GateLevel Minimization


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1 MEC520 디지털공학 GateLevel Minimization JeeHwan Ryu School of Mechanical Engineering GateLevel MinimizationThe Map Method Truth table is unique Many different algebraic expression Boolean expressions may be simplified by algebraic means But, awkward due to the lack of specific rules Karnaugh Map or Kmap method Pictorial form of truth table A simple and straight forward procedure 2/29
2 Why Need to be Simple? Produces a circuit diagram with a minimum number of gates and the minimum number of inputs to the gate Simplest expression is not unique 3/29 TwoVariable Map m1+m2+m3 = x'y +xy' +xy = x +y 4/29
3 ThreeVariable Map Not in a binary sequence, but in a sequence similar to Gray code m m m + m 7 2 = xy z + xyz = xz = x y z + x yz = ( y + y ) = xz x z ( y + y) = x z 5/29 Examples Ex 31) Simplify the Boolean function, F(x, y, z) = Σ(2, 3, 4, 5) F = x'y + xy' Ex 34) Simplify the Boolean Function, F(x, y, z) = Σ(0, 2, 4, 5, 6) F = z + xy x yz y x z 6/29
4 Example Ex 34) Given Boolean function, F = A'C + A'B +AB C +BC a) express it in sum of minterms F(x, y, z) = Σ(1, 2, 3, 5, 7) b) find the minimal sum of products F = C + A'B 7/29 FourVariable Map Ex 35) Simplify the Boolean function, F(w, x, y, z) = Σ(0, 1, 2, 4, 5, 6, 8, 9, 12, 13, 14) F = y'+ w'z' + xz' 8/29
5 Examples 1. Simplify the Boolean function F(x, y, z) = Σ(3, 4, 6, 7) yz + xz 2. Simplify the Boolean function F(x, y, z) = Σ(0, 2, 4, 5, 6) z + xy 9/29 Prime Implicants Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map. F(A,B,C,D) = Σ(0,2,3,5,7,8,9,10,11,13,15) F=BD+B'D'+CD+AD =BD+B'D'+CD+AB' =BD+B'D'+B'C+AD =BD+B'D'+B'C+AB' 10/29
6 FiveVariable Map 11/29 Example Ex 37) Simplify the Boolean function, F(A,B,C,D,E)=Σ(0,2,4,6,9,13,21,23,25,29,31) F = A'B'E' + BD'E + ACE 12/29
7 Examples 1. Simplify the following Boolean functions by first finding the essential prime implicants: F(A,B,C,D)=Σ(0,2,3,5,7,8,10,11,14,15) i) find the essential prime implicants CD+B D ii) find the non essential prime implicants AC+A BD iii) simplify function F CD+B D + AC+A BD 2. Simplify the following Boolean functions, using fivevariable maps: F(A,B,C,D,E)=Σ(0,1,4,5,16,17,21,25,29) Ans) A B D +B C D +AD E 13/29 Product of Sums Simplification Ex 38) Simplify the Boolean function, F(A, B, C, D) = Σ(0, 1, 2, 5, 8, 9, 10) a) sum of products F = B'D' + B C' +A'C'D b) product of sum F' = AB + CD + BD' F = (A' +B')(C' +D')(B' +D) 14/29
8 Product of Sums Simplification F( x, y, z) = Σ(1, 3, 4, 6) = Π(0, 2, 5, 7) F = x'z +xz' F' = xz +x'z' F = (x'+z )(x + z) 15/29 Examples Simplify the following Boolean functions in product of sums: 1. F( w, x, y, z) = Σ(0, 2, 5, 6, 7, 8, 10) Ans) (w +x )(x+z )(x +y+z) 2. F(A,B,C,D) = Π(1,3,5,7,13,15) Ans) (B +D )(A+D ) 16/29
9 Don tcare Conditions Ex 39) Simplify the Boolean function, F(w, x, y, z) = Σ(1,3,7,11,15) Don tcare conditions, d(w, x, y, z) = Σ(0, 2, 5) F(w, x, y, z) = yz + w'x' = Σ(0, 1, 2, 3, 7, 11, 15) F(w, x, y, z) = yz + w'z = Σ(1, 3, 5, 7, 11, 15) 17/29 NAND and NOR Implementation Digital circuits are frequently constructed with NAND or NOR gates rather than AND and OR gates NAND and NOR gates are easier to fabricate with electronic components Basic gates used in all IC digital logic families 18/29
10 NAND Circuits NAND Circuit Single input NAND gate = Complement NAND gate is a universal gate 19/29 TwoLevel Implementation F = ((AB)'(CD)')' = AB + CD Ex 310) Implement the following Boolean function with NAND gates: F(x, y, z) = Σ(1, 2, 3, 4, 5, 7) = xy' + x'y + z 20/29
11 Multilevel NAND Circuits Convert all AND to NAND with NANDinverter Convert all OR to NAND with inverternand Check all the inverter in the diagram. For every inverter that is not compensated by another circle along the same line, insert an inverter (oneinput NAND gate) or complement the input literal 21/29 ExampleMultilevel NAND Circuits 22/29
12 ExampleMultilevel NAND Circuits ( AB + A B)( C + D ) F = A B A B F C D (a) ANDOR gates A B A B F C D (b) NAND gates 23/29 NOR Implementation 24/29
13 NOR Operation is the Dual of the NAND OR gates to NOR gates with NORinvert AND gates to NOR gates with invertnor Any Inverter that is not compensated by another inverter along the same line needs an inverter or the complementation of the input literal 25/29 Example F = (AB' + A'B)(C + D') 26/29
14 ExclusiveOR Function XOR : x y = xy + x y XNOR : ( x y) ' = xy + x y x 0 = x x 1 = x x x = 0 x x = 1 x y = x y = ( x y) 27/29 Parity Generation and Checking Same circuit for checking and generation P=0 28/29
15 HDL(Hardware Description Language) //HDL Example 31 //Description of the simple circuit of Fig module smpl_circuit(a,b,c,x,y); input A,B,C; output x,y; wire e; and g1(e,a,b); not g2(y, C); or g3(x,e,y); endmodule 29/29
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