$.$ Consider the sequence x$[$n$]$ given by x$[$n$]=$ nu$[$n $1]$ where $|$$|<1$ and

u$[$n$]$ is the unit step sequence, u$[$n$]=1,$ n $0$ and u$[$n$]=0,$ n $<0.$ A

periodic sequence x$[$n$]$ is constructed from x$[$n$]$ in the following way

x$[$n$]=$

X

r$=$

x$[$n $+$ rN$].$

a$.$ Determine the Fourier transform X$($e j$)$ of x$[$n$].$

$[8$ marks$]$

b$.$ Let X$[$k$],$ k $=0,1,...,$ N $1,$ be the discrete Fourier transform of

x$[$n$],$ n $=0,1,...,$ N $1.$ How is X$[$k$]$ related to X$($e j$)?$

$[9$ marks$]$

c$.$ Determine X$[$k$],$ k $=0,1,...,$ N $1,$ the discrete Fourier transform of

x$[$n$],$ n $=0,1,...,$ N $1.$

$[8$ marks$]$