elec 2200-002 digital logic circuits fall 2008 finite state machines (fsm) (chapter 7-10) vishwani...
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ELEC 2200-002Digital Logic Circuits
Fall 2008Finite State Machines (FSM)
(Chapter 7-10)Vishwani D. Agrawal
James J. Danaher ProfessorDepartment of Electrical and Computer Engineering
Auburn University, Auburn, AL 36849http://www.eng.auburn.edu/~vagrawal
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 11
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Two Types of Digital Circuits1. Output depends uniquely on inputs:
Contains only logic gates, AND, OR, . . . No feedback interconnects
2. Output depends on inputs and memory: Contains logic gates, latches and flip-flops May have feedback interconnects Contents of flip-flops define internal state; N flip-flops
provide 2N states; finite memory means finite states, hence the name “finite state machine (FSM)”.
Clocked memory – synchronous FSM No clock – asynchronous FSM
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Textbook Organization
Chapter 6: Sequential devices – latches, flip-flops.
Chapter 7: Modular sequential logic – registers, shift registers, counters.
Chapter 8: Specification and analysis of FSM.
Chapter 9: Synchronous (clocked) FSM design.
Chapter 10: Asynchronous (pulse mode) FSM design.
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 33
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Mealy and Moore FSMMealy machine: Output is a function of inputs and the present state.
Moore machine: Output is a function of the present state alone.
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 44
S0 S1
1/1
1/0
0/0 0/1
Mealy machine
S0/1 S1/0
1/1
1/0
0/1 0/0
Moore machine
G. H. Mealy, “A Method for Synthesizing Sequential Circuits,” BellSystems Tech. J., vol. 34, pp. 1045-1079, September 1955.E. F. Moore, “Gedanken-Experiments on Sequential Machines,” Annals ofMathematical Studies, no. 34, pp. 129-153 ,1956, Princeton Univ. Press, NJ.
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Example 8.17: Robot Control
A robot moves in straight line, encounters obstacle and turns right or left until path is clear; on alternate obstacle encounters use right and left turn strategies.
Define input: One bitX = 0, no obstacle
X = 1, an obstacle encountered
Define outputs: Two bitsZ1, Z2 = 00 no turn
Z1, Z2 = 01 turn right by a predetermined angle
Z1, Z2 = 10 turn left by a predetermined angle
Z1, Z2 = 11 output not used
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Example 8.17: Robot Control (Continued . . . 2)
Because turning strategy depends on the action for the previous obstacle, the robot must remember the past.
Therefore, we define internal memory states:State A = no obstacle detected, last turn was left
State B = obstacle detected, turning right
State C = no obstacle detected, last turn was right
State D = obstacle detected, turning left
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Realization of FSMThe general hardware architecture of an FSM, known as Huffman model, consists of:
Flip-flops for storing the state.
Combinational logic to generate outputs and next state from inputs and present state.
Clock to synchronize state changes.
Initialization hardware to set the machine in a known state.
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 77
Combinational logic
Flip-flops
OutputsInputs
Presentstate
Nextstate
ClockClear
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Example 8.17: Robot Control (Continued . . . 3)
Construct state diagram.
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 88
A
D C
B
A: no obstacle, last left turnB: obstacle, turn rightC: no obstacle, last right turnD: obstacle, turn left
Input: X = 0, no obstacleX = 1, obstacle
Outputs:Z1, Z2 = 00, no turnZ1, Z2 = 01, right turnZ1, Z2 = 10, left turn
0/001/01
0/000/00
0/00
1/01
1/101/10
X Z1 Z2
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Example 8.17: Robot Control (Continued . . . 4)
Construct state table.
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A
D C
B
0/001/01
0/000/00
0/00
1/01
1/101/10
X Z1 Z2
A/00
C/00
C/00
A/00
B/01
B/01
D/10
D/10
XPresent 0 1state
A
B
C
D
Nextstate
OutputsZ1, Z2
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XY1 Y2 0 1
00
01
11
10
Example 8.17: Robot Control (Continued . . . 5)
State assignment: Need log24 = 2 binary state variables for 4 to represent 4 states.
Let memory variables be Y1,Y2:
A: Y1, Y2 = 00; B: Y1, Y2 = 01; C: Y1, Y2 = 11, D: Y1, Y2 = 10
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 1010
A/00
C/00
C/00
A/00
B/01
B/01
D/10
D/10
XPresent 0 1state
A
B
C
D
00/00
11/00
11/00
00/00
01/01
01/01
10/10
10/10
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XY1 Y2 0 1
00
01
11
10
Example 8.17: Robot Control (Continued . . . 6)
Construct truth tables for outputs, Z1 and Z2, and excitation variables, Y1 and Y2.
