An unknown plant can be “identified” using the stochastic gradient algorithm by using the connections shown below (a) Develop the stationary solution to this problem. (b) Develop the stochastic...

An unknown plant can be “identified” using the stochastic gradient algorithm by using the connections shown below

(a) Develop the stationary solution to this problem.

(b) Develop the stochastic gradient and LMS iterators.

(c) Suppose that the “unknown” model is given by an FIR model

Assume that x, n are white with Rxx = 1/12, Rnn = 1/12. Simulate this system and obtain the optimal stationary and LMS solutions. Calculate the minimum mean-squared error in this case. Compare the solutions.

(d) FIR filters can also be designed in this manner by specifying the unknown system coefficients and allowing the adaptive algorithm to learn the response. Start with a second-order Butterworth low-pass response (fc = 10 Hz) and excited the filter with pseudorandom noise. Design a 16, 32, and 64 weight filter and “teach” the adaptive filter. Discuss your results.