Homework 5 (Due: October 10)

[PDF]

Reading

Madhow:

1. Chapter 3: sections 3.4 and 3.5. Note, the textbook in section 3.5 covers synchronization in continuous time while we treat it in discrete time in class. The latter is a little easier and allows MATLAB experiments. Nevertheless, you need to read section 3.5.

Problems

1. Madhow: Problem 3.21
2. Use MATLAB (or Octave) to conduct the following experiments with the adaptQuadratic.m (URL: http://www.spec.gmu.edu/~pparis/classes/resources_460/adaptQuadratic.m) MATLAB script that we used in class.
   1. Vary the step size parameter μ to take the values 0.01, 0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95, 0.99. For each of these values, measure the number of iterations n until |x[n]-2| < 10^-5 for the first time. Present your findings in a plot or table. Discuss any observations that you deem interesting.
   2. What did you observe for μ = 0.5? Can you explain your observation based on the iterative update rule x[n + 1] = x[n] ⋅ (1 - 2μ) + 4μ?
   3. What do you observe when μ = 1? Can you explain your observation based on the update rule?
   4. What happens if μ < 0 or μ > 1? Can you explain your observations from the update rule?
3. Use MATLAB (or Octave) to conduct the following experiments with the adaptivePhaseTracking.m (URL: http://www.spec.gmu.edu/~pparis/classes/resources_460/adaptivePhaseTracking.m) MATLAB script that we used in class.

   For this problem, disable frequency offset (set do_freq_offset = false) and disable the I part of the controller by setting eta = 0.

   1. Set the standard deviation of the additive noise to sigma=0.2;. Vary the step size parameter μ to take on the values 0.01, 0.03, 0.1, 0.3, 1. For each of these values of μ measure
      • the iteration number N₀ such that the phase error for the corrected signal r[n] is less than 0.01, i.e., |phase(r)| < 10^-2, for the first time.
      • the variance of the phase error Var(phase(r)) = ∑ _n=N0+1^N₀+M|phase(r)|², with M = 100.
   2. Plot N₀ versus the step size μ and plot var_θ versus the step size μ. Discuss your findings and make a well justified recommendation for how the step size μ should be selected.
   3. Repeat the experiment for two other noise levels (e.g., sigma=0.05; and sigma=0.8;. Would you change your choice of μ in either of these cases?
4. Repeat the preceding problem with frequency offset and integral control: set do_freq_offset = true and eta = mu/100.