# AMAT 415 — Week 9 problems

50) Find the $z$-transform of the sequence $x=(\ldots,0,0,\overset{\downarrow}1,0,1,0,1,0,\ldots)$. (Give a closed-form expression.)

51) Let $x$ be as in the previous problem.

•  Find two signals $u$ and $v$ such that $u*v=x$.  (You might consider letting $u$ be a $\delta$-spike.)
• If $u=\delta_m$, can you find a $v$ such that $u*v=x$?
• Can you a solution with causal signals find $u$ and $v$ such that both $u$ and $v$ have infinitely many nonzero components and $u*v=x$?  (You might consider $z$-transforms for this last part.)

52) Find a simple expression for $\mathcal{Z}(x)$, where

$\displaystyle{x=(\ldots,\frac{1}{4!},\frac{1}{3!},\frac{1}{2!},\overset{\downarrow}1,1,1,1,\ldots)}$.

53) Show that the $z$-transform of the sequence

$\displaystyle{x=(\ldots,\frac{1}{8},\frac{1}{4},\frac{1}{2},\overset{\downarrow}1,1,1,1,\ldots)}$

is $\displaystyle{\frac{-z}{(1-2z)(1-z)}}$.

(Problems 52 and 53 are from Chapter 5.1 of Transform methods in applied mathematics, by Lancaster and Salkauskas.)

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