homework5_soln_2009_corrected

homework5_soln_2009_corrected - Homework
#5
Solutions


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Unformatted text preview: Homework
#5
Solutions
 ECE
15A,
Winter
2009
 1)
F(a,b,c,d)
=
Σm
(
2,3,4,5,6,7,9,12,15)
+
Σd(1,
10,13)

 
 
 
 abcd
 F
 m0
 0000
 
 m1
 0001
 x
 m2
 0010
 1
 m3
 0011
 1
 m4
 0100
 1
 m5
 0101
 1
 m6
 0110
 1
 m7
 0111
 1
 m8
 1000
 
 m9
 1001
 1
 m10
 1010
 x
 m11
 1011
 
 m12
 1100
 1
 m13
 1101
 x
 m14
 1110
 
 m15
 1111
 1
 
 Size
1
Implicants
 1
1’s
 m1
 0001
 m2
 0010
 m 4
 0100
 2
1’s
 m3
 0011
 m5
 0101
 m6
 0110
 m9
 1001
 m10
 1010
 m12
 1100
 3
1’s
 m7
 0111
 m13
 1101
 4
1’s
 m15
 1111
 
 Size
2
Implicants
 1
1’s
 m(1,
3)
 00‐1
 m(1,
5)
 0‐01
 m(1,
9)
 ‐001
 m(2,
3)
 001‐
 m(2,
6)
 0‐10
 m(2,
10)
 ‐010
 m(4,
5)
 010‐
 m(4,
6)
 01‐0
 m(4,
12)
 ‐100
 2
1’s
 m(3,
7)
 0‐11
 m(5,
7)
 01‐1
 2
1’s
 m(3,
7)
 m(5,
7)
 m(6,
7)
 m(5,
13)
 m(9,
13)
 m(12,
13)
 3
1’s
 m(7
,
15)
 
 m(13,
15) Size
4
implicants
 
 1
1’s
 0‐11
 01‐1
 011‐
 ‐101
 1‐01
 110‐
 ‐111
 1 1 ‐1 
 0‐‐1
 ‐‐01
 0 ‐1 ‐
 01‐‐
 ‐10‐
 ‐1 ‐1 
 
 
 
 
 2
1’s
 
 Prime
Implicant
Chart
 
 m(1,
3,
5,
7)
 m(1,
5,
9,
13)
 m(2,
3,
6,
7)
 m(4,
5,
6,
7)
 m(4,
5,
12,
13)
 m(5
,
7,
13,
15)
 
 2
 
 
 x
 
 
 
 m(1,
3,
5,
7)
 m(1,
5,
9
13)
 m(2,
3,
6,
7)
 m(4,
5,
6,
7)
 m(4
,
5,
12,
13)
 m(5,
7,
13,
15)
 3
 x
 
 x
 
 
 
 4
 
 
 
 x
 x
 
 5
 x
 x
 
 x
 x
 x
 6
 
 
 x
 x
 
 
 7
 x
 
 x
 x
 
 x
 9
 
 x
 
 
 
 
 12
 
 
 
 
 x
 
 15
 
 
 
 
 
 x
 m(2,
3,
6,
7),
m(4,
5,
12,
13),
m(5,
7,
13,
15),
m(1,
5,
9,
13)
are
all
essential
 F
=
a’c
+
bc’
+
bd
+
c’d
 
 
 2)
F(a,b,c,d)
=
Σm
(3,6,7,15)
+
Σd(1,2,4,5,11,13,14)

 
 Prime
Implicant
table
 
 
 3
 6
 7
 15
 A
 m(1,
3,
5,
7)
 x
 
 x
 
 B
 m(2,
3,
6,
7)
 x
 x
 x
 
 C
 m(3,
7,
11,
15)
 x
 
 
 x
 D
 m(4,
5,
6,
7)
 
 x
 x
 
 E
 m(5,
7,
13,
15)
 
 
 x
 x
 F
 m(6,
7,
14,
15)
 
