Topic: Computing a z-transform

## Question

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

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

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

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}

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

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

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.

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

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

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

## Alumni Liaison

Prof. Math. Ohio State and Associate Dean
Outstanding Alumnus Purdue Math 2008

Jeff McNeal