Mastering Digital Electronics: From Binary Basics to Verilog Design
Mastering Digital Electronics: From Binary Basics to Verilog Design Description Explore the core concepts of digital electronics starting from binary numbers to number system conversions, complements, signed arithmetic, and logic simplification techniques like K-maps — all in a simplified manner for learners. Introduction Digital electronics is the backbone of all modern devices—from smartphones to spacecraft. Whether you’re an engineering student just starting your journey, a faculty member preparing course content, or a VLSI enthusiast building your first project, a solid grasp of binary logic, number systems, and circuit design is essential. This blog walks you through key concepts with clear explanations and step-by-step examples, making learning intuitive and hands-on. Perfect for anyone looking to strengthen their foundation in digital design or preparing for practical lab sessions and design challenges. Digital Systems: A digital system is a system that processes discrete (separate) values, typically using binary signals (0s and 1s). Unlike analog systems that deal with continuous signals, digital systems operate using logic levels that are either on (1) or off (0) Binary values are a base- 2 numeral system and are represented by digits 0 and 1. Digital systems operate using binary. But we also use other number systems to simplify representation: Octal (base-8) for compact 3-bit grouping. Decimal (base-10) for human-friendly interaction. Hexadecimal (base-16) for 4-bit grouping in digital design. Types of Number Systems in Digital Electronics Number System Base Digits Used Binary 2 0, 1 Octal 8 0–7 Decimal 10 0–9 Hexadecimal 16 0–9, A(10)–F(15) The six letters (in addition to the 10 integers) in hexadecimal represent: 10, 11, 12, 13, 14, and 15, respectively. Binary Arithmetic: Addition: Rules of binary addition are as follows 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0, and carry 1 to the next higher significant bit Subtraction: Rules of binary subtraction are as follows 0 – 0 = 0 0 – 1 = 1, and borrow from the next higher significant bit 1 – 0 = 1 1 – 1 = 0 Number Base Conversions Number base conversion is the process of converting numbers from one base to another (e.g., Decimal to Binary, Binary to Hexadecimal, etc.). This is essential in digital systems for communication between humans and machines. 1) Decimal to binary Method: Divide by 2 repeatedly and record the remainders in reverse. Example: Convert 13 to binary→ 13 ÷ 2 = 6, remainder 1→ 6 ÷ 2 = 3, remainder 0→ 3 ÷ 2 = 1, remainder 1→ 1 ÷ 2 = 0, remainder 1Answer: 1101₂ 2) Binary to decimal Method: Multiply each bit by 2ⁿ (starting from right, n = 0) and sum. Example:Convert 1011₂ to decimal→ (1×2³) + (0×2²) + (1×2¹) + (1×2⁰)→ 8 + 0 + 2 + 1 = 11 3) Binary to Octal Method: Group binary digits in sets of 3 (from right), then convert. Example:Binary: 110110→ Group: 110 110→ Octal: 6 6 → 66₈ 4) Binary to Hexadecimal (Base-2 → Base-16) Method: Group binary digits in sets of 4 (from right), then convert. Example:Binary: 11101001→ Group: 1110 1001→ Hex: E9 → E9₁₆ 5) Hexadecimal to Binary (Base-16 → Base-2) Method: Replace each hex digit with 4-bit binary. Example:Hex: 2F→ 2 = 0010, F = 1111→ Binary: 00101111 Decimal to Binary (with fractional part) Steps: Convert the integer part to binary using repeated division by 2. Convert the fractional part using repeated multiplication by 2. Combine both parts with a binary point (.) in between. Example: Convert 10.625 to binary Step 1: Integer part (10)Divide by 2 and record the remainders (bottom to top): 10 ÷ 2 = 5 → remainder 0 5 ÷ 2 = 2 → remainder 1 2 ÷ 2 = 1 → remainder 0 1 ÷ 2 = 0 → remainder 1 → Binary: 1010 Step 2: Fractional part (0.625)Multiply by 2 and take the integer parts: 0.625 × 2 = 1.25 → 1 0.25 × 2 = 0.5 → 0 0.5 × 2 = 1.0 → 1 → Binary: .101 Final Answer:10.625 (decimal) = 1010.101 (binary) Binary to Decimal (with fractional part) Steps: Convert the integer part using powers of 2 from right to left. Convert the fractional part using negative powers of 2 from left to right. Example: Convert 1010.101 to decimal Step 1: Integer part (1010) 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 8 + 0 + 2 + 0 = 10 Step 2: Fractional part (.101) 1×2⁻¹ + 0×2⁻² + 1×2⁻³ = 0.5 + 0 + 0.125 = 0.625 Final Answer:1010.101 (binary) = 10.625 (decimal) Complements of Numbers In digital systems and arithmetic, complements are used to simplify subtraction and handle negative numbers in binary systems. There are two main types: 1’s Complement 2’s Complement 1’s Complement (One’s Complement) Definition:The 1’s complement of a binary number is formed by flipping all the bits — changing 1 to 0 and 0 to 1. Example:Binary: 101100111’s complement: 01001100 This is equivalent to a logical NOT operation. 2’s Complement (Two’s Complement) Definition:The 2’s complement is found by adding 1 to the 1’s complement of a number. Steps to Find 2’s Complement: Take the binary number. Find its 1’s complement (invert bits). Add 1 to the result. Example:Binary: 00010100 (20 in decimal)→ 1’s complement: 11101011→ Add 1: 111011002’s complement: 11101100 (Represents -20 in 8-bit 2’s complement form) Why Are Complements Important? Used in binary subtractionA – B = A + (2’s complement of B) Helps in representing negative numbers in binary. Simplifies hardware design for arithmetic units. Signed Binary Numbers In digital systems, numbers can be positive or negative. Negative numbers in binary are represented using signed binary representations. Unlike unsigned binary (which represents only positive numbers), signed binary formats allow for both positive and negative values. For an 8-bit unsigned binary number, the range is 0 to 255.For an 8-bit signed binary number (using 2’s complement), the
