$36.$ The following $8-$point sequences are defined over $0$n $7.$ Without computing

their DFTs$,$ determine which have real$-$valued $8-$point DFTs and which have $8-$point

imaginary$-$valued DFTs$.$ If DFTs are complex valued, explain why.

$($a$)$ x$1[$n$]=\{0,3,1,2,0,2,1,3\}.$

$($b$)$ x$2[$n$]=\{5,2,9,4,7,4,9,2\}.$

$($c$)$ x$3[$n$]=\{8,3,1,2,6,2,1,3\}.$

$($d$)$ x$4[$n$]=\{0,1,3,2,5,2,3,1\}.$

$($e$)$ x$5[$n$]=\{10,5,7,4,5,4,7,5\}.$