Lecture 21: Definition of the Two-Sided Laplace Transform and Region of Convergence
7The Laplace Transform
So far, we have studied the Fourier series and the Fourier transform for the analysis of periodic and aperiodic signals, and LTI systems. These tools are useful because they allow us to analyze continuous-time signals and systems in the frequency domain. In particular, signals can be represented as linear combinations of periodic complex exponentials, which are eigenfunctions of LTI systems. But if we replace by the more general complex variable in the Fourier transform equations, we obtain the Laplace transform, a generalization of the Fourier transform.
The Fourier transform was defined only for stable systems or signals that taper off at infinity (signals of finite energy or absolutely integrable.) On the other hand, the Laplace transform of an unbounded signal or of an unstable impulse response is defined. The Laplace transform can also be used to analyze differential LTI systems with nonzero initial conditions.
7.1Definition of the Two-Sided Laplace Transform
The Laplace transform of is defined as follows:
(6.1)
where is a complex variable. Notice that the Fourier transform is given by the same equation, only for the Fourier transform.
Let the Laplace variable be written as . Then the Laplace transform can be interpreted as the Fourier transform of the signal :
(6.2)
Given , this integral may or may not converge, depending on the value of (the real part of ).
Example: Find the Laplace transform of .
(6.3)
This Laplace transform converges only for values of in the open half-plane to the right of . This half plane is the region of convergence (ROC) of the Laplace transform. It is represented as follows:
Recall that the Fourier transform of converges only for (decaying exponential), whereas its Laplace transform converges for any (even for growing exponentials), as long as . In other words, the Fourier transform of converges for .
Example: Find the Laplace transform of .
(6.4)
This Laplace transform converges only in the ROC which is the open half-plane to the left of .
Important note:The ROC is an integral part of a Laplace transform. It must be specified.
Without it, you can't tell what the corresponding time-domain signal is.
7.1.1Inverse Laplace Transform
The inverse Laplace transform is in general given by
(6.5)
This contour integration equation is rarely used because we are mostly dealing with linear systems and standard signals whose Laplace transforms are found in tables of Laplace transform pairs. Thus, in this course we will use the partial fraction expansion technique to find the continuous-time signal corresponding to a Laplace transform.
7.2Convergence of the Two-Sided Laplace Transform
As mentioned above, convergence of the integral in Equation (6.1) depends on the value of the real part of the complex Laplace variable. Thus the region of convergence in the complex plane or "s"-plane is either a vertical half-plane, a vertical strip, or nothing. We have seen two examples above that led to open half-plane ROCs. Here is a signal for which the Laplace transform only converges in a vertical strip.
Example: Consider the signal . Its Laplace transform is given by
(6.6)
The ROC is a strip between the real parts -2 and 1.
7.3Poles and Zeros of Rational Laplace Transforms
Complex or real exponential signals have rational Laplace transforms. That is, they are ratios of a numerator polynomial and a denominator polynomial.
(6.7)
The zeros of the numerator are called the zeros of the Laplace transform. If is a zero, then . The zeros of the denominator are called the poles of the Laplace transform. If is a pole, then .
Example: has poles at .
For differential LTI systems, the zeros of the characteristic polynomial are equal to the poles of the Laplace transform of the impulse response.
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