Consider the signal [Fig. 1-6(a)]x(1) = e-a'u(t) a > 0where u(t) is the unit step function defined byS1 t>0и()10 t<0Find the Fourier transform of x (t)

Answered: Consider the signal [Fig. 1-6(a)] x(t)…

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

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Consider the signal [Fig. 1-6(a)]
x(1) = e-a'u(t) a > 0
where u(t) is the unit step function defined by
S1 t>0
и()
10 t<0
Find the Fourier transform of x (t)

Expert Solution

Step 1

For any continuous-time function x(t), the Fourier transform of x(t) can be expressed as

$X (ω) = \int_{- \infty}^{+ \infty} x (t) e^{- j ω t} d t - - - > (1)$

Plugging the expression of the given function x(t) into equation (1), the following can be obtained:

$X (ω) = \int_{0}^{+ \infty} e^{- a t} e^{- j ω t} d t = > X (ω) = \int_{0}^{+ \infty} e^{- (a + j ω) t} d t = > X (ω) = - \frac{1}{(a + j ω)} \times [e^{- (a + j ω) \infty} - e^{- (a + j ω) 0}] = > X (ω) = - \frac{1}{(a + j ω)} \times [0 - 1] = > X (ω) = \frac{1}{(a + j ω)}$

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