digital logic tutorial combinational logic

digital logic tutorial combinational logic

• 1.
  Charles Kime &Thomas Kaminski © 2004 Pearson Education, Inc. Terms of Use (Hyperlinks are active in View Show mode) Chapter 2 – Combinational Logic Circuits Part 1 – Gate Circuits and Boolean Equations Logic and Computer Design Fundamentals
• 2.
  Chapter 2 -Part 1 Overview  Part 1 – Gate Circuits and Boolean Equations • Binary Logic and Gates • Boolean Algebra • Standard Forms  Part 2 – Circuit Optimization • Two-Level Optimization • Map Manipulation • Multi-Level Circuit Optimization  Part 3 – Additional Gates and Circuits • Other Gate Types • Exclusive-OR Operator and Gates • High-Impedance Outputs
• 3.
  Chapter 2 -Part 1 Binary Logic and Gates  Binary variables take on one of two values.  Logical operators operate on binary values and binary variables.  Basic logical operators are the logic functions AND, OR and NOT.  Logic gates implement logic functions.  Boolean Algebra: a useful mathematical system for specifying and transforming logic functions.  We study Boolean algebra as foundation for designing and analyzing digital systems!
• 4.
  Chapter 2 -Part 1 Binary Variables  Recall that the two binary values have different names: • True/False • On/Off • Yes/No • 1/0  We use 1 and 0 to denote the two values.  Variable identifier examples: • A, B, y, z, or X1 for now • RESET, START_IT, or ADD1 later
• 5.
  Chapter 2 -Part 1 Logical Operations  The three basic logical operations are: • AND • OR • NOT  AND is denoted by a dot (·).  OR is denoted by a plus (+).  NOT is denoted by an overbar ( ¯ ), a single quote mark (') after, or (~) before the variable.
• 6.
  Chapter 2 -Part 1  Examples: • is read “Y is equal to A AND B.” • is read “z is equal to x OR y.” • is read “X is equal to NOT A.” Notation Examples  Note: The statement: 1 + 1 = 2 (read “one plus one equals two”) is not the same as 1 + 1 = 1 (read “1 or 1 equals 1”).  B A Y  y x z   A X 
• 7.
  Chapter 2 -Part 1 Operator Definitions  Operations are defined on the values "0" and "1" for each operator: AND 0 · 0 = 0 0 · 1 = 0 1 · 0 = 0 1 · 1 = 1 OR 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1 NOT 1 0 0 1
• 8.
  Chapter 2 -Part 1 0 1 1 0 X NOT X Z Truth Tables  Truth table  a tabular listing of the values of a function for all possible combinations of values on its arguments  Example: Truth tables for the basic logic operations: 1 1 1 0 0 1 0 1 0 0 0 0 Z = X·Y Y X AND OR X Y Z = X+Y 0 0 0 0 1 1 1 0 1 1 1 1
• 9.
  Chapter 2 -Part 1  Using Switches • For inputs:  logic 1 is switch closed  logic 0 is switch open • For outputs:  logic 1 is light on  logic 0 is light off. • NOT uses a switch such that:  logic 1 is switch open  logic 0 is switch closed Logic Function Implementation Switches in series => AND Switches in parallel => OR C Normally-closed switch => NOT
• 10.
  Chapter 2 -Part 1  Example: Logic Using Switches  Light is on (L = 1) for L(A, B, C, D) = and off (L = 0), otherwise.  Useful model for relay circuits and for CMOS gate circuits, the foundation of current digital logic technology Logic Function Implementation (Continued) B A D C
• 11.
  Chapter 2 -Part 1 Logic Gates  In the earliest computers, switches were opened and closed by magnetic fields produced by energizing coils in relays. The switches in turn opened and closed the current paths.  Later, vacuum tubes that open and close current paths electronically replaced relays.  Today, transistors are used as electronic switches that open and close current paths.
• 12.
  Chapter 2 -Part 1 Logic Gates (continued)  Implementation of logic gates with transistors (See Reading Supplement  CMOS Circuits)  Transistor or tube implementations of logic functions are called logic gates or just gates  Transistor gate circuits can be modeled by switch circuits • F +V X Y +V X +V X Y • • • • • • • • • • • • (a) NOR G = X +Y (b) NAND (c) NOT X .Y X • • •
• 13.
  Chapter 2 -Part 1 (b) Timing diagram X 0 0 1 1 Y 0 1 0 1 X · Y (AND) 0 0 0 1 X 1 Y (OR) 0 1 1 1 (NOT) X 1 1 0 0 (a) Graphic symbols OR gate X Y Z 5 X 1 Y X Y Z 5 X · Y AND gate X Z 5 X Logic Gate Symbols and Behavior  Logic gates have special symbols:  And waveform behavior in time as follows:
• 14.