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00/00
11/00
11/00
00/00
01/01
01/01
10/10
10/10
NextState, Y1*, Y2*
OutputsZ1, Z2
InputPresent
state Outputs Next state
X Y1 Y2 Z1 Z2 Y1* Y2*
0 0 0 0 0 0 0
0 0 1 0 0 1 1
0 1 0 0 0 0 0
0 1 1 0 0 1 1
1 0 0 0 1 0 1
1 0 1 0 1 0 1
1 1 0 1 0 1 0
1 1 1 1 0 1 0
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Example 8.17: Robot Control (Continued . . . 7)
Synthesize logic functions, Z1, Z2, Y1*, Y2*.
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InputPresent
state Outputs Next state
X Y1 Y2 Z1 Z2 Y1* Y2*
0 0 0 0 0 0 0
0 0 1 0 0 1 1
0 1 0 0 0 0 0
0 1 1 0 0 1 1
1 0 0 0 1 0 1
1 0 1 0 1 0 1
1 1 0 1 0 1 0
1 1 1 1 0 1 0
Z1 = XY1Y2 + XY1 Y2 = XY1
Z2 = XY1Y2 + XY1 Y2 = XY1
Y1* = XY1 Y2 + . . .
Y2* = XY1 Y2 + . . .
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Example 8.17: Robot Control (Continued . . . 8)
Synthesize logic functions, Z1, Z2, Y1*, Y2*.
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1 1
1 1X
Y1
Y2
1 1
1 1X
Y2
1 1X
Y1
Y2
1 1X
Y1
Y2
Y1
Z1
Z2
Y1*
Y2*
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Example 8.17: Robot Control (Continued . . . 9)
Synthesize logic and connect memory elements (flip-flops).
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Y2
Y1
Y1
Y2
XZ1
Z2
Y1*
Y2*
CK
CLEAR
Combinational logic
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Steps in FSM SynthesisExamine specified function to identify inputs, outputs and memory states.
Draw a state diagram.
Minimize states (see Section 9.1).
Assign binary codes to states (Section 9.4).
Derive truth tables for state variables and output functions.
Minimize multi-output logic circuit.
Connect flip-flops for state variables. Don’t forget to connect clock and clear signals.
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Architecture of an FSMThe Huffman model, containing:
Flip-flops for storing the state.
Combinational logic to generate outputs and next state from inputs and present state.
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Combinational logic
Flip-flops
OutputsInputs
Presentstate
Nextstate
ClockClear
D. A. Huffman, “The Synthesis of Sequential Switching Circuits,J. Franklin Inst., vol. 257, pp. 275-303, March-April 1954.
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State Minimization
An FSM contains flip-flops and combinational logic:
Number of flip-flops, Nff = log2 Ns , Ns = #states
Size of combinational logic depends on state assignment.
Examples:
1.Ns = 16, Nff = log2 16 = 4
2.Ns = 17, Nff = log2 17 = 4.0875 = 5
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Ceiling operator
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Equivalent StatesTwo states of an FSM are equivalent (or indistinguishable) if for each input they produce the same output and their next states are identical.
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Si
Sj
Sm
Sn
1/0
1/0
0/0
0/0
Si,j
Sm
Sn
1/0
0/0
Si and Sj are equivalent andmerged into a single state.
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Minimizing StatesExample: States A . . . I, Inputs I1, I2, Output, Z
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Present state
Next state, output (Z)
InputI1 I2
A D, 0 C, 1
B E, 1 A, 1
C H, 1 D, 1
D D, 0 C, 1
E B, 0 G, 1
F H, 1 D, 1
G A, 0 F, 1
H C, 0 A, 1
I G, 1 H, 1
A and D are equivalent
A and E produce same output.Can they be equivalent?
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Implication Table Method
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A B C D E F G H
B
C
D
E
F
G
H
I
√BDCG
ADCF
√
CDAC
EHAD
EHAD
EGAH
Present state
Next state, output (Z)
InputI1 I2
A D, 0 C, 1
B E, 1 A, 1
C H, 1 D, 1
D D, 0 C, 1
E B, 0 G, 1
F H, 1 D, 1
G A, 0 F, 1
H C, 0 A, 1
I G, 1 H, 1
ADCF
CDAC
BCAG
BDCG
ACAF
GHDH
GHDH
ABFG
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Implication Table Method (Cont.)
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A B C D E F G H
B
C
D
E
F
G
H
I
√BDCG
ADCF
√
CDAC
EHAD
EHAD
EGAH
ADCF
CDAC
BCAG
BDCG
ACAF
GHDH
GHDH
Equivalent states:
S1: A, D, G
S2: B, C, F
S3: E, H
S4: IABFG
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Minimized State Table
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Present state
Next state, output (Z)
InputI1 I2
A D, 0 C, 1
B E, 1 A, 1
C H, 1 D, 1
D D, 0 C, 1
E B, 0 G, 1
F H, 1 D, 1
G A, 0 F, 1
H C, 0 A, 1
I G, 1 H, 1
Present state
Next state, output (Z)
InputI1 I2
S1 = (A, D, G) S1, 0 S2, 1
S2 = (B, C, F) S3, 1 S1, 1
S3 = (E, H) S2, 0 S1, 1
S4 = I S1, 1 S3, 1
Original Minimized
Number of flip-flops is reducedfrom 4 to 2.
