A low-pass filter having a frequency response $$H(j\omega )$$ = $$A(\omega ){e^{j\Phi (\omega )}}$$, does not product any phase distortion if

The minimum sampling frequency (in samples /sec) required to reconstruct the following signal from its samples without distortion $$x(t) = 5{\left( {{{\sin \,\,2\,\pi \,1000\,t)} \over {\pi \,t}}} \right)^3} + 7{\left( {{{\sin \,\,2\,\pi \,1000\,t} \over {\pi \,t}}} \right)^2}$$

would be

A signal m(t) with bandwidth 500 Hz is first multiplied by a signal g(t) where $$g(t)\, = \,\,\sum\limits_{k = - \infty }^\infty {{{( - 10)}^k}\,\delta (t - 0.5x{{10}^{ - 4}}k)} $$
The resulting signal is then passed through an ideal low pass filter with bandwidth 1 kHz. The output of the low pass filter would be

A solution for the differential equation $$\mathop x\limits^. $$(t) + 2 x (t) = $$\delta (t)$$ with intial condition $$x({0^ - }) = 0$$ is