Finite State Machine Design - SNU DALdal.snu.ac.kr/~kchoi/class/lc_intro/fsm_des.pdf · Finite State Machine Word Problems Finite String Pattern Recognizer A finite string recognizer

Finite State Machine Design

Adapted from slides by R. H. Katz

• Counters: Sequential Circuits where State = Output

• Generalizes to Finite State Machines:– Outputs are Function of State (and Inputs)

– Next States are Functions of State and Inputs

– Used to implement circuits that control other circuits

– "Decision Making" logic

Concept of Finite State Machine


Computer Hardware = Datapath + Control

RegistersCombinational Functional

Units (e.g., ALU)Busses

FSM generating sequencesof control signals

Instructs datapath what todo next

"Puppet"

"Puppeteer who pulls thestrings"

Qualifiers

Control

Control

Datapath

State

ControlSignalOutputs

QualifiersandInputs


Example: Odd Parity Checker

Even [0]

Odd [1]

Reset

0

0

1 1

Assert output whenever input bit stream has odd # of 1's

StateDiagram

Present State Even Even Odd Odd

Input 0 1 0 1

Next State Even Odd Odd Even

Output 0 0 1 1

Symbolic State Transition Table

Output 0 0 1 1

Next State 0 1 1 0

Input 0 1 0 1

Present State 0 0 1 1

Encoded State Transition Table


Example: Odd Parity Checker

Next State/Output Functions

NS = PS xor PI; OUT = PS

D

R

Q

Q

Input

CLK PS/Output

\Reset

NS

D FF Implementation

T

R

Q

Q

Input

CLK

Output

\Reset

T FF Implementation

Timing Behavior: Input 1 0 0 1 1 0 1 0 1 1 1 0

Clk

Output

Input 1 0 0 1 1 0 1 0 1 1 1 0

1 1 0 1 0 0 1 1 0 1 1 1

Basic Design Approach

Six Step Process

1. Understand the statement of the Specification

2. Obtain an abstract specification of the FSM(state table or state diagram)

3. Perform a state minimization

4. Perform state assignment

(# state variables) ≥ log2 (#states)

5. Choose FF types to implement FSM state registerand build excitation table

6. Combinational circuit minimization

Basic Design Approach

Example: Vending Machine FSM

General Machine Concept:deliver package of gum after 15 cents deposited

single coin slot for dimes, nickels

no change

Block Diagram

Step 1. Understand the problem:

Vending Machine

FSM

N

D

Reset

Clk

OpenCoin

SensorGum

Release Mechanism

Draw a picture!

Vending Machine Example

Tabulate typical input sequences:three nickelsnickel, dimedime, nickeltwo dimestwo nickels, dime

Draw state diagram:

Inputs: N, D, reset

Output: open

Step 2. Map into more suitable abstract representation

Reset

N

N

N

D

D

N D

[open]

[open] [open] [open]

S0

S1 S2

S3 S4 S5 S6

S8

[open]

S7

D


Step 3: State Minimization

Reset

N

N

N, D

[open]

