Question 5 [18 points]: Short answer questions
(a) [5 points] Consider a linear memoryless system the produces the following output
y(t) = \begin{cases} 1+t & t \in [-1,0] \\ 1-t & t \in [0, 1] \\ 0 & o.w. \end{cases}
when the input is x(t) = 1 \forall t.
Compute the output of the system when the input is $x_1(t) = \sin(2\pi t)$.
(b) [5 points] Suppose the input-output relationship of a system is characterized by the following relation:
y(t) = f(x(t)).
Is the system necessarily
\begin{itemize}
\item memoryless
\item linear
\item time-invariant
\item Further if the function $f(x)$ is strictly increasing, then is the system invertible?
\end{itemize}Justify your answers for each part.
(c) [2 points] A continuous, non-constant, signal x(t) with period T = 1 has all the odd Fourier-Series coefficients to be zero. Is T = $\frac{1}{2}$ a period of x(t). (Justify your answer).
(d) [6 points] Determine all real-valued functions x(t) with period $2\pi$ that satisfies $x''(t) + x(t) = 0$. (Hint: use Fourier-Series) Here $x''(t)$ denotes the second derivative of x(t).
