Let $$X \in \left\{ {0,1} \right\}$$ and $$Y \in \left\{ {0,1} \right\}$$ be two independent binary random variables.

If $$P\left( {X\,\, = 0} \right)\,\, = p$$ and $$P\left( {Y\,\, = 0} \right)\,\, = q,$$ then $$P\left( {X + Y \ge 1} \right)$$ is equal to

A zero mean white Gaussian noise having power spectral density $${{{N_0}} \over 2}$$ is passed through an $$ LTI $$ filter whose impulse response $$h(t)$$ is shown in the figure. The variance of the filtered noise at $$t = 4$$ is

Let $$Q\left( {\sqrt y } \right)$$ be the BER of a BPSK system over an AWGN channel with two - sided noise power spectral density N₀/2. The parameter 𝛾 is a function of bit energy and noise power spectral density. A system with two independent and identical AWGN channels with noise power spectral density N₀/2 is shown in the figure. The BPSK demodulator receives the sum of outputs of both the channels

If the BER of this system is $$Q\left( {b\sqrt y } \right),$$ then the value of b is -----------.

Consider a random process $$X\left( t \right) = \sqrt 2 \sin \left( {2\pi t + \varphi } \right),$$ where the random phase $$\varphi $$ is uniformly distributed in the interval $$\left[ {0,\,\,2\pi } \right].$$ The auto - correlation $$E\left[ {X\left( {{t_1}} \right)X\left( {{t_2}} \right)} \right]$$ is