Topic: Computing a z-transform

( Practice Question 2, ECE438 Fall 2010, Prof. Boutin )

## Question

Compute the z-transform of the discrete-time signal

$x[n]= 4^n \left(u[n+3]-u[n-4] \right)$.

Note: there are two tricky parts in this problem. Do you know what they are?

## Solution 1

x[n] = 4nu[n + 3] − 4nu[n − 4]

$x[n] = \sum_{n=-\infty}^{\infty} 4^n u[n+3] z^{-n} - \sum_{n=-\infty}^{\infty} 4^n u[n-4] z^{-n}$

${\color{red}\not}x {\color{red}\not}[n] {\color{red}X(z)} = \sum_{n=3}^{\infty} 4^n z^{-n} - \sum_{n=-\infty}^{4} 4^n z^{-n}$

${\color{red}\not}x {\color{red}\not}[n] {\color{red}X(z)}= \sum_{n=0}^{\infty} (\frac{4}{z})^n - 1 - 4^1z^{-1} - 4^2z^{-2} - 4^3z^{-3} - \sum_{n=4}^{\infty} (\frac{4}{z})^n$

this is the mistake I made on my exam - could you please clarify my work, professor?

• Certainly! This is a very common mistake: splitting a sum that converges for most z's into two sums that diverge for most z's. The key is to notice that the first sum above has a finite number of terms: so convergence of the entire sum is guaranteed, unless one (or more) of the terms of the sum diverge (for example, 1/z diverges when z=0). Observe that, by splitting the sum this way, you get an empty ROC. The correct ROC for this z-transform is actually all the finite complex plane except zero. -pm
• Another thing I see is the manipulation of the sum with negative indices, namely :
${\color{green}\sum_{n=-\infty}^{4} 4^n z^{-n } = \sum_{n=4}^{\infty}(\frac{4}{z})^n }$
which is incorrect. The correct way to manipulate it is the following:
\begin{align} \sum_{n=-\infty}^{4} 4^n z^{-n } &= \sum_{k=\infty}^{-4} 4^{-k} z^{k } \text{ (letting }k=-n), \\ &= \sum_{k=-4}^{\infty} 4^{-k} z^{k } \text{ (since the order of the terms in the sum does not matter)}, \\ &= 4^{4}z^{-4}+4^{3}z^{-3}+4^{2}z^{-2}+4^{1}z^{-1}+ \sum_{k=0}^{\infty} 4^{-k} z^{k } \end{align}
Hope that helps! -pm

$X(z) =\sum_{n=0}^{\infty} (\frac{4}{z})^n - 1 - 4z - 4^2z^{2} - 4^3z^{3} - (4^{4}z^{-4}+4^{3}z^{-3}+4^{2}z^{-2}+4^{1}z^{-1}+ \sum_{k=0}^{\infty} 4^{-k} z^{k })$

Or better yet:
$X(z) =\sum_{n=-3}^{3} (\frac{4}{z})^n$ -pm

Note: although the signal given looks very similar to

${\color{blue} x_1[n]= 4^n u[n+3]- 2^n u[n-4] }$.

and to

${\color{blue} x_2[n]= 4^n u[n+3]- 2^n u[-n-4] }$.

the computation of the z-transform is very different. -pm