Orthogonal Expansion Any matrix that possesses a complete set of eigenvectors can be expanded in...

Orthogonal Expansion Any matrix that possesses a complete set of eigenvectors can be expanded in terms of those eigenvectors. The expansion takes the following form when the eigenvectors are orthonormal:

In the case of the DFf, there are only four distinct eigen-spaces, so the sum can be grouped according lo the different eigenvalues,

where N1 is the set of indices for eigenvectors belonging to and so on. Each term in parentheses is an inner product, requiring N multiplications.

a. Write a M ATLAB function that will compute the DFf via this expansion (2-2), specifically for the N = 16 case. Verify that the correct DFf will be obtained when the real eigenvectors (determined previously) are used, and compare to the output of fft.

b. Possible co111putatio11: Count the total number of operations (real multiplications and additions) needed to compute the DFf via the orthogonal expansion (2-2). Since the eigenvectors can be chosen to be purely real, the computation of the DFf via the orthogonal expansion will simplify when the input vector is purely real. The real part of the transform will depend only on the first two sums, and the imaginary part on the second two.