????????a?? ??f?a?? ?????

1
????????a?? ??f?a?? ?????µata
2
????????a?? ?????µata
Sta a???????a?? ?????µata ? ???d?? z de? e??a?
s????t?s? µ??? t?? e?s?d?? x a??? ?a? t??
?at?stas?? y t?? ?????µat?? p??? t?? efa?µ??? t??
e?s?d?? z f ( x, y)
3
????????a?? ?????µata

• ?a a???????a?? ?????µata d?a??????ta? se
• S?G????? ? efa?µ??? t?? e?s?d??, ? e?t??es? t??
  ?e?t???????, ? ???? t?? e??d?? ???eta? se
  ?a????sµ??e? ???????? st??µ??
• ?S?G????? ?? a??a??? ?at?stas?? de? ?????ta? se
  ?a????sµ????? ??????? a??? se t??a???? ???????
  p?? ?a???????ta? ap? t? ?????µa ?a? µ???

4
????????a?? ?????µata
??a s??????? a???????a?? ?????µa p??pe?
e?????sµ?? ?a ???s?µ?p??e? s?µata
s???????sµ??. ?a s?µata s???????sµ?? ep??e?????
ta st???e?a µ??µ?? t?? ?????µat??, ?ste a?t? ?a
a??????? ?at?stas? se d?a???t?? ????????
st??µ?? ? s???????sµ?? ep?t?????eta? µ?s? µ?a?
?e???t??a? ??????-???????? ? ?p??a t??f?d?te?
t? s?st?µa µe µ?a pe???d??? se??? pa?µ??
???????? ?a st???e?a µ??µ?? µp????? ?a a???????
?at?stas? µ??? ?at? t?? ??e?s? pa?µ?? ????????
5
Flip - Flop
?as??? st???e?a µ??µ?? ?p?????st????
s?st?µ?t?? ???e flip-flop ap????e?e? ??a bit
p????f???a? ??a ?????µa flip-flop µp??e? ?a
d?at????e? se µ?a d?ad??? ?at?stas? ep a???st??,
??? ?t?? ??p??? s?µa p???a??se? a??a??
?at?stas?? ?as??? d?µ??? st???e?? t?? ???
(?ata????t??, ?et??t??, State Machnines)
6
Flip Flop - ?e?t?????a
?as??? ?????µa Flip- Flop
1 0
0 (af?? µ?a e?s?d?? e??a? 1)
1 (af?? ?a? ?? d?? e?s?d?? e??a? 0)
7
Flip Flop - ?e?t?????a
0 (ap? p???)
0 0
0 (af?? µ?a e?s?d?? e??a? 1)
pa?aµ??e? st? 1 (af?? ?a? ?? d?? e?s?d?? e??a? 0)
8
Flip Flop - ?e?t?????a
0 1
1 (af?? ?a? ?? d?? e?s?d?? e??a? 0)
0 (af?? µ?a e?s?d?? e??a? 1)
9
Flip Flop - ?e?t?????a
0 0
pa?aµ??e? 1 (af?? ?a? ?? d?? e?s?d?? e??a? 0)
0 (af?? µ?a e?s?d?? e??a? 1)
10
Flip Flop - ?e?t?????a
S R Q Q
1 0 1 0
0 0 1 0
0 1 0 1
0 0 0 1
1 1 0 0
11
RS - Flip Flop
?? ßas??? flip-flop (RS) µp??e? ?a t??p?p????e?
µe t?? p??s???? µ?a? ep?p??s?et?? e?s?d?? ??a
t??? pa?µ??? ????????
R S Qt Qt1
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 0
1 1 0 ?p??sd.
1 1 1 ?p??sd.
R Q
Clk
S Q
12
T - Flip Flop
T Q
T Qt1
0 Qt
1 Qt
Clk Q
13
D - Flip Flop
D Q
D Qt1
0 0
1 1
Clk Q
14
JK - Flip Flop
J Q
J K Qt Qt1
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 0
Clk
K Q

