Digital Logic Design Gate Level Minimization

Outlines• Introduction• The Karnaugh Map Method– Two-Variable Map– Three-Variable Map– Four-Variable Map– Five-Variable Map• Product of Sums Simplification• Don’t-Care Conditions• NAND and NOR Implementation• Other Two-Level Implementations• Exclusive-OR FunctionDr. Eng. Ahmed H. Madian 2

More ExamplesDr. Eng. Ahmed H. Madian 3

Four-variable Map Simplification• One square represents a mintermof 4 literals.• A rectangle of 2 adjacent squaresrepresents a product term of 3literals.• A rectangle of 4 squaresrepresents a product term of 2literals.• A rectangle of 8 squaresrepresents a product term of 1literal.• A rectangle of 16 squaresproduces a function that is equalto logic 1.Dr. Eng. Ahmed H. Madian 5

Example• Simplify the following Boolean functiong(A,B,C,D) = ∑m(0,1,2,4,5,7,8,9,10,12,13).• First put the function g( ) into the map, and then groupas many 1s as possible.ab000111cd00 01 11 101111111111100 01 11 10101 1 111111111g(A,B,C,D) = b’d’ +c’+a’bdDr. Eng. Ahmed H. Madian 6

Example: 4-variable Karnaugh Map• Example: F(w,x,y,z)= Σ(0,1,2,4,5,6,8,9,12,13,14)wxyz00 01 11 100011101111111111011F(w,x,y,z) = y’ + w’z’ + xz’Dr. Eng. Ahmed H. Madian 7

Simplifying logic Function using 4-variable K-MapF = A’B’C’ + B’CD’ + A’BCD’ + AB’C’= B’D’ + B’C’ + A’CD’Dr. Eng. Ahmed H. Madian 8

Choice of Blocks• We can simplify function by using larger blocks– Do we really need all blocks?– Can we leave some out to further simplify expression?• Function needs to contain special type of blocks– They are called Essential Prime Implicants• Need to define new terms– Implicant– Prime implicant– Essential prime implicantDr. Eng. Ahmed H. Madian 9

• ImplicantTerminology– Any product term in the SOP form– A block of 1’s in a K-map• Prime implicant (PI)– Product term that cannot be further reduced– Block of 1’s that cannot be further increased• Essential prime implicant (EPI)– Prime implicant that covers a 1 (minterm) that is not covered by anyother prime implicant• Quine’s Theorem:– Boolean function can be implemented with only essential primeimplicants (but other solutions exist)– The number of such implicants is minimumcd00 01 11 10Dr. Eng. Ahmed H. Madian 10ab00011110111111111

Systematic Procedure for SimplifyingBoolean Functions1. Generate all PIs of the function.2. Include all essential PIs.3. For remaining minterms not included in theessential PIs, select a set of other PIs to coverthem, with minimal overlap in the set.4. The resulting simplified function is the logicalOR of the product terms selected above.Dr. Eng. Ahmed H. Madian 11

Examples to illustrate termsC0 11 11 00 0BA1 01 01 11 1D6 prime implicants:A'B'D, BC', AC, A'C'D, AB, B'CDessentialminimum cover: AC + BC' + A'B'D5 prime implicants:BD, ABC', ACD, A'BC, A'C'Dessentialminimum cover: 4 essential implicantsC0 01 10 10 1BA1 01 01 10 0DDr. Eng. Ahmed H. Madian 12

Example• f(a,b,c,d) =∑m(0,1,2,3,4,5,7,14,15).• Five grouped terms, not allneeded.• 3 shaded cells covered by onlyone term• 3 EPIs, since each shaded cell iscovered by a different term.• F(a,b,c,d) = a’b’ + a’c’ + a’d +abcab cd111111111Dr. Eng. Ahmed H. Madian 13

Product of Sums Simplification using K- Map• Use sum-of-products simplification on the zerosof the function in the K-map to get F’.• Find the complement of F’, i.e. (F’)’ = F– Recall that the complement of a booleanfunction can be obtained by (1) taking thedual and (2) complementing each literal.– OR, using DeMorgan’s Theorem.Dr. Eng. Ahmed H. Madian 14

Product of Sums Minimization• How to generate a product ofsums from a Karnaugh map?– Use duality of Booleanalgebra (DeMorgan law)• Look at 0s in map instead of 1s– Generate blocks around 0’s– Gives inverse of function– Use duality to generateproduct of sums• Example:– F = Σ(0,1,2,5,8,9,10)– F’ = AB+ CD + BD’– F = (A’+B’)(C’+D’)(B’+D)Dr. Eng. Ahmed H. Madian 15

Example: POS minimizationab cd000100 01 11 1011111 1 1 0110011100000• F’(a,b,c,d) = ab’ + ac’ + a’bcd’• Complement of literals in (F’) to get FF = (a’+b)(a’+c)(a+b’+c’+d)Dr. Eng. Ahmed H. Madian 17

Don't Care Conditions• There may be a combination of input values which– will never occur– if they do occur, the output is of no concern.• The function value for such combinations is called adon't care.• They are usually denoted with x. Each x may bearbitrarily assigned the value 0 or 1 in animplementation.• Don’t cares can be used to further simplify a functionDr. Eng. Ahmed H. Madian 21

Minimization using Don’t Cares• Treat don't cares as if they are 1s to generate PIs.• Delete PI's that cover only don't care minterms.• Treat the covering of remaining don't care minterms asoptional in the selection process (i.e. they may be, butneed not be, covered).Dr. Eng. Ahmed H. Madian 22

Minimization example• F(w,x,y,z) = Σ(1,3,7,11,15) and d(w,x,y,z) = Σ(0,2,5)• What are possible solutions?Dr. Eng. Ahmed H. Madian 23

Another Example• Simplify the functiong(a,b,c,d) whose K-map isshown at right.ab cdx 1 0 01 x 0 x1 x x 10 x x 0• g = a’c’+ abor• g = a’c’+b’dx 1 0 01 x 0 x1 x x 10 x x 0x 1 0 01 x 0 x1 x x 10 x x 0Dr. Eng. Ahmed H. Madian 25