Lecture 12
Finite State Machine Design
Prith Banerjee
ECE C03
Advanced Digital Design
Spring 1998
ECE C03 Lecture 12
1
Outline
• Review of sequential machine design
• Moore/Mealy Machines
• FSM Word Problems
– Finite string recognizer
– Traffic light controller
• READING: Katz 8.1, 8.2, 8.4, 8.5, Dewey 9.1, 9.2
ECE C03 Lecture 12
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Concept of the State Machine
Computer Hardware = Datapath + Control
Qualifiers
Registers
Combinational Functional
Units (e.g., ALU)
Busses
Control
Control
FSM generating sequences
of control signals
Instructs datapath what to
do next
"Puppeteer who pulls the
strings"
State
Control
Signal
Outputs
Qualifiers
and
Inputs
"Puppet"
Datapath
ECE C03 Lecture 12
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Example: Odd Parity Checker
Assert output whenever input bit stream has odd # of 1's
Reset
0
Even
[0]
1
0
1
Odd
[1]
State
Diagram
Present State
Even
Even
Odd
Odd
Input
0
1
0
1
Next State
Even
Odd
Odd
Even
Output
0
0
1
1
Symbolic State Transition Table
Present State
0
0
1
1
Input
0
1
0
1
Next State Output
0
0
1
0
1
1
0
1
Encoded State Transition Table
ECE C03 Lecture 12
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Odd Parity Checker Design
Next State/Output Functions
NS = PS xor PI; OUT = PS
Input
NS
Input
D
Q
CLK
R
Output
Q
CLK
PS/Output
R
Q
Q
\Reset
\Reset
T FF Implementation
D FF Implementation
Input
T
1
0
0
1
1
0
1
0
1
1
1
0
Clk
Output
1
1
1
0
1
1
0
0
1
0
1
Timing Behavior:
Input
1 0120 1 1 0 1 0 1 1 1 0
ECE C03
Lecture
1
5
Timing of State Machines
When are inputs sampled, next state 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., tri-state enable: effective immediately
sync. counter clear: effective at next clock event
ECE C03 Lecture 12
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Timing of State Machine
Example: Positive Edge Triggered Synchronous System
State Time
Clock
On rising edge, inputs sampled
outputs, next state computed
After propagation delay, outputs and
next state are stable
Inputs
Immediate Outputs:
affect datapath immediately
could cause inputs from datapath to change
Outputs
Delayed Outputs:
take effect on next clock edge
propagation delays must exceed hold times
ECE C03 Lecture 12
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Communicating State Machines
One machine's output is another machine's input
X
FSM 2
FSM 1
CLK
Y
FSM 1
Y=0
Y=0
X=0
X=0
A
[1]
C
[0]
B
C
D
D
X
FSM 2
X=1
Y=0,1
A
X=1
Y=1
B
[0]
A
X=0
Y
D
[1]
Machines advance in lock step
Initial inputs/outputs: X = 0, Y = 0
ECE C03 Lecture 12
8
Basic Design Approach
1. Understand the statement of the Specification
2. Obtain an abstract specification of the FSM
3. Perform a state mininimization
4. Perform state assignment
5. Choose FF types to implement FSM state register
6. Implement the FSM
1, 2 covered now; 3, 4, 5 covered later;
4, 5 generalized from the counter design procedure
ECE C03 Lecture 12
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Example: Vending Machine FSM
General Machine Concept:
deliver package of gum after 15 cents deposited
single coin slot for dimes, nickels
no change
Step 1. Understand the problem:
Draw a picture!
