Chapter 3: Gate-Level Minimization. Karnaugh Maps ( K-Maps )

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1 Chapter 3: Gate-Level Minimization G. W. Cox Spring 2 Karnaugh Maps ( K-Maps ) Base on the Theorem: xa + xa = x(a + A) = x This means that two prouct terms that iffer only in one literal (the literal is complemente in one term an uncomplemente in the other) can be groupe to form a single term without that literal. Call terms like this logically ajacent. examples: w x yz + wx yz = x yz ab + ab = a xyz + xy z + xy z + x y z = xz + y z G. W. Cox Spring 2

2 Extening the theorem The theorem extens to any number of terms that is a power of 2 Notice that all possible combinations of A an B are inclue here xa B + xa B + xab + aab = xa (B + B) + xa(b + B) = xa + xa = x G. W. Cox Spring 2 Logically-ajacent minterms The Karnaugh Map ( K-Map ) is a iagram that arranges the minterms so that if two Minterms are Logically Ajacent, they are physically ajacent in the iagram. m x y z m4 xy z m x y z m5 xy z For 3 variables m3 x yz m7 xyz m2 x yz m6 xyz This gives a convenient way to combine logically-ajacent terms G. W. Cox Spring 2

3 2-variable Karnaugh Map x y f x y G. W. Cox Spring 2 Grouping x y We can group 2 ajacent terms on the K-Map. The resulting term contains those literals that are common to all of the groupe terms. In the example above, we groupe minterms xy an xy The result is x G. W. Cox Spring 2

4 Grouping x y Note that it s OK to use a minterm more than once. G. W. Cox Spring 2 The K-Map gives the same results as algebraic simplification (faster) Algebraically: x y f f(x,y) = x y + xy + xy = x y + xy + xy + xy = x y + xy + xy + xy = (x +x)y + x(y +y) = y + x x y f(x,y) = y + x G. W. Cox Spring 2

5 3-variable K-Maps x yz m m m 3 m 2 m 4 m 5 m 7 m 6 En cells are ajacent G. W. Cox Spring 2 Example f(x,y,z) = Σm(2-5) x yz 3 2 f(x,y,z) = xy + x y G. W. Cox Spring 2

6 Example 2 f(x,y,z) = Σm(3,4,6,7) x yz 3 2 f(x,y,z) = xz + yz G. W. Cox Spring 2 Example 3 f(x,y,z) = Σm(,, 3, 4, 5) x yz 3 2 f(x,y,z) = y + x z G. W. Cox Spring 2

7 Mapping SOP expressions onto the K-Map For each prouct term, write in every cell that is in the intersection of the literals in the term x yz 3 2 x yz 3 2 x yz y + x z yz x x yz 3 2 x yz 3 2 xz xy + x z + y z G. W. Cox Spring 2 Note: the bigger the groupings, the simpler the expression x yz 3 2 f(x,y,z) = y + yz x yz 3 2 f(x,y,z) = y + z G. W. Cox Spring 2

8 Note: We must cover all terms, but covering terms more than once increases complexity x yz 3 2 f(x,y,z) = y z + xz x yz 3 2 f(x,y,z) = y z + xz + xy G. W. Cox Spring 2 4 variable K-Maps yz wx Sies are ajacent m m m 3 m 2 m 4 m 5 m 7 m 6 m 2 m 3 m 5 m 4 m 8 m 9 m m Top an Bottom Are ajacent Note that The 4 corners are ajacent to each other G. W. Cox Spring 2

9 Grouping f(w, x,y,z) = Σm(,,,, 4, 5) yz wx f(w, x,y,z) = w x y + wy G. W. Cox Spring 2 Crossing the sies f(w, x,y,z) = Σm(-2, 4-6, 8, 9, 2-4) yz wx f(w, x,y,z) = y + w z + xz G. W. Cox Spring 2

10 Crossing top an bottom, grouping corners (continue) F(A,B,C,D) = A B C + B CD +A BCD + AB C CD AB F(A,B,C,D) = B C + B D + A CD G. W. Cox Spring 2 Definitions: - Prime Implicant = Group Grouping efinitions - Essential Prime Implicant = a Prime Implicant that inclues at least one minterm that is not inclue in any other Prime Implicant CD AB B D an BD are Essential Prime Implicants CD an B C are Prime Implicants, but not Essential Prime Implicants G. W. Cox Spring 2

11 . Select all Essential Prime Implicants A grouping process 2. Select the minimum number of aitional Prime Implicants that cover all remaining minterms CD AB f(a,b,c,d) = B D + BD + CD -ORf(A,B,C,D) = B D + BD + B C Same number of literals G. W. Cox Spring 2 Complementing with K-Maps To simplify the complemente expression, group the s f(w, x,y,z) = Σm(,,,, 4, 5) yz wx f (w, x,y,z) = w y + w x + wy G. W. Cox Spring 2

12 Classroom Examples G. W. Cox Spring 2 Don t cares G. W. Cox Spring 2

13 Don t Cares Sometimes we on t care if a the result of evaluating a Boolean expression for a particular minterm gives a or. This often happens because the combination of literals represente by the minterm cannot occur. Example: f(abcd)= if the BCD igit ABCD is even (not ) A B C D f f(abcd) = Σm(2,4,6,8) + Σ(-5) G. W. Cox Spring 2 Possible to consier on t cares in grouping: () Treat them as if each is f(abcd) = Σm(2,4,6,8) + Σ(-5) If we treate all on t cares as : CD AB f(abcd) = BD + AB + AD + CD (8 literals) G. W. Cox Spring 2

14 Possible ways to consier on t cares in grouping: (2) Treat them as if each is f(abcd) = Σm(2,4,6,8) + Σ(-5) If we treate all on t cares as : CD AB f(abcd) = A CD + A BD + AB C D ( literals) G. W. Cox Spring 2 The best way to consier on t cares in grouping f(abcd) = Σm(2,4,6,8) + Σ(-5) The best approach: - treat s as where they help make larger groups for the real s - treat them as where they o not. CD AB f(abcd) = CD + BD + AD (6 literals) G. W. Cox Spring 2

15 yz wx Example (3.9 in text) f(wxyz) = Σm(,3,7,,5) + Σ(,2,5) f(wxyz) = yz + w x yz wx f(wxyz) = yz + w z G. W. Cox Spring 2 Classroom Examples G. W. Cox Spring 2