In this problem, consider the pulse train of width $c$ specified by the periodic function $f$ of period $2\pi$ $$f(x) = \begin{cases}0, & -\pi \leq x < -c/2 \cr 1, & -c/2 \leq x < c/2 \cr 0, & c/2 \leq x < \pi. \cr\end{cases}$$ (a) Suppose that $f$ is the pulse train of width 0.4.
(i) What fraction of the energy of $f$ is contained in the constant term of its Fourier series?
(ii) What fraction of the energy of $f$ is contained in the constant term and the first harmonic together?
(iii) Find a formula for the energy of the $k$th harmonic of $f$.
$E_k =$
(iv) What fraction of the energy of $f$ is captured by the constant term and the first five harmonics?
fraction =
(The constant term and the first 13 harmonics are needed to capture 90 percent of the energy of $f$.)

(b) Now suppose that $f$ is the pulse train of width 2. Answer the same questions as you did for the case of the train of width 0.4, that is:
(i) What fraction of the energy of $f$ is contained in the constant term of its Fourier series?
(ii) What fraction of the energy of $f$ is contained in the constant term and the first harmonic together?
(iii) Find a formula for the energy of the $k$th harmonic of $f$. $E_k =$
(iv) How many terms of the Fourier series of $f\ $ are needed to capture 90 percent of the energy of $f$? number =

Notice how the amount of energy in the earlier harmonics changes as the width of the pulse train changes!