Consider random process $$X(t) = 3V(t) - 8$$, where $$V$$ $$(t)$$ is a zero mean stationary random process with autocorrelation $${R_v}\left( \tau \right) = 4{e^{ - 5\left| \tau \right|}}$$. The power of $$X(t)$$ is _______.

An antenna pointing in a certain direction has a noise temperature of 50K. The ambient temperature is 290K. The antenna is connected to a pre-amplifier that has a noise figure of 2dB and an available gain of 40 dB over an effective bandwidth of 12 MHz. The effective input noise temperature T_e for the amplifier and the noise power P_ao at the output of the preamplifier, respectively, are

A random binary wave $$y(t)$$ is given by $$$y\left( t \right) = \sum\limits_{n = - \infty }^\infty {{X_n}p\left( {t - nT - \phi } \right)} $$$

where $$p(t) = u(t) - u(t - T)$$, $$u(t)$$ is the unit step function and $$\phi $$ is an independent random variable with uniform distribution in $$[0, T]$$. The sequence $$\left\{ {{X_n}} \right\}$$ consists of independent and identically distributed binary valued random variables with $$P\left\{ {{X_n} = + 1} \right\} = P\left\{ {{X_n} = - 1} \right\} = 0.5$$ for each $$n$$.

The value of the autocorrelation $${R_{yy}}\left( {{{3T} \over 4}} \right)\underline{\underline \Delta } E\left[ {y\left( t \right)y\left( {t - {{3T} \over 4}} \right)} \right]\,\,$$

equals ------------ .

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