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计算机学院 陈媛媛 [email protected] 第三章 组合逻辑原理. 组合逻辑的定义. 逻辑电路中没有从输出到输入的反馈,且由功能完全的门系列构成,就称为 组合逻辑电路 。. ·. ·. Combinational Logic Functions. Inputs. ·. ·. Outputs. ·. ·. 真值表问题. 1. 开关方程与标准形式. 多变量卡诺图化简. 多输出函数. 2. 4. 6. Content. 卡诺图. 3. 混合逻辑组合电路. 5.

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第三章 组合逻辑原理

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Chenyuanyuan@scu edu cn

[email protected]


5453072

Combinational Logic Functions

Inputs

Outputs


Content

1

2

4

6

Content

3

5


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a,b1210

s10

m10

M10


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    • y=2x

      x=y=


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S3

m3

S2

m2

m4

S1

m1

  • 3-4123123


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S3

m3

S2

m2

m4

S1

m1

s1,s2,s312310

m1,m2,m3,m4:10


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  • 1ABCDA12

  • 1A,B,C,D

    10

    S10.


Content1

1

2

4

6

Content

3

5


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  • 1 1 01


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  • m7= abms

    m11=abms

    m15=abms

    M=abms+abms+abms

    M=bms+abms


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  • m7= abms

    m11=abms

    m15=abms

    M=abms+abms+abms

    M=bms+abms

    bms, abms

    M=bms+abms

    m7m11m15

    M=m7+m11+m15

(1)(2)

(3)


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  • 0 010


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  • M0=a+b+m+s; M1=a+b+m+s;

    M2=a+b+m+s; M3=a+b+m+s;

    M4=a+b+m+s; M5=a+b+m+s;

    M6=a+b+m+s;M8=a+b+m+s;

    M9=a+b+m+s; M10=a+b+m+s;

    M12=a+b+m+sM13=a+b+m+s;

    M14=a+b+m+s;

    M=M0M1M2M3M4M5M6M8M9M10M12M13M14

    M=(a+b)(a+b+m)(a+b+m+s)(a+m)(a+m+s)


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  • M0=a+b+m+s; M1=a+b+m+s;

    M2=a+b+m+s; M3=a+b+m+s;

    M4=a+b+m+s; M5=a+b+m+s;

    M6=a+b+m+s;M8=a+b+m+s;

    M9=a+b+m+s; M10=a+b+m+s;

    M12=a+b+m+sM13=a+b+m+s;

    M14=a+b+m+s;

    M=M0M1M2M3M4M5M6M8M9M10M12M13M14

    M=(a+b)(a+b+m)(a+b+m+s)(a+m)(a+m+s)

    a+b, a+b+m

    (a+b)(a+b+m)(a+b+m+s)(a+m)(a+m+s)

    M0M1

    M=M0M1M2M3M4M5M6M8M9M10M12M13M14

(1)(2)(3)


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  • 1

    • M=abms+abms+abms=m7+m11+m15

      =m(7,11,15)

  • 0

    • M=(a+b+m+s)(a+b+m+s)(a+b+m+s)(a+b+m+s)(a+b+m+s)(a+b+m+s)(a+b+m+s)(a+b+m+s)(a+b+m+s)(a+b+m+s)(a+b+m+s)

      (a+b+m+s)(a+b+m+s)

      =M0M1M2M3M4M5M6M8M9M10M12M13M14

      =M(0,1,2,3,4,5,6,8,9,10,12,13,14)


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  • S=m11+m13 + m14 + m15= abcd+abcd+abcd+abcd

  • S=M0M1M2M3M4M5M6M7M8M9M10M12


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  • step1

    step2xy(z+z)

    step3xyz+xyz.

  • step1

    step2x+y+zz

    step3(x+y+z)(x+y+z).


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    • 1)10abcd:1010

      2)(1010)2=(10)10

      3)mk(k)

    • 1)01x+y+z:011

      2)(011)2=(3)10

      3)Mk(k)


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  • 1.P=f(a,b,c)=ab+ac+bc ()

    step1: ab:c; ac:b; bc:a

    step2: P=ab(c+c)+a(b+b)c+(a+a)bc

    step3: P=abc+abc+abc+abc+abc+abc

    step4: P=m5+m4+m6+m7+m3

    =m(3,4,5,6,7)


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2.Y(a,b,c,d)=abcd+bcd+ad ()

step1: abcd:; bcd:a; ad:b,c

step2: Y=abcd+(a+a)bcd+a(b+b) (c+c)d

step3:

Y=abcd+abcd+abcd+abcd+abcd+abcd+abcd

step4: Y=m9+m15+m7+m3+m13

=m(3,7,9,13,15)


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  • 3.T=f(a,b,c)=(a+b)(b+c) ()

    step1 a+b:cb+c: a

    step2: T=(a+b+cc)(aa+b+c)

    step3 T=(a+b+c)(a+b+c)(a+b+c)(a+b+c)

    step4 T=M2M3M6=M(2,3,6)


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4.Y(a,b,c,d)=(a+b)(b+c+d) ()

step1: a+b: c,d b+c+d:a;

step2: T=(a+b+cc+dd)(aa+b+c+d)

step3:

T=(a+b+c+d)(a+b+c+d)(a+b+c+d)(a+b+c+d)(a+b+c+d)(a+b+c+d)

step4:T=M0+M1+M2+M3+M7+M15=M(0,1,2,3,7,15)


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  • :

    P=f(w,x,y,z)=wx+yz

    T=f(a,b,c,d)=(a+b+c)(a+d)

  • Ans:

    P=f(w,x,y,z)=wxyz+wxyz+wxyz+wxyz+wxyz+wxyz+wxyz

    =m(2,4,5,6,7,10,14)

    T=f(a,b,c,d)= (a+b+c+d) (a+b+c+d) (a+b+c+d) (a+b+c+d) (a+b+c+d) (a+b+c+d)

    =M(4,5,8,10,12,14)


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    • step1

    • step2step1

    • step3step2


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  • f1(a,b,c)= abc + abc + abc + abc = m1 + m2 + m4 + m6

    = (1,2,4,6) = (0,3,5,7) = (a+b+c)(a+b+c)(a+b+c)(a+b+c)


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Content2

1

2

4

6

Content

3

5


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01

A

02B

13B

23A


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AB

A

B

0

1

00

01

11

10

0

m0

m2

m0

m2

m6

m4

0

1

m1

m3

1

m1

m3

m7

m5

AB

CD

00

01

11

10

00

m0

m4

m12

m8

C

01

m1

m5

m13

m9

m3

11

m7

m15

m11

10

m2

m6

m14

m10


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2

0

1

0

0

1

B

A

1

1

1

0

0

1

0

2

3

1


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    • step1

    • step2

    • step3

    • step410


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AB

10

00

01

11

C

0

1

  • 1.

  • 2.

  • 1

1

1

1

1

1


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BC

10

00

01

11

A

0

1

  • 2

1

1

1

1

1


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ab

10

00

01

11

c

0

1

  • f(a,b,c) = ac + abc + bc

    step1

    step2


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ab

10

00

01

11

c

0

1

step3

f(a,b,c) = ac + abc + bc

=ac(b+b)+abc+(a+a)bc

=abc+abc+abc+abc+abc

=abc+abc+abc+abc

step4:

1

1

1

1


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XY

XY

10

00

01

11

10

00

01

11

Z

Z

1

1

1

1

1

0

0

1

1

1

1

1

1

  • f1(x,y,z)= m(2,5,6,7)

  • f2(x, y, z)=m(0,1,2,3,6)


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ab

10

00

01

11

c

0

1

1

1

1

1

1


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1248


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YZ

00

01

11

10

WX

m0

m1

m3

m2

00

m4

m5

m7

m6

01

m12

m13

m15

m14

11

m8

m9

m11

m10

10

2

4

8

161.


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1

1

1

1

1

1

1

1

1

1

1

  • (A,B,C,D) = m(0,1,2,3,4,5,6,7,8,10,13).

    step1

    step2

A

BD

AB

00

01

11

10

CD

00

1

1

1

BCD

1

1

1

01

1

1

11

1

1

1

10

g(A,B,C,D) = A+BD+BCD


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ab

10

00

01

11

c

1

1

1

0

1

1

  • :

    f(a,b,c) = ac + abc + bc

    =a(b+b)c+abc+(a+a)bc

    =abc+abc+abc+abc+abc

    =abc+abc+abc+abc

    = ac+ab+bc

abc

bac

cab


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ab

10

00

01

11

c

0

1

  • :

ACB

1

1

1

1

1

BAC


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xy

z

00

01

11

10

0

1

1

1

1

1

xy

z

1

1

1

1

1

  • f1(x,y,z)= m(2,5,6,7)

    f2(x, y, z)=m(0,1,2,3,6)

f1(x, y, z) = yz + xz

f2(x, y, z) = x+yz


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1

1

1

1

1

1

1

1

1

1

1

  • (1) 1

  • (2) 12n1

  • (3) ()

  • (4)

  • (5)


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  • :

  • (PI):

    ACDABCD,BCD

  • (EPI):

    ABCBD

AB

CD

00

01

11

10

1

00

1

01

1

1

11

1

1

10

1

BCD

ABC

ABCD

ACD

BD


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b

ad

ab

00

01

11

10

cd

1

1

00

1

1

1

01

1

1

1

11

ab

1

1

1

10

acd

acd

  • ab,acd,acd,ad,b

  • abb.

  • acd ad.

  • b, ad, acd

  • b, ad, acd .


