Consider a continuous-time signal defined as $$x(t) = \left( {{{\sin \,(\pi t/2)} \over {(\pi t/2)}}} \right)*\sum\limits_{n = - \infty }^\infty {\delta (t - 10n)} $$ Where ' * ' denotes the convolution operation and t is in seconds. The Nyquist sampling rate (in samples/sec) for x(t) is __________________.

Consider the function $$g(t) = {e^{ - t}}\,\,\,\sin (2\pi t)\,u(t)$$ where u(t) is the unit step function. The area under g(t) is_______________.

The complex envelope of the bandpass signal $$x(t) = - \sqrt 2 \left( {{{\sin \,(\pi t/5)} \over {\pi t/5}}} \right)\,\sin \,\left( {\pi t - {\pi \over 5}} \right)$$, centered about $$\,f = {1 \over 2}\,\,Hz$$, is

The value of $$\sum\limits_{n = 0}^\infty n {\left( {{1 \over 2}} \right)^n}$$ is ________________.