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# Properties of Laplace Transform

The Laplace transform has a set of properties in parallel with that of the Fourier transform. The difference is that we need to pay special attention to the ROCs. In the following, we always assume

• Linearity

( means set contains or equals to set , i.e,. is a subset of , or is a superset of .)

It is obvious that the ROC of the linear combination of and should be the intersection of the their individual ROCs in which both and exist. But also note that in some cases when zero-pole cancellation occurs, the ROC of the linear combination could be larger than , as shown in the example below.

Example: Let

then

We see that the ROC of the combination is larger than the intersection of the ROCs of the two individual terms.

• Time Shifting

• Shifting in s-Domain

Note that the ROC is shifted by , i.e., it is shifted vertically by (with no effect to ROC) and horizontally by .

• Time Scaling

Note that the ROC is horizontally scaled by , which could be either positive () or negative () in which case both the signal and the ROC of its Laplace transform are horizontally flipped.

• Conjugation

Proof:

• Convolution

Note that the ROC of the convolution could be larger than the intersection of and , due to the possible pole-zero cancellation caused by the convolution, similar to the linearity property.

Example Assume

then

• Differentiation in Time Domain

This can be proven by differentiating the inverse Laplace transform:

In general, we have

Again, multiplying by may cause pole-zero cancellation and therefore the resulting ROC may be larger than .

Example: Given

we have:

• Differentiation in s-Domain

This can be proven by differentiating the Laplace transform:

Repeat this process we get

• Integration in Time Domain

This can be proven by realizing that

and therefore by convolution property we have

Also note that as the ROC of is the right half plane , the ROC of is the intersection of the two individual ROCs , except if pole-zero cancellation occurs (when with ) in which case the ROC is the entire s-pane.

Next: Laplace Transform of Typical Up: Laplace_Transform Previous: Properties of ROC
Ruye Wang 2012-01-28