AMAT 415 — Week 5 problems

P20) Find all the solutions of the following equations: (a) z^2=-9i, (b) z^3=i, (c) z^4=-2.

P21) Suppose y=(7.00000, y_1, 0.61803, y_3, -1.61803) is the discrete Fourier transform of a real signal x. Find y_1, y_3, and x.

P22) Suppose x=(x_0,x_1,x_2,x_3)\in \mathbf{C}^4 be a real signal with Fourier transform \mathcal{F}x=(2,-4,6,-4). Find \mathcal{F}y and \mathcal{F}z, where y=(x_2,x_3,x_0,x_1) and z=(x_2,x_1,x_0,x_3). (Hint: Relate y and z to x using shifts and time-reversals.)

P23) Without explicitly calculating any Fourier transforms, find \mathcal{F}(\mathcal{F}x), where x=(2,3,5,7,11,13,17,19).

P24) Let \delta=(1,0,\ldots,0) and let 0<k<\ell<N. Let y_n=\delta_{n-k} and let z_n=\delta_{n-\ell}. Compute the imaginary part of \hat{y}_n + \hat{z}_n. Bonus: Under what condition is this imaginary part equal to zero?

P25) Let N be even and suppose 0<m<N/2. Find \mathcal{F}\{\cos(2\pi mn/N)\}, \mathcal{F}\{\sin(2\pi mn/N)\}, and \mathcal{F} \{\cos(2\pi mn/N+\varphi)\}.

P26) Find the complex and real Fourier series f(t) where f(t) is the function of period 2\pi satisfying f(t)=\cos(t/2), -\pi<t\leq\pi.

P27) Suppose 0<\alpha<\pi. Find the complex and real Fourier series of the function f(t) where f(t) is the function of period 2\pi satisfying f(t)=1 if |t|\leq\alpha and f(t)=0 if \alpha<|t|\leq \pi.  (Start by drawing a picture of f(t).)

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