Final answer:
The student's question involves approximating a periodic signal using the Fourier series expansion, plotting the approximated signal and coefficients, the line spectra, and computing the approximation percentage by analyzing the series' mean-square error compared to the signal's power.
Step-by-step explanation:
The question involves the Fourier series expansion of a periodic signal characterized by a piecewise function which is sinusoidal in the first half of its period and zero in the second half. To approximate this signal, one would typically calculate the Fourier coefficients and construct the approximation by summing sine and cosine terms (or complex exponentials in the case of the exponential form of the Fourier series) according to the standard Fourier series formulations. This requires integrating the product of the given function and the corresponding basis functions over one period of the signal.
Plotting the approximate signal over four periods would reveal how well the finite series captures the behavior of the original function, and plotting the Fourier series coefficients shows the magnitude and phase of each harmonic component of the signal. The line spectra would display the distribution of energy across the different harmonic frequencies. For the approximation percentage, this would involve calculating the mean-square error over one period and expressing it as a percentage of the total signal power.