Question

5. Compute the Fourier transform of the following signals: (a) x(t) = \delta(t - 4) (b) x(t) = e^{-2t}u(t), where u(t) is the unit step function. (c) x(t) = 3 + \sin(2\pi t) - 4\cos(4\pi t) Utilize the results that the Fourier transform of \cos(2\pi f_0 t) is \frac{1}{2}[\delta(f - f_0) + \delta(f + f_0)], and the Fourier transform of \sin(2\pi f_0 t) is \frac{1}{2j}[\delta(f - f_0) - \delta(f + f_0)].

          5. Compute the Fourier transform of the following signals:
(a) x(t) = \delta(t - 4)
(b) x(t) = e^{-2t}u(t), where u(t) is the unit step function.
(c) x(t) = 3 + \sin(2\pi t) - 4\cos(4\pi t)
Utilize the results that the Fourier transform of \cos(2\pi f_0 t) is \frac{1}{2}[\delta(f - f_0) + \delta(f + f_0)],
and the Fourier transform of \sin(2\pi f_0 t) is \frac{1}{2j}[\delta(f - f_0) - \delta(f + f_0)].

5. Compute the Fourier transform of the following signals:
(a) x(t) = δ(t - 4)
(b) x(t) = e^-2tu(t), where u(t) is the unit step function.
(c) x(t) = 3 + sin(2πt) - 4cos(4πt)
Utilize the results that the Fourier transform of cos(2πf0 t) is (1)/(2)[δ(f - f0) + δ(f + f0)],
and the Fourier transform of sin(2πf0 t) is (1)/(2j)[δ(f - f0) - δ(f + f0)].

Added by Frank C.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Ajay Singhal

Step 1

The Fourier transform of a time-shifted signal a(t - t0) is given by e^(-2πifft0)A(f), where A(f) is the Fourier transform of a(t). In this case, t0 = 4, so the Fourier transform of a(t) = (t-4) is e^(-2πif*4)A(f). (b) To compute the Fourier transform of b(t) = Show more…

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5. Compute the Fourier transform of the following signals: (a) a(t) = (t-4) (b) b(t) = e^(-2t)(t), where u(t) is the unit step function c(t) = 3 + sin(2Ï€rt - 4cos(4Ï€rt) Utilize the results that the Fourier transform of cos(2Ï€fot) is [(f - fo) + (f + fo)] and the Fourier transform of sin(2Ï€fot) is 2i[(f - fo) - (f + fo)].