1. FINITE STATE MACHINES (FSMs)
Dr. Konstantinos Tatas

2. ACOE161 - Digital Logic for Computers - Frederick University
Finite State Machine
A generic model for sequential circuits used in sequential circuit design
ACOE161 - Digital Logic for Computers - Frederick University

3. Finite state machine block diagram
State memory: Set of n flip-flops that hold the state of the machine (up to 2^n distinct states)
Next state logic: Combinational circuit that determines the next state as a function of the current state and the input
Output logic: Combinational circuit that determines the output as a function of the current state and the input
ACOE161 - Digital Logic for Computers - Frederick University

4. Finite State Machine types
Mealy machine: The output depends on the current state and input
Moore machine: The output depends only on the current state
State = output state machine: A Moore type FSM where the current state is the output
ACOE161 - Digital Logic for Computers - Frederick University

5. ACOE161 - Digital Logic for Computers - Frederick University
State diagram
A state diagram represents the states as circles and the transitions between them as arrows annotated with inputs and outputs
ACOE161 - Digital Logic for Computers - Frederick University

6. Analysis of FSMs with D flip-flops
Determine the next state and output functions
Use the functions to create a state/output table that specifies every possible next state and output for any combination of current state and input
ACOE161 - Digital Logic for Computers - Frederick University

7. ACOE161 - Digital Logic for Computers - Frederick University
EXAMPLE
ACOE161 - Digital Logic for Computers - Frederick University

8. Next state equations and state table for exampley 1
A+=Ax+Bx
B+=A΄x
Y=(A+B)x΄
ACOE161 - Digital Logic for Computers - Frederick University

9. ACOE161 - Digital Logic for Computers - Frederick University
A+=Ax+Bx
B+=A΄x
Y=(A+B)x΄ A B x A+ B+ y 1
ACOE161 - Digital Logic for Computers - Frederick University

10. Sequential circuit design methodology
From the description of the functionality or the state/timing diagram find the state table
Encode the states if the state table contains letters
Find the necessary number of flip-flops
Select flip/flop type
From the state table, find the excitation tables and output tables
Using Karnaugh maps find the flip-flop input logic expressions
Draw the circuit logic diagram
ACOE161 - Digital Logic for Computers - Frederick University

11. Example: Design the sequential circuit of the following state diagram
ACOE161 - Digital Logic for Computers - Frederick University

12. State/excitation tableDA DB JA KA JB KB 1
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13. Karnaugh maps for combinational circuit
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14. ACOE161 - Digital Logic for Computers - Frederick University
Circuit logic diagram
ACOE161 - Digital Logic for Computers - Frederick University

15. ACOE161 - Digital Logic for Computers - Frederick University
Example: counter
ACOE161 - Digital Logic for Computers - Frederick University

16. Self-correcting state machines
The previous example did not include two possible states “011” and “111”. If the counter unexpectedly falls into one of those states there are two possibilities:
The counter will recover by entering a valid state after a finite number of cycles (self-correcting)
The counter will stay in a non-valid state until the f/fs are reset (not self-correcting)
Finite state machines should be designed to be self correcting by assigning non-valid states to a valid next state (no don’t cares in the excitation table)
ACOE161 - Digital Logic for Computers - Frederick University

17. ACOE161 - Digital Logic for Computers - Frederick University
Example
Design a self-correcting one-digit BCD counter
ACOE161 - Digital Logic for Computers - Frederick University

18. State minimization/assignment
Often the state of the circuit is not also the output and therefore the states are named abstractly
State minimization and state assignment are then required
State minimization is the simplification of the state diagram so that a circuit with less states produces the same output sequence
State assignment is the process of assigning a binary number to each state
ACOE161 - Digital Logic for Computers - Frederick University

19. ACOE161 - Digital Logic for Computers - Frederick University
Example
State values are not important just the input/output sequence
States are symbolized with letters
ACOE161 - Digital Logic for Computers - Frederick University

20. ACOE161 - Digital Logic for Computers - Frederick University
State table
Current state
Input
Next
state
Output a 1 b c d e f g
ACOE161 - Digital Logic for Computers - Frederick University

21. ACOE161 - Digital Logic for Computers - Frederick University
State equivalence
Current state
Input
Next
state
Output a 1 b c d e f g
Two states are equivalent if for any element of the input set both produce the same output and send the circuit to the same state or an equivalent one.
One of the two equivalent states can be eliminated from the state diagram
ACOE161 - Digital Logic for Computers - Frederick University

22. ACOE161 - Digital Logic for Computers - Frederick University
State minimization
Current state
Input
Next
state
Output a 1 b c d e f g d d e
ACOE161 - Digital Logic for Computers - Frederick University

23. ACOE161 - Digital Logic for Computers - Frederick University
State assignment
Since we don’t care about the actual flip-flop values for each state we can assign each state to any binary number we like as long as each state is assigned a unique binary number
If we use 3 bits to encode the states, we have
possible encodings
state
Encoding 1 (binary)
Encoding 2
(Gray)
Encoding 3 a
000 b
001
010
100 c
011 d
101 e
111
ACOE161 - Digital Logic for Computers - Frederick University

24. ACOE161 - Digital Logic for Computers - Frederick University
One-hot encoding
One flip-flop per state encoding
Leads to greater number of flip-flops than binary encoding but possibly to simpler logic
state
Encoding
1 (binary)
2 (Gray)
Encoding 3 (one-hot) a
000
00001 b
001
010
00010 c
011
00100 d
101
01000 e
100
111
10000
ACOE161 - Digital Logic for Computers - Frederick University