We would like to find the Fourier series of a saw-tooth periodic signal x(t) of period T₀ =1. The period of x(t) is

x₁(t) = r(t) − r(t − 1) − u(t − 1)

(a) Sketch x(t) and compute the Fourier coefficients X_k using the integral definition.

(b) An easier way to do this is to use the Laplace transform of x₁(t). Find X_k this way.

(c) Obtain a trigonometric Fourier series x̂(t) consisting of the dc term and 40 harmonics to approximate x(t). Use MATLAB to find the values of x̂(t)for t = 0 to 10 in steps of 0.001. How does it compare with x(t)? Use MATLAB to plot the signal x̂ and its magnitude line spectra.