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### Unit 04

• 1. Department of Communication Engineering, NCTU 1 Unit 4 Application of Boolean Algebra
• 2. Department of Communication Engineering, NCTU 2 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  Three main steps in designing a single-output combinational logic circuit  Find a switching function that specifies the desired behavior of the circuit  Find a simplified algebraic expression for the function  Realized the simplified function using available logic elements  Goals:  How to specify circuit behaviors  How to design a combinational logic circuit
• 3. Department of Communication Engineering, NCTU 3 4.1 Conversion of English Sentences to Boolean Equations
• 4. Department of Communication Engineering, NCTU 4 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  For simple problems, go directly from a word description of the desired circuit behavior to an algebra expression  Mary watches TV if it is Monday night and she has finished her homework. F = A˙B  The alarm will ring iff the alarm switch is turned on and the door is not closed, or it is after 6 P.M. and the window is not closed. Z = AB' + CD'
• 5. Department of Communication Engineering, NCTU 5 4.2 Combinational Logic Design Using a Truth Table
• 6. Department of Communication Engineering, NCTU 6 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  In general, a truth table to design logic circuits  First, list a true table  E.g.  Derive an algebraic expression for f from the table f = A'BC + AB'C' + AB'C + ABC' + ABC (4-1) = A+BC
• 7. Department of Communication Engineering, NCTU 7 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  In stead of writing f in terms of the 1’s of the function, we may also write f in terms of the 0’s of the function  E.g. f ' = (A+B+C)(A+B+C')(A+B'+C) (4-3) = (A+B)(A+B'+C) = A + BC
• 8. Department of Communication Engineering, NCTU 8 4.3 Minterm and Maxterm Expansions
• 9. Department of Communication Engineering, NCTU 9 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  Each term in (4-1) is referred to as a minterm  f = A'BC + AB'C' + AB'C + ABC' + ABC (4-1)  A function written as a sum of minterms is referred to as a minterm expansion or a standard SOP  Each term in (4-3) is referred to as a maxterm  f = (A+B+C)(A+B+C')(A+B'+C) (4-3)  A function written as a product of maxterms is referred to as a maxterm expansion or a standard POS
• 10. Department of Communication Engineering, NCTU 10 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  A minterm of n variables is a product of n literals in which each variable appears exactly once in either true or complemented form  The decimal notation of minterm expansion e.g. f = m (3,4,7)
• 11. Department of Communication Engineering, NCTU 11 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  A maxterm of n variables is a sum of n literals in which each variable appears exactly once in either true or complemented form  The decimal notation of maxterm expansion E.g. f = M(0,,2)  Given the minterm or maxterm expansions for f , the minterm or maxterm expansions for the complement of f are easy to obtain  E.g.  Or 0 1 2 3 4 5 6 7 (0,1,2) (3,4,5,6,7) f m m m m f M M M M M M         0 1 2 0 1 2 0 1 2( )f M M M M M M m m m      
• 12. Department of Communication Engineering, NCTU 12 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  A general switching expansion can be converted to minterm or maxterm expansion either using a truth table or algebraically  For algebraic method, first write the expansion as a sum of products and then introduce the missing variables in each term by applying the theorem X + X’=1  Example f(a,b,c,d) = a’(b’+d) + acd’  1> SOP: f= a’b’+a’d+acd’  Introduce missing variables f= a’b’(c+c’)(d+d’)+a’(b+b’)(c+c’)d’+ a(b+b’)cd’ = a’b’c’d’+a’b’c’d+a’b’cd’+a’b’cd+a’b’c’d+a’b’cd + a’b’cd + a’bcd + abcd’+ ab’cd’ = m (0,1,2,3,5,7,10,14)
• 13. Department of Communication Engineering, NCTU 13 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  General minterm and maxterm expansions  A general minterm expansion f = a0m0 + a1m1+  + a7m7 = ai mi ai = 0 or 1  mi is not present if ai = 0  A general maxterm expansion f = (a0 + m0)(a1 + m1)  (a7 + m7) = (ai + mi) ai = 0 or 1  mi is not present if ai = 1  Equality ai mi = (ai + mi)
• 14. Department of Communication Engineering, NCTU 14 4.5 Incompletely Specified Functions
