1. ECE2030 Introduction to Computer Engineering Lecture 16: Finite State Machines Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering Georgia Tech

2. 2 Mealy and Moore Machines Combinational circuits Inputs X(t)Outputs Z(t) Storage Element S(t) MEALY MACHINE Z(t) =  {S(t), X(t)} Combinational circuits Inputs X(t) Outputs Z(t) Storage Element S(t) MOORE MACHINE Z(t) =  {S(t)}

3. 3 State and State Diagram A state represents the machine snapshot at a given clock period A clock is typically used to synchronize the state transition A graph consists of a set of –Circles: Each represents a state Use double circle to represent the initial state –Directed arc: each represents a state transition –Inputs/outputs Mealy machine: –Label input/output along each arc Moore machine: –Label input along each arc –Label output inside the circle (i.e. state)

4. 4 State Diagrams Example: State: S(t)  {S k, S j } Inputs: X(t)  {a, b} Outputs: Z(t)  {p, q} Initial state: S(0) = S k A Mealy machine exampleA Moore machine example SkSk SjSj a/p b/q b/p a/q a b S k /p S j /q b a

5. 5 State Diagram Examples (Mealy) S0S0 S1S1 0/0 1/1 1/0 0/0 S0S0 S1S1 0/0, 1/1 0/0 1/0

6. 6 State Diagram Examples (Moore) S 0 /1 S 1 /0 0 1 1 0 0, 1 0 1 S 0 /0S 1 /1

7. 7 Design Example: Sequence Recognizer A sequential circuit that recognizes the occurrence of a particular bit sequence Input: X(t)  {0, 1} Output: Z(t)  {0, 1}

8. 8 Sequence Recognizer Time 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 X(t) 1 0 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 Z(t) 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1

9. 9 Sequence Recognizer Time 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 X(t) 1 0 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 Z(t) 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 S0S0 S1S1 1/0 0/0 S2S2 1/0 S3S3 0/0 1/0 0/0 1/1 A Meanly Machine

10. 10 State Table 00 1/0 0/0 1/0 0/0 1/0 0/0 1/1 011011 Present State InputX Next State OutputZ P1P0N1N0 000000 001010 010000 011100 100110 101100 110000 111011

11. 11 Logic Circuits Design Steps Generate a Boolean function for –Each external output –Each state encoded bit Simplify the Boolean functions Draw a D F/F (or register) for each state encoded bit Draw logic circuits for –External outputs –Each inputs of state encoded bits –Input of state encoded bits = the next state –Output of state encoded bits = the current state

12. 12 Logic Circuits Design Present State X Next State Z P1P0N1N0 000000 001010 010000 011100 100110 101100 110000 111011 00011110 0 0001 1 0101 X P1P0 N1 00011110 0 0001 1 1010 X P1P0 N0

13. 13 Logic Circuits Design D0 F/F 11 22 P0N0 D1 F/F 11 22 P1N1 Z D0 F/F 11 22 P0N0 X

14. 14 Example 2 Input: X(t)  {a, b, c} Output: Z(t)  {q, p}

15. 15 State Diagram S EO /q A Moore Machine b S EE /p S OO /p S OE /p a a c b c a C b a b c

16. 16 State Table 01/q b 00/p 10/p 11/p a a c b c a C b a b c Present StateInput (a, b, c)Next StateOutput P1P0X1X0N1N0Z 0000110 0001010 0010000 0100101 0101001 0110011 1000010 1001110 1010100 1100000 1101100 1110110 S EE = 00 S EO = 01 S OO = 10 S OE = 11 a = 00 b = 01 c = 10 p = 0 q = 1

17. 17 Logic Circuit Design Present StateInputNext StateOutput P1P0X1X0N1N0Z 0000110 0001010 0010000 0100101 0101001 0110011 1000010 1001110 1010100 1100000 1101100 1110110 00011110 00 10X0 01 10X0 11 01X1 10 01X1 P1P0 X1X0 N1 00011110 00 11X0 01 00X1 11 00X1 10 11X0 P1P0 X1X0 N0 00011110 00 00X0 01 11X1 11 00X0 10 00X0 P1P0 X1X0 Z

18. 18 Logic Circuit Design D1 F/F 11 22 P1N1 D0 F/F 11 22 P0N0 X1 X0 Z

19. 19 Vending Machine State Machine Dispense a Coke when depositing 15 ¢ Inputs –5 = a nickel –10 = a dime –BC = bad coin (including quarters in this example) Outputs –R = reject –C = coke –N = no coke

20. 20 State Diagram 0 ¢ BC/R 5 ¢ 5/N 10 ¢ 10/N 5/C 10/C 5/N BC/R 10/C BC/R

21. 21 State Table 0 ¢ (00) BC/R 5/N 10/N 5/C 10/C 5/N BC/R 10/C BC/R Present State (0 ¢, 5 ¢, 10 ¢ ) Input (5 ¢, 10 ¢, BC) Next State (0 ¢, 5 ¢, 10 ¢ ) Output (C, N, R) P1P0X1X0N1N0C1C0 00000100 00011000 00100010 01001000 01010001 01100110 10000001 10010101 10101010 XX11XXXX 11XXXXXX 5 ¢ (01) 10 ¢ (10) 5: 00 10: 01 BC: 10 N: 00 C: 01 R: 10

22. 22 Logic Circuits Design Present State InputNext StateOutput P1P0X1X0N1N0C1C0 00000100 00011000 00100010 01001000 01010001 01100110 10000001 10010101 10101010 XX11XXXX 11XXXXXX 00011110 00 01X0 01 10X0 11 XXXX 10 00X1 P1P0 X1X0 N1 00011110 00 10X0 01 00X1 11 XXXX 10 01X0 P1P0 X1X0 N0

23. 23 Logic Circuits Design Present State InputNext StateOutput P1P0X1X0N1N0C1C0 00000100 00011000 00100010 01001000 01010001 01100110 10000001 10010101 10101010 XX11XXXX 11XXXXXX 00011110 00 00X1 01 00X1 11 XXXX 10 00X1 P1P0 X1X0 C1 00011110 00 00X0 01 01X0 11 XXXX 10 11X0 P1P0 X1X0 C0

24. 24 Logic Circuits of the Vending Machine D1 F/F 11 22 P1N1 D0 F/F 11 22 P0N0 X1X0P0P1 C0 C1