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STATE DIAGRAM AND STATE TABLES
DERIVATION OF STATE GRAPHS
Problem Statement specifies the desired relationship between the input and output sequences. Sometimes called the specification.
First step is to translate this specification into a state table or state graph.
In the HDL world, there is a style that allows creation of the next state specification that does not require either a state graph or state table.
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STATE DIAGRAM State transition diagram
a circle: a state a directed lines connecting the circles: the
transition between the statesEach directed line is labeled
“inputs/outputs”
state: A Binput: x
FLIP-FLOP INPUT EQUATIONS The part of circuit
that generates the inputs to flip-flops Also called excitation
functions DA = Ax +Bx DB = A'x
The output equations to fully describe the
sequential circuit y = (A+B)x'
Ax +Bx
Ax
Bx
A 'x
A+B
ANALYSIS WITH D FLIP-FLOPS The input equation
DA=A⊕x⊕y The state equation
A(t+1)=A⊕x⊕y
ANALYSIS WITH JK FLIP-FLOPSDetermine the flip-flop input function in
terms of the present state and input variables
Used the corresponding flip-flop characteristic table to determine the next state
Fig. 5-18Sequential circuit with JK flip-flop
JA = BKA= Bx'JB = x ' KB = A'x + Ax '
STATE TABLE FOR FIG. 5-18
JA = B, KA= Bx'JB = x ', KB = A'x + Ax '
State Transition Diagram for Fig. 5-18
( 1)
( 1)
A t JA K A
B t JB K B
State equation for A and B :
( 1) ( )A t BA Bx A A B AB Ax
The characteristic equation of JK FF is
Method 1
( 1) ( )B t x B A x B B x ABx A Bx , ,
State Transition Diagram for Fig. 5-18
x AB00 01 11 10
0
1A(t +1)
0 1 0 1
0 1 1 1
A’B
AB’
Ax
Using K-map, we also can derive A(t+1).
A(t +1)=A ’B+AB ’+Ax
Method 2
ANALYSIS WITH T FLIP-FLOPS The characteristic equation
Q(t+1)= T⊕Q = TQ'+T'Q
A SEQUENCE DETECTOR EXAMPLE
The specification The circuit will examine a string of 0’s and 1’s
applied serially, once per clock, to the X input and produce a 1 only when the prescribed input sequence occurs. Any sequence ending in 101 will produce and output of Z=1 coincident with the last 1 input. The circuit does not reset when a 1 output occurs so when ever a 101 is in the data stream a 1 is output coincident with the last 1.
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GENERAL FORM OF THE CIRCUIT
The circuit has the general form X – serial input stream Z – serial output stream Clk – the clock
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START CONSTRUCTION OF THE GRAPH.
Choose a starting state and a meaning for that state. The starting state is typically a reset state. Here meaning of starting state can be
The system has been reset and this is the initial state
A sequence of 2 or more 0’s has been received
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ADD THE NEXT STATE
Now add state S1 Meaning – a sequence of 0…01 has been
received when coming from state S0 Meaning – the first 1 has been received.
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TRANSITIONS FROM S1
What happens when in S1 A 0 input causes transition to a new state S2
with new meaning A 1 keeps you in S1 where the first 1 of a
possible 101 sequence has occurred.
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STATE S2
State S2 – what is the meaning of being here? When transition is from S1 it means we have
receive an input stream of xxx10.
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TRANSITIONS FROM S2
Are currently in S2 A 1 arrives and now have a sequence of 101
Action – Output a 1 and have the first 1 of a new sequence, i.e., transition to S1
A 0 arrives – now have a sequence of 100Action – Move back to state S0 where you do not even have the start of a sequence, i.e., one or more 0 inputs.
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THE FULL STATE DIAGRAM
The now completed state diagram
This can now be used to generate a state table – more on that later
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ANOTHER EXAMPLE
Problem Statement: The circuit has the same form as before and shown below. The circuit will detect input sequences that end in 010 or 1001. When a sequence is detected the output Z is 1, otherwise Z is 0.
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THE INITIAL STATE
The RESET state – have no inputs yet Then if you have a 0 input the output is 0 –
transition to S1 If you have a 1 input the output is 0 and
transition to S2
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MEANING OF STATES
S0 – Reset S1 – 0 but not 10 S4 – 1 but not 01
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MORE STATES
Add S2 having meaning that a 01 sequence has been received.
Add S3 having meaning that the sequence 10 has been received
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MEANING OF STATES AFTER S2 S3
S0 – Reset S1 – 0 but not 10 S2 – Sequence of 01 S3 – Sequence of 10 S4 – 1 but not 01
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CONSIDER INPUTS WHEN IN S2, S3
In S2 (01) and get a 0 – Transition to S3 (10) – output a 1
In S3 (10) and get a 1 – Transition to S2 (01)
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ADD A NEW STATE S5
S5 – Have received input sequence 100
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WHEN IN S5
In S5 Input of a 1 means you have
had a input of 1001 so transition to S2 as the input sequence now ends in 01 while Z is 1.
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ADD OTHER TRANSITIONS
Complete the transitions not yet covered
Each state should have an output transition for both a 0 and a 1.
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THE MEANING OF THE STATES
S0 – Reset S1 – 0 (but not 10) S2 – Sequence of 01 S3 – Sequence of 10 S4 – 1 (but not 01) S5 – Sequence of 100
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GUIDELINES
Guidelines for Construction of State Graphs First, construct some sample input and output
sequences to make sure you understand the problem (ref slides 5 and 13)
Determine under what conditions the circuit is in reset state.
