Revision as of 09:04, 11 November 2011

Practice Problem on Z-transform computation

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

$ x[n]= n u[n]-n u[n-3] $

(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!

Instructor's comments: Note that it is not a good idea to write the z-transform as two infinite geometric sums... -pm

Answer 1

Begin with the definition of a Z-Transform.

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

Simplify a little. (pull out the n and realize $ u[n]-u[n-3] $ is only non-zero for 0, 1, and 2.)

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

Then we have a simple case of evaluating for 3 points.

$ \begin{align} X(z) &= 0 z^{-0} + 1 z^{-1} + 2 z^{-2} \\ &= \frac{z+2}{z^2} \end{align} $

TA's comments: What about the ROC?

Instructor's comments: Don't forget to check wether z=infinity is part of the ROC. -pm

Answer 2

$ Z(x[n])= \sum_{n=-\infty}^{\infty}x[n]z^{-n}= \sum_{n=-\infty}^{\infty}n(u[n]- u[n-3])z^{-n} $

when n=0,1,2, x[n] is n; otherwise x[n]=0. So:

$ x(z)=0z^{-0}+1z^{-1}+2z^{-2}=\frac{1}{z}+\frac{2}{z^2} $ with ROC=all finite complex number except 0.

test for infinity:

$ X(\frac{1}{z})=z+z^2 $

when z=0,$ X(\frac{1}{z}) $converges

X(z) converges at $ z=\infty $

so ROC of X(z) is all complex number except 0.

Answer 3

First the axiom need to be prove:

$ Z(\delta [n- n_0]) = \sum_{n=-\infty}^{\infty}\delta[n-n_0]z^{-n} = \sum_{n=-\infty}^{\infty}\delta[n-n_0]z^{-n_0} = z^{-n_0}, ROC = C/[0] $

Observe the original function

$ x\left[ n \right]= n u[n]-n u[n-3] = n(u[n] - u[n-3]) = n(\delta[n] + \delta[n-1] + \delta[n-2]) = 0\delta[n] + 1\delta[n-1] + 2\delta[n-2] $

so by two axioms proved above, with the linearity property,

$ X(z) = Z\left(x[n]\right) =Z\left(\delta[n-1]+2\delta[n-2]\right) = Z\left(\delta[n-1]\right)+Z\left(2\delta[n-2]\right) = z^{-1}+2z^{-2}, ROC = C/[0] $

Instructor's comments: When you write "C" do you mean the finite z-plane only? Note that you need to check convergence at the point z=infinity separately. -pm

Answer 4

$ X[n] = nu[n] - nu[n-3] $

$ X(z) = \sum_{n=0}^{2}n z^{-n} $ = 0 + z^{-1} + 2*Z^{-2}

 ROC z not equal to 1

Instructor's comments: How about z=infinity? Is that point in the ROC? -pm

Back to ECE438 Fall 2011 Prof. Boutin