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Теория управления / Л4-Миль-Мур-СА / pics / 6-sequential logic-31

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example

 input 0 1 0 1 old new state state output 1 2 3 0 1 2 1 5 1 0 3 5 8 0 1 4 5 7 1 1 5 6 2 1 0 6 5 1 0 1 7 4 7 1 1 8 2 6 0 1 A.J. Han Vinck

Step 1:

Group together states with same outputs

(1,3,6,8) (2,5) (4,7)

Step 2: further subdivide groups into subgroups with same transition

(1,3,6,8) (2,5) (4) (7)

STOP:

4 Representants: 1, 2, 4, 7

41

Example: reduced table

 input 0 1 0 1 old new state state output 1 2 1 0 1 2 1 2 1 0 4 2 7 1 1 7 4 7 1 1
 A.J. Han Vinck 42

Another example

 Old state new state output input = 0 input = 1 A B C 0 B A C 0 C D C 0 D D E 1 E A F 0 F B G 0 G A E 0

Step 1: Group together: same output

[ A B C E F G] [D]

Step 2: group according to transition

[A,B,E,F,G] [C] [D]

Step 2: group according to transition

[A,B] [E,F,G] [C] [D]

Step 2: group according to transition [A,B] [E,F,G] [C] [D]

no change: END

DEF: two states are equivalent if and only if, for any input of length k, k > 0, they give rise to the same output.

 A.J. Han Vinck 43

A famous computer scientist

http://www-cs-faculty.stanford.edu/~knuth/

Donald E. Knuth

Professor Emeritus of The Art of Computer Programming at Stanford University

The Art of Computer Programming (TAOCP)

Famous quote: Beware of bugs in the above code; I have only proved it correct, not tried it.

 A.J. Han Vinck 44

example

Check for sub-string 100 in a binary sequence

 Example: input 00101000100000 output 00001000100000 time: Binary input sequence –Output = 1 if 100 detected –Output = 0 otherwise

clock

Every time unit, the system

Output = 1 if true

Output = 0 otherwise

 A.J. Han Vinck 45

example

Keep 2 bits in memory and look for the next input:

 Example: input 00101000100000 output 00001000100000 time:

State table

 Old state ( xt-1 xt-2) new state(xt xt-1 ) output = xt xt-1‘ xt-2‘ input xt input xt 0 0 1 1 00 00 10 0 1 10 01 11 0 0 01 00 10 0 0 11 01 11 0 0

10 equivalent to 11

 A.J. Han Vinck 46

Example implementation

input xt xt-1 xt-2

Output = xt xt-1xt-2

Homework:

Consider the table with reduced number of state

What are the consequences for the implementation?

Draw the Markov state diagram(also in the reduced form)

 A.J. Han Vinck 47

Example, state diagram

State: last 2 incoming bits

Output: 0 or 1

 0 00 10 1 0 Reduced state diagram Output: 0 or 1 0 0 00 10/11 1

0

0

0

 01 11 0

0

0

 0 Q: how many 1‘s do you expect in the output? 01 Q: does it depend on the 0 probability of 1 and 0 in the input sequence?
 A.J. Han Vinck 48

We end with shift registers and counters

1.basic registers (serial-parallel)

2.some applications

3.counters

 A.J. Han Vinck 49

A basic register

 A.J. Han Vinck 50

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