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 _________.

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____________________

The response of the system $$G\left(s\right)\;=\;\frac{s\;-\;2}{\left(s\;+\;1\right)\left(s\;+\;3\right)}$$ to the unit step input u(t) is y(t). The value of $$\frac{\mathrm{dy}}{\mathrm{dt}}\;\mathrm{at}\;\mathrm t\;=\;0^+\;\mathrm{is}$$ ___________.

In the feedback system shown below $$$G\left(S\right)\;=\frac1{(s^2\;+\;2s)}$$$ The step response of the closed-loop system should have minimum settling time and have no overshoot. The required value of gain k to achieve this is __________.