In the following problems we want to find the Fourier transform of the signals.

(a) For the signal

find its Fourier transform by using the Fourier transform of

x(t) = 0.5e^€“at u(t) ˆ’ 0.5e^at u(ˆ’t), a > 0

as a → 0.

Use then Y(Ω) to determine the Fourier transform of the unit-step signal which can be written

u(t) = 0.5 + 0.5 sign(t) = 0.5 + y(t)

(b) Consider the constant A being the superposition of Au(t) and Au(ˆ’t), both of which have Laplace transforms. Find these Laplace transforms, including their regions of convergence, and then try to use these results to find the Fourier transform of A. Is it possible?

Transcribed Image Text:

0.5 120 y(t) = 0.5 sign(t) = -0.5 t<0