From: roth@3d.enet.dec.com (Jim Roth)
Newsgroups: rec.audio
Subject: Re: Jitter Simulation Program
Date: 21 Jan 93 22:30:53 GMT
Organization: Digital Equipment Corporation


In article <1993Jan19.230006@trc.amoco.com>, znpt01@trc.amoco.com (Norman P. Tracy) writes...
>In article <1993Jan19.092845.24382@ryn.mro4.dec.com>, roth@3d.enet.dec.com (Jim Roth) writes:
>> 
>> Clock jitter causes low index phase modulation of spectral components of the
>> audio, and the effect will be proportional to both the amplitude and
>> frequency of the component.

>Jim, what is "low index phase modulation"?

Modulation index is the peak phase deviation in radians in frequency or
phase modulation.  It's usually designated beta in books and papers.

In this case the peak phase shift is considerably less than 2 PI radians
so the index is small enough to work with first order approximations.

If you have a cosine phase modulated by another sinusoid, then for low index
there will be a pair of 90 degrees shifted sidebands beta/2 as large
as the carrier.  This is easy to see by looking at phasor diagrams;
you have a little modulation phasor sinusoidally varying at right angles
to the carrier.  Another way is to look at the carrier where it slews thru
it's zero crossings.  The slope is 2*pi*f per second so beta = 2*pi*f*tau.

So the relative level of each sideband will be

    -20 log10 (beta/2) = -20 log10 (pi*f*tau)  dB down

Trying 10 kHz at zero level, 1 nsec peak (2 nsec peak to peak) sinewave
jitter, we have sidebands -90 dB down.  [NOT -84 dB as they show in a plot
claimed to be for 2 nsec peak to peak.  They are off by a factor of 2.
They appear to be using one of the Blackman-Harris windows; probably
a Kaiser-Bessel window with first null 5 bins away would be a better
choice for such a low noise floor.]

A 1 kHz full level signal will have sidebands -110 dB down, and so on.

If they wished to make a meaningful assessment of the contamination due
to jitter alone, they should have measured the jitter, spectrally analyzed
it, and then used each spectral component (taking into account phase as
well as magnitude) to modulate each sinewave in their test signals using
the above relation for sidebands as a function of frequency and jitter.

This is not a lot of processing for the simple signals they used, and
would have shown that most of the processors would have *very* low audio
band contamination from the jitter alone.  In fact for simple sinewaves,
merely notching out the sinewave out of the DAC (to prewhiten the signal)
and spectrally analyzing the resulting residual would show how much
contamination really exists due to all sources.  (Or better, listening
to the residual.)  This kind of measurement of total noise and distortion
is probably more meaningful, and at any rate should be done in conjunction
with the analysis recommended above to show check if sensible results are
being obtained.

By the way, the spectral purity required by digital audio clocks is
not at all stringent compared to what is required for phase locked
loop frequency synthesizers.  Folks designing these things have long
been aware of such gremlins as "logic induced modulation" even if
fancy names haven't been used.  It's not a difficult measurement.
In fact, it's easy to measure clock jitter to far greater precision
than is necessary, even to asymptotically approach perfection.

>So are the above two paragraphs arguing that jitter can never be audible or
>is only possibly audible at high frequencies?  I think the jury is still out
>and view jitter as a hypothesis which may explain some of the effects critical
>listeners have been reporting and I have heard.

Actually, it would be easy to hook a phase lock loop chip up to
a player's DAC clock line that would let you inject adjustable jitter,
either random or even with some of the audio output mixed in.

Just adjust it and listen, and see what kinds of thresholds there are.
You'll probably find it shocking how much distortion can be really
cranked in this way, when you only vary one experimental parameter.
While I haven't done it with DAC clocks, I did such things with crossover
distortion, slew limiting, and harmonic distortion.

>  One of the reasons I consider jitter a
>likely suspect is the literature I've read seems to agree that to get its
>effects below the 16 bit noise floor it has to be held to less than about
>400 pS (from Harris' paper).  Hawksford quotes a few tenths of a nS.  Do the
>papers you cite above give a conclusion as to the required accuracy?

No conclusios, but one can always plug in some numbers and play what if games.
The figures cited are valid in the sense that if jitter is that low
(and random) that the 16 bit theoretical quantization noise floor won't
be seriously compromised in the case of a pure full level 10 kHz sinewave and
measured with a spectrum analyzer.

>  Do you
>think this level of accuracy is typically maintained in $300 CD players?

I don't know, but there is *no reason why not*.  It is emphatically not
an expensive thing to do properly, particularly in a one piece player
where no extra interface chips and what have you are needed.

- Jim