We want to implement a $6-$point decimation$-$in$-$time FFT using a mixed radix approach. One option is to first take three $2-$point DFTs$,$ and then use the results to compute the $6-$point DFT$.$ For this option:

$($a$)$ Draw a flowgraph to show what a $2-$point DFT calculates.

$($b$)$ Also, fill in the parts of the flowgraph in the figure below involved in calculating the DFT values $x_0,x_1,$ and $x_4.$

$($c$)$ How many complex multiplications does this option require? $($Multiplying a number by $-1$ does not count as a complex multiplication.$)$ A second option is to start with two $3-$point DFTs$,$ and then use the results to compute the $6-$point DFT$.$