A modulated signal is $$y\left( t \right)\, = \,\,\,\,\,\,\,\,\,m\left( t \right)\,\cos \left( {40000\pi t} \right),$$ where the baseband signal $$m\left( t \right)\,$$ has frequency components less than 5 kHz only. The minimum required rate (in kHz) at which $$y\,\,\left( t \right)$$ should be sampled to recover $$m\,\,\left( t \right)$$ is ________.

The phase response of a passband waveform at receiver is given by $$\varphi \,(f) = - 2\,\pi \,\alpha \,(f - {f_c}) - \,2\pi \beta \,{f_c}$$

where $${f_c}$$ is the centre frequency, and $$\alpha $$ and $$\beta $$ are positive constants. The actual signal propagation delay from the trasmitter to receiver is