$1.(\backslash (\backslash$mathbf$\{30\}\backslash )$ points$)$ Consider the following sequence

$\backslash [$

x$($n$)=\backslash$cos $\backslash$left$(\backslash$frac$\{\backslash$pi$\}\{3\}$ n$\backslash$right$),\backslash$quad $0\backslash$leq n $\backslash$leq $7$

$\backslash ]$

a$)$ Compute the $8-$point DFT using the in$-$place radix$-2$ Decimation$-$In$-$Frequency Fast Fourier Transform $($DIF FFT$)$ algorithm. Draw the entire diagram for this algorithm. Follow exactly the corresponding signal flow graphs and keep track of all the intermediate quantities by putting them on the diagram. Record the outputs of each stage. Show all the mathematical details.

b$)$ Determine and compare the total number of complex multiplications that you have performed in the above FFT algorithm with the total number of complex multiplications that needed in direct computation of DFT$.$