4. ban do karnaugh v4

Chng 4

NHP MN MCH S

Ba Karnaugh

v Ti u ha mch logic

Ni dung

1. Mch logic s (Logic circuit)

2. Thit k mt mch s

3. Bn Karnaugh

4. Multilevel optimization

5. Cng XOR/XNOR ( XOR/XNOR gate)

2

Dng nh l Boolean n gin hm sau:

Tn Dng AND Dng OR

nh lut thng nht 1A = A 0 + A = A

nh lut khng OA = O 1+ A = 1

nh lut Idempotent AA = A A + A = A

nh lut nghch o

nh lut giao hon AB = BA A + B = B + A

nh lut kt hp (AB)C = A(BC) (A+B)+C = A + (B+C)

nh lut phn b A + BC = (A + B)(A + C) A(B+C) = AB + AC

nh lut hp th A(A + B) = A A + AB = A

nh lut De Morgan

0AA 1 AA

BAAB ABBA

1. Mch logic s (logic circuit)

3

Dng chnh tc v dng chun ca hm Boolean

Tch chun (minterm): mi l cc s hng tch (AND) m tt c cc bin xut hin dng bnh thng (nu l 1) hoc dng b (complement) (nu l 0)

Tng chun (Maxterm): Mi l cc s hng tng (OR) m tt c cc bin xut hin dng bnh thng (nu l 0) hoc dng b (complement) (nu l 1)

4

Dng chnh tc (Canonical Form)

Dng chnh tc 1: l dng tng ca cc tch chun_1 (minterm_1) (minterm_1 l minterm m ti t hp hm Boolean c gi tr 1).

5

6

Dng chnh tc (Canonical Form) (tt)

Dng chnh tc 2: l dng tch ca cc tng chun_0 (Maxterm_0) (Maxterm_0 l Maxterm m ti t hp hm Boolean c gi tr 0).

Trng hp ty nh (dont care)

Hm Boolean theo dng chnh tc:

F (A, B, C) = (2, 3, 5) + d(0, 7) (chnh tc 1)

= (1, 4, 6) . D(0, 7) (chnh tc 2)

A B C F

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

X

0

1

1

0

1

0

X

0 2 5 6 7

( , , ) ( )( )( )( )( )F x y z x y z x y z x y z x y z x y z

M M M M M

V d

Cu hi: Trong cc biu thc sau, biu thc no dng chnh tc?

a. XYZ + XY

b. XYZ + XYZ + XYZ

c. X + YZ

d. X + Y + Z

e. (X+Y)(Y+Z)

Tr li:

b v d

Dng chnh tc (Canonical Forms) (tt)

Tng cc tch chun

Sum of Minterms

Tch cc tng chun

Product of Maxterms

Ch quan tm hng c

gi tr 1

Ch quan tm hng c

gi tr 0

X = 0: vit X X = 0: vit X

X = 1: vit X X = 1: vit X

Dng chun (Standard Form)

Dng chnh tc c th c n gin ho thnh dng chun tng ng dng n gin ho ny, c th c t nhm AND (hoc

OR) v/ hoc cc nhm ny c t bin hn

Dng tng cc tch - SoP (Sum-of-Product) V d:

Dng tch cc tng - PoS (Product-of-Sum) V d :

C th chuyn SoP v dng chnh tc bng cch AND thm

(x+x) v PoS v dng chnh tc bng cch OR thm xx

V d

Cu hi: Trong cc biu thc sau, biu thc no dng chun?

a. XYZ + XY

b. XYZ + XYZ + XYZ

c. X + YZ

d. X + Y + Z

e. (X+Y)(Y+Z)

Tr li:

Tt c Chun

2. Thit k mt mch logic

V d

Thit k mt mch logic s vi

3 u vo

1 u ra

Kt qu l HIGH khi c t 2 u vo tr ln c gi tr HIGH

12

Th tc (procedure) thit k mch logic s

Bc 1: xy dng bng s tht / chn tr

13


Bc 2: chuyn bng s tht sang biu thc logic

A B C X

0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

Cc nhm AND cho mi

trng hp ng ra l 1

Biu thc SOP cho ng ra X:


