This exercise tests the theoretical misadjustment formula (9.75) using the LMS algorithm in a…

This exercise tests the theoretical misadjustment formula (9.75) using the LMS algorithm in a manner similar to Exercise 9.15, that is, with the configuration in Figure 9.9 and with the following signals:

(Note: This exercise involves a total of 1.5 million LMS iterations. If your computer is too slow, use a compiled program to obtain the 16 data points, and then plot these using MATLAB.)

The objective is to start the LMS algorithm at b_opt and obtain misadjustments experimentally at each of 16 values of u:

The experimental misadjustment at u = 0 is zero, because no adjustment is allowed in this case. At each value of u greater than zero, make 10 runs, each with 104 LMS iterations. At the beginning of the mth run, set the weights to b_opt = [ 3, −1] and add a phase shift of 2mπ/10 to the argument of sk so the signal begins at a different point in its cycle. After all 10 runs, compute the MSE and the misadjustment (M) based on the theoretical minimum MSE. Finally, plot M versus u, and obtain a result similar to Figure 9.11.

Exercises 19.15

This exercise illustrates misadjustment using the LMS algorithm with u = 0.05 and the configuration in Figure 9.9 with the following signals:

f.        Set the random number seed to 123, and find the optimum weight vector using actual signals. Compare with the theoretical optimum in Figure 9.2, which was bopt = [ 3 , −1].

g.       Plot the error contours on the weight plane as shown in Figure 9.10.

h.      Run 2000 iterations of the LMS algorithm with u = 0.05, using bopt for the initial weight vector. Plot a point for the weight vector at each iteration, and complete the plot in Figure 9.10.

i.          Compute the theoretical misadjustment for this case.

j.        Compute the misadjustment using the error signal, ek, in this example, and compare the result with the answer to part (d)