Block Diagram
N
Coin
Sensor
D
Reset
Vending
Machine
FSM
Open
Gum
Release
Mechanism
Clk
ECE C03 Lecture 12
10
Vending Machine Example
Step 2. Map into more suitable abstract representation
Reset
S0
Tabulate typical input sequences:
three nickels
nickel, dime
dime, nickel
two dimes
two nickels, dime
Draw state diagram:
N
S1
N
N
S2
N
D
S4
S5
S6
[open]
[open]
[open]
S3
Inputs: N, D, reset
Output: open
D
D
D
S7
S8
[open]
[open]
ECE C03 Lecture 12
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Vending Machine Example
Step 3: State Minimization
Present
State
Reset
0¢
Inputs
D N
0¢
0
0
1
1
0
0
1
1
0
0
1
1
X
N
5¢
D
5¢
N
10¢
D
10¢
N, D
15¢
[open]
15¢
reuse states
whenever
possible
0
1
0
1
0
1
0
1
0
1
0
1
X
Next
State
Output
Open
0¢
5¢
10¢
X
5¢
10¢
15¢
X
10¢
15¢
15¢
X
15¢
0
0
0
X
0
0
0
X
0
0
0
X
1
Symbolic State Table
ECE C03 Lecture 12
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Vending Machine Example
Step 4: State Encoding
Present State Inputs
Q1 Q0
D N
0
0
0
1
1
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Next State
D 1 D0
Output
Open
0 0
0 1
1 0
X X
0 1
1 0
1 1
X X
1 0
1 1
1 1
X X
1 1
1 1
1 1
X X
0
0
0
X
0
0
0
X
0
0
0
X
1
1
1
X
ECE C03 Lecture 12
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Vending Machine Example
Step 5. Choose FFs for implementation
Q1
Q1 Q0
DN
0
D
1
1
0
1
1
1
X
X
X
X
1
1
1
1
0
1
Q0
N
1
1
0
0
0
1
0
0
1
1
0
0
1
0
N
D
X
X
X
X
0
1
1
1
N
D
Q0
K-map for D0
D
CLK
R
Q
Q1
Q
\ Q1
Q1
Q1 Q0
DN
N
Q0
K-map for D1
D1
Q1
Q1 Q0
DN
0
Q1
D
D FF easiest to use
X
X
X
X
0
0
1
0
Q0
K-map for Open
D1 = Q1 + D + Q0 N
\reset
N
\ Q0
Q0
\N
Q1
N
Q1
D
OPEN
D0
D
CLK
R
\reset
Q
Q0
Q
\ Q0
D0 = N Q0 + Q0 N + Q1 N + Q1 D
OPEN = Q1 Q0
8 Gates
ECE C03 Lecture 12
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Alternative State Machine
Representations
Why State Diagrams Are Not Enough
Not flexible enough for describing very complex finite state machines
Not suitable for gradual refinement of finite state machine
Do not obviously describe an algorithm: that is, well specified
sequence of actions based on input data
algorithm = sequencing + data manipulation
separation of control and data
Gradual shift towards program-like representations:
• Algorithmic State Machine (ASM) Notation
• Hardware Description Languages (e.g., VHDL)
ECE C03 Lecture 12
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Alternative State Machine
Representations
Algorithmic State Machine (ASM) Notation
Three Primitive Elements:
• State Box
• Decision Box
• Output Box
State Machine in one state
block per state time
Single Entry Point
Unambiguous Exit Path
for each combination
of inputs
Outputs asserted high (.H)
or low (.L); Immediate (I)
or delayed til next clock
State
Entry Path
State Code
*
State
Name
State
Output List
T
***
State Box
Condition
Condition
Box
Conditional
Output List
F
ASM
Block
Output
Box
Exits to
other ASM Blocks
ECE C03 Lecture 12
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ASM Notation
Condition Boxes:
Ordering has no effect on final outcome
Equivalent ASM charts:
A exits to B on (I0 • I1) else exit to C
A
A
010
I0
T
010
F
I1
T
F
I1
F
F
I0
T
T
B
C
ECE C03 Lecture 12
B
C
17
ASM Example: Parity Checker
Input X, Output Z
Ev en
0
Nothing in output list implies Z not asserted
Z asserted in State Odd
F
X
T
Odd
Present Next
Input State
State Output
F
—
Even
Even
T
—
Even
Odd
F
A
Odd
Odd
T
A
Odd
Even
1
H. Z
F
X
Symbolic State Table:
T
Trace paths to derive
state transition tables
Encoded State Table:
Present Next
Input State
State Output
0
0
0
0
1
0
0
1
0
1
1
1
1
1
1
0
ECE C03 Lecture 12
18
ASM Chart: Vending Machine
0¢
00
10¢
T
D
10
T
D
F
F
F
F
N
N
T
5¢
T
15¢
01
11
H.Open
T
N
F
F
D
Reset
F
T
T
ECE C03 Lecture 12
0¢
19
Moore and Mealy Machine Design
Procedure
Moore Machine
Xi
Inputs
Zk
Outputs
Combinational
Logic for
Outputs and
Next State
State Register
Clock
Outputs are function
solely of the current
state
Outputs change
synchronously with
state changes
State
Feedback
State
Register
Xi
Inputs
Mealy Machine
Comb.
