Consider a discrete-time periodic signal with period N = 5. Let the discrete-time Fourier series (DTFS) representation be $$x[n] = \sum\limits_{k = 0}^4 {{a_k}{e^{{{jk2\pi m} \over 5}}}} $$, where $${a_0} = 1,{a_1} = 3j,{a_2} = 2j,{a_3} = - 2j$$ and $${a_4} = - 3j$$. The value of the sum $$\sum\limits_{n = 0}^4 {x[n]\sin {{4\pi n} \over 5}} $$ is

Let an input $$x[n]$$ having discrete time Fourier transform $$x({e^{j\Omega }}) = 1 - {e^{ - j\Omega }} + 2{e^{ - 3j\Omega }}$$ be passed through an LTI system. The frequency response of the LTI system is $$H({e^{j\Omega }}) = 1 - {1 \over 2}{e^{ - j2\Omega }}$$. The output $$y[n]$$ of the system is

Let $$x(t) = 100\cos (10.5Wt)$$ be passed through an LTI system having impulse response $$h(t) = \pi {\left( {{{\sin Wt} \over {\pi t}}} \right)^2}\cos 10Wt$$. The output of the system is

Let $$\mathrm{x_1(t)=u(t+1.5)-u(t-1.5)}$$ and $$\mathrm{x_2(t)}$$ is shown in the figure below. For $$\mathrm{y(t)=x_1(t)~*~x_2(t)}$$, the $$\int_{ - \infty }^\infty {y(t)dt} $$ is ____________ (rounded off to the nearest integer).