For the system $$\mathop X\limits^ \bullet = \left[ {\matrix{ 2 & 0 \cr 0 & 4 \cr } } \right]X + \left[ {\matrix{ 1 \cr 1 \cr } } \right]u;\,\,\,y = \left[ {\matrix{ 4 & 0 \cr } } \right]X,\,$$ with u as unit impulse and with zero initial state, the output, $$y$$, becomes

Obtain a state variable representation of the system governed by the differential equation: $${{{d^2}y} \over {d{t^2}}} + {{dy} \over {dt}} - 2y = u\left( t \right){e^{ - t}},\,\,\,$$ with the choice of state variables as $${x_1} = y,$$ $${x_2} = \left( {{{dy} \over {dt}} - y} \right){e^t}.$$ Aso find $${x_2}\left( t \right),$$ given that $$u(t)$$ is a unit step function and $${x_2}\left( 0 \right) = 0.$$

For the circuit shown in Fig. the Boolean expression for the output $$Y$$ in terms of inputs $$P,$$ $$Q,$$ $$R$$ and $$S$$ is

The frequency of the clock signal applied to the rising edge triggered $$D$$ flip-flop shown in Fig. is $$10$$ $$kHz.$$ The frequency of the signal available at $$Q$$ is