Bc 3: n gin biu thc logic qua bin i i s

15

Hn ch ca bin i i s

Hai vn ca bin i i s

1. Khng c h thng

2. Rt kh kim tra rng gii php tm ra l ti u hay cha?

Bn Karnaugh s khc phc nhng nhc im ny

Tuy nhin, bn Karnaugh ch gii quyt cc hm Boolean c khng qu 5 bin

16


Bc 4: v s mch logic cho

17

3. Ba Karnaugh

Chi ph to ra mt mch logic

Chi ph (cost) to ra mt mch logic lin quan n:

S cng (gates) c s dng

S u vo ca mi cng

Mt literal l mt bin kiu Boolean hay b (complement) ca n

19

Chi ph to ra mt mch logic

Chi ph ca mt biu thc Boolean B c biu din di dng tng ca cc tch (Sum-of-Product) nh sau:

Trong k l s cc term trong biu thc B

O(B) : s cc term trong biu thc B

PJ(B): s cc literal trong term th j ca biu thc B

20

Chi ph to ra mt mch logic V d

Tnh chi ph ca cc biu thc sau:

21

Bn /Ba Karnaugh

M. Karnaugh, The Map Method for Synthesis of combinatorial Logic Circuits, Transactions of the American Institute of Electrical Engineers, Communications

and Electronics, Vol. 72, pp. 593-599, November 1953.

Ba Karnaugh l mt cng c hnh hc n gin ha cc biu thc logic

Tng t nh bng s tht, ba Karnaugh s xc nh gi tr ng ra c th ti cc t hp ca cc u vo

tng ng.

22

Ba Karnaugh

Ba Karnaugh l biu din ca bng s tht di dng mt ma trn cc (matrix of squares) hay cc cells trong

mi cell tng ng vi mt dng tch chun (minterm)

hay dng tng chun (Maxterm).

Vi mt hm c n bin, chng ta cn mt bng s tht c 2n hng, tng ng ba Karnaugh c 2n (cell).

biu din mt hm logic, mt gi tr ng ra trong bng s tht s c copy sang mt tng ng trong ba K

23

Ba Karnaugh 2 bin

24

Ba Karnaugh 3 bin

25

V d:

(cha ti u)

(ti u)

(i s)

Ba Karnaugh 3 bin

26

Cch 1 Cch 2 Cch 3

Lu : c th s dng cch no biu din ba-K cng c, nhng

phi lu trng s ca cc bin th mi m bo th t cc theo gi

tr thp phn.

Ba Karnaugh 3 bin

27

Ba Karnaugh 3 bin

28

f

(cha ti u)

(ti u)

Ba Karnaugh 3 bin

29

F F

Bn Karnaugh 3 bin

G = F 30

G G

Bn Karnaugh 3 bin

31

Rt gn cha ti u Rt gn ti u

V d:

F = xz + xy + yz F = xz + xy

Ba Karnaugh 3 bin

32

V d:

Ba Karnaugh 4 bin

Khoa KTMT 33

Simplify

F = ac + ab + d

Khc vi slide c

Ba Karnaugh 4 bin

34

Ba Karnaugh 4 bin

35

Hm c t khng y (Incompletely Specified Functions)

Gi thuyt: N1 khng bao gi cho kt qu ABC = 001 v

ABC = 110

Cu hi : F cho ra gi tr g trong trng hp ABC = 001 v ABC = 110 ?

We dont care!!!

36

Trong trng hp trn th chng ta phi lm th no n gin N2?

37

Gi s F(0,0,1) = 0 v F(1,1,0)=0, ta c

biu thc sau:

Hm c t khng y (tt) (Incompletely Specified Functions)

= AC(B + B) + (A + A)BC

= AC1 + 1BC

= AC + BC

F(A,B,C) = ABC + ABC + ABC + ABC

Tuy nhin, nu gi s s F(0,0,1)=1 v F(1,1,0)=1, ta c biu thc sau:

38

So snh vi gi thuyt trc :

F(A,B,C) = AC + BC, gii php no chi ph t hn (tt hn)?


