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 _______________________________________________ time-nuts mailing list -- [email protected] To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