15
JK - Flip Flop .. ?a?a?t???st???? ???a?a? 2
J Q
J K Qt1
0 0 Qt
0 1 0
1 0 1
1 1 Qt
Clk
K Q

16
?????s? ????????a??? ?????µ?t??
17
???s?se?? ?at?stas??
?p? t??? p??a?e? t?? 2 D flip-flop ?atast?????µe
t?? e??s?se?? ep?µe??? ?at?stas?? ?(t1) ?(t)
x(t) B(t) x(t) gt A(t1) Ax Bx B(t1)
A(t) x(t) gt B(t1) Ax G?a t?? ???d? y
e??a? y(t) A(t) B(t) x(t) gt y (AB) x
18
???a?a? ?atast?se??
??????sa ?at?stas? ??????sa ?at?stas? ??s?d?? ?p?µe?? ?at?stas? ?p?µe?? ?at?stas? ???d??
? ? x A B y
0 0 0 0 0 0
0 0 1 0 1 0
0 1 0 0 0 1
0 1 1 1 1 0
1 0 0 0 0 1
1 0 1 1 0 0
1 1 0 0 0 1
1 1 1 1 0 0
19
?????aµµa ?atast?se??
1/0
0/0
0/1
00
10
1/0
0/1
1/0
0/1
01
11
1/0
20
?a??de??µa
?a p?a?µat?p????e? ? a????s? t?? ??t???
a???????a??? ?????µat??. ?? ???e? a?t? t? ?????µa
?
A
Q Q
B
Q Q
?
?
21
???t??a Mealy ?a? Moore
???t??? Moore ?? ???d?? e??a? s??a?t?se?? t??
pa???sa? ?at?stas?? µ???
???d??
??s?d??
S??d?ast??? ?????µa ?p?µ. ?at?st.
Flip Flops
S??d?ast??? ?????µa ???d??
22
???t??a Mealy ?a? Moore
???t??? Mealy ?? ???d?? e??a? s??a?t?se?? t?s?
t?? pa???sa? ?at?stas?? ?s? ?a? t?? e?s?d??
???d??
??s?d??
S??d?ast??? ?????µa ?p?µ. ?at?st.
Flip Flops
S??d?ast??? ?????µa ???d??
23
??a??st?p???s? ?atast?se??
??? ?????µata e??a? ?s?d??aµa e?? pa?????? t??
?d?e? e??d??? ??a t?? ?d?e? a???????e? e?s?d??,
?a? a?t? ?s??e? ??a ??e? t?? a???????e?
e?s?d?? ??? ?atast?se?? e??a? ?s?d??aµe? e?? ??a
???e st???e?? t?? s?????? e?s?d?? d????? a???ß??
t?? ?d?a ???d? ?a? µetaf????? t? ?????µa st??
?d?a ?at?stas? e?te se ?s?d??aµ? ?at?stas? ?ta?
d?? ?atast?se?? e??a? ?s?d??aµe? t?te ? µ?a t???
µp??e? ?a apa?e?f?e? ????? ?a a??????? ?? s??se??
e?s?d??-e??d??
24
??a??st?p???s? ?atast?se??
????????a ??s?d?? 01010110100 e?????s? ap? (a)
25
??a??st?p???s? ?atast?se??
?p?µe?? ?at?stas? ?p?µe?? ?at?stas? ???d?? ???d??
?a???sa x 0 x 1 x 0 x 1
a a b 0 0
b c d 0 0
c a d 0 0
d e f 0 1
e a f 0 1
f g f 0 1
g a f 0 1
26
??a??st?p???s? ?atast?se??
?p?µe?? ?at?stas? ?p?µe?? ?at?stas? ???d?? ???d??
?a???sa x 0 x 1 x 0 x 1
a a b 0 0
b c d 0 0
c a d 0 0
d e f 0 1
e a f 0 1
f e f 0 1
27
??a??st?p???s? ?atast?se??
?p?µe?? ?at?stas? ?p?µe?? ?at?stas? ???d?? ???d??
?a???sa x 0 x 1 x 0 x 1
a a b 0 0
b c d 0 0
c a d 0 0
d e d 0 1
e a d 0 1
28
??a??st?p???s? ?atast?se?? - ?s??s?
?a e?a??st?p????e? ? a???µ?? t?? ?atast?se??
?p?µe?? ?at?stas? ?p?µe?? ?at?stas? ???d?? ???d??
?a???sa x 0 x 1 x 0 x 1
a f b 0 0
b d c 0 0
c f e 0 0
d g a 1 0
e d c 0 0
f f b 1 1
g g h 0 1
h g a 1 0
29
???a?e? ????e?s??
?at? t? s?ed?as? a???????a??? ?????µ?t?? s??????
????????µe t? µet?ßas? ap? t?? t?????sa st??
ep?µe?? ?at?stas? ?a? ?????µe ?a ß???µe t??
s?????e? e?s?d?? t?? FFs
T-FF
D-FF
Q(t) Q(t1) T
0 0 0
0 1 1
1 0 1
1 1 0
Q(t) Q(t1) D
0 0 0
0 1 1
1 0 0
1 1 1
30
???a?e? ????e?s??
JK-FF
Q(t) Q(t1) J K
0 0 0 X
0 1 1 X
1 0 X 1
1 1 X 0
SR-FF
Q(t) Q(t1) S R
0 0 0 X
0 1 1 0
1 0 0 1
1 1 X 0
31
S?ed?as? ????????a??? ?????µ?t??

