul 2009 16:38:23 +0200
>From: Magnus Danielson <magnus at rubidium.dyndns.org>
>Subject: Re: [time-nuts] Measuring phase shift between 1 Hz DMTD
>	signals by I+Q processing
>To: Discussion of precise time and frequency measurement
>	<time-nuts at febo.com>
>
>Joe Gwinn wrote:
>>  It occurs to me that there is a possible alternative to the ZCD-chain
>>  approach typical in DMTDs, if one is willing to provide two mixers and
>>  two ADCs per channel, with a 90 degree phase offset between LO signals
>>  provided to the mixers of a channel.  The output of the four ADCs will
>>  be a pair of I+Q signals, one pair per DMTD channel.
>>
>>  The key observation is that if one has two signals, one being a time
>>  delayed replica of the other, if one multiplies one signal by the
>  > complex [conjugate] of the other signal, the result is Exp[j(phase
>  > difference)].  This is true whatever the waveform of the signal, so long
>>  as the only difference in signals is a delay.  The mathematical argument
>>  function of this exponential is the desired phase.
>>
>>  In practice, one will sample far faster than 1 Hz, say 1 MHz, and will
>>  heavily average the resulting stream of products.
>>
>>  Now I have not gone through the math to estimate performance compared to
>>  the traditional ZCD approach, but the complex multiply and average
>>  approach should be quite robust against noise, and is easily implemented
>>  in a DSP or FPGA.
>
>The time-difference between the two sampling points could be minimized
>in such an approach as the phase could be shifted arbitrarily in the
>post-processing such that the effective phase difference between the two
>chains reduces to near zero and hence the correlation between the
>channels for the transfer oscillator would be better in phase and cancel
>the transfer oscillator out better.

you don't have to use a real physical delay out in the analog
hardware.

>The postprocessing would then slowly tune the I/Q phase and keep a phase
>adjustment track such that post-correlation could turn it back for
>proper phase-trace.

But, unlike ZCD-triggered counters, there is no disadvantage or
difficulty if the phase difference is adjusted exactly to zero, where
the two 1 Hz sinewaves coincide.

>An alternative approach is to use the Costas tracking loop as Bruce
>suggested.

A Costas loop is far more complex, but they do work well.  Given near
constant phase delay, don't know if a Costas loop is worth the
trouble.

The Costas loop will not by itself solve the problem of
transfer-oscillator noise.

>Regardless this first stage of digital processing can be done in a FPGA
>frontend and bring the resulting signal bandwidth into very reasonable
>rates, just as for a GPS receiver.

Yes.  Is 0.01 Hz slow enough?

Joe Gwinn

```