Difference between revisions of "Convergence of Z Transform" - Rhea

Latest revision as of 14:53, 8 October 2009

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Hmm... This looks a lot like my course notes... Perhaps you want to write this somewhere, otherwise one might think that you are pretending that you wrote this yourself. --Mboutin 12:26, 23 September 2009 (UTC)

Convergence of Z Transform, ECE 438 Notes Sept. 2, 2009

Definition: A series $ \sum_{\infty}^{n=0} a_n $ is said to converge to a value V if for every $ \epsilon > 0 $, there exists a positive integer M such that $ |\sum_{n=0}^{N-1} a_n - V | < \epsilon, $ for all N > M

For the Z transform, it is customary to talk about the "region of absolute convergence."

Definition: A series $ \sum^{\infty}_{n=0} a_n $ is called "absolutely convergent" when $ \sum_{n=0}^{\infty} |a_n | $ converges.

Fact: If $ \sum|a_n| $ converges, then $ \sum a_n $ converges also, i.e. the region of absolute convergence is included in the region of convergence.

In the literature and here: ROC means "region of absolute convergence"

Reference for Z transform: Chapter 10 of the ECE 301 book

Facts about ROC

X(z) converges absolutely

 $ \leftrightarrow \sum_n |x[n]z^{-n}| $ converges

 $ \leftrightarrow \sum_n |x[n]||z^{-n}| $ converges

Fact 1: ROC is made of rings around the origin. If $ z_0 $ is in the ROC, then any other z with $ |z| = |z_0| $ is also in the ROC

Fact 2: If x[n] is "causal" (i.e. x[n] = 0 for all n < 0) and $ z_0 $ is in the ROC then any z with $ |z|>|z_0| $ is also in the ROC

Proof for z with $ |z|> |z_0| $: $ \sum_{n=- \infty}^{\infty}|x[n]z^{-n}| = \sum_{n=0}^{\infty}|x[n]z^{-n}| = \sum_{n=0}^{\infty}|x[n]|{|z|}^{-n} $

$ \le \sum_{n=0}^{\infty}|x[n]|{|z_0|}^{-n} = \sum_{n=0}^{\infty}|x[n]z_0^{-n}| $ which converges by assumption

$ \rightarrow $ X(z) converges absolutely

Fact 3: If x[n] is anti-causal (i.e. x[n] = 0 for n > 0) and $ z_0 $ is included in ROC, then any z with $ |z|< z_0 $ is also in ROC. Has a similar proof.

Fact 4: If x[n] is "mixed causal" (two-sided signal) and $ z_0 $ is included in ROC then there exist $ r_1, r_2 $ included in all real numbers $ \ge 0, r_1 <|z|< r_2 $ such that X(z) converges for all z with $ r_1 <|z|< r_2 $.

The complex plan contains an important point z = $ \infty $, which cannot be compressed into the sphere.

Definition: We say that X(z) converges at z = $ \infty $ if $ X(\frac{1}{z}) $ converges at 0

Example 1:

$ x_1[n] = \delta [n-1] $

$ X_1(z) = \sum^{\infty}_{n=- \infty} \delta [n-1]z^{-n} = \frac{1}{z} $, which is well-defined everywhere except z = 0

Now $ X_1(\frac{1}{z}) = z $ is well-defined at z = 0 $ \rightarrow z= \infty $ is in ROC

ROC: all of the complex plane except z = 0

Example 2:

$ x_2[n] = \delta [n+1] $

$ X_2(z) = \sum^{\infty}_{n=- \infty} \delta [n+1]z^{-n} =z $, which is well-defined everywhere on the finite complex plane

Now $ X_2(\frac{1}{z}) = \frac{1}{z} $ is infinite at z = 0 $ \rightarrow z= \infty $ is not in ROC

ROC: finite complex plane