Write a computer simulation of the PLL estimation problem. Do this by generating two independent Gaussian random variables to form Z₁ and Z₂ given by (11.196). Thus, for a given (11.201). Call the first value Î¸₀. Estimate the next value of θ, call it θ₁, from the algorithm.

where Z_1,0 and Z_2,0 are the first values of Z₁ and Z₂ generated and É› is a parameter to be varied (choose the first value to be 0.01). Generate two new values of Z₁ and Z₂ (call them Z_1,1 and Z_2,1) and form the next estimate according to 

Continue in this fashion, generating several values of θ₁. Plot the θ_is versus i, the sequence index, to determine if they seem to converge toward zero phase. Increase the value of É› by a factor of 10 and repeat. Can you relate the parameter É› to a phase-lock loop parameter (see Chapter 4)? This is an example of Monte Carlo simulation.

Z20 0, = 0, + e tan %3D '1,0 Z21 02 = 0, + e tan! '2,1