Transcription of 20 The Laplace Transform
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20 The Laplace Transform
20 The Laplace Transform Recommended Problems Consider the signal x(t) = 3e 2'u(t) + 4e3'u(t). (a) Does the Fourier Transform of this signal converge? (b) For which of the following values of a does the Fourier Transform of x(t)e -" converge? (i) a = 1 (ii) a = (iii) a = (c) Determine the Laplace Transform X(s) of x(t). Sketch the location of the poles and zeros of X(s) and the ROC. Determine the Laplace Transform , pole and zero locations, and associated ROC for each of the following time functions. (a) e -"u(t), a > 0 (b) e ~atu(t), a < 0 (c) -e -a tu(-t), a < 0 Shown in Figures to are four pole-zero plots. For each statement in Table about the associated time function x(t), fill in the table with the cor-responding constraint on the ROC.
The Laplace Transform / Problems P20-3 P20.6 (a) From the expression for the Laplace transform of x(t), derive the fact that the Laplace transform of x(t) is the Fourier x(t) weighted by an exponential. (b) Derive the expression for the inverse Laplace transform using the Fourier transform …
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