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Karnaugh
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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)
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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
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Th tc (procedure) thit k mch logic s
Bc 1: xy dng bng s tht / chn tr
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Th tc (procedure) thit k mch logic s
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:
Th tc (procedure) thit k mch logic s
Bc 3: n gin biu thc logic qua bin i i s
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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
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Th tc (procedure) thit k mch logic s
Bc 4: v s mch logic cho
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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
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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
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Chi ph to ra mt mch logic V d
Tnh chi ph ca cc biu thc sau:
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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.
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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
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Ba Karnaugh 2 bin
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Ba Karnaugh 3 bin
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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
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f
(cha ti u)
(ti u)
Ba Karnaugh 3 bin
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F F
Bn Karnaugh 3 bin
G = F 30
G G
Bn Karnaugh 3 bin
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Rt gn cha ti u Rt gn ti u
V d:
F = xz + xy + yz F = xz + xy
Ba Karnaugh 3 bin
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V d:
Ba Karnaugh 4 bin
Khoa KTMT 33
Simplify
F = ac + ab + d
Khc vi slide c
Ba Karnaugh 4 bin
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Ba Karnaugh 4 bin
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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!!!
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Trong trng hp trn th chng ta phi lm th no n gin N2?
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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:
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So snh vi gi thuyt trc :
F(A,B,C) = AC + BC, gii php no chi ph t hn (tt hn)?
Hm c t khng y (tt) (Incompletely Specified Functions)
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.
Hm c t khng y (tt) (Incompletely Specified Functions)
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
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V d
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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
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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)
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Ti thiu biu thc s dng Essential Prime Implicant (EPI) (tt)
Lu xc nh mt minimum SOP s dng K-map
46
Ti thiu biu thc s dng Essential Prime Implicant (EPI) (tt)
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
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
Mch Exclusive OR v Exclusive NOR
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
Mch Exclusive OR v Exclusive NOR
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?