Outline. EECS150 - Digital Design Lecture 5 - Boolean Algebra II. Canonical Forms. Sum of Products (cont.)

Transcription

1 EECS5 Digitl Design Leture 5 Boolen Alger II Septemer, 22 John Wwrznek Outline Cnonil Forms The give us metho to go rom TT to Boolen Equtions Twolevel Logi Simpliition Kmp metho Multilevel Logi NAND/NOR networks EXOR revisite Fll 22 EECS5 Le5Bool2 Pge Fll 22 EECS5 Le5Bool2 Pge 2 Cnonil Forms Stnr orm or Boolen epression unique lgeri epression rom TT. Two Tpes: * Sum o Prouts (SOP) * Prout o Sums (POS) Sum o Prouts (isjuntive norml orm, minterm epnsion). Emple: minterms One prout (n) term or eh in : = = + + Fll 22 EECS5 Le5Bool2 Pge 3 Sum o Prouts (ont.) Cnonil Forms re usull not miniml: Our Emple: = = + + = + ( + = + ) = + = + + = + = ( + ) = ( + ) = + Fll 22 EECS5 Le5Bool2 Pge 4

2 Cnonil Forms Prout o Sums (onjuntive norml orm, mterm epnsion). Emple: mterms One sum (or) term or eh in : = (++)(++ )(+ +) = (+ + )( ++)( ++ )( + +)(++ ) Mpping rom SOP to POS (or POS to SOP): Derive TT then proee. Fll 22 EECS5 Le5Bool2 Pge 5 Twolevel Logi Simplition Ke tool: The Uniting Theorem ( + ) = () = = + = ( +) = vlues hnge within the onset rows vlues on t hnge is eliminte, remins g g = + = ( +) = vlues st the sme vlues hnges remins, eliminte Fll 22 EECS5 Le5Bool2 Pge 6 Boolen Cues Visul tehnique or ientiing when the Uniting Theorem n e pplie ue 2ue Fll 22 EECS5 Le5Bool2 Pge 7 3ue z Suues o on noes n e use or simpliition. Onset ille in noes, oset empt noes g sserte & unhnge vries + ' = FA rr out: out 3vrile ue emple (' + ) ('+) (+') out = + + Wht out lrger suues? = + = Both & hnge, is sserte & remins onstnt. Fll 22 EECS5 Le5Bool2 Pge 8

3 Krnugh Mp Metho Kmp is n lterntive metho o representing the TT n to help visul the jenies. 5 & 6 vrile kmps possile Emples Krnugh Mp Metho = out = + + g = '. Cirle the lrgest groups possile. 2. Group imensions must e power o 2. = Fll 22 EECS5 Le5Bool2 Pge 9 Fll 22 EECS5 Le5Bool2 Pge = '' + Cirling Zeros Kmps (ont.) = + ' + '' (igger groups re etter) w z BCD inrementer emple w z w = = = (' + + )(' + + ')( + + ') Fll 22 EECS5 Le5Bool2 Pge = z = Fll 22 EECS5 Le5Bool2 Pge 2

4 = = Higher Dimensionl Kmps e = = e = = Multilevel Comintionl Logi Emple: reue sumoprouts orm = + e + + e + + e + g implementtion in 2levels with gtes: ost: 7input OR, 6 3input AND => 5 trnsistors el: 3input AND gte el + 7input OR gte el Ftore orm: = ( + +)( + e) + g ost: 3input OR, 2 2input OR, 3input AND => 2 trnsistors el: 3input OR + 3input AND + 2input OR e Whih is ster? g In generl: Using multiple levels (more thn 2) will reue the ost. Sometimes lso el. Sometime treo etween ost n el. Fll 22 EECS5 Le5Bool2 Pge 3 Fll 22 EECS5 Le5Bool2 Pge 4 Multilevel Comintionl Logi Another Emple: F = let = = + = + NANDNAND & NORNOR Networks DeMorgn s Lw: ( + ) = ( ) = + + = ( ) ( ) = ( + ) = = = = No onvenient hn methos eit or multilevel logi simpliition: CAD Tools, emple misii (UCB) 2 eploit some speil struture, emple er push ules or introue in pirs or remove pirs. Are these optimiztions still relevnt or LUT implementtions? Fll 22 EECS5 Le5Bool2 Pge 5 Fll 22 EECS5 Le5Bool2 Pge 6

5 NANDNAND & NORNOR Networks NANDNAND & NORNOR Networks Mpping rom AND/OR to NAND/NAND Mpping AND/OR to NOR/NOR Mpping OR/AND to NOR/NOR ) ) ) ) ) ) ) ) ) ) pir OR/AND to NAND/NAND ' ' ' ' Fll 22 EECS5 Le5Bool2 Pge 7 Fll 22 EECS5 Le5Bool2 Pge 8 Multilevel Networks EXOR Funtion F = ( + ) + ' or nor Prit, ition mo 2 or = + ' ' Convert to NANDs (note nout) Another pproh: i = then else ' Fll 22 EECS5 Le5Bool2 Pge 9 Fll 22 EECS5 Le5Bool2 Pge 2