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  • f2(a,b,c,d),

  • f2bd.

acd

bd

abd

abc

bcd

acd

abc

ab

10

00

01

11

cd

1

00

1

1

1

01

1

1

11

1

1

1

10


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`

  • f(a,b,c,d) = m(0,1,4,5,8,11,12,13,15).

    5

    ac,cd,acd.

  • Ans:

    f(a,b,c,d) = cd + ac + bc + acd

bc

cd

ab

cd

1

1

1

1

ac

1

1

1

1

1

abd

acd


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xy

z

00

01

11

10

1

1

1

0

1

1

1

1

F(x,y,z)=(0,2,3,4,5,7)

6

Ans:

F(x,y,z)=xz+yz+xy

F(x,y,z)=yz+xy+xz

xy

z

00

01

11

10

1

0

1

1

1

1

1

1


5453072

  • f(a,b,c,d) = (0,3,4,5,7,11,13,15)

  • Ans:

    f(a,b,c,d) =acd+cd+bc

ab

cd

1

1

1

1

1

1

1

1


5453072

F(w,x,y,z)=(0,1,4,5,9,11,13,15)

F(a,b,c,d)=(0,1,2,4,5,6,8,9,12,13,14)

F(a,b,c,d)=(1,3,4,5,7,8,9,11,15)

F(w,x,y,z)=(1,5,7,8,9,10,11,13,15)


5453072

F(w,x,y,z)=(0,1,4,5,9,11,13,15)

wx

10

00

01

11

yz

00

1

1

1

1

01

1

1

1

11

1

10

ANS:F(w,x,y,z)=wy+wz


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ab

cd

10

10

00

00

01

01

11

11

ab

cd

00

00

01

01

11

11

10

10

F(a,b,c,d)=(0,1,2,4,5,6,8,9,12,13,14)

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

F(a,b,c,d)=c+ad+bd


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ab

10

00

01

11

cd

00

01

11

10

F(a,b,c,d)=(1,3,4,5,7,8,9,11,15)

1

1

1

1

1

1

1

1

1

ANS:F(a,b,c,d)=cd+ad+abc+abc


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wx

10

00

01

11

yz

00

01

11

10

F(w,x,y,z)=(1,5,7,8,9,10,11,13,15)

1

1

1

1

1

1

1

1

1

F(w,x,y,z)=yz+xz+wx


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  • ()


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  • d()10


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  • (dont care minterms)

    • (d)


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b1b0

00

01

11

10

b3b2

00

0

0

0

0

01

0

1

1

1

11

d

d

d

d

10

1

1

d

d

8421BCD


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A=f(w,x,y,z)=(5,6,7,8,9)+

d(10,11,12,13,14,15)

B=f(w,x,y,z)=(1,2,3,4,9)+

d(10,11,12,13,14,15)

C=f(w,x,y,z)=(0,3,4,7,8)+

d(10,11,12,13,14,15)

D=f(w,x,y,z)=(0,2,4,6,8)+

d(10,11,12,13,14,15)


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A=w+xz+xy

B=xy+xz+xyz

C=yz+yz

D=z


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cd

ab

01

11

00

10

0

1

0

1

00

1

1

0

1

01

0

0

d

d

11

10

1

1

d

d

0

1

0

1

1

1

0

1

0

0

d

d

1

1

d

d

0

1

0

1

1

1

0

1

0

0

d

d

1

1

d

d

f = acd+ab+cd+abc

f = acd+ab+cd+abd


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    • 0

    • 01


5453072


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a+b+c+d

cd

ab

00

01

11

10

00

1

1

1

1

1

1

1

0

01

  • F = (a+b)(a+c)(a+b+c+d)

0

0

1

1

11

0

0

0

0

10

a+c

a+b


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AB

10

00

01

11

CD

00

01

0

0

0

0

11

0

10

0

C+D

F=(C+D)(A+B+D)(A+B+C)

A+B+C

A+B+D


Content3

1

2

4

6

Content

3

5


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  • (Quine-Mcluskey)


Content4

1

2

4

6

Content

3

5


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  • LEDLEDLED


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1

1

F=AB

F=A+B

L:

H:


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  • step1:

    step2:

    step3:


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=

+

=

L

A

B

AB

L

=

AB

&

A

A

1

B

B


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=

=

+

L

A

B

A

B

L

=

A

+

B

A

&

A

1

B

B


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  • step1:

    step2

    step3


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3

1

2


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H

H

H

H=D+C(A+B)


Content5

1

2

4

6

Content

3

5


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AB

AB

00

00

01

01

11

11

10

10

1

1

0

0

1

1

1

1

1

1

C

C

  • . F1=f(a,b,c)=(2,6,7)

    F2=f(a,b,c)=(1,3,7)

F1=ac+bc

F1=ac+abc

F2=bc+abc

F2=bc+ab


5453072

F1=ac+bc

F1=ac+abc

F2=bc+abc

F2=bc+ab


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