1. F?ast??? ?e????af? ?pa?t?se?? (? d????aµµa)
2. ??µ??????a ???a?a ?atast?se??
3. (??a??st?p???s? ?atast?se??)
4. ??d???p???s? ?atast?se?? µe d?ad???? t?µ??
5. ???es? a???µ?? flip-flop ?a? ???µas?a t???
6. ?p????? t?p?? flip-flop
7. ?p? p??a?a ?atast?se?? -gt ???a?e? ????e?s??,
   e??d??
8. ???es? s??a?t?se?? e?s?d?? t?? FF (µet? ap?
   ap??p???s?)
9. ?????? ?????aµµa

32
?a??de??µa
?a s?ed?aste? t? ?????µa p?? ???p??e? t? a???????
d????aµµa
0
?
1
1
?
D
1
0
1
0
C
St? d????aµµa de? a?apa??st??ta? µetaß??t??
e??d??. ? ?d?a ? ?at?stas? t?? FF e??a? ? ???d??
0
33
?a??de??µa
??d???p???s? ? 00 ?
01 C 10
D 00
???a?a? ?atast?se??
??????sa ??s?d?? (x) ?p?µe??
? 0 A
? 1 B
? 0 C
? 1 B
C 0 C
C 1 D
D 0 D
D 1 A
34
?a??de??µa
?p?????? 4 ?atast?se??, ??a apa?t???ta? d?? FF,
?st? ? ?a? ?
??????sa ??????sa ??s?d?? ?p?µe?? ?p?µe??
? ? x ? ?
0 0 0 0 0
0 0 1 0 1
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 1 1
1 1 0 1 1
1 1 1 0 0
35
?a??de??µa
???a?a? ????e?s?? ???p???s? µe JK flip-flop
??????sa ??????sa ??s?d?? ?p?µe?? ?p?µe?? ??s?d?? t?? FF ??s?d?? t?? FF ??s?d?? t?? FF ??s?d?? t?? FF
? ? x ? ? JA KA JB KB
0 0 0 0 0
0 0 1 0 1
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 1 1
1 1 0 1 1
1 1 1 0 0
36
?a??de??µa
???a?a? ????e?s?? ???p???s? µe JK flip-flop
??????sa ??????sa ??s?d?? ?p?µe?? ?p?µe?? ??s?d?? t?? FF ??s?d?? t?? FF ??s?d?? t?? FF ??s?d?? t?? FF
? ? x ? ? JA KA JB KB
0 0 0 0 0 0 X 0 X
0 0 1 0 1 0 X 1 X
0 1 0 1 0 1 X X 1
0 1 1 0 1 0 X X 0
1 0 0 1 0 X 0 0 X
1 0 1 1 1 X 0 1 X
1 1 0 1 1 X 0 X 0
1 1 1 0 0 X 1 X 1
37
?a??de??µa
S??d?ast??? ?????µa
J Q
Clk
K Q

