F1 AB C D ABC+ +

AB (C C) (D D) C D (A A) (B B) ABC (D D)⋅ + ⋅ + + ⋅ + ⋅ + + ⋅ +

AB (CD CD CD C D) CD (AB AB AB A B)⋅ + + + + ⋅ + + + + +ABCD ABCD

ABCD ABCD ABCD ABC D ABCD AB CD ABCD+ + + + + + +

A B C D ABCD ABCD+ + A A A+ =

A B C D ABCD AB CD ABCD ABCD+ + + +

ABC D ABC D ABCD ABCD+ + +

f (x, y, z) (xy z) (y x z)+ +

xy xyz y z x z+ + +

xy (z z) xyz yz (x x) x z (y y)⋅ + + + ⋅ + + +

xyz xyz xyz xyz xyz xyz xy z+ + + + + +

xyz xyz xyz xy z+ + +

(P Q) (P R)+ +

(P Q R R) (P R Q Q)+ + ⋅ + + ⋅

(P Q R) (P Q R) (P Q R) (P Q R)+ + + + + + + + A BC (A B) (A C)+ = + +

(P Q R) (P Q R) (P Q R)+ + + + + + A A A⋅ =

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

F (A, B, C, D) B C BD AC+ +

∴ F (w, x, y, z) w x y xyz w z+ +

∴ F (w, x, y, z) (w x) (w y z)+ + +

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

00 01 11 10

00

01

11

10

ABCD

1 X

X

1 1

1 1

1111

X

wxyz

1 1

1

00 01 11 10

00

01

11

10

w z

w yxX

1 1

1

X X

x y z

w xy z

w + x

w + x

w x+

w + x

y + z

0

XX

X

X X X

y + z y z+ y + z

0

0 0 0 0

w + y + z

w + x

∴ A B DE A B CD A B E A C E ABCD ABDE ABC D+ + + + + +

F (A, B, C, D, E) A B D B C D A D E+ +

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

x

w

x

w

w

yz

F

BCDE

1

1

1

11

00 01 11 10

00

01

11

10

A B D E

A B C D

A EB

1

1

A = 0

BCDE

1

1

1

11

00 01 11 10

00

01

11

10

A EC

A CB D1

A = 1

1

1

A B D E

A B C D

00 01 11 10

00

BCDE

01

11

10

1 1

1 1

For A = 0

00 01 11 10

00

BCDE

01

11

10

1

For A = 1

1

1

1

1

m1 m1

m3 m8

m6 m3

m7 m6

m8 m9

m9 m10

m10 m12

m12 m7

m14 m14

m15 m11

m11 m13

m13 m15

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

(B D)

∴ F (A, B, C, D) BD A BC+ +

m0 m0

m1 m1

m2 m2

m3 m3

m10 m10

m11 m12

m12 m11

m13 m13

m14 m14

m15 m15

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

Primeimplicants

m1

1, 3, 9, 11 (BD)

8, 9, 10, 11, 12, 13, 14, 15

6, 7, 14, 15 (BC)

m3 m6 m9 m10 m12 m14 m15m7 m8 dm11 dm13

F (A, B, C, D) A B AC AB+ +

m0 m0

m1 m1

m2 m2

m8 m8

m9 m9

m15 m17

m17 m24

m21 m21

m24 m25

m25 m15

m27 m27

m31 m31

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

Primeimplicants

m0

0, 1, 2, 3 (AB)

2, 3, 10, 11 (BC)

10, 11, 14, 15 (AC)

12, 13, 14, 15 (AB)

m1 m2 m11 m12 m13 m14m3 m10 m15

F (A, B, C, D, E) A C D B C D AB C E ABCE BCDE AB D E+ + + + +

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

Primeimplicants

m0

0, 1, 8, 9 (ACD)

8, 9, 24, 25 (BCD)

0, 2 (ABCE)

1, 17 (BCDE)

m1 m2 m15 m17 m21 m24 m25m8 m9 m27

17, 21 (ABDE)

17, 25 (ACDE)

25, 27 (ABCE)

15, 31 (BCDE)

