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SJSU CS 147 - 28FCS147L23Revision3

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Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64Slide 65Slide 66Counter Example•3-bit synchronous binary counter using D FF’s•Step (1): State Diagram•Step (2): State Assignment–Use variables (A,B,C) and 3-bit binary assignment•Step (3): State Transition Table•Step (4): Derivation of Next State Equations•Step (5): Logic DiagramSequence Detector •Design sequence detector (for 1101) using J-K FF•Step (1): State Diagram•Steps (2) and (3): State Assignment & State Transition TableT r a n s i t i o n J I n p u t K I n p u t0 - > 00 x0 - > 11 x1 - > 0x 11 - > 1x 0•Step (4): Derivation of Next State and Output Equations•Step (5): Logic DiagramConversion to NAND-Gate CircuitsA + B C + D E ' FABCDE 'FABCDE 'FABCDE 'FSimplification Goals•Goal -- minimize the cost of realizing a switching function•Cost measures and other considerations–Number of gates–Number of levels–Gate fan in and/or fan out–Interconnection complexity–Preventing hazards•Two-level realizations–Minimize the number of gates (terms in switching function)–Minimize the fan in (literals in switching function)Exampleg(A,B,C) = AB + A B + AC Determine the form and the number of terms and literals in each of the following.Two-level form, three products , two sums, six literals.f(X,Y,Z) = X Y(Z + Y X) + Y ZFour-level form, four products, two sums, seven literals. --------------------Minimization Methods•Commonly used techniques–Boolean algebra postulates and theorems–Karnaugh maps–Quine-McCluskey methodMinimum SOP and POS Representations•The minimum sum of products (MSOP) of a function, f, is a SOP representation of f that contains the fewest number of product terms and fewest number of literals of any SOP representation of f.•Example -- f(a,b,c,d) = m(3,7,11,12,13,14,15) = ab + acd + acd = ab + cd•The minimum product of sums (MPOS) of a function, f, is a POS representation of f that contains the fewest number of sum terms and the fewest number of literals of any POS representation of f.•Example -- f(a,b,c,d) = M(0,1,2,4,5,6,8,9,10) = (a + c)(a + d)(a + b + d)(b + c + d) = (a +c)(a + d)(b + c)(b + d)Karnaugh Maps•Karnaugh maps (K-maps) -- convenient tool for representing switching functions of up to six variables.•K-maps form the basis of useful heuristics for finding MSOP and MPOS representations.•An n-variable K-map has 2n cells with each cell corresponding to a row of an n-variable truth table.•K-map cells are labeled with the corresponding truth-table row.•K-map cells are arranged such that adjacent cells correspond to truth rows that differ in only one bit position (logical adjacency). •Switching functions are mapped (or plotted) by placing the function’s value (0,1,d) in each cell of the map.Figure Venn diagram and equivalent K-mapfor two variables( d )A B f ( A B )m0m2m1m3AB0 21 3ABBA0 21 3( a ) ( b ) ( c )( e ) ( f )BA0 21 3( g )0 1010 00 11 01 1BABABAA B A B A Bm2m1m3A Bm00 101Figure Venn diagram and equivalent K-mapfor three variablesm0Bm2m6m1m3m7m50 2 6 41 3 7 5BAA BCAB( a )( d )A B CA B C A B CA B CA B Cm0m1m2m3m4m5m6m7m4CACCA B0 2 6 41 3 7 50 0 0 1 1 1 1 001CA B CA B CA B CCAB( b ) ( c )( e ) ( f )Plotting (Mapping) Functions in Canonical Form on a K-map•Let f be a switching function of n variables where n  6.•Assume that the cells of the K-map are numbered from 0 to 2n where the numbers correspond to the rows of the truth table of f. •If mi is a minterm of f, then place a 1 in cell i of the K-map.•Example -- f(A,B,C) = m(0,3,5)•If Mi is a maxterm of f, then place a 0 in cell i.•Example -- f(A,B,C) = M(1,2,4,6,7)•If di is a don’t care of f, then place a d in cell i.Figure Plotting functions on K-mapsBCACA B10 0 0 1 1 1 1 00 2 6 41 3 7 501B0 0 00 1 0 1CACA B0 0 0 1 1 1 1 00 2 6 41 3 7 501B0 0 00 0CACA B10 0 0 1 1 1 1 00 2 6 41 3 7 5011 1CAB( a ) ( b )( c ) ( d )f(A,B,C) = m(0,3,5) = M(1,2,4,6,7)Figure K-maps for f(a,b,Q,G) (a) Minterm form. (b) Maxterm form.QGa b10 0 0 1 1 1 1 00 4 1 2 81 5 1 3 93 7 1 5 1 12 6 1 4 1 00 00 11 11 0bG11 11 1 1 11 1aQQGa b0 0 0 1 1 1 1 00 4 1 2 81 5 1 3 93 7 1 5 1 12 6 1 4 1 00 00 11 11 0bG0 0aQ0 00 0( a ) ( b )f(a,b,Q,G) = m(0,3,5,7,10,11,12,13,14,15) = M(1,2,4,6,8,9)Figure : f(Q,G,b,a). b aQG0 0 0 1 1 1 1 00 4 1 2 81 5 1 3 93 7 1 5 1 12 6 1 4 1 00 00 11 11 0Ga1 1Qb1 11 11 1 1 1f(Q,G,b,a) = m(0,12,6,14,9,13,3,7,11,15) = m(0,3,6,7,9,11,12,13,14,15)(a) Venn diagram form. (b) Sum of minterms. (c) Maxterms.CA B0 0 0 1 1 1 1 00 2 6 41 3 7 501B00CACA B0 0 0 1 1 1 1 00 2 6 41 3 7 501CAB( b ) ( c )U n i v e r s a l s e tB CABA BA BCB C1 11 00( a )0f(A,B,C) = AB + BC. (a) Maxterms, (b) Minterms, (c) Minterms of f .C DA B00 0 0 1 1 1 1 00 4 1 2 81 5 1 3 93 7 1 5 1 12 6 1 4 1 00 00 11 11 0BD0 00 0 00 0CA( A + C )( B + C + D )C DA B0 0 0 1 1 1 1 00 4 1 2 81 5 1 3 93 7 1 5 1 12 6 1 4 1 00 00 11 11 0BD111 1 1 11 1CA( a ) ( b )C DA B10 0 0 1 1 1 1 00 4 1 2 81 5 1 3 93 7 1 5 1 12 6 1 4 1 00 00 11 11 0BD1 11 1 11 1CAA CB C D( B + C )B C( c )f(A,B,C,D) = (A + C)(B + C)(B + C + D).(a) K-map of f, (b) K-map of f.C DA B0 0 0 1 1 1 1 00 4 1 2 81 5 1 3 93 7 1 5 1 12 6 1 4 1 00 00 11 11 0BD111 11ACC DA B0 0 0 1 1 1 1 00 4 1 2 81 5 1 3 93 7 1 5 1 12 6 1 4 1 00 00 11 11 0BD1 1AC1 11( a ) ( b )111111f(A,B,C,D)= (A+B)(A+C+D)(B+C+D)Simplification of Switching FunctionsUsing K-maps•K-map cells that are physically adjacent are also logically adjacent. Also, cells on an edge of a K-map are logically …


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SJSU CS 147 - 28FCS147L23Revision3

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