$13\mathrm{.}3$ The $802\mathrm{.}11$a and $802\mathrm{.}11$b physical layers make use of data scrambling $($see Appendix $12$B$).$ For $802\mathrm{.}11,$ the scrambling equation is

P$($X$)=1+$ X$^4+$ X$^7$

In this case the shift register consists of seven elements, used in the same manner as the five$-$element register in Figure $12\mathrm{.}17.$ For the $802\mathrm{.}11$ scrambler and descrambler,

a$)$ Show the expression with exclusive$-$or operators that corresponds to the polynomial definition. $$b$)$ Draw a figure similar to Figure $12\mathrm{.}17.$