CS221: Digital Design. Dr. A. Sahu. Indian Institute of Technology Guwahati

Transcription

1 CS221: Digital Design QMLogicMinimization Minimization Dr. A. Sahu DeptofComp.Sc.&Engg. Indian Institute of Technology Guwahati 1

2 Outline Study of Components Logic Implementation Using MUX & Decoder 4 Bit Adder, BCD adder Quine McCluskey(QM) Logic Minimization Examples Writing C/C++ program for QM Method 2

3 Decoders Internal design AND gate for each output to detect input combination Decoder with enable e Outputs all 0 if e=0, Regular behavior if e=1 n input decoder: 2 n outputs d0 0 3 i1 i0 i1 i0 i0 i1 i0 i1i0 i1i0 d0 d1 d2 d3 Covers All Minterms i0 i1 i0 i1 d1 d2 0 0 e d3 1 1 e 0 d0 d1 d d3 0

4 Boolean Function Implementation using Decoders Usingan to 2ndecoderandORgatesany n and gates any functions of n variables can be implemented. Example: S(x,y,z)= Σ(1,2,4,7), C(x,y,z)=Σ(3,5,6,7) Functions S and C can be implemented using a 3 to 8 decoder and two 4 input OR gates Decoder: Covers AllMinterms

5 Mux 4 1 i0 i1 d i2 i3 s1 s0 i0 i1 i2 i3 d 4x1 mux s1 s0 Covers All Minterms

6 Implementing logic Function using MUX i0 i1 d i2 i3 s1 s0 A B 4x1 mux F (A, B) = m(0,2) 6

7 Adding Two One bit Operands One bit Half Adder: A B A B Sum Cout C out HA Sum One bit Full Adder: Sum = A B Cout = A.B C in A B Sum Cout A B Sum = A B Cin FA C Cout = AB+BCi A.B B.Cin C out in + A.Cin Sum

8 N Bit Ripple Carry Adder: Series of FA Cells To add two n bit numbers A n-1 B n-1 A 2 B 2 A 1 B 1 A 0 B 0 FA C n... n FA FA FA C 0 S n-1 S 2 S 1 S 0 Adder delay = Tc * n Tc = (C in to C out delay) ofafa A B C out FA C in Sum 8

9 4bitBinaryAdder A 3 B 3 A 2 B 2 A 1 B 1 A 0 B 0 Co FA FA FA FA Ci S 2 S 2 S 1 S 0 B 3 B 2 B 1 B 0 A 3 A 2 A 1 A 0 Ci 4 Bit Adder C 0 S 3 S 2 S 1 S 0 9

10 BinaryAdder(TwoLevel) Treat as 9 input & 5 output functions Generate TthTbl Truth Table for each outputs t Solve each function using KMAP/QM Method Only Two Level: No carry Propagation Ci a1 b1 c1 d1 a2 b2 c2 d2 Adder 2 Level a b c d Co 10

11 DecimalAdder Decimal numbers are represented with BCD code. WhentwoBCDdigitsAandBareadded B are added if A+B<10 result is a valid BCD digit if A+B>9 result will not be valid BCD digit. It must be corrected by adding 6 to the result If A+B >9 add 6 to solve this issue

12 DecimalAdder Carry out K 4 bit binary adder Z 8 Z 4 Z 2 Z 1 Carry in Output Carry C=K+Z 8 Z 4 +Z 8 Z 2 If(C)add6 4 bit binary adder

13 BCDAdder(TwoLevel) Treat as 9 input & 5 output functions Generate TthTbl Truth Table for each outputs t Solve each function using KMAP/QM Method Only Two Level: No carry Propagation Ci a1 b1 c1 d1 a2 b2 c2 d2 Adder 2 Level a b c d Co 13

14 Quine McCluskeyMethod for Minimization KMAP methods was practical for at most 6 variable functions Larger number of variables: need method that can be applied to computer based minimization Quine McCluskey method For example: m(0,1,2,3,5,7,13,15)

15 QMMethod Method PhaseI:findingPis Pis Tabular methods: Grouping and combining PhaseII:CoversminimalPIs PIs 15

16 QM Method QM Method Mintermsthat differ in one variable s value can be combined. Thus we list our mintermsso that they are in groups with each group having the same number of 1s. So the first step is ordering the minterms according to their number of 1s (0 cube list) only mintermsresiding in adjacent groups have the chance to be combined.):

17 QM Method m (0,1,2,3,5,7,13,15 ) 0_Cube

18 QM Method: Combining Adjacent Compare mintermsof a group with those ofanadjacentonetoform1 adjacent one form 1 cube cubelist. When doing the combining, we put checkmark alongside the mintermsin the 0 cubelistthathavebeencombined. that have been combined.

19 QM Method 1_Cube 0_Cube 0 000X 1, X X X01 2, X X X1 34 3, X X111 11X1 4,5

20 QM Method: Combining Adjacent Do same combination of comparing adjacent group minterms To form 2 cubes, 3 cubes and so on. Ol Only mintermsof it adjacent groups have the chance of being combined Which have an X in the same position.

21 QM Method 1_Cube 0 000X 00X0 1 00X1 0X01 001X 2 0X11 01X1 X101 3 X111 11X1 2_Cube 0 00XX * 0XX1 * 1 2 X1X1 *

22 Q MMethod:CoverPIs Method: PIs : terms left without checkmarks. After identifying our PIs, we list them against the mintermsneeded to be covered m (0,1,2,3,5,7,13,15 )

23 QMMethod:Covers Method To find a minimal cover, we first need to find essential lpis To do this we need to find columns that only haveone checkmark in them, the according row will thus show the essential PI. AfteridentifyingessentialPIs essential PIs, thatareare necessarily part of the cover, we cover any remaining mintermsusing a minimal set of PIs. In this example: F ( a, b, c, d) = ab + bd

24 Thanks 24