An information source generates a binary sequence $$\left\{ {{\alpha _n}} \right\}.{\alpha _n}$$ can take one of the two possible values −1 and +1 with equal probability and are statistically independent and identically distributed. This sequence is pre-coded to obtain another sequence $$\left\{ {{\beta _n}} \right\},$$ as $${\beta _n} = {\alpha _n} + k{\mkern 1mu} {\alpha _{n - 3}}$$ . The sequence $$\left\{ {{\beta _n}} \right\}$$ is used to modulate a pulse $$g(t)$$ to generate the baseband signal

$$x\left( t \right) = \sum\limits_{n = - \infty }^\infty {{\beta _n}g\left( {t - nT} \right),} $$ where $$g\left( t \right) = \left\{ {\matrix{ {1,} & {0 \le t \le T} \cr 0 & {otherwise} \cr } } \right.$$

If there is a null at $$f = {1 \over {3T}}$$ in the power spectral density of $$X(t)$$, then $$k$$ is _________.

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 ideal band-pass channel 500Hz-2000 Hz is deployed for communication. A modem is designed to transmit bits at the rate of 4800 bits/s using 16-QAM. The roll-off factor of a pulse with a raised cosine spectrum that utilizes the entire frequency band is____________________

A discrete memoryless source has an alphabet $$({a_1},\,{a_2},\,{a_3},\,{a_4})\,$$ with corresponding probabilities$$\left( {{1 \over 2}\,\,,{1 \over 4},\,{1 \over 8},\,\,{1 \over 8}\,} \right)$$. The minimum required average codeword length in bits to represent this source for error-free reconstruction is__________________________