Finite State Machine Basics

1
Finite State Machine Basics

• ECE-331, Digital Design
• Dr. Ron Hayne
• Electrical and Computer Engineering

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Automata vs Combinational

• Combinational Circuits
• No feedback
• Boolean Algebraic mapping of input binary values
  to output value(s)
• Finite State Machines
• Feedback
• State and input dependent output

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Automata Model
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Finite State Machine

• Combinational Circuit Generates
• Excitation variables for synchronous FSMs, or
• Next state variables for asynchronous FSMs
• Memory
• Feedback from output to input
• Can be as simple as time delay for asynch FSM
• Usually flip-flops for synch FSM

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Abstract Model of FSM

• Machine is a quintuple of sets
• M ( S, I, O, ?, ? )
• S Finite set of states
• I Finite set of inputs
• O Finite set of outputs
• ? State transition function
• ?/? Mealy/Moore output function

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FSM and Clocks

• Synchronous FSMs may change state only when a
  unique input, the clock, occurs
• Asynchronous FSMs may change state when input
  changes
• Next state depends on present input and present
  state for both Moore and Mealy

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Synchronous Moore Machine
Input
Comb Ckt
Output
Present State
Next State
Output Is Only a Function of Present State
Clock
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Synchronous Mealy Machine
Input
Comb Ckt
Output
Present State
Next State
Output Is Function of Present State AND Present
Input
Clock
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Primitive State Diagram, Moore
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Moore Machine State Diagram
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Primitive State Diagram, Mealy
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Mealy Machine State Diagram
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Unsolved FSM Design Problem

• Mapping of Primitive States to Binary Numeric
  Representation Is Not Unique
• Some Mappings Are Better Than Others Since They
  Can Be Implemented in Less Hardware and Operate
  Faster
• Usual Solution Is to Make Large FSM Out of
  Smaller FSMs and Try All Mappings for Smaller
  Machines (ECE-548)

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It can be shown that ...

• Any Mealy Machine Can Be Converted to a
  Behaviorally Equivalent Moore Machine and
  Vice-Versa
• An Incompletely Specified Machine Can Be
  Converted to a Completely Specified Machine
  Through the Addition of States

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FSM Design Approaches

• One-Hot
• One flip-flop is used to represent each state
• Costly in terms of hardware, but trivial to
  design
• Binary Coded State
• n flip-flops used to store 2n states
• Most efficient
• Need to account for unused states

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Mealy Machine, e.g.

• Rabin-Scott Machine
• Language Recognizer
• Detect the sequence 0101
• String input left to right
• Continues to function if longer string
• detects last four characters
• non-resetting

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Mealy 0101 Detector

• M ( S, I, O, ?, ? )
• S A, B, C, D
• I 0, 1
• O 0, 1 not detected, detected
• ? next slides
• ? next slides

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0101 State Diagram
0 detected
A
B
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0101 State Diagram
0/0
0 detected
1/0
A
B
0/0
1/0
C
01 det
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0101 State Diagram
0/0
0 detected
1/0
A
B
0/0
1/0
1/0
D
C
0/0
01 det
010 det
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0101 State Diagram
0/0
0 detected
1/0
A
B
0/0
1/0
0/0
1/0
1/1
D
C
0/0
01 det
010 det
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Mealy Transition/Output Table
Next State/Output
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Moore 0101 Detector

• M ( S, I, O, ?, ? )
• S A, B, C, D, E
• I 0, 1
• O 0, 1 not detected, detected
• ? next slide
• ? next slide

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Moore Transition/Output Table
Next State
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Moore 0101 State Diagram
0
1
0 detected
0
A/0
B/0
0
1
1
1
0101 det
C/0
0
0
01 det
1
010 det
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Moore/Mealy

• More States in Moore Than in Behavioral
  Equivalent Mealy
• Mealy
• Usually pulse output (during transition)
• Moore
• Usually level output (during state)

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Summary

• FSM
• Synchronous
• Asynchronous
• Moore
• Mealy
• Example