15¢

0¢

5¢

10¢

D

D

reuse stateswheneverpossible

Symbolic State Table

Present State

0¢

5¢

10¢

15¢

D

0 0 1 1 0 0 1 1 0 0 1 1 X

N

0 1 0 1 0 1 0 1 0 1 0 1 X

Inputs Next State

0¢5¢10¢X 5¢10¢15¢X

10¢15¢15¢X

15¢

Output Open

0 0 0 X 0 0 0 X 0 0 0 X 1


Step 4: State Encoding

Next State D 1 D 0

0 0 0 1 1 0 X X 0 1 1 0 1 1 X X 1 0 1 1 1 1 X X 1 1 1 1 1 1 X X

Present State Q 1 Q 0

0 0

0 1

1 0

1 1

D

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

N

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

Inputs Output Open

0 0 0 X 0 0 0 X 0 0 0 X 1 1 1 X


Step 5. Choose FF type for implementation

J-K FF

excitation tableNext State

D 1 D 0

0 0 0 1 1 0 X X 0 1 1 0 1 1 X X 1 0 1 1 1 1 X X 1 1 1 1 1 1 X X

Present State Q 1 Q 0

0 0

0 1

1 0

1 1

D

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

N

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

Inputs K 1

X X X X X X X X 0 0 0 X 0 0 0 X

K 0

X X X X 0 1 0 X X X X X 0 0 0 X

J 1

0 0 1 X 0 1 1 X X X X X X X X X

J 0

0 1 0 X X X X X 0 1 1 X X X X X

Output Open

0 0 0 X 0 0 0 X 0 0 0 X 1 1 1 X

K-map for K1K-map for J1

K-map for K0K-map for J0

J1 = D + Q0 N

K1 = 0

J0 = Q0 N + Q1 D

K0 = Q1 N

OPEN = Q1 Q0

7 Gates

OPEN Q 1

\ Q 0

N

Q 0 J

K R

Q

Q

J

K R

Q

Q

Q 0

\ Q 1

\ Q 1

\ Q 0

Q 1

\reset

D

D

N

N

CLK

CLK


Step 6. Combinational circuit minimization

Q1 Q0D N

Q1

Q0

D

N

00 01 11 10

00

01

11

10

X X

1

X X X X

1 1 X X

X X

Q1 Q0D N

Q1

Q0

D

N

00 01 11 10

00

01

11

10

0 0

X

X X X X

X X 0 0

0 0

X

X

X

Q1 Q0 Q1

Q0

D

N

00 01 11 10

00

01

11

10

X 0

X

X X X X

0 X X 1

X 1

0

1

X

Q1 Q0 Q1

Q0

D

N

00 01 11 10

00

01

11

10

0 X

1

X X X X

X 0 0 X

0 X

X

X

0

0 0

0

D N D N


D FF easiest to use

D1 = Q1 + D + Q0 N

D0 = N Q0 + Q0 N + Q1 N + Q1 D

OPEN = Q1 Q0

8 Gates

CLK

OPEN

CLK

Q 0

D

R

Q

Q

D

R

Q

Q

\ Q 1

\reset

\reset

\ Q 0

\ Q 0

Q 0

Q 0

Q 1

Q 1

Q 1

Q 1

D

D

N

N

N

\ N

D 1

D 0

K-map for OpenK-map for D0 K-map for D1

Q1 Q0D N

Q1

Q0

D

N

Q1 Q0D N

Q1

Q0

D

N

Q1 Q0D N

Q1

Q0

D

N

00 01 11 10 00 01 11 10 00 01 11 10

00

01

11

10

00

01

11

10

00

01

11

10

1 1

1

X X X X

1 1 1 1

1 1

1 1

1 1 1

X X X X

1 1 1

X X X X

1

1

1

Moore and Mealy Machine Design Procedure

Definitions

Moore Machine

Outputs are functionsolely of the current

state

Outputs change synchronously with

state changes

Mealy Machine

Outputs depend onstate AND inputs

Input change causesan immediate output

change

Asynchronous signals

State Register Clock

State Feedback

Combinational Logic for

Outputs and Next State

X Inputs

i Z Outputs

k

Clock

state feedback


Next State (Flip-flop Inputs)

State Register

Comb. Logic for Outputs

Z Outputs

k

X Inputs

i

Moore and Mealy Machines

State Diagram Equivalents

Outputs are associated with State

Outputs are associated with Transitions

Reset/0

N/0

N/0

N+D/1

15¢

0¢

5¢

10¢

D/0

D/1

(N D + Reset)/0

Reset/0

Reset/1

N D/0

N D/0

MooreMachine Reset

N

N

N+D

[1]

15¢

0¢

5¢

10¢

D

[0]

[0]

[0]

D

N D + Reset

Reset

Reset

N D

N D

MealyMachine

Moore and Mealy MachinesStates vs. Transitions

Mealy Machine typically has fewer states than Moore Machinefor same output sequence

Same I/O behavior

Different # of states

1

1

0

1

2

0

0

[0]

[0]

[1]

1/0

0

1

0/0

0/0

1/1

1

0

Moore and Mealy Machines

Synchronous Mealy Machine

latched state AND outputs

avoids glitchy outputs!

State Register Clock

Clock


Outputs and Next State

state feedback

X Inputs

i Z Outputs

k

• Mealy model of a serial adder

Q D

Q C

ai

bi

si

couti

cini

glitchglitch

CLRclock

reset

• State table

• State diagram

present input ai, bi

presentstate cini

01

00 01 11 10

0, 00, 1

0, 11, 0

1, 01, 1

0, 11, 0

next state couti, output si

S0cini=0

reset

S1cini=1

01/010/011/1

00/001/110/1

11/0

00/1

• Mealy to Moore conversion

S0cini=0

reset

S1cini=1

01/010/011/1

00/001/110/1

11/0

00/1

S00cini=0si=0

reset

S10cini=1si=0

011000

11

00

S01

cini=0si=1

S11

cini=1si=1

11

000110

0110 00

11

11 0110

present input ai, bi

presentstate cini

S00S01S10S11

00 01 11 10

S00S00S01S01

S01S01S10S10

S10S10S11S11

S01S01S10S10

next state

output si

0101

State assignment :State assignment :

S = (cini si) S00 = 00S01 = 01S10 = 10S11 = 11

4 states : 2 FFs

Q D

Q C

ai

bi

si

couti

cini

glitchglitch

CLRclock

reset

Q D

Q CCLR

no glitch

delayed by1 clock period

more states (FFs)

