In the signal flow diagram given in the figure, $${u_1}$$ and $${u_2}$$ are possible inputs whereas $${y_1}$$ and $${y_2}$$ are possible outputs. When would the $$SISO$$ system derived from this diagram be controllable and observable?

$$f\left( {A,\,B,\,C,\,D} \right) = \Pi M\left( {0,1,3,4,5,7,9,11,12,13,14,15} \right)$$ is a Maxterm representation of a Boolean function $$f(A,B,C,D)$$ where $$A$$ is the $$MSB$$ and $$D$$ is the $$LSB$$. The equivalent minimized representation of this function is

In the $$4 \times 1$$ multiplexer, the output $$F$$ is given by $$F = A \oplus B.$$ Find the required input $${{\rm I}_3}{{\rm I}_2}{{\rm I}_1}{{\rm I}_0}.$$

The figure shows a digital circuit constructed using negative edge triggered $$J-K$$ flip flops. Assume a starting state of $${Q_2}\,{Q_1}\,{Q_0} = 000.$$ This state $${Q_2}\,{Q_1}\,{Q_0} = 000$$ will repeat after _____ number of cycles of the clock $$CLK.$$