The signal

has DFT X given by

X $=[$ D$0$ D$1$ D$2$ D$3$ D$4$ D$5$ D$6$ D$7$ D$8$ D$9$ D$10$ D$11$

$]$T

$($i$)(4$ pts$.)$ For which indices k is Dk necessarily equal to zero?

$($ii$)(4$ pts$.)$ If

x$(1)=[$ a b c $]$T $,$

express the DFT X$(1)$ in terms of nonzero Dk$$s$.$

$($iii$)(4$ pts$.)$ If

x$(2)=[$ a b c a b c a b c $]$T $,$

express the DFT X$(2)$ in terms of nonzero Dk$$s$.$

$($iv$)(4$ pts$.)$

x$(3)=[$ a jb $$c $000$a $$jb c $000]$T

has DFT X$(3)$ given by

X$(3)=[$ E$0$ E$1$ E$2$ E$3$ E$4$ E$5$ E$6$ E$7$ E$8$ E$9$ E$10$ E$11$

$]$T

express each E$`$ in terms of nonzero Dk$$s$.$

$($v$)(4$ pts$.)$ If the time$-$domain signal x$(4)$ has DFT

X$(4)=[000$ a b c $]$T $,$

express the entries of x$(4)$ in terms of nonzero Dk$$s$.$