[time-nuts] Down-conversion to IF and sampling

[Stephan Sandenbergh]

Hi,

A happy New Year to you all!

Also, thank you to everyone who replied in such detail. It is always a
privilege being able to bounce ideas off of the time-nuts community.

I have found that it is often the case that the error introduced by ADC
sampling is ignored. However, there is an error introduced during
down-conversion and the ADC (with its imperfect time base) then samples
both the resultant IF signal and the IF error term. The result is a
complicated error when frequency offset, drift and random effects are
included.

I plotted the result for a few oscillator drift rate values. It seems that
the 'extra' error introduced by the imperfect ADC time base would be
negligible for many applications for OCXO drift rates or better. This is
likely the reason why it is often ignored.

I'm glad I understand it better now.

Regards,

Stephan.



On Mon, Dec 25, 2017 at 3:11 AM Attila Kinali <attila at kinali.ch> wrote:

> On Sat, 23 Dec 2017 21:00:39 +0000
> Stephan Sandenbergh <ssandenbergh at gmail.com> wrote:
>
> > I guess I'm just worried that I might be missing something obvious here?
> > And I know there is no better place to ask such a question other than on
> > time-nuts :)
>
> No, your derivation is correct. Though conventionally, it would be
> written as: V_{IF}(t) = sin[(ω_{RF} - ω_{LO})t - Δφ_{RF-LO}(t)]
> Hence, the sampled signal becomes:
> V_{IF}[nT_s] = sin[(ω_{RF}*(nT_s) - ω_{LO}*(nT_s) - Δφ_{RF-LO}[nTS]]
>
> In this notation, it is a bit more obvious what's going on. Assuming
> both RF and LO frequency are constant, then the sampled voltage only
> depends on the difference of the frequencies, and the initial phase offset
> at time t = 0*T_s. Be aware that this only holds true if you either
> use a low pass filter after the mixer or use complex down conversion.
> In all other cases you have to account for the (ω_{RF} + ω_{LO}) component
> as well.
>
> Using x(t) = x_0 + y_0*t confuses things a bit, as this means
> that you are modulating the phase with a frequency of y_0, which you
> probably do not intend.
>
> Any phase noise you have in the system, you can fold into φ_{RF-LO}(t).
>
> Please note, the above has the implicit assumption, that:
> ω_{IF} is < 0.5 * ω_{LO}, ie that the IF signal is in the first
> Nyquist zone. Otherwise you have to treat the ADC as another mixer stage,
> with it's own ω_{LO_{ADC}} and φ_{LO_{ADC}}.
>
> As Tim Shoppa mentioned, you do not want to have a ratio with small
> integers
> between the LO frequency and the sampling frequency, as any feedthrough of
> the LO and its harmonics will lead to a DC offset and spurs. The amplitude
> of both will depend on the exact phase relation between the LO frequency
> and the sampling frequency, which is usually stable, but not time-nuts
> stable.
>
>
>                                 Attila Kinali
> --
> It is upon moral qualities that a society is ultimately founded. All
> the prosperity and technological sophistication in the world is of no
> use without that foundation.
>                  -- Miss Matheson, The Diamond Age, Neil Stephenson
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