  Chapter 2 -Part 1 Logic Diagrams and Expressions  Boolean equations, truth tables and logic diagrams describe the same function!  Truth tables are unique; expressions and logic diagrams are not. This gives flexibility in implementing functions. X Y F Z Logic Diagram Equation Z Y X F   Truth Table 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 X Y Z Z Y X F   
• 15.
  Chapter 2 -Part 1 1. 3. 5. 7. 9. 11. 13. 15. 17. Commutative Associative Distributive DeMorgan ’s 2. 4. 6. 8. X . 1 X = X . 0 0 = X . X X = 0 = X . X Boolean Algebra  An algebraic structure defined on a set of at least two elements, B, together with three binary operators (denoted +, · and ) that satisfies the following basic identities: 10. 12. 14. 16. X + Y Y + X = (X + Y) Z + X + (Y Z) + = X(Y + Z) XY XZ + = X + Y X . Y = XY YX = (XY)Z X(YZ) = X + YZ (X + Y)(X + Z) = X . Y X + Y = X + 0 X = + X 1 1 = X + X X = 1 = X + X X = X
• 16.
  Chapter 2 -Part 1  The identities above are organized into pairs. These pairs have names as follows: 1-4 Existence of 0 and 1 5-6 Idempotence 7-8 Existence of complement 9 Involution 10-11 Commutative Laws 12-13 Associative Laws 14-15 Distributive Laws 16-17 DeMorgan’s Laws  If the meaning is unambiguous, we leave out the symbol “·” Some Properties of Identities & the Algebra  The dual of an algebraic expression is obtained by interchanging + and · and interchanging 0’s and 1’s.  The identities appear in dual pairs. When there is only one identity on a line the identity is self-dual, i. e., the dual expression = the original expression.
• 17.
  Chapter 2 -Part 1  Unless it happens to be self-dual, the dual of an expression does not equal the expression itself.  Example: F = (A + C) · B + 0 dual F = (A · C + B) · 1 = A · C + B  Example: G = X · Y + (W + Z) dual G =  Example: H = A · B + A · C + B · C dual H =  Are any of these functions self-dual? Some Properties of Identities & the Algebra (Continued)
• 18.
  Chapter 2 -Part 1  There can be more that 2 elements in B, i. e., elements other than 1 and 0. What are some common useful Boolean algebras with more than 2 elements? 1. 2.  If B contains only 1 and 0, then B is called the switching algebra which is the algebra we use most often. Some Properties of Identities & the Algebra (Continued) Algebra of Sets Algebra of n-bit binary vectors
• 19.
  Chapter 2 -Part 1 Boolean Operator Precedence  The order of evaluation in a Boolean expression is: 1. Parentheses 2. NOT 3. AND 4. OR  Consequence: Parentheses appear around OR expressions  Example: F = A(B + C)(C + D)
• 20.
  Chapter 2 -Part 1 Example 1: Boolean Algebraic Proof  A + A·B = A (Absorption Theorem) Proof Steps Justification (identity or theorem) A + A·B = A · 1 + A · B X = X · 1 = A · ( 1 + B) X · Y + X · Z = X ·(Y + Z)(Distributive Law) = A · 1 1 + X = 1 = A X · 1 = X  Our primary reason for doing proofs is to learn: • Careful and efficient use of the identities and theorems of Boolean algebra, and • How to choose the appropriate identity or theorem to apply to make forward progress, irrespective of the application.
• 21.
  Chapter 2 -Part 1  AB + AC + BC = AB + AC (Consensus Theorem) Proof Steps Justification (identity or theorem) AB + AC + BC = AB + AC + 1 · BC ? = AB +AC + (A + A) · BC ? = Example 2: Boolean Algebraic Proofs
• 22.
  Chapter 2 -Part 1 Example 3: Boolean Algebraic Proofs  Proof Steps Justification (identity or theorem) = Y X Z ) Y X (   ) Z X ( X Z ) Y X (     Y Y
• 23.
  Chapter 2 -Part 1 x y  y      Useful Theorems    n inimizatio M y y y x y y y x         tion Simplifica y x y x y x y x           Absorption x y x x x y x x       Consensus z y x z y z y x                    z y x z y z y x          Laws s DeMorgan' x x    x x x x x x x x y x  y
• 24.
  Chapter 2 -Part 1 Proof of Simplification    y y y x y y y x       x  x
• 25.