Logic for
Outputs
Combinational
Logic for
Next State
(Flip-flop
Inputs)
Zk
Outputs
Clock
state
feedback
Outputs depend on
state AND inputs
Input change causes
an immediate output
change
Asynchronous signals
ECE C03 Lecture 12
20
Equivalence of Moore and Mealy
Machines
Moore
Machine
N D + Reset
(N D + Reset)/0
Reset/0
Reset
0¢
0¢
Mealy
Machine
[0]
Reset/0
Reset
N/0
5¢
5¢
N D/0
N
ND
D/0
D
[0]
N
N/0
10¢
10¢
D
D/1
[0]
N+D
N D/0
N+D/1
ND
15¢
15¢
[1]
Reset/1
Outputs are associated
with State
Reset
Outputs are associated
with Transitions
ECE C03 Lecture 12
21
States vs Transitions
Mealy Machine typically has fewer states than Moore Machine
for same output sequence
0
0/0
0
0
[0]
Same I/O behavior
0
0
1
Different # of states
1/0
0/0
1
1
1/1
[0]
1
2
[1]
S0
00
S0
IN
S1
0
IN
01
IN
S2
1
S1
1
Equivalent
ASM Charts
IN
10
H.OUT
IN
H.OUT
ECE C03 Lecture 12
22
Analyze Behavior of Moore Machines
Reverse engineer the following:
X
X
\B
J
Q
C
KR Q
FFa
A
\A
Input X
Output Z
State A, B = Z
\Reset
Clk
X
X
\A
J
Q
C
KR Q
FFb
Z
\B
\Reset
Two Techniques for Reverse Engineering:
• Ad Hoc: Try input combinations to derive transition table
• Formal: Derive transition by analyzing the circuit
ECE C03 Lecture 12
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Ad Hoc Reverse Engineering
Behavior in response to input sequence 1 0 1 0 1 0:
100
X
Clk
A
Z
\Reset
Reset
X=1
X=0
X=1
X=0
X=1
X=0
X=0
AB = 00 AB = 00 AB = 11 AB = 11 AB = 10 AB = 10 AB = 01 AB = 00
A B
0 0
Partially Derived
State Transition
Table
0 1
1 0
1 1
ECE C03 Lecture 12
X
0
1
0
1
0
1
0
1
A+
?
1
0
?
1
0
1
1
B+
?
1
0
?
0
1
1
0
Z
0
0
1
1
0
0
1
1
24
Formal Reverse Engineering
Derive transition table from next state and output combinational
functions presented to the flipflops!
Ja = X
Jb = X
Ka = X • B
Kb = X xor A
Z=B
FF excitation equations for J-K flipflop:
A+ = Ja • A + Ka • A = X • A + (X + B) • A
B+ = Jb • B + Kb • B = X • B + (X • A + X • A) • B
Next State K-Maps:
A+
State 00, Input 0 -> State 00
State 01, Input 1 -> State 01
B+
ECE C03 Lecture 12
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Complete ASM Chart of Moore Machine
00
S0
11
S3
H.Z
0
1
X
0
X
1
S1
01
S2
10
H.Z
0
X
1
1
X
0
Note: All Outputs Associated With State Boxes
No Separate Output Boxes — Intrinsic in Moore Machines
ECE C03 Lecture 12
26
Behavior of Mealy Machines
Clk
D
A
Q
DA
C
\A
X
\A
R
\X
Q
J
C
K
\Reset
A
X
B
Q
R
Q
\B
\Reset
DA
\X
B
B
Z
\X
X
A
Input X, Output Z, State A, B
State register consists of D FF and J-K FF
ECE C03 Lecture 12
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Ad Hoc Reverse Engineering
Signal Trace of Input Sequence 101011:
100
Note glitches
in Z!
