1 2 : 4 5. Why Digital Systems? Lesson 1: Introduction to Digital Logic Design. Numbering systems. Sample Problems 1 5 min. Lesson 1-b: Logic Gates

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1 Leon : Introduction to Digitl Logic Deign Computer ided Digitl Deign EE 39 meet Chvn Fll 29 Why Digitl Sytem? ccurte depending on numer of digit ued CD Muic i digitl Vinyl Record were nlog DVD Video nd udio mp3 (qulity depend on mpling/mount of it) Relile Error Correction Cpilitie Dicrete Vlue with Lrge Noie Mrgin Technology cn e implemented ft, chep CMOS emiconductor Numering ytem inry Code, CD clock exmple Wetern World Deciml or e The ytem tht we ll know nd tke for grnted proly picked ecue of the numer of finger on humn hnd Myn Vigeiml or e 2 hd concept of zero nd hd modern poitionl nottion llowed for repreenting lrge rnge (very mll to very lrge numer) poitionl nottion llow for long rithmetic Computer nd Digitl Sytem inry or e 2 ey to implement phyiclly high or low voltge on wire llow for ue of oolen mth nd philoopher logic true or fle = high or low = one or zero = on or off Hexdeciml Sytem e 6 Cn quickly convert lrge inry numer to hexdeciml nd ck y inpection One Hexdeciml digit repreent four inry digit ( 9, F) inry Coded Deciml ( weighted code) ued with imple LED diply (wtch diply, etc) 2 : 4 5 : vlue through not legl vlue Smple Prolem 5 min Convert. 2 to deciml Leon -: Logic Gte Convert 2 to Hex Convert to inry

2 ic Opertion - Inverter ic Opertion Logicl ND Inverion opertion (K the complement) opertion performed on only ingle vrile indicted y prime ( ) or overr (prime i eier to ue) the inverion of i nd the inverion of i inverter conit of two trnitor in CMOS (don t need to know thi for tet) ND function opertion performed on two or more oolen vrile output i one iff (if nd only if) oth input re one indicted y multipliction ymol (lthough not multipliction) * cn e ued, or two djcent vrile re umed to e NDed ic Opertion Logicl OR ic Opertion Logicl NND OR function opertion performed on two or more oolen vrile output i one if either or oth of the input i one indicted y ddition ymol (lthough not ddition) + cn e ued NND function Invere of ND function output i zero iff (if nd only if) oth input re one Smple Prolem 2 5 min Excluive Or Logic gte Convert NND gte into INVERTER Hint: no gte necery X X X Y = X Y + X Y 2

3 Excluive NOR Logic gte (XNOR) Smple Prolem 3 5 min X Y = X Y + X Y = (X Y) Convert XOR gte into INVERTER Hint: no gte necery = lo known n Equivlence Opertion or it Compre Smple Prolem 4 5 min Logic Network Convert XOR gte into UFFER Hint: no gte necery = F(,,c) = + c + c c c F Truth Tle Smple Prolem 5 5 min F(,,C) = + C C * F F(,,C) = + C.Truth Tle 2.Logic Network Circuit Digrm 3

4 Leon -c: Mxterm nd Minterm Deign y Truth Tle (ed on of tle) C = C + C + C = + C C C C Minterm expnion in m-nottion Mxterm expnion C = C + C + C rewritten in m-nottion (,,c) = m 3 + m 6 + m 7 C = m 3 C = m 6 C = m 7 = (++c) (++c ) (+ +c) ( ++c) ( ++c ) C rewritten mxterm expnion (,,c) = M M M 2 M 4 M 5 ( + + c) = M ( + + c ) = M ( + + c) = M 2 ( + + c) = M 4 ( + + c ) = M 5 Leon -d: oolen lger Theorem Theorem ic Theorem: X + = X X * = X X + = X * = Idempotent Lw: X + X = X Involution Lw: (X ) = X Lw of Complementrity X * X = X X + X = X * X = 4

5 Theorem (2) Theorem (3) - Simplifiction Communitive Lw: X * Y = Y * X ocitive Lw: X * (Y * ) = (X * Y) * Ditriutive Lw: X + Y = Y + X (X + Y) + = X + (Y+) X * (Y + ) = X * Y + X * X + Y * = (X + Y) * (X + ) Demorgn Lw: (X + X2 + X3) = X * X2 * X3 (X * X2 * X3) = X + X2 + X3. X Y + X Y = X 2. X + X Y = X 3. (X + Y ) Y = X Y (X + Y) (X + Y ) = X X (X + Y) = X XY + Y = X + Y Smple Prolem 6 5 min F = C + C + Leon -e: 4-it dder/sutrcter X Y Y X Y + Y = X + Y F= +C + L - dder Cell inry ddition dd three it numer two numer one crry-in from previou tge provide it um nd crry out Cin dder Sum Cout Cin Sum Cout ddition Tle + =, c= + =, c= + =, c= + =, c= ut Crry One to next column + 2 = 6 = + = 8 5

6 4-it inry dder uing dder 4-it inry dder S4 S3 S2 S S + dder Sum C3 3 3 dder C2 2 2 dder C dder C dder Crry ut Crry One to next column S4 S3 S2 S S inry Sutrction Sign nd Mgnitude ytem Sutrction Tle - = - = - = - = - ut orrow One from next column 2 = 6 = - = 6 ign it = - = + 2 = -6 mgnitude (me in thi ce) 2 = 6 One complement Two complement 2 = 6 2 = -6 forml converion => N = (2 n ) N Exmple (2 n -) N = 6 - imple converion => flip ll it 2 = 6 2 = -6 forml converion => N* = N + Exmple (2n-) N = 6 - dd + 6

7 inry Sutrction 4-it inry Sutrctor - 2 = 6 = 2 Complement N = 6 - N = dd = 6 Ignore the lt crry C3 S4 3 3 dder S3 C2 2 2 dder S2 Ignore the lt crry out C dder S C dder S dder/sutrctor Hint: Comine dder nd utrctor with one control input dd/utrct = dd with dd/utrct = Sutrct from dd/sutrct 4-it dder / Sutrctor S4 S3 S2 S S Ue EX-OR Opertion dd/su (S) dd/u (S) dder/sutrcter L Intruction: 4 4 [3:] [3:] dd Crete new project for l dder dder dder dder Sum[4:] 5 When dding Symol -> ue the ymol nme (firt l) 7