 
 x
 x
 
 =
(A
+
B
+
C)(B
+
D)(A
+
B
+
D
+
E
+
F)(D
+
E
+
F)
 =
(AB
+
BB
+
BC+
AD
+
BD
+
CD)(A
+
B
+
D
+
E
+
F)(D
+
E
+
F)
 =
(B
+
AD
+
CD)(A
+
B
+
D
+
E
+
F)(D
+
E
+
F)
 =
(AB
+
BB
+
BD
+
BE
+
BF
+
AAD
+
ABD
+
ADD
+
ADE
+
ADF
+
ACD
+
BCD
+
CDD
+
CDE
+
 CDF)(D
+
E
+
F)
 =
(B
+
AD
+
CD)(D
+
E
+
F)
 =
(BD
+
BE
+
BF
+
ADD
+
ADE
+
ADF
+
CDD
+
CDE
+
CDF)
 =
AD
+
BD
+
BE
+
BF
+
CD
 Minimal
Solutions
 AD
=
a’d
+
a’b
 BD
=
a’c
+
a’b
 BE
=
a’c
+
bd
 BF
=
a’c
+
bc
 CD
=
cd
+
a’b
 
 3)
F(a,b,c,d)
=
Σm
(0,1,5,6,7,12,13,14)
 Prime
Implicant
Table
 
 0
 1
 5
 6
 7
 12
 m(0,
1)
 x
 x
 
 
 
 
 m(1,
5)
 
 x
 x
 
 
 
 m(5,
7)
 
 
 x
 
 x
 
 m(5,
13)
 
 
 x
 
 
 
 m(6,
7)
 
 
 
 x
 x
 
 m(6,
14)
 
 
 
 x
 
 
 m(12,
13)
 
 
 
 
 
 x
 
 m(0,
1),
m(12,
13),
m(6,
14)
are
all
essential,
leaving
5
and
7
left
to
cover.
 
 Minimal
Solution
 f
=
m(0,
1)
+
m(5,
7)
+
m(6,
14)
+
m(12,
13)
=
a’b’c’
+
a’bd
+
bcd’
+
abc’
 
 4)
F(a,b,c,d,e)
=
Σm
(0,2,3,7,8,9,10,11,13,14,16,23,24,27,28)
+
Σd
(1,6,21)

 
 Size
1
implicants
 0
1’s
 1
1’s
 m0
 m1
 m2
 m8
 m16
 m3
 m6
 m9
 m10
 m24
 m7
 m11
 m13
 m14
 m21
 00000
 00001
 00010
 01000
 10000
 00011
 00110
 01001
 01010
 11000
 00111
 01011
 01101
 01110
 10101
 13
 
 
 
 x
 
 
 x
 14
 
 
 
 
 
 x
 
 2
1’s
 3
1’s
 4
1’s
 m28
 m23
 m27
 11100
 10111
 11011
 
 Size
2
implicants
 0
1’s
 m(0,
1)
 m(0,
2)
 m(0,
8)
 m(0,
16)
 m(1,
3)
 m(1,
9)
 m(2,
3)
 m(2,
6)
 m(2,
10)
 m(8,
9)
 m(8,
10)
 m(8,
24)
 m(16,
24)
 m(3,
7)
 m(3,
11)
 m(9,
11)
 m(9,
13)
 m(10,
11)
 m(10,
14)
 m(24,
28)
 m(6,
7)
 m(6,
14)
 m(7,
23)
 m(21,
23)
 m(11,
27)
 0000‐
 000‐0
 0‐000
 ‐0000
 000‐1
 0‐001
 0001‐
 00‐10
 0‐010
 0100‐
 010‐0
 ‐1000
 1‐000
 00‐11
 0‐011
 010‐1
 01‐01
 0101‐
 01‐10
 11‐00
 0011‐
 0‐110
 ‐0111
 101‐1
 ‐1011
 1
1’1
 2
1’s
 3
1’s
 
 Size
4
implicants
 0
1’s
 m(0,
1,
2,
3)
 m(0,
1,
8,
9)
 m(0,
2,
8,
10)
 m(0,
8,
16,
24)
 m(1,
3,
9,
11)
 m(2,
3,
10,
11)
 m(2,
3,
6,
7)
 m(2,
6,
10,
14)
 m(8,
9,
10,
11)
 000‐‐
 0‐00‐
 0 ‐0 ‐0 
 ‐‐000
 0 ‐0 ‐1 
 0‐01‐
 00‐1‐
 0‐‐10
 010‐‐
 1
1’s
 