Q D

Q C

ai

bi

si

couti

cini

glitchglitch

CLRclock

reset

QD

QCCLR

no glitch

Synchronous Mealy machineSynchronous Mealy machine

Timing Control and Clocks

clock

input combinationallogic

output

FF

tc

tff

tsu th

tw

T

T > tffmax + tc

max + tsumax

th < tffmin + tc

min

(usually th < tffmin)

tw > twmin

tPLH = tPHL = 2 nstc

max = 6 ns (use timing analyzer)tff

max = tsumax = 5 ns

E = 4 ns (tolerance)T = tff

max + tcmax + tsu

max + E = 20 ns (50 MHz clock)tw

min = 3 ns

tsu = 5 nstff = 5 ns

tw = 10 ns

T = 20 ns

tc = 6 ns E = 4 ns

Finite State Machine Word Problems

Finite String Pattern Recognizer

A finite string recognizer has one input (X) and one output (Z).The output is asserted whenever the input sequence …010 …has been observed, as long as the sequence 100 has never beenseen.

Step 1. Understanding the problem statement

Sample input/output behavior:

X: 00101010010 …Z: 00010101000 …

X: 11011010010 …Z: 00000001000 …


Finite String Recognizer

Step 2. Draw State Diagrams for the strings that must berecognized. i.e., 010 and 100.

Moore State DiagramReset signal places

FSM in S0

Outputs 1 Loops in State

S0 [0]

S1 [0]

S2 [0]

S3 [1]

S4 [0]

S5 [0]

S6 [0]

Reset

0

1

0

1

0

00,1

Finite State Machine Word ProblemsFinite String Recognizer

Exit conditions from state S3: have recognized …010if next input is 0 then have …0100!if next input is 1 then have …0101 = … 01 (state S2)

S0 [0]

S1 [0]

S2 [0]

S3 [1]

S4 [0]

S5 [0]

S6 [0]

Reset

0

1

0

1

0

00,1

1

0

… 01

… 010… 100


Exit conditions from S1: recognizes strings of form … 0 (no 1 seen)loop back to S1 if input is 0

Exit conditions from S4: recognizes strings of form … 1 (no 0 seen)loop back to S4 if input is 1

S0 [0]

S1 [0]

S2 [0]

S3 [1]

S4 [0]

S5 [0]

S6 [0]

Reset

0 1

0

1

0

1

0

00,1

1

0

… 01

… 010… 100


S2, S5 with incomplete transitions

S2 = …01; If next input is 1, then string could be prefix of (01)1(00)S4 handles just this case!

S5 = …10; If next input is 1, then string could be prefix of (10)1(0)S2 handles just this case!

Final State Diagram

S0 [0]

S1 [0]

S2 [0]

S3 [1]

S4 [0]

S5 [0]

S6 [0]

Reset

0 1

0

1

0

1

0

00,1

1

0

… 01

… 010… 100

1

1

… 10


Finite String RecognizerReview of Process:

• Write down sample inputs and outputs to understand specification

• Write down sequences of states and transitions for the sequencesto be recognized

• Add missing transitions; reuse states as much as possible

• Verify I/O behavior of your state diagram to insure it functionslike the specification

Finite State Machine Word ProblemsTraffic Light Controller

A busy highway is intersected by a little used farmroad. DetectorsC sense the presence of cars waiting on the farmroad. With no caron farmroad, light remain green in highway direction. If vehicle onfarmroad, highway lights go from Green to Yellow to Red, allowing the farmroad lights to become green. These stay green only as long as a farmroad car is detected but never longer than a set interval. When these are met, farm lights transition from Green to Yellow to Red, allowing highway to return to green. Even if farmroad vehicles are waiting, highway gets at least a set interval as green.

Assume you have an interval timer that generates a short time pulse(TS) and a long time pulse (TL) in response to a set (ST) signal. TSis to be used for timing yellow lights and TL for green lights.


Picture of Highway/Farmroad Intersection:

Highway

Highway

Farmroad

Farmroad

HL

HL

FL

FL

C

C


• Tabulation of Inputs and Outputs:

Input SignalresetCTSTL

Output SignalHG, HY, HRFG, FY, FRST

Descriptionplace FSM in initial statedetect vehicle on farmroadshort time interval expiredlong time interval expired

Descriptionassert green/yellow/red highway lightsassert green/yellow/red farmroad lightsstart timing a short or long interval

• Tabulation of Unique States: Some light configuration imply others

StateS0S1S2S3

DescriptionHighway green (farmroad red)Highway yellow (farmroad red)Farmroad green (highway red)Farmroad yellow (highway red)


State diagram:

S0: HG

S1: HY

S2: FG

S3: FY

Reset

TL + C

S0TL•C/ST

TS

S1 S3

S2

TS/ST

TS/ST

TL + C/ST

TS

TL • C

Finite State Machine Design - SNU DALdal.snu.ac.kr/~kchoi/class/lc_intro/fsm_des.pdf · Finite State Machine Word Problems Finite String Pattern Recognizer A finite string recognizer

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