Question

A hardwired $CPU$ uses $10$ control signals $S1$ to $S10$ in various time steps $T1$ to $T5$ to implement $4$ instructions $I1$ to $I4$ as shown below.

	$T1$	$T2$	$T3$	$T4$	$T5$
$I1$	$S1,S3,$ $S5$	$S2,S4,$ $S6$	$S1,S7$	$S10$	$S3,S8$
$I2$	$S1,S3,$ $S5$	$S8,S9,$ $S10$	$S5,S6,$ $S7$	$S6$	$S10$
$I3$	$S1,S3,$ $S5$	$S7,S8,$ $S10$	$S2,S6,$ $S9$	$S10$	$S1,S3$
$I4$	$S1,S3,$ $S5$	$S2,S6,$ $S7$	$S5,S10$	$S6,S9$	$S10$

Which of the following pairs of expressions represent the circuit for generating control signals $S5$ and $S10$ respectively
[(IJ+Ik)Tn indicates that the control signal should be generated in time step $Tn$ if the instruction being executed is IJ or IK]?

• A. $S5=T1+(I2+I4).T3$ and $S10=(I2+I3).T2+I4.T3+(I1+I3).T4+(I2+I4).T5$
• B. $S5=T1+(I2+I4).T3$ and $S10=(I2+I3+I4).T2+(I1+I3).T4+(I2+I4).T5$
• C. $S5=T1+I2.T3$ and $S10=(I1+I3).T4+(I2+I4).T5$
• D. $S5=T1+(I2+I4).T3$ and $S10=(I1+I3).T4+(I2+I4).T5$

Answer 1

The correct option is A $S5=T1+(I2+I4).T3$ and $S10=(I2+I3).T2+I4.T3+(I1+I3).T4+(I2+I4).T5$

First we need to find hardware control signal $S5$:

In instruction $I1,S5$ present for $T1$ only, so $T1,I1$
In instruction $I2,S5$ present for $T1,T3,$ so $T1I2+T3I2$
In instruction $I3,S5$ present for $T1$ only, so $T1I3$
In instruction $I4,S5$ present for $T1,T3,$ so $T1I4+T3I4$

So, logic function is $SOP$ form is:
$=T1I1+T1I2+T3I2+T1I3+T1I4+T3I4$
$=T1(I1+I2+I3+I4)+T3I2+T3I4$
$=T1+T3I2+T3I4(\because I1+I2+I3+I4=1)$
$=T1+T3(I2+I4)$

Similarly for signal $S10,SOP$ form will be:
$=T4I1+T2I2+T5I2+T2I3+T4I3+T3I4+T5I4$
$=T4(I1+I3)+T2(I2+I3)+T5(I2+I4)+T3I4$