A periodic signal x(t) has a period x 1 (t) = r(t) 2r(t 1) + r(t 2),

A periodic signal x(t) has a period

x₁(t) = r(t) ˆ’ 2r(t ˆ’ 1) + r(t ˆ’ 2),   T₀ = 2

(a) Find the Fourier series of z(t) = d²x(t)/dt² using the Laplace transform. Then use the derivative property to find the Fourier trans-form of x(t).

(b) To check your results figure out what the dc value of x(t) and z(t) should be, and whether the Fourier coefficients should be real, purely imaginary, or complex.

(c) Show that the following equation can be used to compute the Fourier transform of the periodic signal z(t):

where Z₁(Ω) is the Fourier transform of z₁(t) = d²x₁(t)/dt². Find its inverse z(t).

Transcribed Image Text:

2π Σ δ(Ω- kD0) Ζ(Ω)Z, (Ω) Το k=-00