Introduction to State Machine

• Combinational circuit design process had two important things:1. A formal way to describe desired circuit

behavior• Boolean equation, or truth table

2. A well-defined process to convert that behavior to a circuit

• We need those things for sequential circuit design

• Finite-State Machine (FSM)– A way to describe desired behavior of

sequential circuit• Akin to Boolean equations for combinational

behavior

The state machineDefinition : A state machine is a system that can be used to

describe the system in terms of set of states that the system goes through.

In this type of system, memory capability of system is a must. Also, the state machine must have a set of inputs and outputs.

Clocked synchronous FSM• Clocked:

all storage elements employ a clock input (i.e. all storage elements are flip-flops)

• Synchronous:all of the flip flops use the same clock signal

• FSM state machine is simply another name for sequential

circuits. Finite refers to the fact that the number of states the circuit can assume if finite

• A synchronous clocked FSM changes state only when a triggering edge (or tick) occurs on the clock signal

Clocked synchronous FSM structure

• States: determined by possible values in sequential storage elements

• Transitions: change of state• Clock: controls when state can change by controlling storage

elements

Storage Elements

CombinationalLogic

Outputs

Next StateCurrent State or State

Inputs

Clock

FSM Types• There are two main types of FSMs

– Mealy (output is function of state and inputs)– Moore (output is only function of state)

Mealy machine

Next state = F1(current state, inputs)

Output = G1(current state, inputs)

Moore machine

Next state = F2(current state, inputs)

Output = G2(current state)

Comparison of Mealy and Moore FSM• Mealy machines have less states

– outputs are on transitions (n2) rather than states (n)• Moore machines are safer to use

– outputs change at clock edge (always one cycle later)– in Mealy machines, input change can cause output change

as soon as logic is done – a big problem when two machines are interconnected – asynchronous feedback may occur if one isn’t careful

• Mealy machines react faster to inputs– react in same cycle – don't need to wait for clock– outputs may be considerably shorter than the clock cycle– in Moore machines, more logic may be necessary to

decode stateinto outputs – there may be more gate delays after clock edge

Mealy and Moore example

D Q

QB

A

clock

out D Q

Q

D Q

Qclock

outA

B

B

Aout

Mealy or Moore ?

Not a state machine

Moore:output =

(state) Moore:output = (state)

Mealy and Moore example (cont’d)

Mealy or Moore ?

Moore:output = (state)

Mealy:output = (state,

input)

D Q

Q

D Q

Q

D Q

Q

D Q

Q

A

B

clock

out

D Q

Q

D Q

Q

A

B

clock

out

Analysis of state machine

How to design a circuit when output waveform of the system is known. Let’s take the two outputs Q1 and Q2.

• The output changes only on the negative clock transitions, thus signal can be generated by using clocked flip flop.

If D flip flop is to be used, then using the excitation table of D flip flop, determine the inputs of the D flip flops D1 and D2 whose outputs will be Q1 and Q2 respectively.

• Sometimes it may happen that we are given with set of input conditions and output is to be obtained.

• First of all, find the set of inputs to the flip flop under consideration using excitation table. Lets take D flip flop for this case

• Now , the inputs of the D flip flop is to be determined with the help of input combination.

• The K-Map for the D1 and D2 as a function of X and Y are derived as follows

• Circuit diagram

• For the given set of inputs and output, design the system using JK flip flop

Solution

Finite state machine• A FSM is an abstract model describing the

synchronous sequential machine.

• FSM design involves drawing state diagram for the problem which is also known as word problem.

• One state diagram is drawn, the next steps involves reduction, state assignments and its realization or design.

Some examples of FSM

• Positive transition detector i.e. system which gives output 1, whenever the inputs to the system changes from 0 to 1.

• Vending machine problems• Serial Adder• Serial Code Converters (BCD to Excess-3)• TLC( Traffic Light Controller)• Sequence detectors

Positive transition detector

• Suppose we have to design a sequential system which is having one serial input IN and will produce output OUT=1 whenever input IN changes from 0 to 1. Input is received one bit per cycle.

Step 1: Understand the problemStep 2: Draw state diagramStep3: reduction , state assignments and

design

• Step 1: The system is to have one input line and one output line. Since the possible transitions can 0 to 1 and 1 to 0, so we can assume that we have only two state. Out of these two state one state will represent that current input is 1 and other state will represent current input as 0.

