$10\mathrm{.}6.$ Given a DSP system with a sampling rate set up to be $8,000$ samples per second, implement an adaptive filter with $5$ taps for system modeling.

As shown in Figure $10\mathrm{.}22,$ assume that the unknown system transfer function is

$H(z)=\frac{0\mathrm{.}25+0\mathrm{.}25z^{-1}}{1-0\mathrm{.}5z^{-1}}$

Determine the DSP equations using the LMS algorithm

$y(n)=$

$e(n)=$

$w(i)=$

for $i=0,1,2,3,4$; that is$,$ write the equations for all adaptive coefficients:

$w(0)=$

$w(1)=$

$w(2)=$

$w(3)=$

$w(4)=$