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Lecture #3 EGR 277 – Digital Logic. Reading Assignment: Chapter 2 in Digital Design, 3 rd Edition by Mano. Example: Find truth table for F and for F’ for the function below. Complement of a Function A function F’ has the exact opposite truth table as function F.
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Lecture #3 EGR 277 – Digital Logic Reading Assignment: Chapter 2 in Digital Design, 3rd Edition by Mano Example: Find truth table for F and for F’ for the function below. Complement of a Function A function F’ has the exact opposite truth table as function F. • Canonical and Standard forms • Boolean functions are commonly expressed using the following forms: • Canonical forms: • Sum of minterms • Product of maxterms • Standard forms: • Sum of products • Product of sums
Lecture #3 EGR 277 – Digital Logic Minterm – (also called a standard product) A minterm is a term containing all n variables (complemented or uncomplemented) ANDed together. Example:f(A,B) has 4 possible minterms. List them. Each minterm represents one n-bit word where: Primed variable 0 Unprimed variable 1 Minterm designation: for function f(x,y,z) the input combination 000 represents minterm x’y’z’ and is designated m0. Example:Show all 8 possible minterms and the shorthand designations for f(x, y, z).
Lecture #3 EGR 277 – Digital Logic Key Point: A Boolean function F may be represented by a sum (ORed together) of its minterms. They represent the input combinations needed to yield F = 1. So minterms represent the 1’s in the truth table for F. Example:Pick a truth table for some function f(x,y,z) and represent f as a sum of minterms. Maxterm – (also called a standard sum) A maxterm is a term containing all n variables (complemented or uncomplemented) ORed together. Example:f(A,B) has 4 possible maxterms. List them.
Lecture #3 EGR 277 – Digital Logic Each maxterm represents one n-bit word where: Primed variable 1 (note that this is opposite of the notation used for minterms) Unprimed variable 0 Maxterm designation: for function f(x,y,z) the input combination 000 represents maxterm (x + y + z) and is designated M0. Example:Show all 8 possible maxterms and the shorthand designations for f(x, y, z). Key Point: A Boolean function F may be represented by a product (ANDed together) of its maxterms. They represent the input combinations needed to yield F = 0. So maxterms represent the 0’s in the truth table for F.
Lecture #3 EGR 277 – Digital Logic Example:Pick a truth table for some function f(x,y,z) and represent f as a product of maxterms. Relationship between minterms and maxterms Show that and
Lecture #3 EGR 277 – Digital Logic Conversion between forms Since minterms represent where F = 1 and maxterms represent where F = 0, all terms are either minterms or maxterms. So if F is expressed as a sum of minterms, then F is a product of the maxterms (the terms that were not minterms). So it is simple to convert between forms. Example:Convert to the other canonical form 1. F(A, B) = (0, 1) 2. F(x, y, z) = (0, 1) 3. F(x, y, z) = (4, 5, 6) 4. F(a, b, c, d, e) = (0-4, 8, 13-18)
Lecture #3 EGR 277 – Digital Logic Conversion to sum of minterms or product of maxterms forms from other forms Possible approaches include Boolean algebra and truth tables. Example:Represent each function below as a sum of minterms: 1. F(A, B) = A 2. F(x, y, z) = xy + z Examples: Represent each function below as a product of maxterms: 1. F(A, B) = A’B + AB’ 2. F(x, y, z) = x’ + y’
Lecture #3 EGR 277 – Digital Logic Standard Forms Canonical forms are not minimized and are not useful for many circuit implementations. Standard forms are more useful. Functions are typically minimized into one of the two standard forms: 1. Sum of Products (SOP) : F = sum of ANDed terms (but not necessarily minterms) 2. Product of Sums (POS) : F = product of ORed terms (but not necessarily maxterms) Example: List several examples of SOP expressions. Example: List several examples of POS expressions.
Lecture #3 EGR 277 – Digital Logic Example: Function F(A,B,C) has the following truth table. Express F in each of the following forms: 1. Sum of minterms 2. Product of maxterms 3. Minimal SOP 4. Minimal POS
Lecture #3 EGR 277 – Digital Logic Standard Forms: 2-Level Implementations Standard forms are referred to as “2-level implementations” because they can be implemented with two levels gates (and thus only two gate delays). Note that this does not include initial inverters. Example: Implement a SOP expression using logic gates to illustrate that it is a 2-level implementation. Example: Implement a POS expression using logic gates to illustrate that it is a 2-level implementation.
Lecture #3 EGR 277 – Digital Logic Non-standard forms: 4 commonly used forms have been covered (sum of minterms, product of maxterms, SOP, and POS). These forms will be used commonly throughout the course. There are, however, other forms. Example: List examples of non-standard expressions and implement at least one of them using logic gates.
Lecture #3 EGR 277 – Digital Logic Basic functions/gates and their truth tables: We have previously defined two functions with two or more inputs: AND and OR. How many possible 2-input logic functions could be defined (consider the diagram shown below)? How many correspond to actual gates (commercially available)? List possible truth tables: 6 commonly defined 2-input logic functions/gates: 1. AND 4. NOR 2. OR 5. XOR (Exclusive-OR) 3. NAND 6. XNOR (Exclusive-NOR or Equivalence)
Lecture #3 EGR 277 – Digital Logic NAND: Show logic symbol, truth table, and logic expressions: NOR: Show logic symbol, truth table, and logic expressions:
Lecture #3 EGR 277 – Digital Logic XOR: Show logic symbol, truth table, and logic expressions: XNOR: Show logic symbol, truth table, and logic expressions:
