Both radians and cycles are unitless quantities. So from a dimensional analysis perspective, rads/s, cycles/s, and /s are all equivalent.

Jitter Man

Let me try again. What I should have said was the radians and cycles are dimensionless. Thus, rads²/Hz or cycles²/Hz both have dimensions of seconds. The reason I am distingushing units from dimensions, is that it is very common for dimensionless units such a radians to get lost in a derivation. Thus, dimensional analysis often simply neglects dimensionless units such as radians. If you are sufficiently determined, you can track them down and keep them balanced, but it is generally more trouble than its worth.

Jitter Man

Of course, but you do not use dimensional analysis to find errors in factors of 2 pi.

Jitter Man

Here's an example of why you need to keep track between radians and cycles.  VCO output frequency vs. input voltage gains (commonly referred to as KVCO) can either be given in units of rad/s/V or cycles/s/V.  Both are exceptable and easily interchangeable since 1 cycle=2pi radians.

Now, if you were designing a PLL that used this VCO and were working the phase transfer functions (for determining loop stability) in units of radians then you would want to be certain that the KVCO value you used were in units of rad/s/V.  If you slipped in the value in terms of cycles/s/V, your open loop gain would appear 6x (2*pi) less and you would have a much less stable PLL.

By the same reasoning, the only unit of noise I know that can be extracted from a phase noise L(f) is rad^2/Hz (as I explained in my last message) but this is not does not fit with Eq. 73.  

Therefore, the unit conversion was either handled in some way that I didn't see, 10^(L(f)/10) does not have units of rad^2/Hz, or Eq. 73 has an error.  This may seem like mathematical minutea, but we're talking about a potential 6x (2*pi) error in a jitter calculation.

Thanks again for your opinions.

Ken,

I entirely agree that L(f) literally has units of 1/Hz which is arrived at from the ratio of the noise voltage power (V^2/Hz) to the signal voltage power (V^2).

What is still a little unclear is how L(f) is being converted to a single side band phase noise density (Sphi(f)).  As it is, L(f) is a dimensionless density.  In order to relate it to phase, we need a power density that carries units of rad^2/Hz or cycles^2/Hz.  When that power density is integrated over a particular bandwidth, it need to indicate phase in either radians or cycles.  Both radians and cycles are technically dimensionless since they both describe a portion of a circle, but obviously with a 2pi scaling difference.

From narrow band modulation theory,
Vout(t) = A * sin(2pi*fout*t+phi(t)) is approximately equal to
Vout(t) = A*(sin(2pi*fout*t)+phi(t)*cos(2pi*fout*t))

L(f) = [Sphi/Hz]/1^2 = Sphi/Hz

From this basic relationship, the phase noise induced voltage magnitude should be phi(t) times the signal amplitude.  Therefore, L(f) = Sphi/1^2.  What I draw from this is that L(f), while still technically 1/Hz, should equal the single sideband phase noise power density and that the units that one gets with this are in rad^2/Hz.  This tracks with most the literature (Hajimiri for example) that computes period jitter as:

sigma^2 = 1/2pi/fo*sqrt(2 * integrate(Sphi(f)*sin^2(2pi*f*1/fout), from 0 to infinity)

The key feature to that equation is that the quantity generated by the integration is rad^2.

From Eq. 19 (page 12), Sphi is being carried through in cycles/Hz and from Eq. 27 (page 15) it has the same dimensions as L(f).

So, the big question is: How are you moving from a dimensionless power spectral density L(f) to Sphi(f) with dimensions of phase^2/Hz?

Thanks.