X
Clk
Outputs valid at
following falling
clock edge
A
B
Z
\Reset
Reset
AB=00
Z =0
X =1
AB=00
Z =0
X =0
AB=00
Z =0
X =1
AB=01
Z =0
X =0
AB=11
Z=1
A B
0 0
X
0
1
Partially completed
0 1
0
state transition table
1
based on the signal
1 0
0
trace
1
1 1
0
ECE C03 Lecture 121
X =1
AB=10
Z =1
A+
0
0
?
1
?
0
1
?
X =1
AB=01
Z =0
B+
1
0
?
1
?
1
0
?
Z
0
0
?
0
?
1
1
?
28
Formal Reverse Engineering
A+ = B • (A + X) = A • B + B • X
B+ = Jb • B + Kb • B = (A xor X) • B + X • B
=A•B•X + A•B•X + B•X
Z
=A•X + B•X
A+
Missing Transitions and Outputs:
State 01, Input 0 -> State 01, Output 1
State 10, Input 0 -> State 00, Output 0
State 11, Input 1 -> State 11, Output 1
B+
Z
ECE C03 Lecture 12
29
ASM Chart of Mealy Machine
S0 = 00, S1 = 01, S2 = 10, S3 = 11
S0
1
H. Z
00
10
S2
0
X
0
S1
X
1
H. Z
S3
01
11
H.Z
0
X
1
1
X
0
NOTE: Some Outputs in Output Boxes as well as State Boxes
This is intrinsic in Mealy Machine implementation
ECE C03 Lecture 12
30
Synchronous Mealy Machines
Clock
Xi
Inputs
Zk
Outputs
Combinational
Logic for
Outputs and
Next State
State Register
Clock
state
feedback
latched state AND outputs
avoids glitchy outputs!
ECE C03 Lecture 12
31
Finite State Machine Word Problems
Mapping English Language Description to Formal Specifications
Case Studies:
• Finite String Pattern Recognizer
• • Traffic Light Controller
We will use state diagrams and ASM Charts
ECE C03 Lecture 12
32
Finite String Pattern Recognizer
A finite string recognizer has one input (X) and one output (Z).
The output is asserted whenever the input sequence …010…
has been observed, as long as the sequence 100 has never been
seen.
Step 1. Understanding the problem statement
Sample input/output behavior:
X: 00101010010…
Z: 00010101000…
X: 11011010010…
Z: 00000001000…
ECE C03 Lecture 12
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Finite String Recognizer
Step 2. Draw State Diagrams/ASM Charts for the strings that must be
recognized. I.e., 010 and 100.
S0
[0]
Outputs 1
Reset
S1
[0]
S4
[0]
S2
[0]
S5
[0]
S3
[1]
S6
[0]
Moore State Diagram
Reset signal places
FSM in S0
Loops in State
ECE C03 Lecture 12
34
Finite String Recognizer
Exit conditions from state S3: have recognized …010
if next input is 0 then have …0100!
if next input is 1 then have …0101 = …01 (state S2)
S0
[0]
Reset
S1
[0]
S4
[0]
S2
[0]
S5
[0]
S3
[1]
S6
[0]
ECE C03 Lecture 12
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Finite String Recognizer
Exit conditions from S1: recognizes strings of form …0 (no 1 seen)
loop back to S1 if input is 0
Exit conditions from S4: recognizes strings of form …1 (no 0 seen)
loop back to S4 if input is 1
S0
[0]
Reset
S1
[0]
S4
[0]
S2
[0]
S5
[0]
S3
[1]
S6
[0]
ECE C03 Lecture 12
36
Finite String Recognizer
S2, S5 with incomplete transitions
S2 = …01; If next input is 1, then string could be prefix of (01)1(00)
S4 handles just this case!
S5 = …10; If next input is 1, then string could be prefix of (10)1(0)
S2 handles just this case!
