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# Properties of ROC

The existence of Laplace transform of a given depends on whether the transform integral converges

which in turn depends on the duration and magnitude of as well as the real part of (the imaginary part of determines the frequency of a sinusoid which is bounded and has no effect on the convergence of the integral).

Right sided signals: may have infinite duration for , and a positive tends to attenuate as .

Left sided signals: may have infinite duration for , and a negative tends to attenuate as .

Based on these observations, we can get the following properties for the ROC:

• If is absolutely integrable and of finite duration, then the ROC is the entire s-plane (the Laplace transform integral is finite, i.e., exists, for any ).

• The ROC of consists of strips parallel to the -axis in the s-plane.

• If is right sided and is in the ROC, then any to the right of (i.e., ) is also in the ROC, i.e., ROC is a right sided half plane.

• If is left sided and is in the ROC, then any to the left of (i.e., ) is also in the ROC, i.e., ROC is a left sided half plane.

• If is two-sided, then the ROC is the intersection of the two one-sided ROCs corresponding to the two one-sided components of . This intersection can be either a vertical strip or an empty set.

• If is rational, then its ROC does not contain any poles (by definition dose not exist). The ROC is bounded by the poles or extends to infinity.

• If is a rational Laplace transform of a right sided function , then the ROC is the half plane to the right of the rightmost pole; if is a rational Laplace transform of a left sided function , then the ROC is the half plane to the left of the leftmost pole.

• A signal is absolutely integrable, i.e., its Fourier transform exists (first Dirichlet condition, assuming the other two are satisfied), if and only if the ROC of the corresponding Laplace transform contains the imaginary axis or .

Example 1: Consider the Laplace transform of a two-sided signal :

The Laplace transform of the two components can be obtained from the two examples discussed above. From example 1, we get

and let in example 2, we have

Combining the two components, we have

Whether exists or not depends on . If , i.e., decays exponentially as , then the ROC is the strip between and and exists. But if , i.e., grows exponentially as , then the ROC is an empty set and does not exist.

Example 2: Given the following Laplace transform, find the corresponding signal:

There are three possible ROCs determined by the two poles and :
• The half plane to the right of the rightmost pole , with the corresponding right sided time function

• The half plane to the left of the leftmost pole , with the corresponding left sided time function

• The vertical strip between the two poles , with the corresponding two sided time function

In particular, note that only the first ROC includes the -axis and the corresponding time function has a Fourier transform. Fourier transform does not exist in the other two cases.

Next: Properties of Laplace Transform Up: Laplace_Transform Previous: Zeros and Poles of
Ruye Wang 2012-01-28