Logic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 1 Gate Circuits and Boolean Equations
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1 Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 28 Pearson Education, Inc. (Hperlinks are active in View Show mode) Overview Part Gate Circuits and Boolean Equations Binar Logic and Gates Boolean Algebra Standard Forms Part 2 Circuit Optimization Two-Level Optimization Map Manipulation Practical Optimization (Espresso) Multi-Level Circuit Optimization Part 3 Additional Gates and Circuits Other Gate Tpes Eclusive-OR Operator and Gates High-Impedance Outputs Chapter 2 - Part 2
2 Binar Logic and Gates Binar variables take on one of two values. Logical operators operate on binar values and binar variables. Basic logical operators are the logic functions AND, OR and NOT. Logic gates implement logic functions. Boolean Algebra: a useful mathematical sstem for specifing and transforming logic functions. We stud Boolean algebra as a foundation for designing and analzing digital sstems! Chapter 2 - Part 3 Binar Variables Recall that the two binar values have different names: True/False On/Off Yes/No / We use and to denote the two values. Variable identifier eamples: A, B,, z, or X for now RESET, START_IT, or ADD later Chapter 2 - Part 4 2
3 Logical Operations The three basic logical operations are: AND OR NOT AND is denoted b a dot ( ). OR is denoted b a plus (). NOT is denoted b an overbar ( ), a single quote mark (') after, or (~) before the variable. Chapter 2 - Part 5 Notation Eamples Eamples: Y = A B is read Y is equal to A AND B. z = is read z is equal to OR. X = A is read X is equal to NOT A. Note: The statement: = 2 (read one plus one equals two ) is not the same as = (read or equals ). Chapter 2 - Part 6 3
4 Operator Definitions Operations are defined on the values "" and "" for each operator: AND = = = = OR = = = = NOT = = Chapter 2 - Part 7 Truth Tables Truth table a tabular listing of the values of a function for all possible combinations of values on its arguments Eample: Truth tables for the basic logic operations: X AND Y Z = X Y X Y OR Z = XY X NOT Z = X Chapter 2 - Part 8 4
5 Logic Function Implementation Using Switches For inputs: logic is switch closed logic is switch open For outputs: logic is light on logic is light off. Switches in parallel => OR Switches in series => AND NOT uses a switch such that: Normall-closed switch => NOT logic is switch open C logic is switch closed Chapter 2 - Part 9 Logic Function Implementation (Continued) Eample: Logic Using Switches B C A D Light is on (L = ) for L(A, B, C, D) = and off (L = ), otherwise. Useful model for rela circuits and for CMOS gate circuits, the foundation of current digital logic technolog Chapter 2 - Part 5
6 Logic Gates In the earliest computers, switches were opened and closed b magnetic fields produced b energizing coils in relas. The switches in turn opened and closed the current paths. Later, vacuum tubes that open and close current paths electronicall replaced relas. Toda, transistors are used as electronic switches that open and close current paths. Optional: Chapter 6 Part : The Design Space Chapter 2 - Part Logic Gate Smbols and Behavior X Y Logic gates have special smbols: AND gate Z X Y X Y OR gate (a) Graphic smbols Z X Y And waveform behavior in time as follows: X X Z X NOT gate or inverter Y (AND) X Y (OR) X Y (NOT) X (b) Timing diagram Chapter 2 - Part 2 6
7 Gate Dela In actual phsical gates, if one or more input changes causes the output to change, the output change does not occur instantaneousl. The dela between an input change(s) and the resulting output change is the gate dela denoted b t G : Input t G t G t G =.3 ns Output.5.5 Time (ns) Chapter 2 - Part 3 Logic Diagrams and Epressions Truth Table X Y Z F = X Y Z X Y Z Equation F = X Y Z Logic Diagram Boolean equations, truth tables and logic diagrams describe the same function! Truth tables are unique; epressions and logic diagrams are not. This gives fleibilit in implementing functions. F Chapter 2 - Part 4 7