  Chapter 2 -Part 1 Proof of DeMorgan’s Laws  y x x  y  y x y x 
• 26.
  Chapter 2 -Part 1 Boolean Function Evaluation x y z F1 F2 F3 F4 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 0 1 z x y x F4 x z y x z y x F3 x F2 xy F1       z yz  y 
• 27.
  Chapter 2 -Part 1 Expression Simplification  An application of Boolean algebra  Simplify to contain the smallest number of literals (complemented and uncomplemented variables): = AB + ABCD + A C D + A C D + A B D = AB + AB(CD) + A C (D + D) + A B D = AB + A C + A B D = B(A + AD) +AC = B (A + D) + A C 5 literals     D C B A D C A D B A D C A B A
• 28.
  Chapter 2 -Part 1 Complementing Functions  Use DeMorgan's Theorem to complement a function: 1. Interchange AND and OR operators 2. Complement each constant value and literal  Example: Complement F = F = (x + y + z)(x + y + z)  Example: Complement G = (a + bc)d + e G = x  z y z y x
• 29.
  Chapter 2 -Part 1 Overview – Canonical Forms  What are Canonical Forms?  Minterms and Maxterms  Index Representation of Minterms and Maxterms  Sum-of-Minterm (SOM) Representations  Product-of-Maxterm (POM) Representations  Representation of Complements of Functions  Conversions between Representations
• 30.
  Chapter 2 -Part 1 Canonical Forms  It is useful to specify Boolean functions in a form that: • Allows comparison for equality. • Has a correspondence to the truth tables  Canonical Forms in common usage: • Sum of Minterms (SOM) • Product of Maxterms (POM)
• 31.
  Chapter 2 -Part 1 Minterms  Minterms are AND terms with every variable present in either true or complemented form.  Given that each binary variable may appear normal (e.g., x) or complemented (e.g., ), there are 2n minterms for n variables.  Example: Two variables (X and Y)produce 2 x 2 = 4 combinations: (both normal) (X normal, Y complemented) (X complemented, Y normal) (both complemented)  Thus there are four minterms of two variables. Y X XY Y X Y X x
• 32.
  Chapter 2 -Part 1 Maxterms  Maxterms are OR terms with every variable in true or complemented form.  Given that each binary variable may appear normal (e.g., x) or complemented (e.g., x), there are 2n maxterms for n variables.  Example: Two variables (X and Y) produce 2 x 2 = 4 combinations: (both normal) (x normal, y complemented) (x complemented, y normal) (both complemented) Y X  Y X  Y X  Y X 
• 33.
  Chapter 2 -Part 1  Examples: Two variable minterms and maxterms.  The index above is important for describing which variables in the terms are true and which are complemented. Maxterms and Minterms Index Minterm Maxterm 0 x y x + y 1 x y x + y 2 x y x + y 3 x y x + y
• 34.
  Chapter 2 -Part 1 Standard Order  Minterms and maxterms are designated with a subscript  The subscript is a number, corresponding to a binary pattern  The bits in the pattern represent the complemented or normal state of each variable listed in a standard order.  All variables will be present in a minterm or maxterm and will be listed in the same order (usually alphabetically)  Example: For variables a, b, c: • Maxterms: (a + b + c), (a + b + c) • Terms: (b + a + c), a c b, and (c + b + a) are NOT in standard order. • Minterms: a b c, a b c, a b c • Terms: (a + c), b c, and (a + b) do not contain all variables
• 35.
  Chapter 2 -Part 1 Purpose of the Index  The index for the minterm or maxterm, expressed as a binary number, is used to determine whether the variable is shown in the true form or complemented form.  For Minterms: • “1” means the variable is “Not Complemented” and • “0” means the variable is “Complemented”.  For Maxterms: • “0” means the variable is “Not Complemented” and • “1” means the variable is “Complemented”.
• 36.
  Chapter 2 -Part 1 Index Example in Three Variables  Example: (for three variables)  Assume the variables are called X, Y, and Z.  The standard order is X, then Y, then Z.  The Index 0 (base 10) = 000 (base 2) for three variables). All three variables are complemented for minterm 0 ( ) and no variables are complemented for Maxterm 0 (X,Y,Z). • Minterm 0, called m0 is . • Maxterm 0, called M0 is (X + Y + Z). • Minterm 6 ? • Maxterm 6 ? Z , Y , X Z Y X
• 37.
  Chapter 2 -Part 1 Index Examples – Four Variables Index Binary Minterm Maxterm i Pattern mi Mi 0 0000 1 0001 3 0011 5 0101 7 0111 10 1010 13 1101 15 1111 d c b a d c b a    d c b a d c b a    d c b a d c b a    d c b a    d c b a d c b a    d b a d c b a d c b a    ? ? ? ? c
• 38.
