The Fourier series of a signal x n and its coefficients X k are both periodic of the same value Nand as such can be written (a) To find the x n , 0 curren n curren N 1, given X k , 0 curren k curren N 1, write a set of N linear equations Indicate how you would find the x n from the matrix equation ...

a Equation ii can be written in matrix form as X Sx where X X 0 X 1 X N 1 T is an N vector containin ...

The Fourier series of a signal x[n] and its coefficients X k are both periodic of the

Question:

The Fourier series of a signal x[n] and its coefficients X_kare both periodic of the same value Nand as such can be written

N-1 0 <n< N – 1 (1) x[n] = Xe2a nk/N k=0 N-1 >x[n]ej2#nk/N 0<k<N- 1 (ii) X: n=0

(a) To find the x[n], 0 ‰¤ n ‰¤ N ˆ’ 1, given X_k, 0 ‰¤ k ‰¤ N ˆ’ 1, write a set of N linear equations. Indicate how you would find the x[n] from the matrix equation. There is duality in the Fourier series and its coefficients, so consider the reverse problem: how would you solve for the X_k given the x[n]?

(b) Let x[n] = n for n = 0, 1, 2, and 0 for n = 3, be a period of a periodic signal x[n] of fundamental period N = 4, use the above method to solve for the Fourier series coefficients X_k, 0 ‰¤ k ‰¤ 3. Use MATLAB to find the inverse of the complex exponential matrix.

(c) Suppose that when computing the X_k for the x[n] signal given above, you separate the sum into 2 sums, one for the even values of n, i.e., n = 0, 2 and the other for the odd values of n, i.e., n = 1, 3. Write an equivalent matrix expression for the X_k.

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