8 Boolean Algebra An algebraic structure defined on a set of at least two elements, B, together with three binar operators (denoted, and ) that satisfies the following basic identities: X = X X = X X = X X X = X = X X. = X X. = X. X = X X. X = X Y = Y X (X Y) Z = X (Y Z) X(Y Z) = XY XZ X Y = X. Y XY (XY) Z = YX = X(YZ) X YZ = (X Y)(X Z) X. Y = X Y Commutative Associative Distributive DeMorgan s Chapter 2 - Part 5 Some Properties of Identities & the Algebra If the meaning is unambiguous, we leave out the smbol The identities above are organized into pairs. These pairs have names as follows: -4 Eistence of and 5-6 Idempotence 7-8 Eistence of complement 9 Involution - Commutative Laws 2-3 Associative Laws 4-5 Distributive Laws 6-7 DeMorgan s Laws The dual of an algebraic epression is obtained b interchanging and and interchanging s and s. The identities appear in dual pairs. When there is onl one identit on a line the identit is self-dual, i. e., the dual epression = the original epression. Chapter 2 - Part 6 8
9 Some Properties of Identities & the Algebra (Continued) Unless it happens to be self-dual, the dual of an epression does not equal the epression itself. Eample: F = (A C) B dual F = (A C B) = A C B Eample: G = X Y (W Z) dual G = Eample: H = A B A C B C dual H = Are an of these functions self-dual? Chapter 2 - Part 7 Some Properties of Identities & the Algebra (Continued) There can be more that 2 elements in B, i. e., elements other than and. What are some common useful Boolean algebras with more than 2 elements?. Algebra of Sets 2. Algebra of n-bit binar vectors If B contains onl and, then B is called the switching algebra which is the algebra we use most often. Chapter 2 - Part 8 9
10 Boolean Operator Precedence The order of evaluation in a Boolean epression is:. Parentheses 2. NOT 3. AND 4. OR Consequence: Parentheses appear around OR epressions Eample: F = A(B C)(C D) Chapter 2 - Part 9 Eample : Boolean Algebraic Proof A A B = A (Absorption Theorem) Proof Steps Justification (identit or theorem) A A B = A A B X = X = A ( B) X Y X Z = X (Y Z)(Distributive Law) = A X = = A X = X Our primar 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 identit or theorem to appl to make forward progress, irrespective of the application. Chapter 2 - Part 2
11 Eample 2: Boolean Algebraic Proofs AB AC BC = AB AC (Consensus Theorem) Proof Steps Justification (identit or theorem) AB AC BC = AB AC BC? = AB AC (A A) BC? = Chapter 2 - Part 2 Eample 3: Boolean Algebraic Proofs ( X Y)Z XY = Y(X Z) Proof Steps Justification (identit or theorem) ( X Y )Z X Y = Chapter 2 - Part 22
12 2 Chapter 2 - Part 23 Useful Theorems ( )( ) n Minimizatio = = ( ) tion Simplifica = = ( ) Absorption = = Consensus z z z = ( ) ( )( ) ( ) ( ) z z z = DeMorgan's Laws = = Chapter 2 - Part 24 Proof of Simplification ( )( ) = =
13 Proof of DeMorgan s Laws = = Chapter 2 - Part 25 Boolean Function Evaluation F = z F2 = z F3 = z z F4 = z z F F2 F3 F4 Chapter 2 - Part 26 3
14 Epression Simplification An application of Boolean algebra Simplif to contain the smallest number of literals (complemented and uncomplemented variables): A B A CD A BD A C D A BCD = 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 Chapter 2 - Part 27 Complementing Functions Use DeMorgan's Theorem to complement a function:. Interchange AND and OR operators 2. Complement each constant value and literal z z Eample: Complement F = F = ( z)( z) Eample: Complement G = (a bc)d e G = Chapter 2 - Part 28 4
15 Overview Canonical Forms What are Canonical Forms? Minterms and Materms Inde Representation of Minterms and Materms Sum-of-Minterm (SOM) Representations Product-of-Materm (POM) Representations Representation of Complements of Functions Conversions between Representations Chapter 2 - Part 29 Canonical Forms It is useful to specif Boolean functions in a form that: Allows comparison for equalit. Has a correspondence to the truth tables Canonical Forms in common usage: Sum of Minterms (SOM) Product of Materms (POM) Chapter 2 - Part 3 5
