For the signals $$x(t)$$ and $$y(t)$$ shown in the figure, $$z(t)=x(t)*y(t)$$ is maximum at $$t=T_1$$. Then $$T_1$$ in seconds is __________ (Round off to the nearest integer)

The period of the discrete-time signal $$x[n]$$ described by the equation below is $$N=$$ __________ (Round off to the nearest integer).

$$x[n] = 1 + 3\sin \left( {{{15\pi } \over 8}n + {{3\pi } \over 4}} \right) - 5\sin \left( {{\pi \over 3}n - {\pi \over 4}} \right)$$

The discrete-time Fourier transform of a signal $$x[n]$$ is $$X(\Omega ) = (1 + \cos \Omega ){e^{ - j\Omega }}$$. Consider that $${x_p}[n]$$ is a periodic signal of period N = 5 such that

$${x_p}[n] = x[n]$$, for $$n = 0,1,2$$

= 0, for $$n = 3,4$$

Note that $${x_p}[n] = \sum\nolimits\limits_{k = 0}^{n - 1} {{a_k}{e^{j{{2\pi } \over N}kn}}} $$. The magnitude of the Fourier series coeffiient $$a_3$$ is __________ (Round off to 3 decimal places).

A signal $$x(t) = 2\cos (180\pi t)\cos (60\pi t)$$ is sampled at 200 Hz and then passed through an ideal low pass filter having cut-off frequency of 100 Hz.

The maximum frequency present in the filtered signal in Hz is ___________ (Round off to the nearest integer).