PROBLEM 2.2. Let $s(t)$ be the real-valued signal from HW Prob. 1.2, whose Fourier transform is:
$S(f) = \begin{cases} 8\cos^2(0.5\pi f), & \text{for } |f| < 1 \text{ Hz},\\ 0, & \text{for } |f| > 1 \text{ Hz}, \end{cases}$
as sketched below:
(a) How fast would $s(t)$ need to be sampled to ensure that
$s(t)$ can be perfectly reconstructed from these samples?
(b) Define $s_k = s(kT)$ as the k-th sample of $s(t)$, when the sampling rate is $f_s = 1/T$
samples per second. Let $y(t) = \sum_{k=-\infty}^{\infty} s_k g(t - kT)$ be an attempt at reconstruction,
where $g(t) = \frac{\sin(\pi t/T)}{\pi t/T}$ is the interpolating sinc function of an ideal DAC.
(Reconstruction is successful, $y(t) = s(t)$, when $f_s$ satisfies the conditions of part (a).)
Find $y(t)$ when $f_s = 1$ Hz. (Hint: this is not fast enough, so $y(t) \neq s(t)$).
