<< Chapter < Page Chapter >> Page >
Explains how to find the region of convergence for continuous-time linear time-invariant systems.

With the Laplace transform , the s-plane represents a set of signals ( complex exponentials ). For any given LTI system, some of these signals may cause the output of the system toconverge, while others cause the output to diverge ("blow up"). The set of signals that cause the system's output to convergelie in the region of convergence (ROC) . This module will discuss how to find this region of convergence forany continuous-time, LTI system.

Recall the definition of the Laplace transform,

Laplace transform

H s t h t s t
If we consider a causal , complex exponential, h t a t u t , we get the equation,
t 0 a t s t t 0 a s t
Evaluating this, we get
-1 s a t s a t 1
Notice that this equation will tend to infinity when t s a t tends to infinity. To understand when this happens, we take one more step by using s σ j ω to realize this equation as
t j ω t σ a t
Recognizing that j ω t is sinusoidal, it becomes apparent that σ a t is going to determine whether this blows up or not. What we find is that if σ a is positive, the exponential will be to a negative power, which will cause it to go to zero as t tends to infinity. On the other hand, if σ a is negative or zero, the exponential will not be to a negative power, which will prevent it from tending to zero and the systemwill not converge. What all of this tells us is that for a causal signal, we have convergence when

Condition for convergence

s a

Although we will not go through the process again for anticausal signals, we could. In doing so, we would find that thenecessary condition for convergence is when

Necessary condition for anti-causal convergence

s a

Graphical understanding of roc

Perhaps the best way to look at the region of convergence is to view it in the s-plane. What we observe is that for asingle pole, the region of convergence lies to the right of it for causal signals and to the left for anti-causal signals.

The Region of Convergence for a causal signal.
The Region of Convergence for an anti-causal signal.

Once we have recognized this, the natural question becomes: What do we do when we have multiple poles? The simple answeris that we take the intersection of all of the regions of convergence of the respective poles.

Find H s and state the region of convergence for h t a t u t b t u t

Breaking this up into its two terms, we get transfer functions and respective regions of convergence of

H 1 s 1 s a , s a
H 2 s -1 s b , s b
b s a . If a b , we can represent this graphically. Otherwise, there will be no region of convergence.

The Region of Convergence of h t if a b .
Got questions? Get instant answers now!

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask