# TWO-LEVEL COMBINATIONAL LOGIC

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1 TWO-LEVEL COMBINATIONAL LOGIC OVERVIEW Canonical forms To-level simplification Boolean cubes Karnaugh maps Quine-McClusky (Tabulation) Method Don't care terms Canonical and Standard Forms Minterms and Maxterms For to binary variables Eand F combined ith an AND operation, the minterms or standard products are: EF EF EF, and EF. That is, to binary variables provide # 8 # % possible combinations (minterms.) 8 variables have # minterms. Each minterm has each variable being primed if the corresponding bit of the binary number is a unprimed if a " Similarly, to binary variables Eand F combined ith an OR operation, the maxterms or standard sums are: EF EF EF, and EF. To binary variables also # 8 provide # % possible combinations (maxterms) and 8 variables have # maxterms. Each maxterm has each variable being primed if the corresponding bit of the binary number is a " and unprimed if a!! and A maxterm is the complement of its corresponding minterm, and vice versa. Sum of Products (or Minterms) A Boolean function can be expressed as a sum of minterms. The minterms hose sum defines the Boolean function are those that give the 1's of the function in a truth table. Product of Sums (or Maxterms) A Boolean function can be expressed as a product of maxterms. The maxterms hose sum defines the Boolean function are those that give the 0's of the function in a truth table. Minterms and Maxterms for Three Binary Variables 1

2 Minterms Maxterms \ ] ^ Term Designation Term Designation!!! \]^ 7 \]! ^ Q!!! " \]^ 7 " \]^ Q "! "! \]^ 7 # \] ^ Q #! " " "!! "! " " "! " " " \]^ 7 \] ^ 7 \] ^ 7 \]^ 7 \]^ 7 \$ % & ( \] ^ Q \ ]^ Q \ ]^ Q \ ] ^ Q \ ] ^ Q \$ % & ( Example: Given a three-variable truth table as follos: \ ] ^ Function J Function J " #!!!!!!! " "!! "!!!! " "! " "!! "! "! "! " " "!! " " " " " " J and J " # can be expressed as a sum of products as follos: 2

3 J" \]^\] ^ \]^7" 7% 7( J# \]^\] ^\]^ \]^7\$ 7& 7 7( J and J " # can also be expressed as a product of sums as follos: J" \ ]^ \ ] ^ \ ] ^ \ ]^ \ ] ^ QQQQQ! # \$ & J# \ ]^ \ ]^ \ ] ^ \ ]^ QQQQ! " # % Boolean functions expressed as a sum of products or product of sums are said to be in canonical form A convenient ay to express these function is by using a short notation, decimal form: J \ ] ^! 7 " % ( and J \ ] ^! 7 \$ & ( " # or J \ ] ^ NQ! # \$ % & and J \ ] ^ NQ! " # % " # Standard forms A Boolean function is said to be in standard form if the function contains one, to or any number of literals. For example: J ] \] \]^ or J \ ] ^ \ ]^ [ " # A Boolean function may be expressed in a nonstandard form. For example, the function J [\]^ [ \]^ Examples: 1) Given the folloing truth table. Express J in a canonical minterms and maxterms. 3

4 \ ] ^ J!!!!!! " "! "!!! " "! "!! " "! " " "! 1 1 " " " " 2) Design a digital logic circuit that ill activate an alarm if a door or indo is open during non-business hours. Assume that Clock G! (non-business hours) " (business hours) Door H! (closed) " (opened) Windo [! (closed) " (opened) Alarm E! (off) " (on) Conversion beteen canonical form 1) To convert from a sum of products to a product of sums: rerite the minterm canonical form in a shorthand notation then replace the existing term numbers by the missing numbers. For example: J \ ] ^! 7 " \$ ( NQ! # % & 2) To convert from a product of sums to a sum of products: rerite the maxterm canonical form in a shorthand notation then replace the existing term numbers by the missing numbers. For example: J \ ] ^ N Q! # % &! 7 " \$ ( 4

5 3) To obtain the minterm (or maxterm) canonical form of the complement, given the Boolean function in a sum of products (or product of sums) form : list the term numbers that are missing in J For example: J \ ] ^! 7! # % & J \ ] ^! 7 " \$ ( J \ ] ^ NQ " \$ ( J \ ] ^ NQ! # % & Positive and Negative Logic Truth Table Positive Logic Negative Logic \ ] ^ \ ] ^ \ ] ^ lo lo lo lo high lo high lo lo high high high

6 Truth Table Positive Logic Negative Logic \ ] ^ \ ] ^ \ ] ^ lo lo high lo high lo high lo lo high high lo Example: Traffic lights -- to define three signals Functionally Complete Operation Sets A functionally complete operation set is a set of logic functions from hich any combinational logic expression can be realized. For example, AND, OR, and NOT can be realized using NAND or NOR. Not using NAND gates 6

7 NOT realized using NAND AND using NAND gates DBC abcb OR using NAND gates DBC abc b XOR using NAND gates DBC BC a b abc b abcbabc b Note that EFE F EF ae Fb E FEF E F aefb AND, NOR equivalent logic B C B C B C BC BC B C!! " "!! " "! " "!!!!! "!! "!!!! " "!! " "!! 7

8 OR, NAND equivalent logic B C B C BC B C BC B C!! " "!! " "! " "! " " " " "!! " " " " " " "!! " "!! Conversion from AND/OR to NAND/NAND Conversion from AND/OR to NOR/NOR 8