• Step 2: Draw state diagram

• Step 3: No reduction needed.• Step 4: State Assignments: Since we have only two

states, so we need only one bit to code the two states. In our case lets assign 0 to state Zero and 1 to state One.• Step 5: Designs the circuit diagram for this form state table as shown below.

Present state QD

Input (IN) Next state QD+1

Output (OUT) DA

0 0 0 0 0

0 1 1 1 1

1 0 0 0 0

1 1 1 0 1

Forming K – Map for DA and OUT we have QD

IN

0 1

0 0 0

1 1 1

QD

IN

0 1

0 0 0

1 1 0

OUT=Q’D.IN DA= IN

• Circuit Diagram:

Vending Machine Problem

• The vending machine delivers package of gum after receiving minimum of 15 cents of coin. The machine has a single slot that accepts 10 cent coin or 5 cent coin one at a time. A mechanical sensor indicates whether 10 cent coin or 5 cent coin have been inserted.

• Step 1: Understand problemAssuming that sensor gives two outputs which

sense the coins. Lets take T signal for 10 cent coin and F signal for 5.

As per given condition the gum is delivered only when the coin inserted is minimum 15.

• There is 5 possible best way of getting 15 cent minimum.

1. 5 > 5 > 5 ( 3 continuous 5 cent coin)

2. 5 > 10 (5 cent follows with 10 cent coin)

3. 10 > 5 (10 cent follows with 5 cent coin) 4. 10 > 10 (2 continuous 10 cent coin)

5. 5 > 5 > 10 (2 continuous 5 cent coin follows

with one 10 cent coin)

• Since once a package is delivered, the machine should be in initial state for the next customer.

• Also T=1 indicates 10 cent coin and F=1 indicates 5 cent coin.

• We can’t have T=1 and F=1 at a same time.

• Lets start drawing state diagram.

Step 2: State Diagram• Initially we will start with the state where nothing is

being sensed i.e. T=0 and F=0.• The system will be in the initial state until T or F

is not 1. Once T or F is detected we can start with states. If T=1 means 10 cent coin is being received so this will be indicated by state S1. If initially F=1 means 5 cent coin is received, so we will indicate this as state S2.

• After S1, if T=1 means (case 4 ) i.e. our condition is satisfied and package will be delivered so it will be indicated by state SOPEN.

Step 3: State Reduction• The details techniques for the state

reduction will be discussed later.• Here we can directly draw the reduced

state diagram by care full observations. 5 cent coin follows by 10 cent coin will be

having same state when 10 cent coin follows by 5 cent coin.

Similarly 10 cent coin state received from starting state is equivalent to two consecutive 5 cent coin.

Reduced state diagram

State Assignments and designPresent State Q0 Q1

T F Next State Q0 Q1

Output Z

COMMENT

0 0 0 0 0 0 0 NO CHANGE

0 0 0 1 0 1 0 STATE 01

0 0 1 0 1 0 0 STATE 10

0 0 1 1 X X 0 NOT ALLOW STATE

0 1 0 0 0 1 0 NO CHANGE

0 1 0 1 1 0 0 STATE 10

0 1 1 0 1 1 1 STATE 11

0 1 1 1 X X 0 NOT ALLOW STATE

1 0 0 0 1 0 0 NO CHANGE

1 0 0 1 1 1 1 STATE 11

1 0 1 0 1 1 1 STATE 11

1 0 1 1 X X 0 NOT ALLOW STATE

1 1 0 0 0 0 0 TO RESET

1 1 0 1 0 0 0 TO RESET

1 1 1 0 0 0 0 TO RESET

1 1 1 1 00 0 TO RESET

FSM for serial adder

• Suppose we have to add two inputs X and Y where each bits of the inputs are coming serially as shown below

State Diagram for serial adder

• Since carry generated from first bit addition must be added with the next bits, so care must be taken.

• Lets we have two states A,B. State A indicating carry=0 and the state B ,indicating carry=1.

