Consider two continuous time signals $x(t)$ and $y(t)$ as shown below

If $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is ______.

A continuous time signal $x(t) = 2 \cos(8 \pi t + \frac{\pi}{3})$ is sampled at a rate of 15 Hz. The sampled signal $x_s(t)$ when passed through an LTI system with impulse response

$h(t) = \left( \frac{\sin 2 \pi t}{\pi t} \right) \cos(38 \pi t - \frac{\pi}{2})$

produces an output $x_o(t)$. The expression for $x_o(t)$ is ______.

The radian frequency value(s) for which the discrete time sinusoidal signal $x[n] = A \cos(\Omega n + \pi/3)$ has a period of 40 is/are __.

The relationship between any N-length sequence $x[n]$ and its corresponding N-point discrete Fourier transform $X[k]$ is defined as

$X[k] = \mathcal{F}\{x[n]\}$.

Another sequence $y[n]$ is formed as below

$y[n] = \mathcal{F}\{ \mathcal{F}\{ \mathcal{F}\{ \mathcal{F}\{x[n]\}\}\}\}\}$.

For the sequence $x[n] = \{1, 2, 1, 3\}$, the value of $Y[0]$ is _________.