CSE 140: Components and Design Techniques for Digital Systems
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1 Lecture 4: Four Input K-Maps CSE 4: Components and Design Techniques for Digital Systems CK Cheng Dept. of Computer Science and Engineering University of California, San Diego
2 Outlines Boolean Algebra vs. Karnaugh Maps Algebra: variables, product terms, minterms, consensus theorem Map: planes, rectangles, cells, adjacency Definitions: implicants, prime implicants, essential prime implicants Implementation Procedures 2
3 An example id A B C D input K-map CD AB 3
4 id A B C D input K-map CD AB 4
5 Arrangement of variables Adjacency and partition 4-input K-map CD AB
6 K-Map vs. Boolean Expression Boolean Expression Variable x i and complement x i Product term P = Π i x i Each minterm Two adjacent minterms Each minterm has n adjacent minterms Boolean algebra K-Map Half planes Rx i and Rx i Rectangle R p = i P x i One element cell Two neighboring cells Each cell has n neighbors Two dimensional tool? D Don t Care set handling 6
7 Procedure for finding the minimal function via K-maps (layman terms). Convert truth table to K-map 2. Group adjacent ones: In doing so include the largest number of adjacent ones (Prime Implicants) CD AB 3. Create new groups to cover all ones in the map: create a new group only to include at least one cell (of value ) that is not covered by any other group (Essential Prime Impliants) 4. Select the groups that result in the minimal sum of products (we will formalize this because its not straightforward) 7
8 Reading the reduced K-map CD AB m(2,3,6,7) m(5,7) m(8,9) m(,2,8,) = AC + ABD + ABC + BD 8
9 Definitions: implicant, prime implicant, essential prime implicant Implicant: A product term that has non-empty intersection with on-set F and does not intersect with off-set R. Prime Implicant: An implicant that is not covered by any other implicant. Essential Prime Implicant: A prime implicant that has an element in on-set F but this element is not covered by any other prime implicants. 9
10 Examples of Primes and Essential Primes
11 Implicant, Prime, Essential Prime CD AB Prime m(2,3,6,7) m(5,7) m(8,9) m(,2,8,) = AC + ABD + ABC + BD
12 Definition: Prime Implicant. Implicant: A product term that has non-empty intersection with on-set F and does not intersect with off-set R. 2. Prime Implicant: An implicant that is not covered by any other implicant. CD AB Q: How about this one? Is it a prime implicant? A. es B. No 2
13 Definition: Essential Prime Essential Prime Implicant: A prime implicant that has an element in on-set F but this element is not covered by any other prime implicants. AB CD Q: Is the blue group an essential prime? A. es B. No 3
14 Definition: Essential Prime Q: Which of the following product term(s) is (are) not an essential prime for the given K-map? ab cd A. bc d B. d b C. ac D. ab E. ad m(5,3) m(,2,8,) m(,,4,5) m(2,3,4,5) m(8,,2,4)
15 Procedure for finding the minimal function via K-maps (formal terms). Convert truth table to K-map 2. Include all essential primes AB CD 3. Include non essential primes as needed to completely cover the onset (all cells of value one) 5
16 K-maps with Don t Cares A B C D AB CD 6
17 K-maps with Don t Cares A B C D AB CD 7
18 K-maps with Don t Cares A B C D AB CD = A + BD + C 8
19 Reducing Canonical expressions Given F(a,b,c,d) = Σm (,, 2, 8, 4) D(a,b,c,d) = Σm (9, ). Draw K-map cd ab 9
20 Reducing Canonical Expressions Given F(a,b,c,d) = Σm (,, 2, 8, 4) D(a,b,c,d) = Σm (9, ). Draw K-map cd ab
21 Reducing Canonical Expressions Given F(a,b,c,d) = Σm (,, 2, 8, 4) D(a,b,c,d) = Σm (9, ). Draw K-map ab cd
22 . Draw K-map 2. Identify Prime implicants 3. Identify Essential Primes Reducing Canonical Expressions ab cd PI Q: How many primes (P) and essential primes (EP) are there? A. Four (P) and three (EP) B. Three (P) and two (EP) C. Three (P) and three (EP) D. Four (P) and Four (EP) 22
23 Reducing Canonical Expressions. Prime implicants: Σm (,, 8, 9), Σm (, 2, 8, ), Σm (, 4) 2. Essential Primes: Σm (,, 8, 9), Σm (, 2, 8, ), Σm (, 4) ab cd PI Q: Do the E-primes cover the entire on set? A. es B. No 23
24 Reducing Canonical Expressions. Prime implicants: Σm (,, 8, 9), Σm (, 2, 8, ), Σm (, 4) 2. Essential Primes: Σm (,, 8, 9), Σm (, 2, 8, ), Σm (, 4) 3. Min exp: Σ (Essential Primes)=Σm (,, 8, 9) + Σm (, 2, 8, ) + Σm (, 4) f(a,b,c,d) =? ab cd PI Q: Do the E-primes cover the entire on set? A. es B. No 24
25 Reducing Canonical Expressions. Prime implicants: Σm (,, 8, 9), Σm (, 2, 8, ), Σm (, 4) 2. Essential Primes: Σm (,, 8, 9), Σm (, 2, 8, ), Σm (, 4) 3. Min exp: Σ (Essential Primes)=Σm (,, 8, 9) + Σm (, 2, 8, ) + Σm (, 4) f(a,b,c,d) = b c + b d + acd ab cd PI Q: Do the E-primes cover the entire on set? A. es B. No 25
26 Another example Given F(a,b,c,d) = Σm (, 3, 4, 4, 5) D(a,b,c,d) = Σm (,, 3).Draw the K-Map cd ab 26
27 Another example Given F(a,b,c,d) = Σm (, 3, 4, 4, 5) D(a,b,c,d) = Σm (,, 3) ab cd
28 Reducing Canonical Expressions. Prime implicants: Σm (, 4), Σm (, ), Σm (, 3), Σm (3, ), Σm (4, 5), Σm (, 5), Σm (3, 5) 2. Essential Primes: Σm (, 4), Σm (4, 5) ab cd
29 Reducing Canonical Expressions. Prime implicants: Σm (, 4), Σm (, ), Σm (, 3), Σm (3, ), Σm (4, 5), Σm (, 5), Σm (3, 5) 2.Essential Primes: Σm (, 4), Σm (4, 5) 3.Min exp: Σm (, 4), Σm (4, 5), (Σm (3, ) or Σm (,3) ) 4. f(a,b,c,d) = a c d + abc+ b cd (or a b d) ab cd
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