Digital Circuits Numbering Systems

• Digital circuits involve numbering systems and codes that are common in digital technology like decimal, binary, and hexadecimal systems
• Digital systems operate using binary numbers
• Hexadecimal systems make it easier for humans to manage binary numbers

Decimal Numbering System

• Decimal counting ranges from 0 up to 103, 199, 999 and 1000.
• The number 2745.214 can be expressed as (2 × 10³) + (7 × 10²) + (4 × 10¹) + (5 × 10⁰) + (2 × 10⁻¹) + (1 × 10⁻²) + (4 × 10⁻³).
• Numbers represent a sum of the products of each digit's value and its positional value (weight)

Binary Numbering System

• Binary counting, using 23, 22, 21, and 20 weights, yields the decimal equivalents from 0 to 7
• A binary number like 1011.101 can be shown with positional values
• 1011.101₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) + (1 × 2⁻¹) + (0 × 2⁻²) + (1 × 2⁻³) = 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125 = 11.625₁₀.

Binary to Decimal Conversion

• Example 1: Convert 11011 to decimal: 12⁴ + 12³ + 0 + 12¹+ 12⁰ = 16 + 8 + 2 + 1 = 27₁₀
• Example 2: Converting 10110101 to decimal is 12⁷ + 0 + 12⁵ + 12⁴ + 0 + 12² + 0 + 1*2⁰ = 181₁₀

Binary to Decimal (Double-Dabble Method)

• Binary 1101 converts to decimal as follows: 1 * 2 = 2, +1 = 3 * 2 = 6, +0 = 6 * 2 = 12, +1 = 13 * 2 = 26, +1 = 27₁₀.
• Double-dabble conversion: Binary 10110101, with the double dabble method, turns into 181₁₀.

Decimal to Binary Conversion

• Divide successively by 2 and note the remainder for decimal to binary conversion.
• Example: Converting 25 to binary involves successive divisions by 2, the result is 11001₂
• Decimal to binary uses the factorization of powers of 2, with an algorithm

Hexadecimal Numbering System

• Hexadecimal has values from 0-9, and uses letters A-F, with 16 as the base
• Table 2.1 displays equivalencies among Hexadecimal, Decimal, and Binary

Decimal to Hex Conversion

• Requires division by 16
• 423 to Hex conversion example outputs 1A7₁₆
• 214 to Hex conversion example outputs D6₁₆

Hexadecimal to Decimal Conversion

• 16⁴, 16³, 16², 16¹, 16⁰, 16⁻¹, 16⁻²⁻³, and 16⁻⁴ are weights of hexadecimal digits
• Convert 356₁₆ to decimal as follows: 3 * 16² + 5 * 16¹ + 6 * 16⁰ = 768 + 80 + 6 = 854₁₀.
• Convert 2AF₁₆ to decimal as follows: 2 * 16² + 10 * 16¹ + 15 * 16⁰ = 512 + 160 + 15 = 687₁₀

Hexadecimal to Binary Conversions

• Each hex digit is converted into a 4-bit binary equivalent.
• As an example 9F2₁₆ --> 1001 1111 0010₂
• Hexadecimal numbering serves as a compact representation of binary numbers, which allows for easy conversion

Binary to Hex Conversion.

• Binary numbers are represented for numbers with 4 columns

BCD Codes

• A decimal number represented by its binary equivalent yields pure binary encoding
• Digits are converted with pure binary equivalent, while disregarding significance such as 874 is 1000 0111 0100
• The direct representation of 4-bit binary allows for fast Decimal/BCD conversions

BCD for Decimal

• 0110 1000 0011 1001 is 6839
• Binary codes exceeding 9 (decimal) are prohibited and indicate an error in the BCD number.
• The group referring to a prohibited code causes an error in that BCD number

Gray Code

• Bits may not simultaneously transition in a circuit and have incorrect readings in short time intervals
• Table 2.2 provides binary and Gray code equivalencies for three bits

Converting Binary to Gray

• There are different steps to converting binary to graycode, different results will be shown

Converting Gray to Binary

• There are different steps to converting graycode to binary, different results will be shown

Gray Code Applications

• Gray code has position encoders with three bits splitting 360° with 45° precession
• Gray code has position encoders with 9 bits splitting 360° with approximately 0.7° precession

ASCII Code

• Standard ASCII codes include character, Hex, and Decimal values

Parity Bit Error Detection

• A parity bit is an added bit appended that consists of bits for error detection

Parity Bit

• The bit of parity can be odd bit, or even bit

Parity Even

• Parity bit is set, so that the number of 1's is set as even bit

Parity Odd

• Parity bit is set, so that the number of 1's is set as odd bit
• An example of sending a character is "#"

Nibbles, Bytes, and Words

• A nibble contains 4 bits, an example is 1,0,1,0
• A byte has 8 bits equal to 2 nibbles like 0,1,1,0,1,0,1,0
• A Word (or Palavra) has groups of bits that represents a single information unit
• Word size depends on the number of binary word bits a system handles like 32 or 64