State Machine Analysis and Design

Z. Jerry ShiComputer Science and Engineering

University of ConnecticutUniversity of Connecticut

Thank John Wakerly for providing his slides and figures.

Clocked synchronous sequential circuits

• A.k.a. “state machines”• Use edge-triggered flip-flopsUse edge triggered flip flops• All flip-flops are triggered from the same master clock signal, and

therefore all change state together

• Feedback sequential circuitsN li i fli fl d i f db k l– No explicit flip-flops; state stored in feedback loops• Example: edge-triggered D flip-flop itself (4 states)• Sections 7.9 (advanced courses)

Describing Sequential Circuits

• State table– For each current-state, specify next-states as function of inputsFor each current state, specify next states as function of inputs– For each current-state, specify outputs as function of inputs

• State diagram– Graphical version of state table State/output table

State-machine structure (Mealy)

output depends onstate and inputp

typically edge-triggered D flip-flopsp ops

State-machine structure (Moore)

output dependson state onlyy

typically edge-triggered D flip-flopsp ops

State-machine structure (pipelined)

• It is desirable that the outputs do not change during a cycle– Output-coded state assignmentp g

• Outputs taken directly from flip-flops, valid sooner after clock edge– Pipelined outputs

• Outputs depends on the input and state in the previous cycleOutputs depends on the input and state in the previous cycle

Notation, characteristic equations

• Q∗ means “the next value of Q.”• “Excitation” is the input applied to a device that determines theExcitation is the input applied to a device that determines the

next state.• “Characteristic equation” specifies the next state of a device as a

function of its excitation.

Latch and flip-flop characteristic equations

State-machine analysis steps

Assumption: Starting point is a logic diagram.1 Determine next-state function F and output function G1. Determine next state function F and output function G.2a. Construct state table

– For each state/input combination, determine the excitation value.

– Using the characteristic equation, determine the corresponding next-state values (trivial with D flip-flop’s).next state values (trivial with D flip flop s).

2b. Construct output table– For each state/input combination, determine the output value.

(Can be combined with state table.)3. (Optional) Draw state diagram

Example state machine

Excitation equations

Transition equations

• Excitation equations

• Characteristic equationsq

• Substitute excitation equations into characteristic equations

Transition and state tables

(Transition equations)

(Output equation)

Transition table State table State/output tableTransition table State table State/output table

State diagram

• Circles for states• Arrows for transitions (note output information)• Arrows for transitions (note output information)

Modified state machine

MAXS Q0 Q1

MAXS

MAXS = Q0 ⋅ Q1

• Moore machine

Updated state/output table, state diagram

Timing diagram for state machine

• State transitions take place if EN = 1• MAX may be changed by the input EN in a cycle• MAXS does not depend on the input ENp p

State-machine design and synthesis

The creative part like Turning the crank like aThe creative part, like writing a program

Turning the crank, like a compiler does

• Example: Design a combination lock with two inputs X1 and X2• Example: Design a combination lock with two inputs, X1 and X2. Open for the sequence X1, X2, X2 (one input per clock).

Example: a combination lock

• Design a combination lock with two inputs, X1 and X2. Open for the sequence X1, X2, X2.

State X1 X2-------------------------- -----------------------------------------------Meaning Name 00 01 10 11 UNLOCK-------------------------- ------------------------------------------------------------------------- -----------------------------------------------Start A A A B A 0Got X1 B A C A A 0(C)

(B)Got X1 B A C A A 0

Got X1,X2 C A D A A 0

G t X1 X2 X2 D A A B A 1(D)

(D)

(C)

Got X1,X2,X2 D A A B A 1(D)

State Assignment

• Can minimize number of states (see text), but hardly anyone bothers anymore.y

• Need to assign state-variable combinations to states.– Minimum number of variables for n states is ⎡log2 n⎤– Using more than minimum number may be advantageous in some

situations, e.g., one variable per state (“one-hot”) (see text).– Example -- 4 states, 2 state variables (Q1,Q2):p , (Q ,Q )

A ==> 00B ==> 01C ==> 10 Up to this point is “art”, the rest C ==> 10D ==> 11 is just “turning the crank.”

Transition table

• Substitute state-variable combinations for states in the state table.

State X1 X2-------------------------- -----------------------------------------------Meaning Q1 Q2 00 01 10 11 UNLOCK-------------------------- -----------------------------------------------St t 0 0 0 0 0 0 0 1 0 0 0Start 0 0 0 0 0 0 0 1 0 0 0Got X1 0 1 0 0 1 0 0 0 0 0 0

Got X1,X2 1 0 0 0 1 1 0 0 0 0 0

Got X1,X2,X2 1 1 0 0 0 0 0 1 0 0 1----------------------------------------------

Q1∗ Q2∗

Transition equations; circuit

• Transition table specifies each state variable (Q1∗, Q2∗) as a combinational logic function of Q1, Q2, X1, X2.g Q , Q , ,– Find a realization of each function by your favorite means -- ad hoc,

minimal sum-of-products, etc.B ild th i it• Build the circuit.

