# AMAT 415 — Weeks 10-11 problems

54) Find a signal with $z$-transform $\displaystyle{\frac{2z-1}{z^2+4}}$.

55) Find the $z$-transform of $x=\{\sin(\pi n/3)\}$. Find the transfer function and impluse response of a filter $F$ such that $F(x)=\delta_0$.

56) Find the impulse response, transfer function, and frequency response of the following filters. Then plot their amplitude and phase responses. (Feel free to use Matlab for the plots.)

• $y_n=x_n + 2x_{n-1} + x_{n-2}$
• $y_n=x_{n+2} + x_{n-2}$

57) Find the amplitude response of the filter with transfer function $\displaystyle{B(z)=\frac{2}{2-z}}$. (Your answer should be evidently real.) Sketch the graph.

58) Find the transfer function of a filter $F$ such that $F$ kills the 400 Hz component of a signal sampled at 1000 Hz. Find such an $F$ whose frequency response is real and even.

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