Practice Question on "Digital Signal Processing"
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] $
(Write enough intermediate steps to fully justify your answer.)
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
