ECE 301 Digital Electronics

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1 ECE 301 Digital Electronics Derivation of Flip-Flop Input Equations and State Assignment (Lecture #24) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6 th Edition, by Roth and Kinney, and were used with permission from Cengage Learning.

2 Sequential Circuit Design 1. Understand specifications 2. Draw state graph (to describe state machine behavior) 3. Construct state table (from state graph) 4. Perform state reduction (if necessary) 5. Assign a binary value to each state (state assignment) 6. Create state transition table 7. Select type of Flip-Flop to use 8. Derive Flip-Flop input equations and FSM output equation(s) 9. Draw circuit diagram Spring 2011 ECE Digital Electronics 2

3 Derivation of Flip-Flop Input Equations Spring 2011 ECE Digital Electronics 3

4 Derivation of FF Input Equations Example #2: Derive the Flip-Flop input equations for the FSM described by the following state table. Assume that JK Flip-Flops are used in the design. Excitation Table: Q Q + J K x x 1 0 x x 0 Spring 2011 ECE Digital Electronics 4

5 Example #2: FF Input Equations Present Next State Output State X = 0 X = 1 X = 0 X = 1 S0 S1 S2 0 1 S1 S2 S3 0 0 S2 S3 S0 1 0 S3 S0 S1 0 1 State Table Spring 2011 ECE Digital Electronics 5

6 Example #2: FF Input Equations 1. Assign a binary value to each state. 2. Construct the state transition table. A + B + J A K A J B K B AB X = 0 X = 1 X = 0 X = 1 X = 0 X = Spring 2011 ECE Digital Electronics 6

7 Example #2: FF Input Equations 3. Construct K-maps for Flip-Flop inputs. 4. Derive the minimized FF input equation. J A = K A = Spring 2011 ECE Digital Electronics 7

8 Example #2: FF Input Equations 3. Construct K-maps for Flip-Flop inputs. 4. Derive the minimized FF input equation. J B = K B = Spring 2011 ECE Digital Electronics 8

9 Derivation of FF Input Equations Example #3: Derive the Flip-Flop input equations for the FSM described by the following state table. Assume that SR Flip-Flops are used in the design. Excitation Table: Q Q + S R x x 0 Spring 2011 ECE Digital Electronics 9

10 Example #3: FF Input Equations Present Next State Output State X = 0 X = 1 X = 0 X = 1 S0 S1 S2 0 1 S1 S2 S3 0 0 S2 S3 S0 1 0 S3 S0 S1 0 1 State Table Spring 2011 ECE Digital Electronics 10

11 Example #3: FF Input Equations 1. Assign a binary value to each state. 2. Construct the state transition table. A + B + S A R A S B R B AB X = 0 X = 1 X = 0 X = 1 X = 0 X = Spring 2011 ECE Digital Electronics 11

12 Example #3: FF Input Equations 3. Construct K-maps for Flip-Flop inputs. 4. Derive the minimized FF input equation. S A = R A = Spring 2011 ECE Digital Electronics 12

13 Example #3: FF Input Equations 3. Construct K-maps for Flip-Flop inputs. 4. Derive the minimized FF input equation. S B = R B = Spring 2011 ECE Digital Electronics 13

14 State Assignment Spring 2011 ECE Digital Electronics 14

15 State Assignment After the number of states in the state table has been reduced A binary value must be assigned to each of the states. State assignment (or state encoding) Binary value = state of Flip-Flops The cost of the logic required to realize the FSM is dependent on the state assignment. Spring 2011 ECE Digital Electronics 15

16 State Assignment Given: A FSM with three states. Requires: Two Flip-Flops (A and B) Can implement a maximum of four states. There are 4 x 3 x 2 = 24 possible state assignments. Spring 2011 ECE Digital Electronics 16

17 State Assignment For a FSM realized using symmetrical Flip- Flops (i.e. JK and SR) 3 unique state assignments for 3-state FSM 3 unique state assignments for 4-state FSM Binary Gray Code Spring 2011 ECE Digital Electronics 17

18 State Assignment Spring 2011 ECE Digital Electronics 18

19 Guidelines for State Assignment The author provides a set of guidelines by which the optimal state assignment can be selected. Spring 2011 ECE Digital Electronics 19

20 One-Hot State Assignment Sometimes, reducing the next-state logic is more important than reducing the number of Flip-Flops. One-hot state assignment may result in minimal next-state logic. Uses one Flip-Flop per state. Exactly one Flip-Flop is set to 1 for each state. Spring 2011 ECE Digital Electronics 20

21 Example: State Assignments For a 4-state FSM, three possible state assignments are: State Binary Gray-code One-hot S S S S # of FF Spring 2011 ECE Digital Electronics 21

22 Example: State Assignments Binary state-assignment: Present Next State State X = 0 X = 1 S0 S1 S2 S1 S2 S3 S2 S3 S0 S3 S0 S1 State Table A + B + AB X = 0 X = State Transition Table Spring 2011 ECE Digital Electronics 22

23 Example: State Assignments Gray-code state-assignment: Present Next State State X = 0 X = 1 S0 S1 S2 S1 S2 S3 S2 S3 S0 S3 S0 S1 State Table A + B + AB X = 0 X = State Transition Table Spring 2011 ECE Digital Electronics 23

24 Example: State Assignments One-hot state-assignment: Present Next State State X = 0 X = 1 S0 S1 S2 S1 S2 S3 S2 S3 S0 S3 S0 S1 State Table A + B + C + D + ABCD X = 0 X = State Transition Table Spring 2011 ECE Digital Electronics 24

25 Questions? Spring 2011 ECE Digital Electronics 25

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