EECS150 Digital Design Lecture 19 Combinational Logic Circuits A Deep Dive March 28 2011 John Wawrzynek Spring 2011 EECS150 Lec19 cl Page 1 Outline Review of three representations for combinational logic truth tables graphical logic gates and algebraic equations Relationship among the three Adder example Laws of Boolean Algebra Canonical Forms Boolean Simplification Spring 2011 EECS150 Lec19 cl Page 2 Combinational Logic CL Defined yi fi x0 xn 1 where x y are 0 1 Y is a function of only X If we change X Y will change immediately well almost There is an implementation dependent delay from X to Y Spring 2011 Page 3 EECS150 Lec19 cl CL Block Example 1 Boolean Equation y0 x0 AND not x1 OR not x0 AND x1 y0 x0x1 x0 x1 Truth Table Description Gate Representation How would we prove that all three representations are equivalent Spring 2011 EECS150 Lec19 cl Page 4 Boolean Algebra Logic Circuits Why are they called logic circuits Logic The study of the principles of reasoning The 19th Century Mathematician George Boole developed a math system algebra involving logic Boolean Algebra His variables took on TRUE FALSE Later Claude Shannon father of information theory showed in his Master s thesis how to map Boolean Algebra to digital circuits Primitive functions of Boolean Algebra Spring 2011 EECS150 Lec19 cl Page 5 Relationship Among Representations Theorem Any Boolean function that can be expressed as a truth table can be written as an expression in Boolean Algebra using AND OR NOT How do we convert from one to the other Spring 2011 EECS150 Lec19 cl Page 6 CL Block Example 2 4 bit adder Truth Table Representation R A B c is carry out In general 2n rows for n inputs 256 rows Is there a more efficient compact way to specify this function Spring 2011 Page 7 EECS150 Lec19 cl 4 bit Adder Example Motivate the adder circuit design by hand addition Add a1 and b1 as follows Add a0 and b0 as follows carry to next stage r a XOR b a b c a AND b ab Spring 2011 r a b ci co ab aci bci EECS150 Lec19 cl Page 8 4 bit Adder Example In general ri ai bi cin cout aicin aibi bicin cin ai bi aibi Full adder cell Now the 4 bit adder ripple adder Spring 2011 Page 9 EECS150 Lec19 cl 4 bit Adder Example Graphical Representation of FAcell ri ai bi cin cout aicin aibi bicin Spring 2011 Alternative Implementation with 2 input gates ri ai bi cin cout cin ai bi aibi EECS150 Lec19 cl Page 10 Defined as Spring 2011 Boolean Algebra EECS150 Lec19 cl Page 11 Logic Functions Do the axioms hold Ex communitive law 0 1 1 0 Spring 2011 EECS150 Lec19 cl Page 12 Other logic functions of 2 variables x y Look at NOR and NAND Theorem Any Boolean function that can be expressed as a truth table can be expressed using NAND and NOR Proof sketch How would you show that either NAND or NOR is sufficient Spring 2011 EECS150 Lec19 cl Page 13 Laws of Boolean Algebra Duality A dual of a Boolean expression is derived by interchanging OR and AND operations and 0s and 1s literals are left unchanged Any law that is true for an expression is also true for its dual Operations with 0 and 1 1 x 0 x x 1 x 2 x 1 1 x 0 0 Idempotent Law 3 x x x x x x Involution Law 4 x x Laws of Complementarity 5 x x 1 x x 0 Commutative Law 6 x y y x x y y x Spring 2011 EECS150 Lec19 cl Page 14 Laws of Boolean Algebra cont Associative Laws x y z x y z x y z x y z Distributive Laws x y z x y x z x y z x y x z Simplification Theorems x y x y x x xy x x y x y x x x y x DeMorgan s Law x y z x y z x y z x y z Theorem for Multiplying and Factoring x y x z x z x y Consensus Theorem x y y z x z x y y z x z x y x z x y x z Spring 2011 EECS150 Lec19 cl Page 15 Proving Theorems via axioms of Boolean Algebra Ex prove the theorem x y x y x x y x y x y y distributive law x y y x 1 x 1 x complementary law identity Ex prove the theorem x xy x xy x x 1 x y identity x 1 x y x 1 y distributive law x 1 y x 1 identity x 1 identity Spring 2011 x EECS150 Lec19 cl Page 16 DeMorgan s Law x y x y x y x y Spring 2011 Exhaustive Proof Exhaustive Proof EECS150 Lec19 cl Page 17 Relationship Among Representations Theorem Any Boolean function that can be expressed as a truth table can be written as an expression in Boolean Algebra using AND OR NOT How do we convert from one to the other Spring 2011 EECS150 Lec19 cl Page 18 Canonical Forms Standard form for a Boolean expression unique algebraic expression directly from a true table TT description Two Types Sum of Products SOP Product of Sums POS Sum of Products disjunctive normal form minterm expansion Example minterms a b c a b c a bc a bc ab c ab c abc abc Spring 2011 abc 000 001 010 011 100 101 110 111 f f 01 01 01 10 10 10 10 10 One product and term for each 1 in f f a bc ab c ab c abc abc f a b c a b c a bc EECS150 Lec19 cl Page 19 Sum of Products cont Canonical Forms are usually not minimal Our Example f a bc ab c ab c abc abc xy xy x a bc ab ab a bc a x y x y x a bc f a b c a b c a bc a b a bc a b bc a b c Spring 2011 EECS150 Lec19 cl Page 20 Canonical Forms Product of Sums conjunctive normal form maxterm expansion Example maxterms a b c a b c a b c a b c a b c a b c a b c a b c abc 000 001 010 011 100 101 110 111 f f 01 01 01 10 10 10 10 10 One sum or term for each 0 in f f a b c a b c a b c f a b c a b c a b c a b c a b c Mapping from SOP to POS or POS to SOP Derive truth table then proceed Spring 2011 EECS150 Lec19 cl Page 21 Algebraic Simplification Example Ex full adder FA carry out function in canonical form Cout a bc ab c abc abc Spring 2011 EECS150 Lec19 cl Page 22 Algebraic Simplification Cout a bc ab c abc abc a bc ab c abc abc …
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