x
J Q
Clk
K Q

38
?a??de??µa
JA
Bx
00 01 11 10
A
0 0 0 1
X X X X
0 1
JA Bx
39
?a??de??µa
KA
Bx
00 01 11 10
A
X X X X
0 0 1 0
0 1
KA Bx
40
?a??de??µa
JB
Bx
00 01 11 10
A
0 1 X X
0 1 X X
0 1
JB x
41
?a??de??µa
KB
Bx
00 01 11 10
A
X X 0 1
X X 1 0
0 1
KB Ax Ax A x

42
?a??de??µa
43
S?ed?as? (?s??s?)

• ?a s?ed?aste? a???????a?? ?????µa µe T flip-flop
  ?a? µ?a e?s?d?, t?t??? ?ste ?ta? x0 ? ?at?stas?
  t?? ?????µat?? ?a pa?aµ??e? ? ?d?a. ?ta? x1 st?
  ?????µa ?a p?a?µat?p?????ta? µetaß?se?? ap? 00 se
  01, se 11, se 00 ?.?.?

44
?ata????t??

• ??a? ?ata????t?? e??a? µ?a ?µ?da ap? d?ad???
  ??tta?a ap????e?s?? (?µ?da ap? flip-flops ?a???a
  ap? ta ?p??a ap????e?e? ??a bit p????f???a?)
• ??a? ?ata????t?? t?? n bits ap?te?e?ta? ap? n
  flip-flops
• ??t?? t?? flip-flops, µp??e? ?a pe???aµß??e? ?a?
  p??e? ??a t?? e?t??es? ?e?t??????? epe?e??as?a?
  ded?µ????
• ? ap???ste??? t?p?? ?ata????t? ap?te?e?ta? ap?
  D-flip-flops, ????? e??te????? p??e?
• ?a ded?µ??a e?s?d?? e????f??ta? st?? ?ata????t?
  ?at? t?? efa?µ??? e??? ?????? pa?µ??
• ?pe?d? ??a ta bits t?? ?ata????t? f??t????ta?
  ta?t?????a (µe ??a µ??? pa?µ?), ? f??t?s?
  ???µ??eta? pa???????

45
?ata????t?? 4-bit
46
?ata????t?? ???s??s??

• ??a? ?ata????t?? st?? ?p??? ?? p????f???e? p??
  pe??????ta?, e??a? d??at?? ?a ???s?a????? p??? t?
  µ?a ? p??? t?? ???? ?ate????s? ???µ??eta?
  ?ata????t?? ???s??s?? (shift-register)
• ??a? ?ata????t?? ???s??s?? ap?te?e?ta? ap? µ?a
  a??s?da flip-flop s??dedeµ???? st? se???, ?ts?
  ?ste ? ???d?? t?? e??? ?a t??f?d?te? t?? e?s?d?
  t?? ep?µe???
• ? ap???ste??? ?ata????t?? ???s??s??, ap?te?e?ta?
  µ??? ap? D flip-flop. ?e ???e pa?µ? t?? ????????,
  t? pe??e??µe?? t?? ?ata????t? ???s?a??e? ?at? µ?a
  ??s? p??? ta de???

47
?ata????t?? ???s??s??
48
?et??t??

• ??a? µet??t?? ap?te?e?ta? ap? ??a? a???µ?
  ?at?????a s??dedeµ???? flip-flop ta ?p??a
  µetaß?????? t? pe??e??µe?? t???, s?????? ?at?
  ??a, ???e f??? p?? st?? e?s?d? t?? µet??t?
  efa?µ??eta? ??a ??? s?µa (p.?. pa?µ?? ????????)
• ? s?ed?as? t?? µet??t?? p?a?µat?p????ta?
  a????????ta? t? µe??d?????a s?ed?as??
  a???????a??? ?????µ?t??
• ?a??de??µa ?a s?ed?aste? µet??t?? ? ?p???? ?a
  apa???µe? st? d?ad??? s?st?µa ap? 0 ??? 7

49
??d??e? ?????e?s?? Sfa?µ?t??