S0
[0]
Reset
S1
[0]
S4
[0]
S2
[0]
S5
[0]
S3
[1]
S6
[0]
ECE C03 Lecture 12
Final State Diagram
37
Review of Design Process
• Write down sample inputs and outputs to understand specification
• Write down sequences of states and transitions for the sequences
to be recognized
• Add missing transitions; reuse states as much as possible
• Verify I/O behavior of your state diagram to insure it functions
like the specification
ECE C03 Lecture 12
38
Traffic Light Controller
A busy highway is intersected by a little used farmroad. Detectors
C sense the presence of cars waiting on the farmroad. With no car
on farmroad, light remain green in highway direction. If vehicle on
farmroad, highway lights go from Green to Yellow to Red, allowing
the farmroad lights to become green. These stay green only as long
as a farmroad car is detected but never longer than a set interval.
When these are met, farm lights transition from Green to Yellow to
Red, allowing highway to return to green. Even if farmroad vehicles
are waiting, highway gets at least a set interval as green.
Assume you have an interval timer that generates a short time pulse
(TS) and a long time pulse (TL) in response to a set (ST) signal. TS
is to be used for timing yellow lights and TL for green lights.
ECE C03 Lecture 12
39
Traffic Light Controller
Picture of Highway/Farmroad Intersection:
Farmroad
C
HL
FL
Highway
Highway
HL
FL
C
Farmroad
ECE C03 Lecture 12
40
Traffic Light Controller
• Tabulation of Inputs and Outputs:
Input Signal
reset
C
TS
TL
Description
place FSM in initial state
detect vehicle on farmroad
short time interval expired
long time interval expired
Output Signal
HG, HY, HR
FG, FY, FR
ST
Description
assert green/yellow/red highway lights
assert green/yellow/red farmroad lights
start timing a short or long interval
• Tabulation of Unique States: Some light configuration imply others
State
S0
S1
S2
S3
Description
Highway green (farmroad red)
Highway yellow (farmroad red)
Farmroad green (highway red)
Farmroad yellow (highway red)
ECE C03 Lecture 12
41
Traffic Light Controller
Refinement of ASM Chart:
Start with basic sequencing and outputs:
S0
S3
H.HG
H.FR
H.HR
H.FY
S1
H.HY
H.FR
S2
H.HR
H.FG
ECE C03 Lecture 12
42
Traffic Light Controller
Determine Exit Conditions for S0:
Car waiting and Long Time Interval Expired- C • TL
S0
S0
H.HG
H.FR
0
H.HG
H.FR
0
TL
TL • C
1
0
C
1
H.ST
1
H.ST
S1
H.HY
H.FR
S1
H.HY
H.FR
Equivalent ASM Chart Fragments
ECE C03 Lecture 12
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Traffic Light Controller
S1 to S2 Transition:
Set ST on exit from S0
Stay in S1 until TS asserted
Similar situation for S3 to S4 transition
S1
H.HY
H.FR
0
TS
H.ST
S2
H.HR
H.FG
1
ECE C03 Lecture 12
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Traffic Light Controller
S2 Exit Condition: no car waiting OR long time interval expired
S0
H.HG
H.FR
0
H.ST
1
TL • C
S3
H.HR
H.FY
TS
0
1
H.ST
S1
H.HY
H.FR
0
TS
H.ST
H.ST
1
S2
H.HR
H.FG
0
TL + C
1
Complete ASM Chart
forLecture
Traffic
ECE C03
12 Light Controller
45
Traffic Light Controller
Compare with state diagram:
TL + C
Reset
S0
TL•C/ST
S1: HY
TS/ST
TS
S1
S2: FG
S3
TS
TS/ST
S0: HG
S3: FY
TL + C/ST
S2
TL • C
Advantages of State Charts:
• Concentrates on paths and conditions for exiting a state
• Exit conditions built up incrementally, later combined into
single Boolean condition for exit
ECE C03
Lecture
12 as an algorithm
• Easier to understand
the
design
46
Summary
• Review of sequential machine design
• Moore/Mealy Machines
• FSM Word Problems
– Finite string recognizer
– Traffic light controller
• NEXT LECTURE: Finite State Machine
Optimization
• READING: Katz 9.1, 2.2.1, 9.2.2, Dewey 9.3
ECE C03 Lecture 12
47