EECS150 - Digital Design Lecture 5 - Boolean Algebra II

Transcription

1 EECS5 Digitl Design Leture 5 Boolen Alger II Septemer, 22 John Wwrzynek Fll 22 EECS5 Le5Bool2 Pge

2 Cnonil Forms Outline They give us metho to go from TT to Boolen Equtions Twolevel Logi Simplifition Kmp metho Multilevel Logi NAND/NOR networks EXOR revisite Fll 22 EECS5 Le5Bool2 Pge 2

3 Cnonil Forms Stnr form for Boolen expression unique lgeri expression from TT. Two Types: * Sum of Prouts (SOP) * Prout of Sums (POS) Sum of Prouts (isjuntive norml form, minterm expnsion). Exmple: minterms f f One prout (n) term for eh in f: f = f = + + Fll 22 EECS5 Le5Bool2 Pge 3

4 Sum of Prouts (ont.) Cnonil Forms re usully not miniml: Our Exmple: f = = + + = + (x y + x = y + x) = + f = + + = + = ( + ) = ( + ) = + Fll 22 EECS5 Le5Bool2 Pge 4

5 Cnonil Forms Prout of Sums (onjuntive norml form, mxterm expnsion). Exmple: mxterms f f One sum (or) term for eh in f: f = (++)(++ )(+ +) f = (+ + )( ++)( ++ )( + +)(++ ) Mpping from SOP to POS (or POS to SOP): Derive TT then proee. Fll 22 EECS5 Le5Bool2 Pge 5

6 Twolevel Logi Simplition Key tool: The Uniting Theorem x (y + y) = x () = x f f = + = ( +) = vlues hnge within the onset rows vlues on t hnge is eliminte, remins g g = + = ( +) = vlues sty the sme vlues hnges remins, eliminte Fll 22 EECS5 Le5Bool2 Pge 6

7 Boolen Cues Visul tehnique for ientifying when the Uniting Theorem n e pplie ue x 2ue xy y x 3ue Suues of on noes n e use for simplifition. Onset fille in noes, offset empty noes z f g f sserte & unhnge vries + ' = Fll 22 EECS5 Le5Bool2 Pge 7

8 FA rry out: 3vrile ue exmple (' + ) ('+) out Fll 22 EECS5 Le5Bool2 Pge 8 (+') out = + + Wht out lrger suues? = + = Both & hnge, is sserte & remins onstnt.

9 Krnugh Mp Metho Kmp is n lterntive metho of representing the TT n to help visul the jenies. 5 & 6 vrile kmps possile Fll 22 EECS5 Le5Bool2 Pge 9

10 Krnugh Mp Metho Exmples f = g = ' out = + + f =. Cirle the lrgest groups possile. 2. Group imensions must e power of 2. Fll 22 EECS5 Le5Bool2 Pge

11 Kmps (ont.) f = '' + f = + ' + '' (igger groups re etter) Cirling Zeros f = (' + + )(' + + ')( + + ') Fll 22 EECS5 Le5Bool2 Pge

12 w x y z BCD inrementer exmple w y w = x = x z y = z = Fll 22 EECS5 Le5Bool2 Pge 2

13 Higher Dimensionl Kmps = = e = = = ef = Fll 22 EECS5 Le5Bool2 Pge 3

14 Multilevel Comintionl Logi Exmple: reue sumofprouts form x = f + ef + f + ef + f + ef + g implementtion in 2levels with gtes: ost: 7input OR, 6 3input AND => 5 trnsistors ely: 3input AND gte ely + 7input OR gte ely f Ftore form: x = ( + +)( + e)f + g ost: 3input OR, 2 2input OR, 3input AND => 2 trnsistors ely: 3input OR + 3input AND + 2input OR e f g x Whih is fster? In generl: Using multiple levels (more thn 2) will reue the ost. Sometimes lso ely. Sometime treoff etween ost n ely. Fll 22 EECS5 Le5Bool2 Pge 4

15 Multilevel Comintionl Logi Another Exmple: F = let x = y = + f = xy + x y x y f No onvenient hn methos exit for multilevel logi simplifition: CAD Tools, exmple misii (UCB) 2 exploit some speil struture, exmple er Are these optimiztions still relevnt for LUT implementtions? Fll 22 EECS5 Le5Bool2 Pge 5

16 NANDNAND & NORNOR Networks DeMorgn s Lw: ( + ) = ( ) = + + = ( ) ( ) = ( + ) = = = = push ules or introue in pirs or remove pirs. Fll 22 EECS5 Le5Bool2 Pge 6

17 NANDNAND & NORNOR Networks Mpping from AND/OR to NAND/NAND ) ) ) ) Fll 22 EECS5 Le5Bool2 Pge 7

18 NANDNAND & NORNOR Networks Mpping AND/OR to NOR/NOR Mpping OR/AND to NOR/NOR ) ) ) ) ) ) pir OR/AND to NAND/NAND ' ' ' ' Fll 22 EECS5 Le5Bool2 Pge 8

19 F = ( + ) + Multilevel Networks ' f Convert to NANDs (note fnout) Fll 22 EECS5 Le5Bool2 Pge 9

20 EXOR Funtion Prity, ition mo 2 x y xor xnor x xxory =x y+xy x' y x y' y Another pproh: x y if x= then y else y' Fll 22 EECS5 Le5Bool2 Pge 2