  Chapter 2 -Part 1  Review: DeMorgan's Theorem and  Two-variable example: and Thus M2 is the complement of m2 and vice-versa.  Since DeMorgan's Theorem holds for n variables, the above holds for terms of n variables  giving: and Thus Mi is the complement of mi. Minterm and Maxterm Relationship y x y · x   y x y x    y x M2   y x· m2  i m M  i i i M m 
• 39.
  Chapter 2 -Part 1 Function Tables for Both  Minterms of Maxterms of 2 variables 2 variables  Each column in the maxterm function table is the complement of the column in the minterm function table since Mi is the complement of mi. x y m0 m1 m2 m3 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 1 x y M0 M1 M2 M3 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 1 1 0
• 40.
  Chapter 2 -Part 1 Observations  In the function tables: • Each minterm has one and only one 1 present in the 2n terms (a minimum of 1s). All other entries are 0. • Each maxterm has one and only one 0 present in the 2n terms All other entries are 1 (a maximum of 1s).  We can implement any function by "ORing" the minterms corresponding to "1" entries in the function table. These are called the minterms of the function.  We can implement any function by "ANDing" the maxterms corresponding to "0" entries in the function table. These are called the maxterms of the function.  This gives us two canonical forms: • Sum of Minterms (SOM) • Product of Maxterms (POM) for stating any Boolean function.
• 41.
  Chapter 2 -Part 1 x y z index m1 + m4 + m7 = F1 0 0 0 0 0 + 0 + 0 = 0 0 0 1 1 1 + 0 + 0 = 1 0 1 0 2 0 + 0 + 0 = 0 0 1 1 3 0 + 0 + 0 = 0 1 0 0 4 0 + 1 + 0 = 1 1 0 1 5 0 + 0 + 0 = 0 1 1 0 6 0 + 0 + 0 = 0 1 1 1 7 0 + 0 + 1 = 1 Minterm Function Example  Example: Find F1 = m1 + m4 + m7  F1 = x y z + x y z + x y z
• 42.
  Chapter 2 -Part 1 Minterm Function Example  F(A, B, C, D, E) = m2 + m9 + m17 + m23  F(A, B, C, D, E) =
• 43.
  Chapter 2 -Part 1 Maxterm Function Example  Example: Implement F1 in maxterms: F1 = M0 · M2 · M3 · M5 · M6 ) z y z)·(x y ·(x z) y (x F1        z) y x )·( z y x ·(     x y z i M0 M2 M3 M5 M6 = F1 0 0 0 0 0 1 1 1 = 0 0 0 1 1 1 1 1 1 1 = 1 0 1 0 2 1 0 1 1 1 = 0 0 1 1 3 1 1 0 1 1 = 0 1 0 0 4 1 1 1 1 1 = 1 1 0 1 5 1 1 1 0 1 = 0 1 1 0 6 1 1 1 1 0 = 0 1 1 1 7 1         1 1 1 1 = 1 1                        
• 44.
  Chapter 2 -Part 1 Maxterm Function Example   F(A, B,C,D) = 14 11 8 3 M M M M ) D , C , B , A ( F    
• 45.
  Chapter 2 -Part 1 Canonical Sum of Minterms  Any Boolean function can be expressed as a Sum of Minterms. • For the function table, the minterms used are the terms corresponding to the 1's • For expressions, expand all terms first to explicitly list all minterms. Do this by “ANDing” any term missing a variable v with a term ( ).  Example: Implement as a sum of minterms. First expand terms: Then distribute terms: Express as sum of minterms: f = m3 + m2 + m0 y x x f   y x ) y y ( x f    y x y x xy f    v v 
• 46.
  Chapter 2 -Part 1 Another SOM Example  Example:  There are three variables, A, B, and C which we take to be the standard order.  Expanding the terms with missing variables:  Collect terms (removing all but one of duplicate terms):  Express as SOM: C B A F  
• 47.
  Chapter 2 -Part 1 Shorthand SOM Form  From the previous example, we started with:  We ended up with: F = m1+m4+m5+m6+m7  This can be denoted in the formal shorthand:  Note that we explicitly show the standard variables in order and drop the “m” designators. ) 7 , 6 , 5 , 4 , 1 ( ) C , B , A ( F m   C B A F  
• 48.