• 15. Department of Communication Engineering, NCTU 15 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  A large system is usually divided into many subcircuits. The output of module 1 may not generate all possible combinations for the input variables of module 2.  In this case, we don’t care these specific combinations when designing the switch circuit for B
• 16. Department of Communication Engineering, NCTU 16 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  When realizing the function, the don‘t care terms can be assigned 0’s or 1’s  If both X’s are assigned 0 F = A'B'C' + A'BC +ABC = A'B'C' + BC  If first X is assigned 1 and the second 0 F = A'B'C' + A'B'C + A'BC +ABC = A'B' + BC  If we assign 1 to both X’s F = A'B'C' + A'B'C + A'BC + ABC' + ABC = A'B' + BC + AB
• 17. Department of Communication Engineering, NCTU 17 4.5 Examples of Truth Table Construction
• 18. Department of Communication Engineering, NCTU 18 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
• 19. Department of Communication Engineering, NCTU 19 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  Error detector for 6-3-1-1 binary-coded-decimal digits
• 20. Department of Communication Engineering, NCTU 20 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  Switching Expression 
• 21. Department of Communication Engineering, NCTU 21 4.5 Design of Binary Adders
• 22. Department of Communication Engineering, NCTU 22 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  Design a 4-bit binary ripple carry adder  Approach 1: construct a truth table  Approach 2: cascade 4 1-bit Full Adders
• 23. Department of Communication Engineering, NCTU 23 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  Construct the true table for 1-bit full adder  Find the switching expressions ( ) ( ) ( ) ( ) in in in in in in in in in in in Sum X Y C X YC XY C XYC X Y C YC X Y C YC X Y C X Y C X Y C                       ( ) ( ) ( ) out in in in in in in in in in in in in C X YC XY C XYC XYC X YC XYC XY C XYC XYC XYC YC XC XY                 
• 24. Department of Communication Engineering, NCTU 24 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  Implement the functions with logic gates  Overflow occurs if adding two positive numbers gives a negative result, or adding two negative numbers results in a positive number 3 3 3 3 3 3V A B S A B S    
• 25. Department of Communication Engineering, NCTU 25 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  The pros and cons of ripple carry adder  Simple in concept  The carry output at stage i+1 Ci+1 = XiYi + (Xi + Yi) Ci  The carries propagate like a ripple and introduce circuit delays : C0 C1  C2   Ci+1  Ci+1 = f (Xi,Yi, Ci) = f (Xi,Yi,Xi-1,Yi-1,Ci-1) =   Alternative: Carry lookahead adder  To avoid circuit delays due to the propagation of carries  Express Ci+1 in terms of C0 and {X0,Yi  Xi,Yi} only
• 26. Department of Communication Engineering, NCTU 26 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  Re-write the output carry at the ith stage as  Ci+1 = gi + pi Ci  The carry-generate function: gi = XiYi  The carry-propagate function pi = Xi + Yi  Expression the carry bit in terms of gi and pi  C1 = g0 + p0 C0  C2 = g1 + p1 C1 = g1 + p1 g0 + p1 p0 C0  C3 = g2 + p2 C2 = g2 + p2 g1 + p2 p1 g0 + p2 p1 p0 C0    Ci = gi + pi gi-1 + pi pi-1gi-2 +  + pi pi-1 pi-2  g0 + pi pi-1 pi-2  p0C0
• 27. Department of Communication Engineering, NCTU 27 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  The circuit implementation of 4-bit carry lookahead adder block Carry lookahead network
• 28. Department of Communication Engineering, NCTU 28 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  For adders with higher number of bits, the carry lookahead network can get quite large in terms of gates and gate inputs. This also presents a limitation in the realization of a large high speed adder  How to circumvent this problem?  Cascade 4-bit carry lookahead adders to form a lager adder
• 29. Department of Communication Engineering, NCTU 29 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  Partition the operands into blocks E.g. C8 = g7 + p7 g6 + p7 p6g5 + p7 p6p5g4 + p7 p6p5p4g3 + p7 p6p5p4p3g2 + p7 p6p5p4p3p2g1 + p7 p6p5p4p3p2p1g0+ p7 p6p5p4p3p2p1p0C0 = g7 + p7 g6 + p7 p6g5 + p7 p6p5g4 + p7 p6p5p4 (g3 + p3g2 + p3p2g1 + p3p2p1g0) + p7 p6p5p4(p3p2p1p0C0) = G1 + P1G0 + P1P0C0 G1= g7 + p7 g6 + p7 p6g5 + p7 p6p5g4 P1 = p7 p6p5p4 G0= g3 + p3g2 + p3p2g1 + p3p2p1g0 P0 = p3p2p1p0
• 30. Department of Communication Engineering, NCTU 30 Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu  Define a 4-bit carry lookahead generator as G= g3 + p3g2 + p3p2g1 + p3p2p1g0 P = p3p2p1p0
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