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State AssignmentState assignment means assigning distinct binary patterns (codes) to states.
N flip-flops generate 2N codes.
While we are free to assign these codes to represent states in any way, the assignment affects the optimality of the combinational logic.
Rules based on heuristics are used to determine state assignment.
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Criteria for State AssignmentOptimize:
Logic gates, or
Delay, or
Power consumption, or
Testability, or
Any combination of the above
Up to 4 or 5 flip-flops: can try all assignments and select the best.
More flip-flops: Use an existing heuristic (one discussed next) or invent a new heuristic.
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The Idea of AdjacencyInputs are A and B
State variables are Y1 and Y2
An output is F(A, B, Y1, Y2)
A next state function is G(A, B, Y1, Y2)
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1 1
1 1
A
B
Y1
Y2
Karnaugh map ofoutput function ornext state function
Larger clucsersproduce smaller logic function.
Clustering mintermsdiffer in one variable.
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Size of an ImplementationNumber of product terms determine number of gates.
Number of literals in a product term determine number of gate inputs, which is proportional to number of transistors.
Hardware α (number of literals)
Examples of four minterm functions:F1 = ABCD +ABCD +ABCD +ABCD has 16 literals
F2 = ABC +ACD has 6 literals
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Rule 1States that have the same next state for a given input should be assigned logically adjacent codes.
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Combinational logic
Flip-flops
OutputsFixedInputs
Presentstate
Nextstate
ClockClear
Si
Sj
Sk
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Rule 2States that are the next states of the same state under logically adjacent inputs, should be assigned logically adjacent codes.
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Combinational logic
Flip-flops
OutputsAdjacentInputs
Fixedpresent
state
Nextstate
ClockClear
SkSm
Si
I1I2
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Example of State Assignment
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Present state
Next state, output (Z)
Input, X0 1
A C, 0 D, 0
B C, 0 A, 0
C B, 0 D, 0
D A, 1 B, 1
D B
A
C
0/0
0/0
0/0
1/01/0
1/0
1/1
0/1
A adj B(Rule 1)
A adj C(Rule 1)
B adj D(Rule 2)
Figure 9.19 of textbook C adj D(Rule 2)
A B
C D
0 1
0
1
Verify that BC andAD are not adjacent.
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A = 00, B = 01, C = 10, D = 11
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 3030
Present state
Y1, Y2
Next state, outputY1*Y2*, Z
Input, X0 1
A = 00 10, 0 11, 0
B = 01 10, 0 00, 0
C = 10 01,0 11, 0
D = 11 00, 1 01, 1
InputPresent
state Output Next state
X Y1 Y2 Z Y1* Y2*
0 0 0 0 1 0
0 0 1 0 1 0
0 1 0 0 0 1
0 1 1 1 0 0
1 0 0 0 1 1
1 0 1 0 0 0
1 1 0 0 1 1
1 1 1 1 1 0
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Logic Minimization for Optimum State Assignment
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 3131
1 1
1 1 1X
Y1
Y2
1
1 1X
Y2
1
1X
Y1
Y2Y1
Z Y1*
Y2*
Result: 5 products, 10 literals.
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Circuit for Optimum State Assignment
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 3232
Y2
Y1
Y1
Y2
X
Z
Y2*
Y1*
CK
CLEAR
Combinational logic
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Using an Arbitrary State Assignment: A = 00, B = 01, C = 11, D = 10
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 3333
Present state
Y1, Y2
Next state, outputY1*Y2*, Z
Input, X0 1
A = 00 11, 0 10, 0
B = 01 11, 0 00, 0
C = 11 01,0 10, 0
D = 10 00, 1 01, 1
InputPresent
state Output Next state
X Y1 Y2 Z Y1* Y2*
0 0 0 0 1 1
0 0 1 0 1 1
0 1 0 1 0 0
0 1 1 0 0 1
1 0 0 0 1 0
1 0 1 0 0 0
1 1 0 1 0 1
1 1 1 0 1 0
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Logic Minimization for Arbitrary State Assignment
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 3434
1 1
1 1X
Y1
Y2
1 1 1
1X
Y2
1
1X
Y1
Y2Y1
Z Y1*
Y2*
Result: 6 products, 14 literals.
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Circuit for Arbitrary State Assignment
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 3535
Y2
Y1
Y1
Y2
X
Z
Y2*
Y1*
CK
CLEAR
Comb.logic
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Find Out More on FSMState minimization through partioning (Section 9.2.2).
Incompletely specified sequential circuits (Section 9.3).
Further rules for state assignment and use of implication graphs (Section 9.4).
Asynchronous or fundamental-mode sequential circuits (Chapter 10).
Fall 2008, Dec 3Fall 2008, Dec 3 ELEC2200-002 Lecture 13ELEC2200-002 Lecture 13 3636