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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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