16 Minterms Minterms are AND terms with ever variable present in either true or complemented form. Given that each binar variable ma appear normal (e.g., ) or complemented (e.g., ), there are 2 n minterms for n variables. Eample: Two variables (X and Y)produce 2 2 = 4 combinations: (both normal) (X normal, Y complemented) X XY Y XY Y (X complemented, Y normal) X (both complemented) Thus there are four minterms of two variables. Chapter 2 - Part 3 Materms Materms are OR terms with ever variable in true or complemented form. Given that each binar variable ma appear normal (e.g., ) or complemented (e.g., ), there are 2 n materms for n variables. Eample: Two variables (X and Y) produce 2 2 = 4 combinations: X Y X Y X Y X Y (both normal) ( normal, complemented) ( complemented, normal) (both complemented) Chapter 2 - Part 32 6
17 Materms and Minterms Eamples: Two variable minterms and materms. Inde Minterm Materm 2 3 The inde above is important for describing which variables in the terms are true and which are complemented. Chapter 2 - Part 33 Standard Order Minterms and materms are designated with a subscript The subscript is a number, corresponding to a binar 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 materm and will be listed in the same order (usuall alphabeticall) Eample: For variables a, b, c: Materms: (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 Chapter 2 - Part 34 7
18 Purpose of the Inde The inde for the minterm or materm, epressed as a binar number, is used to determine whether the variable is shown in the true form or complemented form. For Minterms: means the variable is Not Complemented and means the variable is Complemented. For Materms: means the variable is Not Complemented and means the variable is Complemented. Chapter 2 - Part 35 Inde Eample in Three Variables Eample: (for three variables) Assume the variables are called X, Y, and Z. The standard order is X, then Y, then Z. The Inde (base ) = (base 2) for three variables). All three variables are complemented for minterm ( X, Y, Z) and no variables are complemented for Materm (X,Y,Z). Minterm, called m is XYZ. Materm, called M is (X Y Z). Minterm 6? Materm 6? Chapter 2 - Part 36 8
19 Inde Eamples Four Variables Inde Binar Minterm Materm i Pattern m i M i abc d a b c d abcd? 3? a b c d 5 abcd a b c d 7? a b c d abcd a b c d 3 abcd? 5 abcd a b c d Chapter 2 - Part 37 Minterm and Materm Relationship Review: DeMorgan's Theorem = and = Two-variable eample: M 2 = and m 2 = Thus M 2 is the complement of m 2 and vice-versa. Since DeMorgan's Theorem holds for n variables, the above holds for terms of n variables giving: M = i mi and m = i Mi Thus M i is the complement of m i. Chapter 2 - Part 38 9
20 Function Tables for Both Minterms of Materms of 2 variables 2 variables m m m 2 m 3 M M M 2 M 3 Each column in the materm function table is the complement of the column in the minterm function table since M i is the complement of m i. Chapter 2 - Part 39 Observations In the function tables: Each minterm has one and onl one present in the 2 n terms (a minimum of s). All other entries are. Each materm has one and onl one present in the 2 n terms All other entries are (a maimum of s). We can implement an function b "ORing" the minterms corresponding to "" entries in the function table. These are called the minterms of the function. We can implement an function b "ANDing" the materms corresponding to "" entries in the function table. These are called the materms of the function. This gives us two canonical forms: Sum of Minterms (SOM) Product of Materms (POM) for stating an Boolean function. Chapter 2 - Part 4 2
21 Minterm Function Eample Eample: Find F = m m 4 m 7 F = z z z z inde m m 4 m 7 = F = = 2 = 3 = 4 = 5 = 6 = 7 = Chapter 2 - Part 4 Minterm Function Eample F(A, B, C, D, E) = m 2 m 9 m 7 m 23 F(A, B, C, D, E) = Chapter 2 - Part 42 2
22 Materm Function Eample Eample: Implement F in materms: F = M M 2 M 3 M 5 M 6 F = ( z) ( z) ( z) ( z) ( z) z i M M 2 M 3 M 5 M 6 = F = = 2 = 3 = 4 = 5 = 6 = 7 = Chapter 2 - Part 43 Materm Function Eample F(A,B,C,D) F(A, B,C,D) = = M 3 M8 M M4 Chapter 2 - Part 44 22
23 Canonical Sum of Minterms An Boolean function can be epressed as a Sum of Minterms. For the function table, the minterms used are the terms corresponding to the 's For epressions, epand all terms first to eplicitl list all minterms. Do this b ANDing an term missing a variable v with a term ( v v ). f = minterms. First epand terms: f = ( ) Then distribute terms: = Eample: Implement f as a sum of Epress as sum of minterms: f = m 3 m 2 m Chapter 2 - Part 45 Another SOM Eample Eample: F = A B C There are three variables, A, B, and C which we take to be the standard order. Epanding the terms with missing variables: Collect terms (removing all but one of duplicate terms): Epress as SOM: Chapter 2 - Part 46 23
