The connection between the Fourier series and the Fourier transform can be seen by considering what happens when the fundamental period of a periodic signal increases to a point at which the periodicity is not clear as only one period is seen. Consider a train of pulses x(t) with fundamental period T₀ = 2, and a period of x(t) is x₁(t) = u(t + 0.5) − u(t − 0.5). Increase T₀ to 4, 8, and 16.

(a) Find the Fourier series coefficient X₀ for each of the values of T₀ and indicate how it changes for the different values of T₀.

(b) Find the Fourier series coefficients for x(t) and carefully plot the line spectrum for each of the values of T₀. Explain what is happening in these spectra.

(c) If you were to let T₀ be very large what would you expect to happen to the Fourier coefficients? Explain.

(d) Write a MATLAB script that simulates the conversion from the Fourier series to the Fourier transform of a sequence of rectangular pulses as the period is increased. The line spectrum needs to be multiplied by the period so that it does not become insignificant. Plot using stem the adjusted line spectrum for pulse sequences with periods from 4 to 62.