Revision as of 02:01, 13 September 2013

Practice Problem on Z-transform computation

Compute the compute the z-transform (including the ROC) of the following DT signal:

$ x[n]= n^2 \left( u[n+3]- u[n-1] \right) $

(Write enough intermediate steps to fully justify your answer.)

Share your answers below

You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!

No need to write your name: we can find out who wrote what by checking the history of the page.

Answer 1

x[n] = n²(u[n + 2] − u[n − 1]).

$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $

$ = \sum_{n=-3}^{0} n^2 z^{-n} $

= 9z³ + 4z² + z

= z³(9 + 4z ^{− 1} + z ^{− 2})

= X(z) = (9 + 4z ^{− 1} + z ^{− 2}) / (z ^{− 3}), for all z in complex plane.

Answer 2

Muhammad Syafeeq Safaruddin

x[n] = n²(u[n + 3] − u[n − 1])

x[n] = n²(δ(n + 3) + δ(n + 2) + δ(n + 1) + δ(n))

$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $

$ X(z) = \sum_{n=-\infty}^{+\infty} n^2(\delta(n+3)+\delta(n+2)+\delta(n+1)+\delta(n)) z^{-n} $

X(z) = 9z³ + 4z² + z + 1 for all z in complex plane

Answer 3

Write it here.

Answer 4

Write it here.

Answer 5

Tony Mlinarich

$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $

X(z) = n²(δ(n + 3) + δ(n + 2) + δ(n + 1) + δ(n) + δ(n − 1))z ^{− n}

X(z) = 9z³ + 4z² + z + 1/z<\span>

Back to ECE438 Fall 2013 Prof. Boutin

Answer 7

Yixiang Liu

$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $

$ X(z) = \sum_{n=-\infty}^{+\infty} n^{2}[{u[n+3]-u[n-1]}]z^{-n} $

This expression equals to zero except n = -3, -2, -1

so X(z) = x[ − 3]z³ + x[ − 2]z² + x[ − 1]z¹

      = 9z^{3} + 4z^{2} + z

Answer 8

Xi Wang

$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $

= X(z) = (9z ^{+ 3} + 4z ^{+ 2} + z. The range of the value of z is from negative infinity to positive infinity

Answer 9

$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $

$ X(z) = \sum_{n=-3}^{+1} x[n] z^{-n} $

= X(z) = 9z ^{+ 3} + 4z ⁺² + z + 1 for all z in complex plane

Answer 10

Cary Wood

$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $

$ X(z) = \sum_{n=-3}^{0} x[n] z^{-n} $

= X(z) = 9z ^{+ 3} + 4z ^{+ 2} + z, for all z in complex plane

Answer 11

Shiyu Wang

x[n] = n²(u[n + 3] − u[n − 1])

x[n] = n^{2   (-3=< n < 1)}

$ X(z) = \sum_{n=-3}^{0} n^2 z^{-n} $ 

x(z)=9z³+4z²+z, for all z in complex plane except z=infinity

Answer 12

Matt Miller

x[n] = n²(u[n+3]-u[n-1])

x[n] = n²u[n+3] - n²u[n-1]

x[n] = n²|⁰_-3

$ X(z) = \sum_{n=-3}^{0} n^2 z^{-n} $ 

X(z) = (-3)²z³ + (-2)²z² + (-1)²z¹ (0)²z⁰

X(z) = 9z³ + 4z² + z

lim z->inf X(1/2) = 0, lim z->0 X(1/2) = inf --> valid for all Z in complex plane.