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.