[time-nuts] Phase modulation question

[Magnus Danielson]

On 09/01/2013 01:00 AM, John Miles wrote:
> Normally any higher-order sidebands are ignored when dealing with PM.  You
> can think of it as NBFM with a very low modulation index -- all of the
> "intelligence" is in the first sideband.
>
> Put another way, the modulation index for FM is the peak frequency deviation
> divided by the highest frequency present in the modulating signal.  In phase
> modulation the carrier frequency is normally considered constant.  So the
> numerator of the fraction you would use to index a table of Bessel functions
> is 0, yielding a result of 1.
>
> It gets more complicated when the PM at a given offset frequency shifts the
> carrier by a radian or more.  This is where the small-angle assumption for
> standard L(f) phase noise measurements becomes invalid.  If you are going to
> measure the phase noise of a Cs standard at offsets of small fractions of a
> Hz, this will eventually be an issue given the steep PN slope close to the
> carrier.  Is that what you're doing?
>
> For finer-grained Bessel resolution it make sense to evaluate the function
> yourself.  I think there's a BESSEL() keyword in Excel, and Matlab or Octave
> could certainly do it.
The carrier will have the Bessel function of J_0(x) where x is the
modulation index and the first side-band will have the Bessel function
of J_1(x). As unfold these in their polynomial form we have

J_0(x) = 1 - x^2/(2^2) + x^4/(2^2*4^2)  -  x^6/(2^2*4^2*6^2) + ...

J_1(x) = x/2 - x^3/(2^2*4) + x^5/(2^2*4^4*6) - x^7/(2^2*4^2*6^2*8) + ...

If we assume a very small phase modulation (as pointed out by Bob,
severe vibration violates this) all the higher terms will be very small,
and so would all the J_2(x) and higher be too, so we can without too
much loss of information approximate them to:

J_0(x) = 1

J_1(x) = x/2

J_2(x) = 0

It is worth to mention that the J_1(x) is the Upper Side Band (USB)
coefficient, where as the Lower Side Band (LSB) will be J_-1(x) = -
J_1(x) = -x/2 so, they will have opposite signs. This property of PM
separate it from AM where the side-bands have same sign, which is of
great help to know when separating them.

So, with these approximations at hand, you are free to quickly form your
PM/FM sidebands as you need.

Cheers,
Magnus