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__________________________

A binary communication system makes use of the symbols “zero” and “one”. There are channel errors. Consider the following events:
$${x_0}$$ : a " zero " is transmitted
$${x_1}$$ : a " one " is transmitted
$${y_0}$$ : a " zero " is received
$${y_1}$$ : a " one " is received

The following probabilities are given:
$$P({x_0}) = \,{3 \over 4},\,\left( {\,\left. {{y_0}} \right|{x_0}} \right) = \,{1 \over 2},\,\,and\,P\,\,\left( {\,\left. {{y_0}} \right|{x_1}} \right) = \,{1 \over 2}$$.
The information in bits that you obtain when you learn which symbol has been received (while you know that a " zero " has been transmitted) 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 __________.