If only one or two sequences lead to a 1 output construct a partial state graph.
OR determine what sequences or groups of sequences must be remembered
When adding transitions see if you transition to a defined state or a new state is to be added
Make sure all state have a transition for both a 0 and a 1 but only 1!
Add annotation or create a table to expound the meaning of each state.
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MOORE MACHINE
MEALY MACHINE VS. MOORE MACHINE
Register-transfer-level block diagram
Modern Design
C: Combinational circuitS: Sequential circuit
C
C
CCS
S
S C
Control UnitDatapath
DQ
DQ
.
.
In OutDX
clk
QX
DY
clk
QY？ ？
next state logic output logic
current state register
nextstate
currentstate
D or JK or T ??
FSM DESIGN
CONSIDER THE SEQUENCE DETECTOR
The same sequence detector to detect a sequence ending in 101 but this time a Moore machine implementation.
Moore machine implementation is much the same except that the output designation is now indicated within the state.
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START IN S0
S0 –a state where you have received a non middle 0 or a long string of 0s. Output is 0.
Output is indicated within the state not on the transition.
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TRANSITIONS FORM STATE 1
On a 0 you stay in state 1 On a 1 you transition to state S1. Meaning of
S1 – have the 1st 1 of the sequence
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TRANSITION FROM S1
On a 1 have the first 1 of a sequence – stay in S1.
On a 0 now have a sequence that ends in 10 so define a new state S2 and transition to it.
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STATE S2
S2 has meaning that you have an input sequence that ends in 10 so far.
Transitions from S2 0 input – Back to S0 1 input – Valid sequence go to new state S3 which outputs a 1
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STATE S3
S3 – have received input sequence that ends in 101.
Next input 0 – end of seq (10 so back to S2) 1 – back to S1 (11 so 1st 1)
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STATE TABLE FROM STATE GRAPH
Easy to convert state graph to state table Moore machine note output is function of the state
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CONTRAST THIS TO MEALY MACHINE
Mealy machine state graph and state table In Mealy machine the output is a function of the state and the current input
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NOW ON TO THE OTHER EXAMPLE
Detect the sequences 010 and 1001 and on those output a 1.
Starting state on reset is S0 On a 0 transition to S1 - output 0
Have a first 0 On a 1 transition to S3 - output 0
Have a first 1
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IN S1/0
State S1 have the first 0 of a possible 010 On a 1 now have 01 Transition to a new state S2/0 with meaning
that you have 01 On a 0 stay in S1/0
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FROM S2/0
S2/0 has meaning that you have 01 so far Input is a 0 – Need a new state S4
with meaning that you have received 010 (so output is a 1) and have a 10 for a start of that string.
Input is a 1 so the input is 011 – Go to S3 where as this is the first 1.
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FROM S3/0
S3/0 has meaning that you have the first 1 of the 1001 sequence. Input is a 0 – Go to S5 – meaning have 10 Input is a 1 – stay in S3
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ADD TRANSITIONS FROM S4/1
S4/1 had meaning that the sequence has been 010 so far. Input is a 0 – Now have
100 – Need a new state with this meaning – S6/0
Input is a 1 – Now have 101 so go back to S2/0
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TRANSITIONS FROM S5/0
S5/0 means you have 10 so far Input is a 0 – transition to S6/0 – have 100 so far Input is a 1 – now have 101 or the 01 which is
the meaning of S2/0
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STATE S6/0
S6/0 has meaning that you have a sequence of 100 so far Input is a 1 so have 1001
– a new state S7/1 to signal the sequence 1001.
Input is a 0 so have 1000 and back to S1 as you have a first 0.
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FROM S7/1
S7 has meaning of 1001 so you also have the 01 for the start of that sequence Input is a 0 so have 010
– go to S4/1 Input is a 1 so have 011
– go to S3 as you have a first 1.
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THE STATE TABLE FOR EACH
For the Mealy Machine
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NEXT STATE OUTPUT Present State X=0 X=1 X=0 X=1
S0 S1 S4 0 0 S1 S1 S2 0 0 S2 S3 S4 1 0 S3 S5 S2 0 0 S4 S3 S4 0 0 S5 S1 S2 0 1
FOR THE MOORE MACHINE
The state table for the Moore machine – output is associated with the state.
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Present State Next State X=0 Next State X=1 Output ZS0 S1 S3 0S1 S1 S2 0S2 S4 S3 0S3 S5 S3 0S4 S6 S2 1S5 S6 S2 0S6 S1 S7 0S7 S4 S3 1
THE NEXT STEP
The next step to implementation is state assignment
In state assignment the binary code for each state is chosen.
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EFFECT OF CHOOSING STATE ASSIGNMENT
Choosing one state assignment versus another can have significant implications for circuit implementation.
But first – how do you reduce the number of states in the state table?
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EXAMPLE THAT HAS SINK STATE
Programmed Example 14.2
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INITIAL STATES
The start of the state graph
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STEP 2
More states
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COMPLETE STATE GRAPH
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CORRESPONDING STATE TABLE
From the state graph the state table can be generated
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LECTURE SUMMARY
Have covered state graphs for Mealy and Moore machines
Have covered how to transition from state graphs to state tables.
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