9 Conversion from OR/AND to NOR/NOR Conversion from OR/AND to NAND/NAND Examples: 1 JE FGH FG convert to a NAND/NAND circuit. 9

10 2) Covert the above circuit to NOR/NOR circuit. 3) JE\ \H \FG 10

11 Assignment#2 p.166: 2.7, 2.8, , 2.24, 2.26, 2.27, 2.29, 2.30, 2.35 Simplification of Boolean Functions Boolean cubes Examples 1) J \ ] \] \] 2) Full adder Karnaugh Map Method To-, Three- and Four- K Maps 11

12 Examples: Simplify each of the folloings Boolean functions: 1) J \ ] ^! 7 # \$ % & # Full adder 3) J \ ] ^! 7 3,4, 6, 7 4) J \ ] ^! 7 0, 2, 4, 5, 6 5) J\]\^\] ^]^ J [ \ ] ^! 7 0, " 2, 4, 5, 6 ) * "# "\$ "% ( J[\] \]^ [\]^ [\] Prime Implicants A prime implicant is a minterm obtained by combining the maximum possible number of adjacent squares in the map. If a minterm in a square is covered by only one prime implicant, that prime implicant is said to be essential. Example: J [ \ ] ^! 7 0, 2, 3, 5, 7 ) * 10, 11, "\$ " 5 Examples of four-variable map To-bit comparator To-bit binary adder Don't Care Conditions Examples: 1 J E F G H! 7 " \$ ( "" "&.! # & 2) J E F G H! 7 % & ) * "! "\$.! ( "& \$) BCD increment by 1 function. 12

13 13 Combinational Logic, Spring 1999

14 Five and Six Variable Functions Minimization Five variable K-maps Example: p. 83 J E F G H I! 7 # & ( ) "! "\$ "& "( "* #" #\$ #% #* \$" 14

15 Example: J E F G H I J! 7 # ) "! ") #% # \$% \$( %# %& &! &\$ &) " The Tabulation (Quine-McCluskey) Method The tabulation method consists of to parts. 1) Find all prime implicants. 2) Find the smallest collection of prime implicants that cover the complete on-set of the function. 15

16 Example: J E F G H I! 7 # & ( ) "! "\$ "& "( "* #" #\$ #% #* \$" I II * * * III * IV * The prime implicants are 0-010, 010-0, , , and Then J E F G H I Examples: EGHI FG HI EF IGI 16

17 1) J [ \ ] ^! 7 " % ( ) * "! "" "# "& # J E F G H I! 7! # \$ % & ( "" "& " ") "* #\$ #( \$" 3) J E F G H! 7 % & ) * "! "\$.! ( "& Assignment#\$ p.236: , , 3.16, 3.23, 3.25, 3.28, 3.33, 3.36, 3.43, 3.47, 3.50 Example: Design a 6311 error detector here each of the decimal digits (0,1,...,9) can be represented by a 4-bit code ith eights 6, 3, 1, 1. 6 A B C D F Combinational Logic Design Procedure The design of combinational logic circuits starts from the verbal outline of the problem and ends in a logic circuit diagram or a set of Boolean functions from hich the logic diagram can be easily obtained. 1. The problem is stated. 2. The number of available input variables and required output variables is determined. 3. The input and output variables are designed letter symbols. 4. The truth table that defines the required relationships beteen inputs and outputs is derived. 5. The simplified Boolean function for each output is obtained. 6. The logic diagram is dran. Adders Half adders A half-adder is a circuit that performs addition of to bits. Let the input variables be \ and ] and output variables be W (sum) and G carry). 17

18 \ ] W G!!!!! " "! "! "! " "! " W\]\] \] \ ] \] \] \] G\] \ ] Full-adders A full adder is a combinational circuit that forms the arithmetic sum of three input bits. 18

19 \ ] ^ W G!!!!!!! " "!! "! "!! " "! " "!! "! "! "! " " "!! " " " " " " W\]^\]^ \] ^ \]^^ \] G\] \^]^ Subtractors Half subtractors A half subtractor is a combinational circuit that subtracts to bits and produces their difference. \ ] H F!!!!! " " " "! "! " "!! 19

20 H\]\] F\] Full subtractors A full subtractor is a combinational circuit that performs a subtraction beteen to bits, taking into account that a 1 may have been borroed by a loer significant stage. \ ] ^ H F!!!!!!! " " "! "! " "! " "! " "!! "! "! "!! " "!!! " " " " " H\]^\]^ \] ^ \]^ F\]\^]^ Code conversion The availability of a large variety of codes for the same discrete elements of information results in the use of different codes by different digital systems. It is sometimes necessary to use the output of one system as the input of another. A conversion circuit must be inserted beteen the to systems if each uses different codes for the same information. Therefore, a code converter is a circuit that makes the to systems compatible even though each uses a different binary code. The folloing is an example of conversion form the BCD to the excess 3 code. 20

21 Input BCD Output Excess-3 Code E F G H [ \ ] ^!!!!!! " "!!! "! "!!!! "!! "! "!! " "! " "!! "!!! " " "! "! " "!!!! " "! "!! "! " " " "! "! "!!! "! " " "!! " " "!! "! "! X X X X "! " " X X X X " "!! X X X X " "! " X X X X " " "! X X X X " " " " X X X X 21

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