Skip to content Skip to navigation

Connexions

You are here: Home » Content » Region of Convergence for the Laplace Transform

Navigation

Content Actions

  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    (0 ratings)

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

In these lenses

  • richb's DSP display tagshide tags

    This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection:"Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Region of Convergence for the Laplace Transform

Module by: Richard Baraniuk

Summary: Explains how to find the region of convergence for continuous-time linear time-invariant systems.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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 to converge, while others cause the output to diverge ("blow up"). The set of signals that cause the system's output to converge lie in the region of convergence (ROC). This module will discuss how to find this region of convergence for any continuous-time, LTI system.

Recall the definition of the Laplace transform,

Laplace Transform

Hs=-ht-stdt H s t h t s t (1)
If we consider a causal, complex exponential, ht=-atut h t a t u t , we get the equation,
0-at-stdt=0-a+stdt t 0 a t s t t 0 a s t (2)
Evaluating this, we get
-1s+alimt-s+at1 -1 s a t s a t 1 (3)
Notice that this equation will tend to infinity when limt-s+at t s a t tends to infinity. To understand when this happens, we take one more step by using s=σ+ω s σ ω to realize this equation as
limt-ωt-σ+at t ω t σ a t (4)
Recognizing that -ωt ω t is sinusoidal, it becomes apparent that -σat σ a t is going to determine whether this blows up or not. What we find is that if σ+a σ a is positive, the exponential will be to a negative power, which will cause it to go to zero as tt tends to infinity. On the other hand, if σ+a 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 system will not converge. What all of this tells us is that for a causal signal, we have convergence when

Condition for Convergence

s>-a s a (5)

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

Necessary Condition for Anti-Causal Convergence

s<-a s a (6)

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 a single pole, the region of convergence lies to the right of it for causal signals and to the left for anti-causal signals.

Figure 1
(a) The Region of Convergence for a causal signal.(b) The Region of Convergence for an anti-causal signal.
Figure 1(a) (laplaceroc1.png)Figure 1(b) (laplaceroc2.png)

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

Example 1

Find Hs H s and state the region of convergence for ht=-atut+-btu-t 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

s,s>-a: H 1 s=1s+a s s a H 1 s 1 s a (7)
and
s,s<-b: H 2 s=-1s+b s s b H 2 s -1 s b (8)
Combining these, we get a region of convergence of -b>s>-a b s a . If a>b a b , we can represent this graphically. Otherwise, there will be no region of convergence.

Figure 2: The Region of Convergence of ht h t if a>b a b .
Figure 2 (laplaceroc3.png)

Comments, questions, feedback, criticisms?

Send feedback