# The Illustrated Laplace Transform

In the pages that follow, a series of *Laplace transform pairs* are presented, each showing a function of time *h*(*t*) on the left and the corresponding pole-zero diagram for *H*(*s*) on the right, where:

Each of the *h*(*t*) may be considered as the impulse response of an LTI system. In each pole-zero diagram, poles are shown with an **X** while zeros are plotted with an **O**. The region of convergence shown shaded in each pole-zero diagram is generally the entire *z*-plane to the right of the rightmost pole.

The inverse Laplace transform *h*(*t*) is only generally found from the pole-zero diagram of *H*(*s*) as constant values or gain factors are not visible when poles and zeros are plotted. Similarly, the sum of two pole-zero diagrams is *not* generally the sum of the corresponding Laplace transforms. In particular, the presence of zeros will be affected, as seen when sine and cosine are added together in Figure B.8 and in the summation of Figure B.12.

The importance of the pole-zero diagram to system stability can also be seen, where poles on the right-hand plane are associated with divergent impulse response functions. If the region of convergence includes the line *σ* = 0, then the related Fourier transform can be found along this line. Otherwise, the signal *h*(*t*) has a Laplace ...

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