• ?at? t?? ap????e?s? ?a? a???t?s? d?ad????
  p????f????? se µ?a µ???da µ??µ?? e??a? d??at?? ?a
  s?µß??? sf??µata
• ? a???p?st?a µ?a? µ???da? µ??µ?? µp??e? ?a
  a????e? µe t?? efa?µ??? ??d???? a????e?s?? ?a?
  d?????s?? ?a???
• ??? ????? s??µa a????e?s?? sfa?µ?t?? e??a? t?
  bit ?s?t?µ?a? (p??s?et? bit se ??a d?ad???
  µ???µa, ?ts? ?ste t? p????? t?? µ???d?? ?a ???e?
  e?te pe??tt? e?te ??t??)
• Ge???t??a ?s?t?µ?a? (parity generator) G?a
  ??t?a ?s?t?µ?a ?a? µ???µa t???? bit P x ? y
  ? z
• ??e??t?? ?s?t?µ?a? (parity checker) ??e??t??
  ??t?a? ?s?t?µ?a? C x ? y ? z ? P

50
??d??a? Hamming

• St?? ??d??a Hamming k bits ?s?t?µ?a?
  p??st??e?ta? st? ???? p????f???a? t?? n bits.
• ??µ?????e?ta? µ?a ???? t?? nk bits ?p?? ??
  ??se?? p?? a?t?st?????? se d???µe?? t?? 2
  p????????ta? ??a ta bits ?s?t?µ?a?.
• ?st? ? ??t?µp?t? ???? p????f???a? 11000100.
  ???s???? 4 bit ?s?t?µ?a?
• 1 2 3 4 5 6 7 8 9
  10 11 12
• P1 P2 1 P3 1 0 0 P4 0 1
  0 0
• P1 XOR t?? bits (3, 5, 7, 9, 11) 0
• P2 XOR t?? bits (3, 6, 7, 10, 11) 0
• P3 XOR t?? bits (5, 6, 7, 12) 1
• P4 XOR t?? bits (9, 10, 11, 12) 1
• ?e???? ???? p??? µet?d?s? 001110010100

51
??d??a? Hamming

• ???e bit ?s?t?µ?a? ?aµß??e? t?t??a t?µ? ?ste ?
  s???????? a???µ?? t?? 1 st?? e?e???µe?e? ??se??
  µa?? µe t? bit ?s?t?µ?a? ?a e??a? ??t???
• ?ta? ta 12 bits d?aßast???, e??????ta? ??a p??a??
  ????. ?a t?sse?a bit e?????? p????pt??? ?? e???
• C1 XOR t?? bits (1, 3, 5, 7, 9, 11)
• C2 XOR t?? bits (2, 3, 6, 7, 10, 11)
• C4 XOR t?? bits (4, 5, 6, 7, 12)
• C8 XOR t?? bits (8, 9, 10, 11, 12)
• ?? C C8 C4 C2 C1 0 de? ??e? p?????e? ?a???a
  ?????
• ?? C ! 0 ? d?ad???? a???µ?? t?? 4 bits e??????
  d????e? t? ??s? t?? sf??µat??

52
??d??a? Hamming

• ?a k bits e?????? (2k t?µ??, µe??? 1 p??
  a?t?st???e? st? µ?d??) p??pe? ?a "de??????" p???
  ap? ta nk bits ??e? ?p?ste? a?????s?.
• ??a 2k 1 gt nk gt 2k 1 k gt n
• ???es? t?? bit ??a t?? pa?a???? ?a? t?? ??e???
  t?? ?s?t?µ?a? (ap? ?at????? d?ad???? t?µ?? ap? 0
  ??? 2k 1)
• O ??d??a? Hamming µp??e? ?a a????e?se? ?a? ?a
  d?????se? ??a ap?? ?????. ????ap?? ???? de?
  a????e???ta?