$2\mathrm{.}3.(15$ points$)$ In many signal processing problems, the desired signal $d(n)$ is a sequence that can

carry useful information about future values of this sequence. $s$ such, it is useful to consider

linear models that use past values of $d(n-j),j1$ as well as current and past input signal

values $x(n_i),i0,$ to estimate the current desired signal sample $d(n).$ To this end, consider

the linear model given by

widehat$(d)(n)={\displaystyle \underset{i=0}{\overset{M}{}}}{b}_i(n)x(n-i)+{\displaystyle \underset{j=1}{\overset{N}{}}}{a}_i(n)d(n-i)$

where $b_i(n),0iM$ and $a_j(n),1jN$ are the adaptive parameters of the filter.

a$)$ Derive an algorithm for adjusting the parameters of this system to minimize the mean$-$

squared error cost $J(n)=E\{(e(n){)}^2\}$ using the method of steepest descent. Use vector

notation to specify the algorithm, and carefully define all matrices and vectors needed to

implement this procedure including definitions of quantities inside of any matrix used.

b$)$ Derive a stochastic gradient algorithm for adjusting the parameters of this system to

minimize the mean$-$squared error cost $J(n)=E\{(e(n){)}^2\}$ via an instantancous gradient

approximation. Explain the advantages of using this algorithm over the approach you

derived in a$).$