  Chapter 2 -Part 1 Canonical Product of Maxterms  Any Boolean Function can be expressed as a Product of Maxterms (POM). • For the function table, the maxterms used are the terms corresponding to the 0's. • For an expression, expand all terms first to explicitly list all maxterms. Do this by first applying the second distributive law , “ORing” terms missing variable v with a term equal to and then applying the distributive law again.  Example: Convert to product of maxterms: Apply the distributive law: Add missing variable z: Express as POM: f = M2 · M3 y x x ) z , y , x ( f   y x ) y (x 1 ) y )(x x (x y x x            z y x ) z y x ( z z y x         v v
• 49.
  Chapter 2 -Part 1  Convert to Product of Maxterms:  Use x + y z = (x+y)·(x+z) with , and to get:  Then use to get: and a second time to get:  Rearrange to standard order, to give f = M5 · M2 Another POM Example B A C B C A C) B, f(A,    B z  ) B C B C )(A A C B C (A f      y x y x x    ) B C C )(A A BC C ( f      ) B C )(A A B C ( f      C) B )(A C B A ( f      A y C), B (A x    C
• 50.
  Chapter 2 -Part 1 Function Complements  The complement of a function expressed as a sum of minterms is constructed by selecting the minterms missing in the sum-of-minterms canonical forms.  Alternatively, the complement of a function expressed by a Sum of Minterms form is simply the Product of Maxterms with the same indices.  Example: Given ) 7 , 5 , 3 , 1 ( ) z , y , x ( F m   ) 6 , 4 , 2 , 0 ( ) z , y , x ( F m   ) 7 , 5 , 3 , 1 ( ) z , y , x ( F M  
• 51.
  Chapter 2 -Part 1 Conversion Between Forms  To convert between sum-of-minterms and product-of- maxterms form (or vice-versa) we follow these steps: • Find the function complement by swapping terms in the list with terms not in the list. • Change from products to sums, or vice versa.  Example:Given F as before:  Form the Complement:  Then use the other form with the same indices – this forms the complement again, giving the other form of the original function: ) 7 , 5 , 3 , 1 ( ) z , y , x ( F m   ) 6 , 4 , 2 , 0 ( ) z , y , x ( F m   ) 6 , 4 , 2 , 0 ( ) z , y , x ( F M  
• 52.
  Chapter 2 -Part 1  Standard Sum-of-Products (SOP) form: equations are written as an OR of AND terms  Standard Product-of-Sums (POS) form: equations are written as an AND of OR terms  Examples: • SOP: • POS:  These “mixed” forms are neither SOP nor POS • • Standard Forms B C B A C B A   C · ) C B (A · B) (A    C) (A C) B (A   B) (A C A C B A  
• 53.
  Chapter 2 -Part 1 Standard Sum-of-Products (SOP)  A sum of minterms form for n variables can be written down directly from a truth table. • Implementation of this form is a two-level network of gates such that: • The first level consists of n-input AND gates, and • The second level is a single OR gate (with fewer than 2n inputs).  This form often can be simplified so that the corresponding circuit is simpler.
• 54.
  Chapter 2 -Part 1  A Simplification Example:   Writing the minterm expression: F = A B C + A B C + A B C + ABC + ABC  Simplifying: F =  Simplified F contains 3 literals compared to 15 in minterm F Standard Sum-of-Products (SOP) ) 7 , 6 , 5 , 4 , 1 ( m ) C , B , A ( F  
• 55.
  Chapter 2 -Part 1 AND/OR Two-level Implementation of SOP Expression  The two implementations for F are shown below – it is quite apparent which is simpler! F B C A
• 56.
  Chapter 2 -Part 1 SOP and POS Observations  The previous examples show that: • Canonical Forms (Sum-of-minterms, Product-of- Maxterms), or other standard forms (SOP, POS) differ in complexity • Boolean algebra can be used to manipulate equations into simpler forms. • Simpler equations lead to simpler two-level implementations  Questions: • How can we attain a “simplest” expression? • Is there only one minimum cost circuit? • The next part will deal with these issues.
• 57.
  Chapter 2 -Part 1 Terms of Use  © 2004 by Pearson Education,Inc. All rights reserved.  The following terms of use apply in addition to the standard Pearson Education Legal Notice.  Permission is given to incorporate these materials into classroom presentations and handouts only to instructors adopting Logic and Computer Design Fundamentals as the course text.  Permission is granted to the instructors adopting the book to post these materials on a protected website or protected ftp site in original or modified form. All other website or ftp postings, including those offering the materials for a fee, are prohibited.  You may not remove or in any way alter this Terms of Use notice or any trademark, copyright, or other proprietary notice, including the copyright watermark on each slide.  Return to Title Page