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Finite State Machines (FSMs)

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Finite State Machines (FSMs)

- Today:
- First Hour: FSM Concept
- Section 8.1 of Katz’s Textbook
- In-class Activity #1

- Second Hour: Design Example w/ FSM
- Section 8.2 of Katz’s Textbook
- In-class Activity #2

A Precursor of Finite State Machines

• Counters: Simple sequential circuits

State = Output

No inputs

Simple single-path sequencing through the states

• 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" or “control” logic

Described by State Diagrams, much the same way that combinational logic circuits are described by Boolean Algebra.

Current

State

[output]

New

State

[output]

Current Input(s)

Change of state happens only on the clocking event

001

001

010

010

000

000

Each circle corresponds to a state

The label inside each circle describes the state

011

011

111

111

Arrows represent state transitions

101

101

100

100

110

110

No labels on arrows, since the counter has no inputs

Reset

0

Even

[0]

1

Symbolic State Transition Table

1

Odd

[1]

0

State

Diagram

Encoded State Transition Table

Asserts output whenever input bit stream (seen so far) has odd # of 1's

Observe that the output in this case depends only

upon the present state, and not upon the input.

QQ+T

000

011

101

110

QQ+D

000

011

100

111

T F/F: Excitation Table

D F/F: Excitation Table

D F/F inputs are identical to the next

state outputs in the state transition table

Excitation/Output Functions

D = PS Input; Output = PS

Output

Input

Q

D

T

Input

Clock

D

Q

PS/Output

Q

Clock

R

Q

R

\Reset

\Reset

D FF Implementation

T FF Implementation

Input

1

0

0

1

1

0

1

0

1

1

1

0

Clock

1

1

1

0

1

1

0

0

1

0

1

1

Output

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

When are inputs sampled, next states computed, outputs asserted?

State Time: Time between clocking events

• Clocking event causes state/outputs to transition, based on inputs

• For set-up/hold time considerations:

Inputs should be stable before clocking event

• After propagation delay, Next State entered, Outputs are stable

NOTE: Asynchronous signals take effect immediately

Synchronous signals take effect at the next clocking event

E.g., 3-state enable: effective immediately

sync. counter clear: effective at next clock event

State Time

Clock

Inputs

Outputs

Positive Edge Triggered Synchronous System

On rising edge: inputs sampled,

outputs & next state computed

After propagation delay: outputs and next state are stable

Immediate Outputs

affect datapath immediately

could cause inputs from datapath to change

Delayed Outputs

take effect on next clock edge

propagation delays must exceed hold times

One machine's output is another machine's input

[0], [1]

outputs

- Could be used to model:
- bus protocols, handshaking,
- 2-way communications, etc.

Machines advance in lock step

Initial inputs/outputs: X = 0, Y = 0

Six Step Process

- 1. Understand the statement of the Specification
- 2. Obtain an abstract specification of the FSM
- 3. Perform a state minimization
- 4. Perform state assignment
- 5. Choose FF types to implement FSM state register
- 6. Implement the FSM

General Machine Concept

- deliver package of gum after 15 cents is deposited
- single coin slot for dimes, nickels
- no change

INPUTSOUTPUTS

Block Diagram

Step 1. Understand the problem

Draw a picture!

Tabulate typical input sequences

three nickels

nickel, dime

dime, nickel

two dimes

two nickels, dime

Draw state diagram

Inputs: N, D, reset

Output: open

Step 2. Map into more suitable abstract representation

reuse states

whenever possible

Symbolic State Table

Step 3: State Minimization

Step 4: State Encoding

How many flip-flops are needed?

Q1

Q1

Q1

Q1 Q0

Q1 Q0

Q1 Q0

D N

D N

D N

N

N

N

D

D

D

Q0

Q0

Q0

K-map for D1

K-map for D0

K-map for Open

Step 5. Choose F/Fs for implementation

D F/F easiest to use

J-K F/F

Present State

Inputs

Next State

J

J

K

K

1

1

0

0

Q

Q

Q1+

Q0+

D

N

1

0

0

0

0

0

0

X

0

X

0

0

0 1

0

X

1

X

0

1

1 0

1

X

0

X

1

0

X X

X

X

X

X

1

1

0 1

0 1

0

X

X

0

0

0

1 0

1

X

X

1

0

1

1 1

1

X

X

0

1

0

X X

X

X

X

X

1

1

1 0

1 0

X

0

0

X

0

0

1 1

X

0

1

X

0

1

1 1

X

0

1

X

1

0

X X

X

X

X

X

1

1

1 1

1 1

X

0

X

0

0

0

1 1

X

0

X

0

0

1

X

1 1

X

0

0

1

0

X

X X

X

X

X

1

1

Remapped encoded state transition table

Step 5. Choose FF for Implementation (continued)

D1 = Q1 + D + Q0 N

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

OPEN = Q1 Q0

Step 6. Implementation: D F/Fs

8 Gates

Q1

Q1

J1 = D + Q0 N

K1 = 0

J0 = Q0 N + Q1 D

K0 = Q1 N

Q1 Q0

Q1 Q0

D N

D N

N

N

D

D

Q0

Q0

K-map for J1

K-map for K1

Q1

Q1

Q1 Q0

Q1 Q0

D N

D N

N

N

D

D

Q0

Q0

K-map for J0

K-map for K0

Step 6. Implementation: J-K F/Fs

7 Gates

- Due: End of Class Today.
- RETAIN THE LAST PAGE(S) (#3 onwards)!!
- For Next Class:
- Bring Randy Katz Textbook, & TTL Data Book
- Required Reading:
- Sec 8.4 of Katz

- This reading is necessary for getting points in the Studio Activity!