[sdiy] jitter in oscillators for music purposes

[Magnus Danielson]

From: "Czech Martin" <Martin.Czech at Micronas.com>
Subject: [sdiy] jitter in oscillators for music purposes
Date: Mon, 12 Jul 2004 11:07:26 +0200
Message-ID: <D9D56E8FA1A73542BE9A5EC7E35D37FFF390E6 at EXCHANGE2.Micronas.com>

Martin,

> I keep on thinking about this. And reading.
> 
> http://arxiv.org/pdf/physics/0204033
> 
> This nice article shows a study where a guy 
> captured the noise of a bunch of op amps
> for three months (!) (page 7).
> The idea was to look after the 1/f component, if
> it will finally saturate. No! Even in such a long timeframe
> the 1/f (or 1/f**a with 1<a<2) law will be visible,
> no saturation of noise level in the low end.

Interesting!

> The author also makes some proposal why the appearent catastropy for
> f=0 is not really such a big problem for a power density function
> like 1/f**a.

Now I really need to look at it.

I expect there to be several non-white (i.e. a > 0) components in action in a
real synthesizer. Even the DCO ones has it, due to those components in the
crystal oscillator. It's always there, question is if it has any 
contributional effect to the fatness effect or not, and if so, which strengths
of which noise-type does it?

> More practically I scetched up a sawtooth vco core.
> We have two major sources of hissing noise.
> 1.) the current source (with certainly a nice amount of noise
>     for the very low currents)

also, consider low current => low frequency... ;O)

> 2.) the comparator (assuming that buffer and comparator is
>     a single block)

I never assume the comparator and buffer is a single block except for some
very simple oscillators. There will be some JFET buffer and then some form
of comparator. 

> I assume , that no other interference is present, because
> residuals of the 50/60 Hz stuff should add problems instead
> of beauty, if they are too visible (locking).
> Perhaps this exclusion is not justified.

Now, don't forget that noise (white, 1/f and 1/f^2) from the power supply does
have an effect on normal oscillators, since they enter both the comparator and
the expo current reference, as well as going through the expo itself (noise
being exponentially converted - go figure on that noise-distribution!).

> Now, as Magnus said, the voltage noise on the comparator input
> certainly leads to frequency modulation. Even if it is white,
> the laws of small frequency modulation will shape the spectrum
> accordingly. So we only need to compute the total noise
> voltage at this point, following traditional methods.

I prefer to use phase modulation, since that is also what really happends.
This is also supperiour in most analysis forms. 

Take a normal sawtooth-oscillator, it is insensitive to the DC level of the
powersupply line, but it fails to cancel the AC components since they undergo
an integration in the phase integrator capacitor, thus causing the AC comming
in from the current reference and that from the reference voltage chain to be
out of phase and thus full cancelation fails to work. As the comparator
voltage deviate, a phase modulation is introduced.

> The input voltage noise source of the comparators
> translates directly, the current noise source will translate 
> into input voltage noise via the oscillator
> capacitor. I.e. the later noise voltage should scale with 1/f, and the
> power accordingly with 1/f**2. 
> 
> So, additionally to the frequency modulation shaping, we
> have a strong low frequency domination for the noise current.

Indeed.

> At the same time we notice, that the partials (or should I say
> partials + sideband) are very closely correlated.
> You can not phase modulate them seperately, they are dependend
> on each other.
> So the example of noise driven bandpass filters is interesting,
> but IMHO not really a good model for this single oscillator
> business.

Exactly my view. Things like that does however help us to understand the
audatorial aspects of fatness better, but fails to model them well.

> The noise current from the oscillator current source will have 
> components of this transitor as well as of uncorrelated 
> previous stages (op amps, summing circuit). Further more
> the noise level will depend on the current. Anyway, the
> spectral shaping due to integration through the timing cap
> will take place.

Exactly.

> All three components will add up (in the average) in a 
> geometric sum expression, given that the have no correlation.
> From this I'd assume that the noise voltage that we sum up
> at the comparator input is certainly NOT WHITE but contains
> a very dominating 1/f*2 and maybe a 1/f region until flattening out 
> in the frequencies of interest (0.001 Hz - 100kHz ?).

Right. One has to recall that standard deviation doesn't bite on 1/f noise,
one needs better analysis tools to handle the 1/f and 1/f^2 noises. I propose
the use of Modified Allen Deviation and Time Deviation in order to be able to
analyse this kind of noise.

> Unfortunately the main interest on investigation of this kind is
> towards on chip plls (well, this is one of the biggest problems
> for SoC). 

The thumb rules for PLLs are simple:

The phase-noise of the PLL oscillator is highpassfiltered with the corner
frequency of the PLL bandwidth.
The phase-noise of the reference signal is lowpassfiltered with the corner
frequency of the PLL bandwidth.

Experience show that high-Q PLLs can be a bit difficult, but there is numerous
tricks-of-trade to handle it.

> btw. when tuning the oscillator to very low frequencies we can
> actually listen to the noise on the buffer or view
> it on scope , from time to time disturbed by a large transient.

Right. We could also remove the cap and put a resistor there, so we could see
the noise contributions. Using another resistor we could redo the same thing
and factor out the resistor-dependent behaivours

> So, a DSP model of an analog vco should not modulate the frequency
> with white noise allone, but with 1/f**2 noise as well.

Indeed. Care also needs to be taken to ensure that 1/f really is 1/f all the 
way down to the frequencies of interest. Normal pink noise filters fails to
properly model that I fear, they concentrate on the audio-frequencies.

Noise is a bitch, we just need to learn even more about it.

Cheers,
Magnus