UNLOCKD QQ1

Q2

Q1Q1∗

D Q

Q2

X1

X2Q2∗

X2

CLKQ2

Another design example (from textbook)

• Design a machine with inputs A and B and output Z that is 1 if:– A had the same value at the two previous ticks, orA had the same value at the two previous ticks, or– B has been 1 since the last time the above was true

Timing diagram and state sequence

State assignment

• There are 6,720 different state assignments of 5 states to 3 variables.– And there are even more using 4 or more variables

• Here are a few “obvious” or “interesting” ones:

Transition/output table (decomposed assignment)

• Simple textual substitution• With D flip-flops, excitation table is identical to transition table.

Develop excitationDevelop excitation equations

• Assume unused states have next-state = 000

Don’t cares for the next-state entries of unused states

D1 = 1

D2 = Q1•Q3’•A’+Q3•A+ Q2•B

D3 = A

Logic diagram

D1 = 1

D2 = Q1•Q3’•A’+Q3•A+ Q2•BD2 = Q1•Q3 •A +Q3•A+ Q2•B

D3 = A

Read Chapter 7.5 for designing state machines using state diagrams

State machine design steps

• Construct a state/output table corresponding to the description or specification– Using mnemonic names for the states– May start with a state diagram

• Minimize the number of states (optional)• State assignmentState assignment

– Choose state variables and assign state-variable combinations to the named states

• Substitute the state variable combinations into the state/output table to create• Substitute the state-variable combinations into the state/output table to create the transition/output table

• Choose a flip-flop typeC t t it ti t bl th t h th i d it ti• Construct an excitation table that shows the required excitation

• Derive excitation equations from the excitation table• Derive output equations• Draw a logic diagram

1s-counting machine

• Design a state machine Two inputs, X and Y, and one output Z. Z h ld b 1 if h b f 1’ X d Y i i l i l f 4Z should be 1 if the number of 1’s on X and Y since reset is a multiple of 4, and 0 otherwise.

Transition/excitation and output table

• State assignment: S0-S3 = (00, 01, 11, 10)– Try a different state assignment in homeworkTry a different state assignment in homework

D1 = Q2 • X′ • Y + Q1′ • X • Y + Q1 • X′ • Y′ + Q2 • X • Y′D1 Q2 • X • Y + Q1 • X • Y + Q1 • X • Y + Q2 • X • Y D2 = Q1′ • X′ • Y + Q1′ • X • Y′ + Q2 • X′ • Y′ + Q2′ • X • Y Z = Q1′ • Q2′

Excitation maps for D1 and D2 inputs

Combinational lock

• Design a state machine

O i XOne input X. Two outputs UNLK and HINT.

UNLK is 1 if and only if X is 0UNLK is 1 if and only if X is 0 and the sequence received on X at the preceding seven clock ticks was 0110111was 0110111.

HINT is 1 if and only if the current value of X is the correct one to move the machine closer to being in the “unlocked” state (where UNLK = 1).

Transition/excitation and output table

• State assignmentA - H : 0 - 7A H : 0 7

K-maps for D1, D2, and D3

D1 = Q1•Q2′•X + Q1′•Q2•Q3•X′ + Q1•Q2•Q3′

D2 Q2′ Q3 X + Q2 Q3′ XD2 = Q2′•Q3•X + Q2•Q3′•X

D3 = Q1•Q2′•Q3′ + Q1•Q3•X′ + Q2′•X′ + Q1′•Q3′•X′ + Q2•Q3′•X

K-maps for UNLK and HINT

UNLK = Q1•Q2•Q3•XQ Q Q

HINT = Q1′•Q2′•Q3′•X′ + Q1•Q2′•X + Q2′•Q3•X + Q2•Q3•X′+ Q2•Q3′•X

Another example

• Design a state machine that recognize a binary number that can be evenly divided by 4– Input: D, providing bits from MSB to LSB, 1 bit per cycle– Output: Y if the bits received so far can be evenly divided by 4

StateD

Y0 1

S00 S00 S1 1

S1 S0 S1 0

S0 S00 S1 0

Design State Machines using State Diagrams

• Sometimes people like graphical approaches– It is simplerIt is simpler

• However, it is also more error prone– A state table is an exhaustive listings of the next state for each

state/input combination• You will not miss a caseYou will not miss a case

– In a stage diagram, every input combination should be covered exactly by one expression

Gi i bi i d l i i• Given an input combination, one and only one expression is true• It is not easy to see

Ambiguity in state diagram

• In a properly constructed state diagram, each input combination is covered exactly once by an expression of an outgoing arcy y p g g

• Ambiguous: double-covered or uncovered• Some input combinations are covered by more than one

expressions (double-covered)– Given such an input combination, you have two or more transitions to

follow– AND of any two expressions should be 0

• Some input combinations are not covered by any expressions (uncovered)(uncovered)– Given such an input combination, you do not have any transitions to

follow– OR of all expressions should be 1

Example of an ambiguous state diagram

1965 Ford Thunderbird tail lights

Flashing sequence for T-bird tail-lights

Left turn Right turn

State diagram for T-bird tail-lights state machine

Corrected and enhanced state diagram for T-bird tail-lights state machine

State assignment for T-bird tail-lights state machine

Transition list for T-bird tail-lights state machine