1. CS M51A/EE M16 Winter’05 Section 1 Logic Design of Digital Systems Lecture 11
February 23
W’05
Yutao He
4532B Boelter Hall

2. Outline Administrative Matters Chapter 7
Specification of Sequential Systems
State Minimization

3. Administrative Matters
Project #2
Is posted on the web
Due on March 2 (Wednesday)
Teamwork is allowed and encouraged
Find your partner as early as possible
Homework #7
Midterm
Will be handed back and discussed on Next Monday

4. Sequential Systems: Overview
Basic Concepts
Synchronous sequential systems
Clocks
States
Finite state machines
Mealy and Moore machines
Specification
Time behavior (I/O sequence)
State transition table
State diagram
Minimization
Analog
Digital
Sync.
Async.
Comb.
Seq.
(Hardware) Systems

5. Definition of Sequential Systems

6. Sync. Vs. Async. Sequential Systems
Synchronous
Asynchronous

7. Clock An independent periodic reference signal Provided by
An internal crystal
An external 60 Hz alternating current
Make sure you know
When is the present (t)
When is the next (t+1)
back to the future
When is the previous (t-1)
forth to the past

8. Time-Behavior Specification
Behavior of a sequential system can be specified by a sequence of input(s)/output(s) pairs with respect to the clock signal
Time t
Input xi(t)
Output zi(t)

9. Example 7.1: Serial Decimal Adder
Addition is performed one digit at a time, starting from the LSB
Output is generated at each time instant
As a result, a 8-digit serial decimal adder needs 8 clock cycles to finish the calculation
s = x + y = t
x(t)
y(t)
s(t)
c(t)

10. State
Introduced to help “memorize” the complete input/output sequences
Usually number of states are finite
Itself is also a time function
Two types of states are defined:
present state (PS): s(t)
next state (NS): s(t+1)

11. State Description of Sequential Systems
A sequential system can be specified as a finite state machine (FSM) by specifying
output function: z(t) = H(s(t), x(t))
state transition function: s(t+1) = G (s(t), x(t))
Output
Function
State transition
Inputs x(t)
Outputs z(t)
Present State s(t)
Next State s(t+1)

12. Example 7.3 - Serial Decimal Adder
Inputs: x(t), y(t)  {0,1, …, 9}
Outputs: z(t)  {0, 1, …, 9}
State: c(t)  {0, 1}
Initial State: c(0) = 0
Functions:
State transition function: c(t+1) =
Output function z(t) = (x(t)+y(t)+c(t)) mod 10
1 if x(t)+y(t)+c(t)  10
0 otherwise

13. State Transition Table
An extended truth table for specifying output function and state transition function in a tabular form
PS Inputs x(t)
NS, Outputs z(t)

14. Example 7.4: Odd/Even Detector
Given a system whose input has two values a and b, and whose output also has two values, 0 and 1. The output at time t is 1 if the number of b’s in the input x(0,t) is even, and 0 otherwise.
Inputs: x(t)  {a,b}
Outputs: z(t)  {0, 1}
State: s(t)  {Even, Odd}
Initial State: s(0) = Even
PS Inputs x(t)
a b
NS, Outputs z(t)
Even Even, Odd, 0
Odd Odd, Even, 1 t
x(t) a b b a b a b a
z(t)

15. State Diagram
A graphical specification of a sequential system

16. Example: State Diagram
PS Inputs x(t)
a b
NS, Outputs z(t)
Even Even, Odd, 0
Odd Odd, Even, 1
b/0
b/1
a/1
a/0
Odd
Even

17. Mealy and Moore Machines
Mealy Machine:
Its output depends upon both input and state
Moore Machine:
Its output depends only upon present state
inputs
outputs
state feedback
reg
combinational logic for next state
logic for outputs
state feedback
inputs
outputs
reg
combinational logic for next state
logic for outputs
Mealy Machine
Moore Machine

18. Example 7.5 - A Moore Machine

19. How to Select State Names
Use integers as state names
Example: A modulo-64 counter
Input: x(t)  {0,1}
Output: z(t)  {0,1,…,63}
State: s(t)  {0,1,…,63}
Initial State: s(0) = 0
Function:
Transition function: s(t+1) = [s(t)+x(t)] mod 64
Output function: z(t) = s(t)

20. How to Select State Names (Cont’d)
Use state-vector approach:
state is represented by a vector s = (sn-1, …, s 0)
Example:
A sequential system that counts the occurrence of 55 different events. When the count of event I is a multiple of 100, the output is z(t) = i, otherwise, z(t) = 0
Input: x(t)  {1, 2, …, 55}
Output: z(t)  {0, 1, 2, …, 55}
State: s(t) = (s55,…,s1), si  {0,1,…,99}
State: s(0) = (0,0,…,0)
Functions:
Transition function: si(t+1) =
Output function: z(t) =
[si(t)+1] mod if x(t) = i
si(t) otherwise
i if x(t) = i and si(t) = 99
0 otherwise

21. Case Study 1: Finite Memory Systems
A sequential system has finite memory of length m is z(t) depends only on the last m input values:
z(t) = F(x(t-m+1), t))
Example 7.12:
z(t) =
Finite memory of length four
All finite-memory machines are FSMs
Not all FSMs are finite-memory
p if x(t-3,t) = aaba
q otherwise
1 if number of 1’s in x(0, t) is even
0 otherwise

22. Case Study 2: Pattern Detector
Detect sub-patterns in the input sequence
Two types:
overlapped and non-overlapped
Example:
Input: x(t)  {0,1}
Output: z(t)  {0,1}
Function: z(t) =
1 if x(t-3,t) = 1101
0 otherwise

23. Case Study 3: Controller
A FSM that produces control signals as the states are traversed.
Control signals determine actions performed by other parts of the system.
Two types
Autonomous
State transitions follow a fixed sequence of states, independent of any inputs except the clock.
Non-autonomous
The transition is decided by external inputs

24. Vending Machine Controller

25. Vending Machine Controller (Cont’d)

26. State Minimization Motivation: Basic concept: Basic Methods:
High-level design may generate many redundant states
Fewer states may mean fewer state variables
To reduce the complexity and cost
Basic concept:
Two states are equivalent if they are impossible to distinguish from the outputs of the FSM, i. e., for any input sequence the outputs are the same
(1) Output must be the same in both states
(2) Must transition to equivalent states for all input combinations
Basic Methods:
Table matching
Implication Chart

27. Table Matching Procedure - Overview
Starting with the state table
Step 1: Row matching with respect to outputs
Step 2: Rename the newly partitioned classes
Step 3: List their next state transitions by using new names
Step 4: Check if partitions are same by column matching within classes
If no, go back to Step 1
If yes, The states are minimal
Write the state table for the minimal states

28. State Minimization - Example 7.14

29. Example 7.14 (Cont’d) Row Matching: Column Matching:
P1 = (A, C, E) (B, D, F)
Column Matching:

30. Example 7.14 (Cont’d) Row Matching: Column Matching: Row Matching:
P2 = (A, C, E) (B, D) (F)
Column Matching:
P3 (A, C) (E) (B, D) (F) a b c
Row Matching:
P3 = (A, C) (E) (B, D) (F)
Column Matching:

31. Example 7.14 (Cont’d) Column Matching: Stop: The states are minimal
P4 = (A, C) (E) (B, D) (F) = P3
Stop: The states are minimal

32. Summary Specification of sequential systems
time-behavior
state-transition table
equation
state diagram
Several common types of sequential systems
pattern detectors
controller
State Minimization

33. Next Lecture
Chapter 8: Sections