Introduction to Computer Engineering EECS 203 dickrp/eecs203/ CMOS transmission gate (TG) TG example

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1 Introduction to Computer Engineering EECS 23 dickrp/eecs23/ CMOS trnsmission gte TG Instructor: Robert Dick Office: L477 Tech Emil: Phone: TA: Nel Oz Office: Tech. Inst. L375 Phone: Emil: C TT: Dvid Bild Office: Tech. Inst. L47 Phone: Emil: A B C 3 R. Dick Introduction to Computer Engineering EECS 23 Other TG digrm TG exmple b b f 4 R. Dick Introduction to Computer Engineering EECS 23 5 R. Dick Introduction to Computer Engineering EECS 23 Impct of control on input Prctice w/ Boolen Minimiztion = = TG s resistnce Simplify this expression so tht it hs the miniml possible literl count: b b cd TG s resistnce s internl resistnce 6 R. Dick Introduction to Computer Engineering EECS 23 8 R. Dick Introduction to Computer Engineering EECS 23 Two-Level Logic nd Cnonicl Forms Cnonicl forms - SOP The previous exmple illustrted one stndrd representtion product of sums. These stndrd forms re known collectively s two-level logic: Product of Sums POS e.g. Sum of Products SOP e.g. f = b b f = b b For Sum of Products SOP the cnonicl form is constructed out of minterms. Product term in which ll vribles pper in complemented or uncomplemented forms once For n n-input function corresponds to one of of 2 n possible input combintions Use binry representtion to enumerte minterms Cn you see why these two re equivlent? Why is this known s two-level logic? 9 R. Dick Introduction to Computer Engineering EECS 23 R. Dick Introduction to Computer Engineering EECS 23

2 Cnonicl forms SOP Using Cnonicl Sum of Products Representtion x y z term symbol m m m 2 m 3 m 4 m 5 m 6 m 7 x y z m x y z m x yz m 2 x yz m 3 xy z m 4 xy z m 5 xyz m 6 xyz m 7 Convenient representtion uses opertor nd minterms: f x,...,x n = m i For given function f, list ll minterms for which the function is true: f,b,c = b c = m 6 m 7 m m 2 = m,2,6,7 R. Dick Introduction to Computer Engineering EECS 23 2 R. Dick Introduction to Computer Engineering EECS 23 Cnonicl forms POS Cnonicl forms POS For Products of Sums POS the cnonicl form is constructed out of mxterms. Sum term in which ll vribles pper in complemented or uncomplemented forms once Use binry representtion to enumerte mxterms note: function is not true for mxterms x y z term symbol M M M 2 M 3 M 4 M 5 M 6 M 7 x y z M x y z M x y z M 2 x y z M 3 x y z M 4 x y z M 5 x y z M 6 x y z M 7 3 R. Dick Introduction to Computer Engineering EECS 23 4 R. Dick Introduction to Computer Engineering EECS 23 Using Cnonicl Products of Sums Representtion More exmples of two-level logic Convenient representtion uses opertor nd minterms: f x,...,x n = M i f,b,c,d = m,4,8,,3,5 For given function f, list ll mxterms for which the function is flse: f,b,c = c b = M M 3 M 4 M 5 = M,3,4,5 f w,x,y,z = M5,3,4 f u,v = u u v 5 R. Dick Introduction to Computer Engineering EECS 23 6 R. Dick Introduction to Computer Engineering EECS 23 Logic minimiztion motivtion Logic minimiztion motivtion Wnt to reduce re, power consumption, dely of circuits Hrd to exctly predict circuit re from equtions Cn pproximte re with SOP cubes Minimize number of cubes nd literls in ech cube Algebric simplifiction difficult Hrd to gurntee optimlity K-mps work well for smll problems Too error-prone for lrge problems Don t ensure optiml prime implicnt selection Quine-McCluskey optiml nd cn be run by computer Too slow on lrge problems Some dvnced heuristics usully get good results fst on lrge problems Wnt to lern how these work nd how to use them? Tke Advnced Digitl Logic Design 7 R. Dick Introduction to Computer Engineering EECS 23 8 R. Dick Introduction to Computer Engineering EECS 23

3 Boolen function minimiztion Krnugh mps K-mps Algebric simplifiction Not systemtic How do you know when optiml solution hs been reched? Optiml lgorithm, e.g., Quine-McCluskey Only fst enough for smll problems Understnding these is foundtion for understnding more dvnced methods Not necessrily optiml heuristics Fst enough to hndle lrge problems Fundmentl ttribute is djcency Useful for logic synthesis Helps logic function visuliztion Generl Ide: Circle groups of output vlues typiclly s Result: Circled terms correspond to minimized product terms 9 R. Dick Introduction to Computer Engineering EECS 23 2 R. Dick Introduction to Computer Engineering EECS 23 Krnugh mps Sum of products SOP - Truth tble b b c b d c b bc b cd b f f = b b 22 R. Dick Introduction to Computer Engineering EECS R. Dick Introduction to Computer Engineering EECS 23 Sum of products SOP - KMp Some definitions Equivlent wy of expressing the sme function: implicnt - product term or sum term which covers/includes one or more minterms or mxterms prime implicnt - implicnt tht cnnot be covered by more generl implicnt i.e. one with fewer literls essentil prime implicnts - cover n output of the function tht no other prime implicnt or sum thereof is ble to cover b b 24 R. Dick Introduction to Computer Engineering EECS R. Dick Introduction to Computer Engineering EECS 23 Implicnts K-mp exmple For now, tret s wildcrd f,b,c bc Minimize f,b,c,d =,3,8,9,,,3 f, b, c, d = b b d c d Prime implicnts re not covered by other implicnts Essentil prime implicnts uniquely cover minterms 26 R. Dick Introduction to Computer Engineering EECS R. Dick Introduction to Computer Engineering EECS 23