State table, state assignment Present state Inputs

X YNext state Output

(Sum)

A 0 0 A 0

A 0 1 A 1

A 1 0 A 1

A 1 1 B 0

B 0 0 A 1

B 0 1 B 0

B 1 0 B 0

B 1 1 B 1

Present state QN

InputsX Y

Next state QN+1

Output (Sum)

DQ

0 0 0 0 0 0

0 0 1 0 1 0

0 1 0 0 1 0

0 1 1 1 0 1

1 0 0 0 1 0

1 0 1 1 0 1

1 1 0 1 0 1

1 1 1 1 1 1

a. State Table b. State Table after state assignment

• K-Maps and circuit design

XYQN

00 01 11 10

0 0 0 1 0

1 0 1 1 1

XY QN

00 01 11 10

0 0 1 0 1

1 1 0 1 0

K-Map for DQ=XY +XQN+YQN K-Map for sum= XΘYΘQN

Where Θ is Exor operation.

Simplified Presentation

Serial code converters

• Binary to Gray code converter

• Gary to binary code converter( self)

• BCD to Excess-3 code converters ( Home Assignment)

Binary to Gray code converter• In these type of system, input bit stream is

coming serially and output is also expected serially.

• Here we have to use the truth table of converter wisely.

Binary Gray

X2 X1 X0 Y2 Y1 Y0

0 0 0 0 0 0 0

1 0 0 1 0 0 1

2 0 1 0 0 1 1

3 0 1 1 0 1 0

4 1 0 0 1 1 0

5 1 0 1 1 1 1

6 1 1 0 1 0 1

7 1 1 1 1 0 0

1. Whenever MSB is 0 corresponding Gary code value is 0 and when it’s 1 the Gray code is 1.

2. Whenever the second bit, after 0 MSB ,is 0 then the corresponding Gary code is 0 else its 1.

3. Whenever the second bit, after 1 MSB ,is 0 then the corresponding Gary code is 1 else its 0.

4. Whenever the last bit, after 00 ,is 0 then the corresponding Gary code is 0 else its 1.

5. Whenever the last bit, after 01 ,is 0 then the corresponding Gary code is 1 else its 0.

6. Whenever the last bit, after 10 ,is 0 then the corresponding Gary code is 0 else its 1.

7. Whenever the last bit, after 11 ,is 0 then the corresponding Gary code is 1 else its 0.

• Draw State diagram using the state table given in the previous slide

• Then using the appropriate state assignment and K-Map realization , design the system.

Sequence Detector

• This is special type of systems which is used whenever we need to check a particular pattern in the input sequence

• Suppose input to the system is serial and we have to design the system such that whenever “101” is detected in the input, output will be 1.

INPUT : 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0

OUTPUT : 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0

• There are two types of sequence detector. (a) overlapping sequence detector (b) non-overlapping sequence detector

In (a), we consider the input bit which may be common or overlapping while deciding a pattern or desired sequence in the input bit stream.

Let’s take example of sequence detector for 101. If input bit sequence is 01010101…. , then output expected is 00010101 since when fourth bit is received , pattern 101 is received first time and output is available.

After that if two bits received are 0 , 1 , then the 1 that is already there while forming pattern of 1 0 1 for the first time will again form patter along with the two bit 0 ,1. so output will be 1 again on 6th input bit, i.e the 1 in the 4th location in the input is common in both 101 pattern before and after this 1.

In (b) we need not to consider the input under overlapping condition ie for obtaining desired sequence, a input bits can not be considered more then once.

How to draw state diagram for sequence detector

The input to the system is serially. Lets input is X and output is Z. 101, sequence detector means out Z=1 whenever in the input stream X, we have 101 pattern ie for 01010101…. , as input stream , then output Z expected is 00010101…….

• Class work : draw state diagram for 1001 sequence detector

Non Overlapping Sequence Detector• In non overlapping type of sequence detector, output will

be decided once the desired pattern is received and no output will be there for any other pattern.

Let draw non-overlapping 101 sequence detector. While drawing state diagram for non overlapping type of system, it’s very easy to draw the state diagram since here each bit is being treated independently as non overlapping. As in the case of 101 detector, for input stream:

1001100101010101010…… Output will be : 0000000000010000010….Chose a pair of three bits and searching for 101. If 101

received then output is 1 otherwise output is zero.

EXAMPLE: Draw state diagram for 3 bit palindrome checker i.e. a system is taking input bit serially and output will be 1 when the three bits form a palindrome sequence.