A B C F0 0 0 10 0 1 X0 1 0 10 1 1 11 0 0 01 0 1 01 1 0 X1 1 1 1

+

F(A,B,C) = ABC + ABC + ABC + ABC + ABC + ABC

= AB 1 + AB 1 + AB 1

= AB(C + C) + AB(C + C) + AB(C + C)

= AB + AB + AB

= AB + AB + AB + AB

= A(B + B) + (A + A)B

= A1 + 1B

= A + B

1

1

Tt c cc 1 phi c khoanh trn, nhng vi c gi tr X th

ty chn, cc ny ch c xem xt l 1 nu chng c s dng

n gin biu thc.


n gin POS(Product of Sum)

Khoanh trn gi tr 0 thay v gi tr 1

p dng nh lut De Morgan chuyn t SOP sang POS

Khoa KTMT 40

Implicant c bn (Prime Implicant)

Implicant: l dng tch chun ca mt hm

Mt nhm cc 1 hoc mt 1 n l trn mt K-map kt hp vi nhau to ra mt dng tch chun

Implicant c bn (prime implicant):

Implicant khng th kt hp vi bt k 1 no khc loi b mt bin

Tt c cc prime implicant ca 1 hm c th t c bng cch pht trin cc nhm 1 trong K-map ln nht c th

41

V d

42

a'b'c, a'cd', ac' l cc prime implicants

a'b'c'd', abc', ab'c' l cc implicants

(nhng khng phi l

prime implicants)

Xc nh tt c cc prime implicants

xc nh cc prime implicants, cc gi tr khng xc nh (dont care) c coi nh l gi tr 1.

Tuy nhin, mt prime implicant ch gm cc gi tr khng xc nh (dont care) th khng cn cho biu thc ng ra.

Khng phi tt c cc prime implicant u cn thit to ra minimum SOP

V d Tt c cc prime implicants: a'b'd, bc', ac,

a'c'd, ab, b'cd (ch gm cc gi tr khng xc nh)

Minimum solution:

F = a'b'd+bc'+ac

Ti thiu biu thc s dng Essential Prime Implicant (EPI)

Ti thiu biu thc s dng Essential Prime Implicant (EPI) (tt)

Essential prime implicant (EPI): prime implicant c t nht 1 khng b gom bi cc prime

implicant khc

44

1. Chn ra tt c EPI

2. Tm ra mt tp nh nht cc prime

implicant ph(gom) c tt c cc

minterm cn li (cc minterm khng

b gom bi cc EPI)

45


Lu xc nh mt minimum SOP s dng K-map

46


V d

Step 1: nh du 14

Step 2: nh du 15

Step 3: nh du 16 EPI => A'B c chn

Step 4: nh du 18

Step 5: nh du 19

Step 6: nh du 110 EPI => AB'D' c chn

Step 7: nh du 113

(ti im ny tt c EPIs c xc

nh)

Step 8: AC'D c chn gom cc s 1 cn li

Bn Karnaugh 5 bin

48

Bn Karnaugh 5 bin

49

Bn Karnaugh 5 bin

50

Bn Karnaugh 5 bin

51

Bn Karnaugh 5 bin

Bin i khc

52

V d 2

(31,30,29,27,25,22,21,20,17,16,15,13,11,9,6,4,1,0)F

Bn Karnaugh 5 bin

V d 2

(31,30,29,27,25,22,21,20,17,16,15,13,11,9,6,4,1,0)F

F = ACDE + BCE + BE + BCD + ABD

Bn Karnaugh 5 bin

4. Multiple-Level Optimization

Multiple-Level Optimization

Mch multi-level l mch c nhiu hn hai level (c thm input v/hoc b o output)

