Simplify the following Boolean expressions and minimize the number of literals:
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1 Boolean Algebra Task 1 Simplify the following Boolean expressions and minimize the number of literals: Task 2 Convert the following expressions into sum of products and product of sums: Task 3 Write the following function in Conjunctive Normal Form and Disjunctive Normal Form ( Hint: Use the truth table)
2 Task Form respective equations for output X and Y shown in the circuit below. 4.2 Construct a truth table based on the output equations. 4.3 Minimize the equations formed with the help of the KV-Map. 4.4 Draw a minimized circuit and compare it to the original circuit below. Discuss the result.
3 Task 5 The figure below shows a multilevel structure of a full adder constructed with two half adder. The underlying figure shows the same circuit in its logic form. Figure a : Block diagram Figure b : Combinatorial circuit The propagation delay of the XOR-, AND- and OR- gate of the 74series are given by: t PLH t PHL 7408 AND 20ns 12ns 7432 OR 10ns 22ns 7486 XOR 25ns 15ns
4 Analyse the full adder in diagram b. 5.1 Complete the following truth table: x i y i c i J K L s i c i Which value must x and y take to activate the path from C in to the sum (c i s i ) and the path from C in to C out (c i c i+1 )?What are the propagation delay ( rising and falling) of these path? Hint:Activating a path means that the changes in input state will trigger a change in the path as well as the level of output state. Consider for example the first two line of the truth table. (x i, y i ) = (0,0) enables the propagation of c i (LH) s i (LH ) as well as c i (HL) s i ( HL ) through the XOR gates. However, this value does not activate the propagation of c i c i+1 (this means that changes in c i does not trigger a change in c i+1 remains the same ). 5.3 What value must c i and y take to activate the path from x to sum (x i s i ) and x to C out (x i c i+1 ).What is the propagation delay ( rising and falling) of these path? 5.4 What is the critical path ( primary input to primary output ) and what is the maximum propagation delay?
5 5.5 Construct an impulse diagram showing the propagation delay along the critical path ( with all auxiliary node). Both the output of the sum as well as the carry should be included. Task 6 (Finite state machine) Develop a FSM for Traffic Light Controller that operates in two different modes: - Day mode: Red Red-Yellow Green Yellow Red - Night mode: Yellow All off Yellow The user can switch between modes by simply pressing a Mode button: - When pressed (M=1) change mode (every change starts with yellow). - When not pressed (M=0) Remain in actual mode Hint: The controller has 6 states to operate, namely 4 day mode states (S 0 =R, S 1 =RY, S 2 =G, S 3 =Y) and 2 night mode states (S 4 =Y, S 5 =Off) 6.1 Complete the following State Transition Diagram! (add more transitions) Hint:: There should always be 2 transitions from every state (for M=0 and M=1).
6 6.2 Complete the following State Table: (5) Present State Mode Input Next State Output S (Q 2,Q 1,Q 0 ) M S + (Q 2 +,Q 1 +,Q 0 + ) (R Y G) S S S S S S S S S S S S S S Extract the Output functions R, Y and G as F(Q 2,Q 1,Q 0 )! 6.3 Extract the Next State functions of Q + 2,Q + 1,Q + 0 as F(Q 2,Q 1,Q 0,M)! Hint: Use KV-Map and don t cares to simplify your results! The FSM implementation is done using JK Flip-flop. S J CP K R 6.4 Based on the characteristic function of JK Flip-flop, find out all the excitation functions of Q + 2,Q + + 1,Q 0 namely: J 2, K 2, J 1, K 1, and J 0, K 0! 6.5 Draw the complete Traffic light Control Circuit implementation! Q _ Q Q + = JQ + KQ n 1 n n
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