Practice Question on "Digital Signal Processing"
Topic: Computing an inverse z-transform
Question
Compute the inverse z-transform of the following signal.
$ X(z)=\frac{1}{1+3z} \mbox{, } \Big|z\Big|<\frac{1}{3} $
Prof. Mimi gave me this problem in class on Friday, so I'm posting it and my answer here. --Cmcmican 22:22, 16 April 2011 (UTC)
Answer 1
$ X(z)=\sum_{k=0}^\infty (-3z)^k=\sum_{k=-\infty}^\infty u[k](-3)^kz^k $
let n=-k
$ =\sum_{n=-\infty}^\infty u[-n](-3)^{-n} z^{-n} $
By comparison with $ \sum_{n=-\infty}^\infty x[n] z^{-n}: $
$ x[n]=(-3)^{-n}u[-n]\, $
--Cmcmican 22:22, 16 April 2011 (UTC)
- TA's comment: Good Job!
- Instructor's comment: You may want to mention where you use the fact that |z|<1/3.
Answer 2
I agree, but for the missing steps on |z|<1/3, you can say
Since |z| < 1/3, |3z| < 1
Therefore, |-3z| < 1
By comparison with the geometric series, where it diverges for |-3z| < 1, you can rewrite the problem as shown in Answer 1.
--Kellsper 16:12, 21 April 2011 (UTC)
- Instructor's comment: Good. You may shorten this explanation a bit when you write it on the exam. Just say
- $ X(z)=\frac{1}{1+3z}=\frac{1}{1-(-3z)}=\sum_{k=0}^\infty (-3z)^k $, since $ |-3z|=|3z|<1 $ when $ |z|<\frac{1}{3} $.
- -pm .
Answer 3
Write it here.
