Problem $5$ FFT $(20$ points$)$

Explain$/$discuss the following:

a$.$ A $2-$point butterfly is rarely, if ever, used in practice $-$ A $4-$point butterfly is used instead. Draw these butterflies, and write the output equation in a manner that shows that no multiplications are required for either.

b$.$ A complex multiplication requires $4$ multiplications and $2$ additions on the digital hardware or the computer

c$.$ You do not ever see a butterfly graph for an inverse DFT$.$

d$.$ A data sequence that is $80-$points long must be FFT$\text{'}$d$.$ List all the options that are available for this computation, and discuss their relative merits $($i$.$e$.$ computational requirements, including memory and number$).$ Create a random sequence $\backslash ($ x$[$n$]\backslash )$ that is $80$ points long in MATLAB, and code the brute force $($direct computation$).$ Make it as efficient as possible, and compare to the FFT routine given in Matlab $($and its possible variants$).$ How do you verify that your brute force method works? Can you compare the computation times? Do it if you can.