Finite State Machines

Prof. Sirer

COMP 303

Koç University

Finite State Machines

An electronic machine which has external inputs externally visible outputs internal state

Output and next state depend on inputs current state

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

A Simple Example

Goal: flash hello on LEDsInputs: clockOutputs: Just one 7-segment LED

Flash “h” then “e” then “l” then “l” then “o” h = <0011101> e = <0111110> l = <0010110> o = <1110111>

o0

o1

o2

o3

o4

o5

o6

HELLObox: State Diagram

Determine the transitions label all edges (transitions) with the inputs that cause

them, unlabeled edges are unconditional transitions show start state

S0 S1 S2 S3 S4

<0011101>

H

<0111110>

E

<0010110>

L

< 0010110 >

L

<1110111>

O

HELLObox: State Table

CurrentState

NextState

Output

S0 S1 0011101

S1 S2 0111110

S2 S3 0010110

S3 S4 0010110

S4 S0 1110111

Build state table rote encoding of

the state diagram

HELLObox: State Assignment 1

Current

State

NextState

Output

000 001 0011101

001 010 0111110

010 011 0010110

011 100 0010110

100 000 1110111

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

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

flip-flops

com

bina

tiona

lci

rcui

t

o0o1o2o3o4o5

o6

HELLObox: State Assignment 2

Current

State

NextState

Output

00111010

01111100

0011101

01111100

00101100

0111110

00101100

00101101

0010110

00101101

11101110

0010110

11101110

00111010

1110111

Assign bit patterns to states that would make the resulting device simpleHere, we use far more bits than necessary to simplify the

combinatorial circuit

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

flip-flops

com

bina

tiona

lci

rcui

t

o0o1o2o3o4o5

o6

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

H

a

A Traffic Light

There is a large avenue and a Hospital ER on a side streetThe avenue traffic light should be green at all timesExcept when an ambulance is waiting on the side street

H

A Traffic Light

There is a large avenue and a Hospital ER on a side streetThe avenue traffic light should be green at all timesExcept when an ambulance is waiting on the side street

H

A Traffic Light

There is a large avenue and a Hospital ER on a side streetThe avenue traffic light should be green at all timesExcept when an ambulance is waiting on the side street

H

A Traffic Light

There is a large avenue and a Hospital ER on a side streetThe avenue traffic light should be green at all timesExcept when an ambulance is waiting on the side street

H

A Traffic Light

There is a large avenue and a Hospital ER on a side streetThe avenue traffic light should be green at all timesExcept when an ambulance is waiting on the side street

H

A Traffic Light

There is a large avenue and a Hospital ER on a side streetThe avenue traffic light should be green at all timesExcept when an ambulance is waiting on the side street

H

A Traffic Light

There is a large avenue and a Hospital ER on a side streetThe avenue traffic light should be green at all timesExcept when an ambulance is waiting on the side street

H

A Traffic Light

One input a

Six outputs (AR, AO, AG, HR, HO, HG)

001100 010100 100110 100001 100010 110100

0

1 0 & 1 0 & 1 0

1

0 & 1

0 & 1

S0 S1 S2 S3 S4 S5

FSM: Serial Adder

Add two input bit streams streams are sent with least-significant-bit

(lsb) first

…10110

…01111…00101

FSM: State Diagram

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

S0 S100/0

10/1 01/1 10/0 01/0

11/1

00/1

11/0

Serial Adder: State Table

Write down all input and state combinations

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

Serial Adder: State Assignment

Two states, so 1-bit is sufficient A single flip-flop will

encode the state

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

Serial Adder: Circuit

Equationsz = abs + abs + abs +

abs

s’ = abs + abs + abs + abs

clk

Q D

za

bcombinational

logic

s s’