Problem 2. Consider the function f in L?(0, 27) defined by f(t) = sin(30t) e2 cos(5t) . Sample the function f at points ti = v 271, where v = 2 9, j = 0,...,v -1, and apply the fast Fourier transform to solve the following problems using either Python or Matlab. (i) Find the inverse fast Fourier transform {ck} for {f(t;)}0 . Plot the two-sided power spectrum for f for k from -100 to 100. By plotting the (single-sided) powers (ck|9 for some q, make an educated guess to find a set of frequencies Koo such that f(t) = Cker Cke akt is the Fourier series for f (t) (since v is very large, ak ~ Ck). (ii) Find S C {ck}kez that has twelve elements, and minimizes |If-ps), where ps(t) = Ekes Cke ikt. Also find KC {ck}kez but with twenty-eight elements that minimizes |If-Pull, where px(t) = EkEX Cke-ikt. (iii) Plot ps(t), px(t) and f(t) on the same graph over the interval [0, 27]. (iv) Compute the norm |If| | and the errors IIf - psi| and IIf - pall