24 Shorthand SOM Form From the previous eample, we started with: F = A B C We ended up with: F = m m 4 m 5 m 6 m 7 This can be denoted in the formal shorthand: F(A, B,C) = Σm(,4,5,6,7) Note that we eplicitl show the standard variables in order and drop the m designators. Chapter 2 - Part 47 Canonical Product of Materms An Boolean Function can be epressed as a Product of Materms (POM). For the function table, the materms used are the terms corresponding to the 's. For an epression, epand all terms first to eplicitl list all materms. Do this b first appling the second distributive law, ORing terms missing variable v with a term equal to and then appling the distributive law again. Eample: Convert to product of materms: f (,, z) = Appl the distributive law: = ( )( ) = ( ) = Add missing variable z: z z = ( z) ( z ) Epress as POM: f = M 2 M 3 v v Chapter 2 - Part 48 24
25 Another POM Eample Convert to Product of Materms: f(a, B,C) = A C BC A B Use z = () (z) with = (A C BC), = A, and z =B to get: f = (A C BC A)(A C BC B) Then use = to get: f = (C BC A)(AC C B) and a second time to get: f = (C B A)(A C B) Rearrange to standard order, f = (A B C)(A B C) to give f = M 5 M 2 Chapter 2 - Part 49 Function Complements The complement of a function epressed as a sum of minterms is constructed b selecting the minterms missing in the sum-of-minterms canonical forms. Alternativel, the complement of a function epressed b a Sum of Minterms form is simpl the Product of Materms with the same indices. Eample: Given F(,, z) = Σm (,3,5,7) F(,, z) = Σm(,2,4,6) F(,, z) = ΠM(,3,5,7) Chapter 2 - Part 5 25
26 Conversion Between Forms To convert between sum-of-minterms and productof-materms form (or vice-versa) we follow these steps: Find the function complement b swapping terms in the list with terms not in the list. Change from products to sums, or vice versa. Eample:Given F as before: F(,, z) = Σm(,3,5,7) Form the Complement: F(,, z) = Σm(,2,4,6) Then use the other form with the same indices this forms the complement again, giving the other form of the original function: F(,, z) = ΠM(,2,4,6) Chapter 2 - Part 5 Standard Forms 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 Eamples: SOP: A B C A B C B POS: (A B) (A B C) C These mied forms are neither SOP nor POS (A B C) (A C) A BC AC(A B) Chapter 2 - Part 52 26
27 Standard Sum-of-Products (SOP) A sum of minterms form for n variables can be written down directl 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 2 n inputs). This form often can be simplified so that the corresponding circuit is simpler. Chapter 2 - Part 53 Standard Sum-of-Products (SOP) A Simplification Eample: F(A, B,C) = Σm(,4,5,6,7) Writing the minterm epression: F = A B C A B C A B C ABC ABC Simplifing: F = Simplified F contains 3 literals compared to 5 in minterm F Chapter 2 - Part 54 27
28 AND/OR Two-level Implementation of SOP Epression The two implementations for F are shown below it is quite apparent which is simpler! A B C A B C A B C A B C A B C F A B C F Chapter 2 - Part 55 SOP and POS Observations The previous eamples show that: Canonical Forms (Sum-of-minterms, Product-of- Materms), or other standard forms (SOP, POS) differ in compleit 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 epression? Is there onl one minimum cost circuit? The net part will deal with these issues. Chapter 2 - Part 56 28
29 Terms of Use All (or portions) of this material 28 b Pearson Education, Inc. Permission is given to incorporate this material or adaptations thereof into classroom presentations and handouts to instructors in courses adopting the latest edition of Logic and Computer Design Fundamentals as the course tetbook. These materials or adaptations thereof are not to be sold or otherwise offered for consideration. This Terms of Use slide or page is to be included within the original materials or an adaptations thereof. Chapter 2 - Part 57 29
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