In this problem we derive the direct matrix inversion algorithm for adjusting the weights of an adaptive antenna array. To do so, we revisit the derivation of the Wiener filter presented.

(a) Show that R_xw = r_xd where R_x is an estimate of the correlation matrix of the input vector x[k] and r_xd is an estimate of the cross-cot-relation vector between x[k] and the reference signal d[k]:

The superscript H in the formula for R_x denotes Hermitian transportation (i.e., transposition and complex conjugation), so x[k]x^H [k] denotes the outer product of x[k] with itself. The summations for both &, and i are performed over a total of k snapshots, with each snapshot being represented by the pair {x(k], d[k]).

(b) Using the formulas of part (a), describe an algorithm for computing the weight vector w, given a data set consisting of K snapshots. Hence demonstrate that the complexity of this algorithm grows as M with the size of the weight vector w denoted by M.

.-xki*" k] K k=1 xdk] K xd