PLL design tutorial questions

[mryndzionek]

Hello everyone!

I'm trying to thoroughly understand the this PLL design tutorial and I have two questions:

1. how is omega_n chosen?
2. in the bilinear transform it seems that pre-warping is done with m equal to 0.5 (the 1/2 before the Tustin's substitution).
   How is this chosen?

The two question are probably related. When I plot the closed-loop gain for the analog and digital filters with the data from the tutorial (omega_n = 0.01, pre-warping with 1/2) I get:

Pre-warping "moves" omega_n to the right, but what is the point of doing pre-warping when I can
change omega_n to 0.1 and have no pre-warping?

And the actual simulation seems to indicate that I get the "lock" faster:

omega_n = 0.01 with pre-warping

omega_n = 0.1, no pre-warping

Shining some light onto this is greatly appreciated.