Laplace Transform: Time Scaling Property

In summary, the time scaling property of Laplace transform is a useful tool in solving differential equations. It states that if a function is scaled in time, its Laplace transform is divided by the same scaling factor. This property can be applied to any function with a Laplace transform, but the function must be well-behaved for the transform to exist. The time scaling property does not change the locations of poles and zeros, but it affects their values. Furthermore, it can also be used to find the inverse Laplace transform by dividing the argument of the transform by the scaling factor, although additional transformation techniques may be needed.
  • #1
asmani
105
0
Hi all

According to the textbook Signal and Systems by Oppenheim (2nd edition) pages 685 and 686, if the Laplace transform of x(t) is X(s) with ROC (region of convergence) R, then the Laplace transform of x(at) is (1/|a|)X(s/a) with ROC R/a.

Consequently, for a>1, there is a compression in the size of the ROC of X(s) by a factor 1/a.

But I think the ROC must be aR and not R/a, as the example x(t)=e-|t| shows. Which is correct?

Thanks
 
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  • #2
aR is correct. . .
 

1. What is the time scaling property of Laplace transform?

The time scaling property of Laplace transform states that if a function f(t) has a Laplace transform F(s), then the Laplace transform of the time-scaled function f(at) is equal to F(s/a). In other words, multiplying the argument of a function by a constant a leads to dividing the argument of its Laplace transform by the same constant.

2. How is the time scaling property useful in solving differential equations?

The time scaling property is useful in solving differential equations because it allows us to easily find the Laplace transform of a function that has been scaled in time. This can simplify the process of solving differential equations, as it reduces the number of transformations needed to find the solution.

3. Can the time scaling property be applied to any function?

Yes, the time scaling property can be applied to any function that has a Laplace transform. However, it is important to note that the function must be sufficiently well-behaved for the Laplace transform to exist.

4. How does the time scaling property affect the poles and zeros of a function?

The time scaling property does not change the locations of the poles and zeros of a function. It only affects the scale of the function on the s-plane, which is the domain of the Laplace transform. The poles and zeros of the function remain in the same locations, but their values may change due to the scaling factor.

5. Can the time scaling property be used to find the inverse Laplace transform?

Yes, the time scaling property can be used to find the inverse Laplace transform. By applying the property in reverse, we can find the inverse Laplace transform of a time-scaled function by dividing the argument of its Laplace transform by the scaling factor. However, it is important to note that other transformation techniques may be required to fully invert the Laplace transform.

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