Sx(f)
|
|
|\
| \
| \ 1/f
| \
| \
| \__________
| .
|_____.______________ freq.
fc
it has a flicker corner at $f_c$.
we now assume that a noise profile with power spectral density of Sx(f) is injected to an LTI system, H(f), that represent the switching behavior of PFD when PLL is in lock condition.
h(t)
|
A | __Tp___ _______ _______
| | | | | | |
|___| |________| |________| |________
........Ts......
\begin{eqnarray}H \left( f \right) &=& 2 \pi \Sigma_{n=-\infty}^{+\infty} \left( \frac{T_p}{T_s} sinc\left( \frac{n T_p}{T_s} \right) \right) \delta \left( f-n f_s \right) \\
&=& 2 \pi \Sigma_{n=-\infty}^{+\infty} \left( T_p f_s sinc\left( n T_p f_s \right) \right) \delta \left( f-n f_s \right)
\end{eqnarray}
the power spectral density of $S_x(f)$ noise after filtering by $H(f)$ is
\begin{eqnarray}
S_y \left( f \right) &=& \left| H \left( f \right) \right|^2 S_x \left( f \right)
\end{eqnarray}
$S_y \left( f \right) $ is increasing 6dB for 2x increase of $f_s$ frequency (because of $\left| H \left( f \right) \right|^2$ factor).
now we need to analyse the impact of folding (PFD switching) on the spectrum, i.e. $\Sigma_{n=-\infty}^{+\infty} \left(. \right)$. In reality, charge pump bandwidth is limited and the spectrum is as shown below:
Sx(f)
|
|
|\
| \
| \ 1/f
| \
| \
| \__________
| . |
|______._________|_________ freq.
fc f0
$f_c$ is flicker corner, and $f_0$ is the bandwidth of analog charge pump. We also assume that $f_0 >> f_s$ and $f_c << f_s$. The key to noise folding analysis is that the tail of $S_x(f)$ noise is thermal and has much less power than the flicker region.
* Thermal noise region ($f>f_c$) folded in flicker region, $f < f_c $, [lets only talk about noise folding in the first Nyquist zone], has almost zero impact on the power of noise in flicker region.
** Thermal noise region ($f>f_c$) folded in thermal noise region increase the noise floor because the power of folding term is comparable to the power of the thermal noise.
the question is: "what is the impact of refrence frequency ($f_s$) on different noise regions?"
by increasing $f_s$, $K$ will be reduced. It means that for larger reference frequency, we have less folding terms in noise thermal region. flicker region doesn't change (significantly) by noise folding mechanism becuase flicker noise power is much higher than thermal noise that is folding in flicker region. For example, if $f_s$ is increased by 2x, output reffered flicker noise increasing by 6dB (as $|H(f)|.^2$ gain does); however, thermal noise region increased by +6dB-3dB (6dB follows $|H(f)|.^2$ gain increase and -3dB is because, increasing $f_s$ introduces 2x less folding terms in thermal region).
Summary: refrence frequency ($f_s$) has the superposition of the following impacts on charge pump output refered phase noise:
1) increasing refernce frequency from $f_{s1}$ to $f_{s2}$, increases the whole CP output reffered noise by $20 \log_{10} \left(\frac{f_{s2}}{f_{s1}} \right)$
2) increasing refernce frequency from $f_{s1}$ to $f_{s2}$, decreases thermal noise of CP by $-10 \log_{10} \left(\frac{f_{s2}}{f_{s1}} \right)$ but doesn't change the flicker noise
