Let $$\,x\,\,\left( t \right)\,\,\, = \,\,\,\cos \,\,\,\left( {10\pi t} \right)\,\, + \,\,\cos \,\,\left( {30\pi t} \right)$$ be sampled at $$20\,\,\,Hz$$ and reconstructed using an ideal low-pass filter with cut-off frequency of $$20\,\,\,Hz$$. The frequency/frequencies present in the reconstructed signal is/are.

Consider two real valued signals, $$x\left( t \right)$$ band - limited to $$\,\left[ { - 500Hz,\,\,500Hz} \right]$$ and $$y\left( t \right)$$ band - limited to $$\,\left[ { - 1\,\,kHz,\,\,1kHz} \right].$$ For $$z\left( t \right)\,\, = \,x\left( t \right).y\left( t \right),$$ the Nyquist sampling frequency $$\left( {in\,\,kHz} \right)$$ is________.

A band-limited signal with a maximum frequency of 5 kHz is to be sampled. According to the sampling theorem, the sampling frequency which is not valid is

Consider a sampled signal $$y\left( t \right) = 5 \times {10^{ - 6}}\,x\left( t \right)\,\,\sum\limits_{n = - \infty }^{ + \infty } {\delta \left( {t - n{T_s}} \right)} $$

where $$x\left( t \right) = 10\,\,\cos \,\left( {8\pi \times {{10}^3}} \right)\,\,t$$ and
$${T_s} = 100\,\,\mu \sec .$$ When $$y\left( t \right)$$ is passed through an ideal low-pass filter with a cutoff frequency of 5 KHz, the output of the filter is