# [time-nuts] 20logN was Re: phase noise questions (long)

Bruce Griffiths bruce.griffiths at xtra.co.nz
Wed Jan 23 19:20:11 EST 2008

```Mike Feher wrote:
> Christophe -
>
> While all of the mathematics to prove the 20logN behavior can be found in
> most elementary texts, the concept of what is really happening may not be so
> obvious from the math, even though the math of course supports the 20logN
> behavior. You, yourself said it in your assumption in 2a. When a carrier is
> phase modulated by a single frequency you do get the +/- sidebands about the
> carrier as stated. The sidebands have two items contributing to the
> sidebands location and their amplitude. Their location is due to the
> frequency of modulation, and their amplitude to the deviation of that
> frequency about the carrier. When doubled, the carrier is doubled but the
> modulation frequency remains the same, yet the deviation is also doubled,
> hence the 6 dB increase relative to the carrier of said sideband amplitude,
> or the level it was before. This was a very common way to get increased
> deviation in WWII military radios. They started out at a low frequency and
> FM modulated it one way or another, however with a resultant low deviation.
> After numerous multiplier stages to get to the final frequency, the
> discriminator, yet the audio frequency was the same as initially injected.
> Regards - Mike
>
>
>
> Mike B. Feher, N4FS
> 89 Arnold Blvd.
> Howell, NJ, 07731
> 732-886-5960
>
>
Mike

Such analysis only applies when the modulation index of the multiplied
signal is also small.
In most cases this is true, but the above simplified analysis doesn't
apply when the multiplication factor is sufficiently large that the
carrier collapses.
Carrier collapse is a significant problem when multiplying OCXO outputs
to the TeraHertz region.
The mathematical approach still works as long as previously neglected
terms are taken into account.

Bruce

```