# PLL in noise-free and noisy channels

I'm trying to evaluate the performance of PLL in compensating the phase noise in the absence and presence of AWGN noise, and the counterintuitive result I consistently get is that PLL performs better in noisy channels. Is this expected? and Why?

The code with which I get this result is a little bit complicated because it's part of a larger project, but what I do basically:

I generate the phase noise, which is symbol-by-symbol. I have the symbols in blocks of length K. So, what I do, I first find the the mean of the magnitude squares of the generated noise, and I find the mean of the magnitude squares of the residual error at the output of the PLL between the estimated phase noise, and the actual phase noise. The latter is larger the former in case of noise-free AWGN channel, but smaller in case of noisy channels.

• "I get the result", " why?" Please show how you get that result. The answer to "why" is exactly there! Instead of letting us get out a textbook and copy something you've probably read elsewhere already, show us how you get that result and we can discuss it exactly at your level of understanding. Sep 14 at 11:49
• OK, the code is a little bit complicated because it's part of a larger project, but what I do basically, I generate the phase noise, which is symbol-by-symbol. I have the symbols in blocks of length $K$. So, what I do, I first find the the mean of the magnitude squares of the generated noise per block, and I find the mean of the magnitude squares of the residual error at the output of the PLL between the estimated phase noise, and the actual phase noise for the same block. The latter is larger the former in case of noise-free AWGN channel, but smaller in case of noisy channels. Sep 14 at 12:02
• I added that information – it's highly relevant – to your question's body. Sep 14 at 12:58
• Shouldn't the error in the PLL be between the output phase and zero phase errror, not "estimated phase noise"? The PLL removes phase noise at the oscillator output, within it's loop bandwidth. Sep 14 at 13:04
• @DanBoschen Thank you for pointing this out. You're right. I was evaluating the performance wrongly. Now I get better performance after PLL in both cases, noisy and noise-free channels. Thanks again! Sep 15 at 11:34

The PLL removes the low frequency components of the phase noise in the output oscillator that is locked to a reference, and passes through the low frequency components and removes the high frequency components of the reference it is locked to. Thus the phase noise of the output oscillator sees a high pass filter with the use of the PLL, and the phase noise of the reference oscillator see a complementary low pass filter. The filtering operation has a slope consistent with the order of the loop.

To evaluate noise in the PLL due to phase noise. It is helpful to first understand phase noise in terms of its frequency performance, and then with that understand how the phase lock loop filters this phase noise. Below is a graphic demonstrating the concept of phase noise components and what the power spectral density of phase noise is showing us. Here we see the power spectral density (PSD) as a typical Phase Noise plot, showing us the power due to phase fluctuations in the signal at each frequency component of those fluctuations. I have taken a slice at a low and high frequency component, to show if we were able to filter the phase noise at those frequency offsets, what the phase fluctuations vs time would be and how they each have a frequency consistent with the frequency offset selected, and a relative magnitude consistent with relative power at that frequency offset.

If we were to lock the oscillator with the phase PSD shown above to a much cleaner reference oscillator (as typically done), using a PLL loop bandwidth of 200 KHz (the plots show “CR Loop BW” as these are graphics I have explaining a carrier recovery loop which similarly tracks out the low frequency phase noise components), the low frequency phase noise components below this bandwidth would be reduced by a high pass filter with a slope consistent with the order of the loop. The resulting slices of phase noise would appear as below. To determine the total noise, the resulting phase noise would be integrated out to the measurement bandwidth of the system (indicated as "Channel BW" in the plots above), consistent with the shaded area in the plot below. Since the dominant noise is in the low frequencies, there will always be a significant improvement in the reduction of total phase noise after the PLL, assuming we are locked to a lower noise reference. 