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. 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. 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.

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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