Vi mt hm cho sn, cc mch multi-level c th gim chi ph cng u vo so vi vi mch two-level (POS v SOP)

Vic ti u ha mch multi-level c th thc thin bng cch p dng cc php bin i (transformation), ng thi nh gi chi ph (evaluating cost) cho cc mch ny

Php bin i (Transformations)

Php t nhn t chung (Factoring) - tm ra mt thnh phn chung (factored form) t biu thc SOP hoc POS

Php phn tch (Decomposition) - biu thc hm c tch thnh mt tp cc hm mi

Php thay th (Substitution) G vo F - Biu thc F c xem nh l mt hm ca bin G v tt c cc bin ban u

Php kh (Elimination) - ngc vi php thay th

Php rt trch (Extraction) - php phn tch (decomposition) p dng cho nhiu hm ng thi

V d: Php t nhn t chung (Factoring)

Hm:

Factoring

Factoring tip:

Factoring tip:

V d: php phn tch (decomposition)

F = ACD + ABC + ABC + ACD G = 16 Nhm AC + AC v B + D c th c nh ngha nh l

hai hm mi H v E

F c phn tch thnh: F = (AC + AC)(B + D):

F = H E, H = AC + AC, E = B + D G = 10

Chui bin i trn lm gim G t 16 xung 10

Mch sau cng c ba cp (level) cng vi cc cng o u vo

V d: php thay th (Substitution)

Thay th E vo F Quay li F bc cui cng ca php bin i factoring

t , v thay th vo F:

php thay th cho kt qu G ging vi php phn tch (decomposition)

G=12

G=10

V d: php kh (Elimination)

Gi s c mt tp cc

hm mi:

Kh X v Y trong Z:

Flattening (bin i thnh SOP):

iu ny lm tng ch ph G nhng to ra mt biu thc SOP mi c th s dng ti u 2 cp (two-level

optimization)

G=10

G=12

G=10

V d: php kh (Elimination)

Ti u 2 cp ( two-level optimization) cho kt qu:

V d ny minh ho:

Ti u c th bt u bng bt k tp ng thc no, khng ch vi dng tch chun (minterm) hoc bng chn tr ( true

table)

Tng chi ph G tm thi trong qu trnh bin i c th to ra mt kt qu sau cng vi chi ph G nh hn

G=4

V d: php rt trch (Extraction)

Cho 2 hm:

Tm phn chung (common factor) v nh ngha n l 1 hm

Thc hin rt trch biu din E v H bng 3 hm

Ch ph G gim l kt qu ca vic chia s logic ca 2 hm E v H

Tm tt Multi-level Optimization

Cc php bin i

Php t nhn t chung (Factoring) tm ra mt thnh phn chung (factored form) t biu thc SOP hoc POS

Php phn tch (Decomposition) biu thc hm c tch thnh mt tp cc hm mi

Php thay th (Substitution) ca biu thc G trong biu thc F Biu thc F c xem nh l mt hm ca bin G v tt c cc bin ban u

Php kh (Elimination): ngc vi php thay th

Php rt trch (Extraction): php phn tch (decomposition) p dng cho nhiu hm ng thi

5. Cng XOR v XNOR

Mch Exclusive OR v Exclusive NOR

Exlusive OR (XOR) cho ra kt qu HIGH khi hai u vo khc nhau

x = AB + AB

Output expression:

XOR Gate Symbol

Exlusive NOR (XNOR) cho ra kt qu HIGH khi hai u vo ging nhau

XOR v XNOR cho ra kt qu ngc nhau


Output expression

x = AB + AB XNOR Gate

Symbol

V d

Thit k mt mch pht hin ra 2 s nh phn 2 bit

c bng nhau hay khng

Lm sao ti u mch

bng cng XNOR


B to v kim tra Parity (Parity generator and checker)

Cng XOR v XNOR rt hu dng trong cc mch vi mc ch pht v kim tra parity

Any question?

4. ban do karnaugh v4

Documents