Finite State Machines

1
Finite State Machines

• Prof. Sirer
• COMP 303
• Koç University

2
Finite State Machines

• An electronic machine which has
• external inputs
• externally visible outputs
• internal state
• Output and next state depend on
• inputs
• current state

3
Designing an FSM

• Draw a state diagram
• Write down state transition table
• Assign numbers to states
• Determine logic equations for all flip-flops and
  outputs

4
A Simple Example

• Goal flash hello on LEDs
• Inputs clock
• Outputs Just one 7-segment LED
• Flash h then e then l then l then o
• h lt0011101gt
• e lt0111110gt
• l lt0010110gt
• o lt1110111gt

5
HELLObox State Diagram
S0
S1
S2
S3
S4
lt0011101gt H
lt0111110gt E
lt0010110gt L
lt 0010110 gt L
lt1110111gt O

• Determine the transitions
• label all edges (transitions) with the inputs
  that cause them, unlabeled edges are
  unconditional transitions
• show start state

6
HELLObox State Table

• Build state table
• rote encoding of the state diagram

CurrentState NextState Output
S0 S1 0011101
S1 S2 0111110
S2 S3 0010110
S3 S4 0010110
S4 S0 1110111
7
HELLObox State Assignment 1

• Assign bit patterns to states that would make the
  resulting device simple
• One option is shown
• Determine logic equations for every bit of output
  and next state, for every flip-flop and output

Current State NextState Output
000 001 0011101
001 010 0111110
010 011 0010110
011 100 0010110
100 000 1110111
8
HELLObox 1

• 10 bits of information (7 outputs 3 bits of
  next state) are computed by the combinatorial
  circuit
• All 10 bits have non-trivial logic equations

combinational circuit
flip-flops
9
HELLObox State Assignment 2

• Assign bit patterns to states that would make the
  resulting device simple
• Here, we use far more bits than necessary
• to simplify the combinatorial circuit

Current State NextState Output
00111010 01111100 0011101
01111100 00101100 0111110
00101100 00101101 0010110
00101101 11101110 0010110
11101110 00111010 1110111
10
HELLObox 2

• 15 bits of information (7 outputs 8 bits of
  next state) are computed by the combinatorial
  circuit
• 8 bits have non-trivial logic equations
• all 7 outputs are simple pass-throughs

combinational circuit
flip-flops
11
A Traffic Light

• There is a large avenue and a Hospital ER on a
  side street
• The avenue traffic light should be green at all
  times
• Except when an ambulance is waiting on the side
  street
• A sensor detects a waiting ambulance
• Design the light controller

a
H
12
A Traffic Light

• There is a large avenue and a Hospital ER on a
  side street
• The avenue traffic light should be green at all
  times
• Except when an ambulance is waiting on the side
  street

H
13
A Traffic Light

• There is a large avenue and a Hospital ER on a
  side street
• The avenue traffic light should be green at all
  times
• Except when an ambulance is waiting on the side
  street

H
14
A Traffic Light

• There is a large avenue and a Hospital ER on a
  side street
• The avenue traffic light should be green at all
  times
• Except when an ambulance is waiting on the side
  street

H
15
A Traffic Light

• There is a large avenue and a Hospital ER on a
  side street
• The avenue traffic light should be green at all
  times
• Except when an ambulance is waiting on the side
  street

H
16
A Traffic Light

• There is a large avenue and a Hospital ER on a
  side street
• The avenue traffic light should be green at all
  times
• Except when an ambulance is waiting on the side
  street

H
17
A Traffic Light

• There is a large avenue and a Hospital ER on a
  side street
• The avenue traffic light should be green at all
  times
• Except when an ambulance is waiting on the side
  street

H
18
A Traffic Light

• There is a large avenue and a Hospital ER on a
  side street
• The avenue traffic light should be green at all
  times
• Except when an ambulance is waiting on the side
  street

H
19
A Traffic Light

• One input a
• Six outputs (AR, AO, AG, HR, HO, HG)

0
1
0 1
1
0 1
0 1
0
0 1
S0
S1
S2
S3
S4
S5
001100
010100
100110
100001
100010
110100
20
FSM Serial Adder

• Add two input bit streams
• streams are sent with least-significant-bit (lsb)
  first

10110
00101
01111
21
FSM State Diagram
00/1
00/0
S0
S1
11/1
11/0
10/1
01/1
10/0
01/0

• Two states S0 (no carry), S1 (carry in hand)
• Inputs a and b
• Output z
• Arcs labelled with input bits a and b, and output
  z

22
Serial Adder State Table
a b state z next state
0 0 S0 0 S0
0 1 S0 1 S0
1 0 S0 1 S0
1 1 S0 0 S1
0 0 S1 1 S0
0 1 S1 0 S1
1 0 S1 0 S1
1 1 S1 1 S1

• Write down all input and state combinations

23
Serial Adder State Assignment
a b s z s
0 0 0 0 0
0 1 0 1 0
1 0 0 1 0
1 1 0 0 1
0 0 1 1 0
0 1 1 0 1
1 0 1 0 1
1 1 1 1 1

• Two states, so 1-bit is sufficient
• A single flip-flop will encode the state

24
Serial Adder Circuit
a

• Equations
• z abs abs abs abs
• s abs abs abs abs

z
combinational logic
b
s
s
Q
D
clk