TOPIC: Finite State Machines State minimization:

McGill University Department of Electrical and Computer Engineering Winter 2011

Course: ECSE 323-Digital Systems Design

ASSIGNMENT #10B Wednesday March. 30, 2011

TOPIC: Finite State Machines State minimization: Use partitioning to reduce the number of states of the following state table: Present State A B C D E F G H

Next State Input X=0 B C B E D F H H

Output Z Input X=1 D H G C E A A A

0 1 1 1 1 1 1 1

Question 1: Derive the resulting minimized state table Question 2: Draw the resulting state diagram.

Solution: P1 = (A) (BCDEFGH) P2 = (A)(BCDE)(FGH) P3 = (A)(BC)(DE)(FGH) P3 = (A)(BC)(DE)(FGH) F=G=H, B=C, D=E State truth table: Present State F B

Next State Input X=0 F B

Output Z Input X=1 A F

1 1

D A

D B

B D

1 0

State diagram:

Question 3: Show that the following state diagram has a minimal number of states.

Solution: P1 = B (AC), consequently => B≠A , B≠C

When X=1 (AC) => (BC), consequently A≠C P2=(A)(B)(C) Question 4: Using the state diagram of Q3, propose an equivalent state diagram that have 6 states. Solution: One of the possible solutions is

Synthesis with JK flip-flops: Question 5: Synthesize the state machine of Q3 using JK flip-flops. Use the following state mapping: A=00, B=01, C=10 Solution: State table: Current State A B C

Next State X=0 A C A

Output Z X=1 B A C

0 1 1

Truth table: Current State A B C D

Next State X=0 J1 K1 0d 1d d0 dd

Output Z X=1

J0 K0 0d d0 0d dd

J1 K1 0d 0d d1 dd

J0 K0 1d d0 0d dd

0 1 1 d

J1 = X’ Q0 K1 = X J0 = X Q1’ K0 = 0 Z = Q0 + Q1

Circuit:

Question 6: Derive the Algorithmic State Machine from the state machine diagram. Solution: