Subject: Re: Phase noise, jitter web site
X-Newsreader: Microsoft Outlook Express 5.50.4920.2300
Date: Fri, 08 Nov 2002 18:20:29 GMT
NNTP-Posting-Date: Fri, 08 Nov 2002 13:20:29 EST
Organization: Cox Communications
"Bob" wrote in message
> Here's what (I think) I mean by the phase of the phase noise.
> If you do Fourier analysis on a time-domain signal you get both amplitude
> and phase information. You need both of these to go backwards and recreate
> the original time-domain signal.
> The dBc/Hz information is only amplitude vs. frequency. There's not (that
> I've seen) phase info, too, published.
> How can the max phase deviation, from a non-jittered signal, be
> reconstructed without this phase information?
Ahhh, I understand now. First, though, to follow up, I tried tracking down
the other article from the trade magazine that I thought I had, and was
unsuccessful. If I come across it, I'll post a link.
The reason there's no phase information is that the Fourier transform you're
looking at is the power spectrum of the signal. If the time domain signal is
f(t), and the frequency domain signal is F(w), then the power spectrum is
S(w) = F(w)F*(w), where * denotes the complex conjugate. In terms of
magnitude and phase, S(w) = |F(w)|exp(jX(w))|F(w)exp(-jX(w)) = |F(w)|^2.
Returning to the time domain for a moment, if you had an unfiltered white
noise signal in the time domain, and you took the Fourier transform, you'd
get random amplitude and phase in the frequency domain. Even if you took the
Fourier transform of filtered noise, you'd get random amplitude and phase
over the passband of the filter.
On the other hand, the signal does have a power spectrum, which shows the
effects of filtering very nicely. The question then is, given that we know
the power spectrum, what does that tell us about the time domain?
The power spectrum can't be used to convert back to a particular time domain
noise sequence, so you can't recover the precise noise signal that the power
spectrum was calculated from. This is noise, though, and (ideally) the time
domain signal is random. What we need is the statistics of the time domain
sequence, not a particular time domain sequence. What Wiener showed was that
the inverse Fourier transform of the power spectrum is the autocorrelation
of the noise sequence. The autocorrelation at time zero, R(0), is the power
of the noise sequence, which is also the variance. The square root of
variance is standard deviation... That's not quite the entire story, since
what you're really interested in is the phase variance between two time
points (one period, for example, if you're looking for the jitter in one
period). Kundert and Drakhlis show how this is done: the autocorrelation of
the phase difference between a starting point and an end point is
2(R(0)-R(T)), which results in an additional sin^2(wT/2) term in the
frequency domain, where T is the period over which you're measuring the
variance. Take the inverse Fourier transform of that, and voila: variance.
Take the square root and you've got jitter. Remember that phase noise
transforms to phase jitter, so multiply by T/2pi to convert to seconds
(assuming you're working in radians).
Hope that helps.
-- Mike --