27, 31 (ABDE)

m31

M d( , , , , , ) ( , , )0 1 4 11 13 15 5 7 8∏ ∏+m d( , , , , , , ) ( , , )2 3 6 9 10 12 14 5 7 8+∑

m2 m2

m3 dm8

m6 m3

m9 m6

m10 m9

m12 m10

m14 m12

dm5 dm5

dm7 m14

dm8 dm7

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

A B C

A C D

A B D

A C

C D

A D

( ) ( ) ( )1 0 0 0 1 1 0- - - - -+ +

∴ ABC AC AD+ +

m0 m0

m1 m1

m4 m4

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

Prime implicants

ABC 8, 9

m2 m3 m6 m9 m10 m12 m14 dm5 dm7 dm8

ACD 8, 12

ABD 5,7

AC 2,3,6,7

CD 2,6,10,14

AD 8,10,12,14

X

ABCD

00 01 11 10

00

01

11

10

X 1

1 1

1

11

X 1

AC

ADABC

m5 m16

m16 m5

m17 m17

m21 m21

m25 m25

m29 m29

x x x1 2 4

x x x2 3 4

x x x2 4 5

x x x1 4 5

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

x x x x x x x x x1 4 5 2 3 4 1 2 4+ +

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

BCD

ACD

BCD

ABD

ABC

AB

ABD

1,9

4,6

6,7

9,11

7,15

11,15

8,9,10,11

m1 m4 m6 m7 m8 m9 m10 m11 m15Prime

implicants

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

S0I3 I2 I1 I0

D1 D0 D1 D0

D1 D0

S S2 : 1MUX 2

2 : 1MUX 1

2 : 1MUX 3

Y

Y2

SS1

Y1 Y0

E

Y Y

a

a

D0 D1 D2 D3 D4 D5 D6 D7

0 1 2 3 4 5 6 7

8 9 10 11 12 13 14 15

1 a a a a a a0

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

D31 7ID30 6ID29 5ID28 4ID27 3ID26 2ID25 1ID24 0I

E S2 S1 S0

8 : 1MUX 4 Y4

D23 7ID22 6ID21 5ID20 4ID19 3ID18 2ID17 1ID16 0I

8 : 1MUX 3

Y3

S2 S1 S0E

D15 7ID14 6ID13 5ID12 4ID11 3ID10 2ID9 1ID8 0I

E S2 S1 S0

8 : 1MUX 2 Y2

D7 7ID6 6ID5 5ID4 4ID3 3ID2 2ID1 1ID0 0I

8 : 1MUX 1

Y1

E

Y

S0 S1 S2

S4 S3

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

a

D0D1D2D3D4D5D6D7

S S S2 1 0

B C D

8 : 1MUX Y

1

0

D0 D1 D2 D3 D4 D5 D6 D7

0 1 2 3 4 5 7

10 11 12 13 14 159

A

A

0A A A A A

6

8

0 0

For F1

D8 D9 D10 D11 D12 D13 D14 D15

0 1 2 3 4

10 11 12 13 14 159

A

A

0A A A A

8

5 6 7

A A A

For F2

D16 D17 D18 D19 D20 D21 D22 D23

0 1 2 3

10 11 12 13 159

A

A

A0 A 0 A

8

5 6 7

1 1 0

4

14

For F3

C m (3, 5, 6, 7)out =∑∑ m (1, 2, 4, 7)

D 1in =

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

D24 D25 D26 D27 D28 D29 D30 D31

0 1 2 3

10 11 12 13 15

A

A

AA A A 1

5 6 7

0 A 0

4

148 9

For F4

1 : 8DEMUX

YYYYYYYY

01234567

S0 S1 S2

A B C

D = 1in

Cout

S

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

1 : 8DEMUX

YYYYYYYY

01234567

S S S0 1 2

A B C

D = 1in

f1

f2

Y0Y1Y2Y3Y4Y5Y6Y7Y8Y9

Y10Y11Y12Y13Y14Y15

F1

F2

A

B

C

D

A

B

C

D4 : 16

Decoder

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

00 00

01 01

00 1101 10 00 1101 10E E1 0

E E1 0 E E1 0

E E1 0E E3 2

E E3 2 E E3 2

E E3 2

X XX X0 0X X

1 11 00 00 1

10 10

11 11

1 11 00 00

X X1 0X XX X

For B0

B =0 0E

For B1

1

00

01

00 1101 10

XX 0X

00 00

10

11

00 0

X1 X

For B3

B = E + E= E + E

1 1 0 1 0

1 0

E E

B = E E + E E E3 3 2 3 1 0

00

01

00 1101 10

XX 0X

00 10

10

11

11 01

X0 XX

For B2

B = + E E E + E E2 2 1 2 1 0 3 1 0E E E

1

X

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

BCD code

Excess -3 code

E3 E2 E1 E0

B0

B1

B2

B3

(G G G G ) G G (G G G G ) G G3 2 3 2 1 0 3 2 3 2 1 0+ + +

+ + + +(G G G G ) G G (G G G G ) G G3 2 3 2 1 0 3 2 3 2 1 0

(G 3 ⊕ G2) G G1 0 G 3 G2 G G1 0

(G 3 ⊕ G2 G G1 0 G 3 G2 G G1 0

⊕ G G1 0 (G G G G )1 0 1 0+

⊕ ⊕

⊕ G G1 0⊕ G G3 2⊕ ⊕

⊕ ⊕ ⊕

G G3 2 (G G G G ) G3 2 3 2 1+

⊕ G1

( ) ( )G G G G G G3 2 1 3 2 1⊕ + ⊕

G G G3 ⊕ ⊕2 1

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

00

01

00 1101 10G G3 2 G G3 2

G G1 0 G G1 0

10 01

01 10

10

11

01 10

10 01

00

01

00 1101 10

1

0

0

1

1

0

0

1

10

11

1 1

1

0

0 1

0

0

For A For B

G G G G3 2 3 2+ G 3

⊕

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

00

01

00 1101 10G G3 2 G G3 2

G G1 0 G G1 0

00 00

11 11

10

11

1 1

00 00

00

01

00 1101 10

0

0

0

0

0

0

0

0

10

11

1 1

1

1

1 1

1

1

For C For D

1 1

G3 G2 G1 G0

Gray code

Binary code

A

B

C

D

2n ≥

∴

Qn Qn+1

QD QC QB QA QD+ QC

+ QB+ QA

+ JD KD JC KC JB KB JA KA

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

0

0 0 1 0

X X X X

X X X X

0 0

Q QA B

Q QC D00 01 11 10

00

01

11

10

0

J = Q Q QD B C D

X

X X X X

0 0 X 1

X X X X

X X

Q QA B

Q QC D00 01 11 10

00

01

11

10

X

K = QD C

For JD For KD

0

X X X X

0 0 X 0

X X X X

1 0

Q QA B

Q QC D00 01 11 10

00

01

11

10

0

J = Q QC C D

For JC

X

0 0 1 0

X X X X

X X X X

X X

Q QA B

Q QC D00 01 11 10

00

01

11

10

X

K = Q QC C D

For KC

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

0 1 X X

0 1 X X

X X X X

1 X X

Q QA B

Q QC D00 01 11 10

00

01

11

10

0

J = QB D

X X 1 0

X X 1X

X X X X

X 1 0

Q QA B

Q QC D00 01 11 10

00

01

11

10

X

K = Q + QB A D

For JB For KB

X X

1 X X 0

X X X X

X

1

1

Q QA B

Q QC D00 01 11 10

00

01

11

10

X

1

1

J = Q + QA A C

X 1 1 X

1 X XX

X X X X

Q QA B

Q QC D00 01 11 10

00

01

11

10

1 1 XX

K = 1A

For JA For KA

JA QA

KA QA

A

QAQC

CLK

JB QB

KB QB

B

QAQD

QD JC QC

KC QC

C

QCQD

JD QD

KD QD

D

QB

QDQC

QC

(LSB)QA QB QC Q (MSB)D

Output

1

TECHNICAL PUBLICATIONS - An up thrust for knowledgeTM

A0 D0

A1 D1

A2 D2

A3 D3

ROM4 16×

Addresslines

Binaryinput

Datalines

Gray codeoutput