COMP-1200 Combinational Logic Notes

Course Overview

Course Code: COMP-1200
Semester: Spring 2025
Institution: Wentworth Institute of Technology
Instructors: Prof. Gyllinsky, Prof. Park, Prof. Firouzbakht, Prof. Anwaruddinta, and others

Combinational Logic Basics

Combinational Logic Circuits

• These circuits are designed such that the output is solely determined by the current values of the input signals.

• They do not take into account any historical values or past input states, meaning they are stateless.

• Examples include adders, multiplexers, and encoders, which perform specific logical operations based on their current inputs.

Sequential Logic Circuits

• In contrast, sequential logic circuits consider both current and past input states, making them dependent on the sequence of inputs received.

• They utilize memory elements such as flip-flops to store past information, allowing them to retain the effect of previous inputs.

• Common applications include counters, registers, and state machines.

Logical Functions and Their Representations

• A logical function can be represented in multiple forms, each of which provides the same functional outcome:

  • Logic Circuit: A graphical representation using logic gates like AND, OR, NOT, etc.

  • Truth Table: A tabular representation that outlines all possible input combinations and their corresponding outputs.

  • Logic Expression: An algebraic representation showing the relationship between the inputs and outputs using logical operators.

• All forms are equivalent in functionality, providing diverse ways to analyze and implement logic functions effectively.

Logic Gates: Basic Notations

• AND (⋅)

  • Represents conjunction; the output is true only if both inputs are true. Sample representation: A AND B can be simplified to A ⋅ B.

• OR (+)

  • Represents disjunction; the output is true if at least one input is true. It is expressed as A + B.

• NOT (') or Overbar

  • Represents negation, where the output is the inverse of the input. For instance, NOT A is noted as A'.

• XOR (⊕)

  • Stands for Exclusive OR; the output is true if exactly one of the inputs is true, denoted as A XOR B, or A ⊕ B.

Logic Operations Precedence

• Proper use of parentheses is essential for ensuring the intended operations are conducted correctly.

• The precedence of operations is as follows:

  1. NOT

  2. AND

  3. OR

• Expression equivalence examples provide clarification on how operations can be restructured without changing their overall result.

Logic Function Example

Decoders

• n-to-2^n Decoder

  • Accepts n binary inputs and produces 2^n unique outputs.

  • Enables the selection of a specific output wire based on the binary input value, allowing for precise control in digital circuits.

• Use Cases:

  • Traffic signal management systems that activate only the appropriate signal light based on the input.

  • Converting memory address inputs into control signals for selecting various opcodes in microprocessors.

Multiplexers (MUX)

• Configuration:

  • A multiplexer has 2ⁿ data inputs and a single output.

  • An n-bit selector is utilized to determine which input to forward to the output.

• Operational Logic:

  • If the selector (SEL) input is 0, the output directly corresponds to Input0; if SEL = 1, the output corresponds to Input1.

Adders

• Half Adder

  • It accepts 2 binary inputs (A and B) and produces two outputs: Sum and Carry-out, like binary addition.

• Full Adder

  • This expands the functionality of the half adder by incorporating a Carry-in (Cin) input.

  • It is formulated as having 3 inputs (A, B, Cin) to produce 2 outputs (Sum S, Carry-out Cout).

  • Full adders are essential for performing addition on multi-bit binary numbers.

• 4-bit Ripple-Carry Adder

  • This circuit adds two sets of 4 bits (designated A[3:0] and B[3:0]), producing a 4-bit output (S[3:0]) and a single Carry-out (Cout).

Propagation Delay

• Delays are intrinsic to logic gate operations, typically ranging from 10 picoseconds to 100 picoseconds depending on the technology used.

• A specific delay example provides clarity on how these delays affect output response timing and overall system performance.

Overflow in Signed Adders

• Conditions to detect overflow in 4-bit signed addition processes include:

  • If the carry input (Cin) to a binary addition is 0 and the carry-out (Cout) is 1, overflow occurs.

  • Alternatively, if Cin = 1 and Cout = 0, overflow is also detected, leading to incorrect result representation.

Boolean Algebra Fundamentals

• Complement Rules

  • Fundamental rules include: A + A' = 1 and A ⋅ A' = 0, which are foundational to simplifying Boolean expressions.

• OR Rules

  • Important rules include: A + 1 = 1, A + 0 = A, and A + A = A.

• AND Rules

  • Essential rules are: A ⋅ 1 = A, A ⋅ 0 = 0, and A ⋅ A = A.

• Distributive Laws apply, providing additional support for simplifying complex logical expressions.

Steps for Converting Truth Tables to Expressions

1. For each row where the output (Q) is true (1), formulate a corresponding Boolean expression based on the input variables.

2. Combine these expressions using logical ORs to construct a comprehensive Sum of Products (SOP) expression, illustrating the logical relationship.

Minimizing Logic Expressions

• This process employs various Boolean algebra rules to simplify complex logical expressions, which enhances the efficiency of implementation, reduces resource costs, and improves operational speed.

• A number of simplification techniques and rules such as Absorption and DeMorgan's Law are beneficial for both minimization and deeper understanding of logical relationships.

Future Topics

• The course will extend into detailed discussions on information representation and further investigations into digital logic systems, encompassing logic gates, K-maps, and advanced logic design methodologies.