Could I please have the worked solution for the question as follows:

An N point discrete Fourier transform, DFT, is given by: X(k)=n=0N1x(n)ejN2kn=n=0N1x(n)WNkn where: WN=ejN2 For a four point transform, develop this expression to show how the DFT of the sequence x(n) may be obtained using the fast Fourier transform (FFT) algorithm. Explain why the Butterfly operation is a useful building block in FFT calculations and sketch the signal flow graph using the butterfly notation for a four point FFT employing the decimation in time al gorithm. ). If a time domain signal is given by the sequence: x(n)={21,1,1,21}, calculate the discrete Fourier transform of this data using the FFT method you have developed in parts (a) and (b)