Page 158 - Phase-Locked Loops Design, Simulation, and Applications

PLL PERFORMANCE IN THE PRESENCE OF NOISE Ronald E. Best 98
The noise spectrum has a bandwidth of B . Note that B is finite in all practical cases, because
i i
there is always a prefilter or preamplifier in the system that limits noise bandwidth. We now
calculate the phase jitter created by the noise at the reference input of the PLL (cf. input u in
1
Fig. 2.1). Assuming the noise spectrum is “white” as explained in the preceding section,

Gardner found the mean square value of input phase jitter to be 1

(4.1)

with P = signal power (W) and P = noise power (W). Note that is simply the square of
s n
the rms value of input phase jitter.
We now define the SNR at the input of the PLL as

(4.2)

For the phase jitter at the input of the PLL, we get the simple relation

(4.3)

that is, the square of the rms value of the phase jitter is inversely proportional to the SNR at
the input of the PLL. We remember that is the mean square of the phase noise that
modulates the carrier frequency f ′ of the PLL (f ′ = ω ′/2π). Consequently, the spectrum
0 0 0
Θ (ω) of input phase jitter is identical with the input noise spectrum shifted down by f ′. 1
n1 0
Fig. 4.4b shows the mean square of input phase jitter

(4.4)

Since the noise spectrum has been assumed to be white, the mean square has the
constant value Φ. Knowing the spectrum of input phase jitter, we can compute the spectrum
of the down-scaled output phase jitter from

(4.5)

Fig. 4.4c plots the Bode diagram of H(ω), and Fig. 4.4d shows the mean square of
the output phase noise spectrum, which is simply the multiplication of curves (b) and (c) in the
figure. To compute the mean square of the output phase jitter , we use Parseval’s

theorem to get