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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