Given the DFT pair

x$[$k$]$DFTX$[$r$],$x$[$k$]\backslash$xleftrightarrow$\{\backslash$text$\{$DFT$\}\}$ X$[$r$],$x$[$k$]$DFT$$X$[$r$],$

for a sequence of length NNN$,$ express the DFT of the following sequences as a function of X$[$r$]$X$[$r$]$X$[$r$]$:

$($Assume that NNN is even. Use G$[$r$]$G$[$r$]$G$[$r$]$ and H$[$r$]$H$[$r$]$H$[$r$]$ in Proposition $12\mathrm{.}1)$

$($i$)$ y$[$k$]=$x$[2$k$]$y$[$k$]=$ x$[2$k$]$y$[$k$]=$x$[2$k$]$;

$($ii$)$ y$[$k$]=\{$x$[0\mathrm{.}5$k$]$k$$even$0$otherwisey$[$k$]=\backslash$begin$\{$cases$\}$ x$[0\mathrm{.}5$k$]$ & k $\backslash$text$\{$ even$\}\backslash \backslash 0$ & $\backslash$text$\{$otherwise$\}\backslash$end$\{$cases$\}$y$[$k$]=\{$x$[0\mathrm{.}5$k$]0$k$$evenotherwise$$;

$($iii$)$ y$[$k$]=$x$[$N$1$k$]$y$[$k$]=$ x$[$N $-1-$ k$]$y$[$k$]=$x$[$N$1$k$]$ for $0$k$$N$10\backslash$leq k $\backslash$leq N $-10$k$$N$1$;

$($iv$)$ y$[$k$]=($x$[$k$]$x$[$k$2])$ej$(10$k$/$N$)$y$[$k$]=($x$[$k$]-$ x$[$k $-2])$ e$^\{$j$(10\backslash$pi k $/$ N$)\}$y$[$k$]=($x$[$k$]$x$[$k$2])$ej$(10$k$/$N$).$