Rectifying a sinusoid provides a way to create a dc source. In this problem we consider the Fourier series of the full and the half-wave rectified signals. The full-wave rectified signal x_f(t) has a fundamental period T₀= 1 and its period from 0 to 1 is

x₁(t) = sin(Ï€ t)                    0 ‰¤ t ‰¤ 1

while the period for the half-wave rectifier signal x_h(t) is

with fundamental period T₁= 2.

(a) Obtain the Fourier coefficients for both of these periodic signals.

(b) Use MATLAB and your analytic results obtained above to plot the magnitude line spectrum of the half-wave signal and use the dc and 40 harmonics to obtain an approximation of the half-wave signal. Plot them.

sin(ët) 0 < t < 1 x2(t) 1