 Size
8
implicants
 0
1’s
 
 
 m(0,
1,
2,
3,
8,
9,
10,
11)
 0‐0‐‐
 
 
 
 
 
 Prime
implicants
table
 
 m(0,
1,
2,
3,
8,
9,
10,
11)
 m(0,
8,
16,
24)
 m(2,
3,
6,
7)
 m(2,
6,
10,
14)
 m(9,
13)
 m(24,
28)
 m(7,
23)
 m(21,
23)
 m(11,
27)
 
 0
 x
 x
 
 
 
 
 
 
 
 2
 x
 
 x
 x
 
 
 
 
 
 3
 x
 
 x
 
 
 
 
 
 
 7
 
 
 x
 
 
 
 x
 
 
 8
 x
 x
 
 
 
 
 
 
 
 9
 x
 
 
 
 x
 
 
 
 
 10
 x
 
 
 x
 
 
 
 
 
 11
 x
 
 
 
 
 
 
 
 x
 13
 
 
 
 
 x
 
 
 
 
 14
 
 
 
 x
 
 
 
 
 
 16
 
 x
 
 
 
 
 
 
 
 23
 
 
 
 
 
 
 x
 x
 
 24
 
 x
 
 
 
 x
 
 
 
 27
 
 
 
 
 
 
 
 
 x
 28
 
 
 
 
 
 x
 
 
 
 Essential
 m(9,
13),
m(2,
6,
10,
14),
m(0,
8,
16,
24),
m(11,
27),
m(24,
28)
 Best
remaining
cover
(3,
7,
23)
 m(0,
1,
2,
3,
8,
9,
10,
11),
m(7,
23)
 
 f
=
a’bd’e
+
a’de’
+
c’d’e’
+
bc’de
+
abd’e’
+
a’c’
+
b’cde

 
 5)
G(a,b,c,d,e,f)
=
Σm
(0,1,2,3,10,17,18,19,32,47,48,63)
+
Σd(12,13,14,16,31,33,49)
 a)
G
=
a’c’d’
+
c’d’e’
+
acdef
+
a’b’cef’
or
 




G
=
a’c’d’
+
c’d’e’
+
acdef
+
a’b’d’ef’
 b)
underlined
terms
 c)
G
=
acdef
+
a’b’d’ef’
+
ac’d’e’f’
+
a’b’c’d’
+
a’c’d’f
+
a’c’d’e
 
 
 6)
F(a,b,c,d)
=
Σm
(3,11,13,15)
+
Σd(2,4,5,7,10,12,14)
 
 Prime
Implicants
Table
 
 A
 B
 C
 D
 E
 
 m(2,
3,
10,
11)
 m(3,
7,
11,
15)
 m(4,
5,
12,
13)
 m(5,
7,
13,
15)
 m(12,
13,
14,
15)
 
 b’c
 cd
 bc’
 bd
 ab
 3
 x
 x
 
 
 
 11
 x
 x
 
 
 
 13
 
 
 x
 x
 x
 15
 
 x
 
 x
 x
 F
 
 m(10,
11,
14,
15)
 ac
 
 x
 
 x
 =
(A
+
B)(A
+
B
+
F)(C
+
D
+
E)(B
+
D
+
E
+
F)
 =
(A
+
B)(C
+
D
+
E)(B
+
D
+
E
+
F)
 =
(AC
+
AD
+
AE
+
BC
+
BD
+
BE)(B
+
D
+
E
+
F)
 =
ABC
+
ACD
+
ACE
+
ACF
+
ABD
+
AD
+
ADE
+
ADF
+
ABE
+
ADE
+
AE
+
AEF
+
BC
+
BCD
+
 BCE
+
BCF
+
BD
+
BDE
+
BDF
+
BE
+
BDE
+
BEF
 Minimal
solutions
 AD
=
b’c
+
bd
 AE
=
b’c
+
ab
 BC
=
cd
+
bc’
 BD
=
cd
+
bd
 BE
=
cd
+
ab
 ...
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