4 K-mp simplifiction technique K-mp simplifiction technique For ll minterms Find mximl groupings of s nd X s djcent to tht minterm. Remember to consider top/bottom row, left/right column, nd corner djcencies. These re the prime implicnts. Revisit the s elements in the K-mp. If covered by single prime implicnt, the prime is essentil, nd prticiptes in finl cover. The s it covers do not need to be revisited. 28 R. Dick Introduction to Computer Engineering EECS R. Dick Introduction to Computer Engineering EECS 23 K-mp simplifiction technique Product of sums POS If there remin s not covered by essentil prime implicnts, Then select the smllest number of prime implicnts tht cover the remining s. This cn be difficult for complicted functions. Will present n lgorithm for this in future lecture. b b 3 R. Dick Introduction to Computer Engineering EECS 23 3 R. Dick Introduction to Computer Engineering EECS 23 POS K-mp techniques POS K-mp exmple Or Direct reding by covering zeros nd inverting vribles Invert function Do SOP Invert gin Apply De Morgn s lws Minimize f,b,c = 2,4,5,6 f,b,c = b c b 32 R. Dick Introduction to Computer Engineering EECS R. Dick Introduction to Computer Engineering EECS 23 SOP from Krnugh mp Six-vrible K-mp exmple z,b,c,d,e,f = 2,8,,8,24,26,34,37,42,45,5,53,58,6 34 R. Dick Introduction to Computer Engineering EECS R. Dick Introduction to Computer Engineering EECS 23

5 Six-vrible K-mp exmple Six-vrible K-mp exmple EF AB = EF AB = EF AB = EF AB = EF AB = EF AB = EF AB = EF AB = z,b,c,d,e,f = d ef de f Cd f 36 R. Dick Introduction to Computer Engineering EECS R. Dick Introduction to Computer Engineering EECS 23 Don t Cre logic Don t Cre vlues All specified Boolen vlues re or However, during design some vlues my be unspecified Don t cre vlues s llow circuit optimiztion, i.e., Incompletely specified functions llow optimiztion Insted, leve these vlues undefined Also clled Don t Cre vlues Allows ny function implementing the specified vlues to be used E.g., could use b b However, best to use simpler 38 R. Dick Introduction to Computer Engineering EECS R. Dick Introduction to Computer Engineering EECS 23 Stisfibility Don t Cres Observbility Don t Cres brke sensor wheel sensor 8 32 decision output pply brke pulse brke invlid output relese brke Input cn never occur This cn hppen within circuit Some modules will not be cpble of producing certin outputs < Output will be ignored for certin inputs 4 R. Dick Introduction to Computer Engineering EECS 23 4 R. Dick Introduction to Computer Engineering EECS 23 Don t cre K-mp exmple is necessry Minimize f w,x,y,z =,3,8,9,,,3 d5,7,5 f w,x,y,z = wx z Some Boolen functions cn not be represented with one logic level b b 42 R. Dick Introduction to Computer Engineering EECS R. Dick Introduction to Computer Engineering EECS 23

6 is sufficient f,b b Two-level well-understood As we will see lter, optiml minimiztion techniques known for two-level However, optiml two-level solution my not be optiml solution Sometimes suboptiml solution to the right problem is better thn the optiml solution to the wrong problem All Boolen functions cn be represented with two logic levels Given k vribles, 2 K minterm functions exist Select rbitrry union of minterms 44 R. Dick Introduction to Computer Engineering EECS R. Dick Introduction to Computer Engineering EECS 23 Two-level sometimes imprcticl f,b,c,d b cd Consider 4-term XOR prity gte: b c d b c d b cd bc d bcd bc d bcd b c d b cd Two-level wekness Two-level representtion is exponentil However, it s simple concept Is n i x i odd? Problem with representtion, not function 46 R. Dick Introduction to Computer Engineering EECS R. Dick Introduction to Computer Engineering EECS 23 Two-level wekness Reding ssignment Two-level representtions lso hve other weknesses Conversion from SOP to POS is difficult Inverting functions is difficult -ing two SOPs or ing two POSs is difficult Neither generl POS or SOP re cnonicl Equivlence checking difficult POS stisfibility N P-complete M. Morris Mno nd Chrles R. Kime. Logic nd Computer Design Fundmentls. Prentice-Hll, NJ, fourth edition, 28 Section 2.6 Also red TTL reference, Don Lncster. TTL Cookbook. Howrd W. Sms & Co., Inc., 974, s needed 48 R. Dick Introduction to Computer Engineering EECS 23 5 R. Dick Introduction to Computer Engineering EECS 23

Digital Design. Chapter 6: Optimizations and Tradeoffs

Digital Design. Chapter 6: Optimizations and Tradeoffs Digitl Design Chpter 6: Optimiztions nd Trdeoffs Slides to ccompny the tetbook Digitl Design, with RTL Design, VHDL, nd Verilog, 2nd Edition, by Frnk Vhid, John Wiley nd Sons Publishers, 2. http://www.ddvhid.com