A period of a periodic signal x(t)with fundamental period T₀ = 2 is

x₁ (t) = cos(t) [u(t) − u(t − 2)].

(a) Plot the signal x(t) and find the Fourier series coefficients {X_k} for x(t) using the integral equation.

(b) Use the Laplace transform to find the Fourier series coefficients {X_k}.

(c) After considering the symmetry of the zero-mean signal x(t) − X₀, can you determine whether the X_k should be real, purely imaginary, or complex? Indicate how you make this determination.

(d) To simplify computing the Laplace transform of x₁(t) consider the following expression:

cos(t) u(t − 2) = [A cos(t − 2) + B sin(t − 2)] u(t − 2)

Find the values of Aand Band show how to compute L[cos (t) u(t − 2)].