The following problems relate to the modulation property of the Fourier transform:

(a) Consider the signal

x(t) = p(t) + p(t) cos(Ï€ t) where p(t) = u(t + 1) ˆ’ u(t ˆ’ 1)

i. Use the modulation property to find the Fourier transform X(Ω)in terms of P(Ω), the Fourier transform of p(t).

ii. Let g(t) = x(t ˆ’ 1). Use the Laplace transform of g(t) to obtain X(Ω). Verify the expression obtained for X(Ω)coincides with the previous one.

(b) Consider the signal z(t) = cos(t) [u(t) ˆ’ u(t ˆ’ Ï€/2)].

i. If Z(s) is the Laplace transform of z(t) under what conditions can you obtain the Fourier transform by letting s = jΩ­?

ii. Is it true that the Fourier transform of z(t) is

Transcribed Image Text:

o-jAN/2 jN +e Z(N) =