enee244-02xx digital logic design lecture 12. announcements hw4 due today hw5 is up on course...
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ENEE244-02xxDigital Logic Design
Lecture 12
Announcements
• HW4 due today• HW5 is up on course webpage. Due on 10/16.• Recitation quiz on Monday, 10/13– Will cover the material from lectures 10,11,12.
Agenda
• Examples of finding minimal sums for 3-variable and 4-variable Boolean functions.
Using Karnaugh Maps to Find Minimal Sums
𝑓 (𝑥 , 𝑦 , 𝑧 )=∑𝑚(2,4,5,6,7)0 1 3 2
4 5 7 6
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
𝑓 (𝑥 , 𝑦 , 𝑧 )=∑𝑚(2,4,5,6,7)0 0 0 1
1 1 1 1
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
• Step 1: Find all prime implicants
• Verify these are the only prime implicants.
0 0 0 1
1 1 1 1
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
• Step 2: Find all essential prime implicants
• Essential 1-cells are in green.• Both prime implicant subcubes contain essential 1-
cells.• So minimal sum is:
0 0 0 1
1 1 1 1
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
𝑓 (𝑥 , 𝑦 , 𝑧 )=∑𝑚(0,1,2,3,4,6,7 )0 1 3 2
4 5 7 6
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
𝑓 (𝑥 , 𝑦 , 𝑧 )=∑𝑚(0,1,2,3,4,6,7 )1 1 1 1
1 0 1 1
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
• Step 1: Find all prime implicants
• Verify these are the only prime implicants.
1 1 1 1
1 0 1 1
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
• Step 2: Find all essential prime implicants
• Essential 1-cells are in green.• All three prime implicant subcubes contain essential 1-
cells.• So minimal sum is:
1 1 1 1
1 0 1 1
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
𝑓 (𝑥 , 𝑦 , 𝑧 )=Π𝑀 (1,4,5,6 )0 1 3 2
4 5 7 6
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
𝑓 (𝑥 , 𝑦 , 𝑧 )=Π𝑀 (1,4,5,6 )1 0 1 1
0 0 1 0
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
• Step 1: Find all prime implicants
• Verify these are the only prime implicants.
1 0 1 1
0 0 1 0
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
• Step 2: Find all essential prime implicants
• Essential 1-cells are in green.• Two of the prime implicant subcubes contain essential
1-cells.• So minimal sum will contain the terms:
1 0 1 1
0 0 1 0
00 01 11 10
𝑦𝑧
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
• Step 3: Check if all 1-cells are covered
• Yes, so Final sum:
1 0 1 1
0 0 1 0
00 01 11 10
𝑥 0
1
Using Karnaugh Maps to Find Minimal Sums
𝑓 (𝑤 ,𝑥 , 𝑦 ,𝑧 )=∑𝑚(0,1,6,7,8,14,15)0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
00 01 11 10
𝑤𝑥
00
01
11
10
Using Karnaugh Maps to Find Minimal Sums
𝑓 (𝑤 ,𝑥 , 𝑦 ,𝑧 )=∑𝑚(0,1,6,7,8,14,15)1 1 0 0
0 0 1 1
0 0 1 1
1 0 0 0
00 01 11 10
𝑤𝑥
00
01
11
10
Using Karnaugh Maps to Find Minimal Sums
• Step 1: Find all prime implicants
• Verify these are the only prime implicants.
1 1 0 0
0 0 1 1
0 0 1 1
1 0 0 0
00 01 11 10
𝑤𝑥
00
01
11
10
Using Karnaugh Maps to Find Minimal Sums
• Step 2: Find all essential prime implicants
• Essential 1-cells are in green.• All three prime implicant subcubes contain essential 1-cells.• So minimal sum is:
1 1 0 0
0 0 1 1
0 0 1 1
1 0 0 0
00 01 11 10
𝑤𝑥
00
01
11
10
Using Karnaugh Maps to Find Minimal Sums
𝑓 (𝑤 ,𝑥 , 𝑦 ,𝑧 )=∑𝑚(0,1,5,6,7,8,15)0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
00 01 11 10
𝑤𝑥
00
01
11
10
Using Karnaugh Maps to Find Minimal Sums
𝑓 (𝑤 ,𝑥 , 𝑦 ,𝑧 )=∑𝑚(0,1,5,6,7,8,15)1 1 0 0
0 1 1 1
0 0 0 0
1 0 0 0
00 01 11 10
𝑤𝑥
00
01
11
10
Using Karnaugh Maps to Find Minimal Sums
• Step 1: Find all prime implicants
• Verify these are the only prime implicants.
1 1 0 0
0 1 1 1
0 0 0 0
1 0 0 0
00 01 11 10
𝑤𝑥
00
01
11
10
Using Karnaugh Maps to Find Minimal Sums
• Step 2: Find all essential prime implicants
• Essential 1-cells are in green.• Only two prime implicant subcubes contain essential 1-cells.• So minimal sum must contain terms:
1 1 0 0
0 1 1 1
0 0 0 0
1 0 0 0
00 01 11 10
𝑤𝑥
00
01
11
10
Using Karnaugh Maps to Find Minimal Sums
• Step 3: Determine optimal choice of remaining prime implicants
• Can cover remaining 1-cells with a single prime implicant: • Final sum:
1 1 0 0
0 1 1 1
0 0 0 0
1 0 0 0
00 01 11 10
𝑤𝑥
00
01
11
10
