In this problem we derive the complex LMS algorithm. Referring to figure and starting with the instantaneous cost function where e[n] is the error signal and M is the number of antenna elements, do the following:

(a) Determine the derivative of the cost function) with respect to the kth elemental weight w_k[n].

(b) Using the instantaneous derivative ∂/∂W_k[n] denoted by ∆J[k] determine the adjustment ∆w_k [n] made to the kth elemental weight in accordance with the rule

(c) Verify the composition of the complex LMS algorithm described in Equations (8.75) to (8.77).

Note that wk[n] is complex valued, and you need to consider its real arid imaginary parts separately.

|e[n]|2 Awr[n] = -J[k]