BCD and Gray Code

Not all binary representations are pure base-2. BCD (Binary-Coded Decimal) and Gray code are specialized encodings that solve specific problems in digital systems—from displaying numbers on calculators to reading rotary encoders without errors.

Binary-Coded Decimal (BCD)

What Is BCD?

BCD represents each decimal digit as a 4-bit binary number:

Key Difference

BCD uses 4 bits per decimal digit; pure binary is more compact but harder to display.

Why Use BCD?

Easy Decimal Display

• Each 4-bit group → one digit on 7-segment display
• No complex binary-to-decimal conversion needed
• Calculators and meters use this

Exact Decimal Values

• Financial calculations need exact decimal
• 0.10 is exact in BCD, approximate in binary floating-point
• Avoids rounding errors in currency

Human-Readable Storage

• Timestamps often stored in BCD
• Easy to read in hex dumps
• BIOS real-time clocks use BCD

BCD Arithmetic

Addition Challenge

BCD addition needs adjustment when result > 9:

• 0101 + 0100 = 1001 (5 + 4 = 9) ✓ Valid BCD
• 0111 + 0101 = 1100 (7 + 5 = 12) ✗ Invalid BCD
• Must add 6 (0110) to correct: 1100 + 0110 = 0001 0010 = 12

Why Add 6?

BCD only uses values 0000-1001 (0-9). Values 1010-1111 (10-15) are invalid. Adding 6 carries into the next digit.

BCD Variants

Packed BCD

• Two digits per byte
• More efficient storage
• Example: 42 = 0100 0010 in one byte

Unpacked BCD

• One digit per byte (upper 4 bits unused/zero)
• Easier processing
• Example: 42 = 00000100 00000010 (two bytes)

Excess-3 BCD

• Add 3 to each digit before encoding
• Simplifies some arithmetic circuits
• 0 = 0011, 9 = 1100

Gray Code

What Is Gray Code?

Gray code is a binary sequence where only one bit changes between consecutive values:

Why Gray Code Matters

The Problem Gray Code Solves

In regular binary, going from 7 (0111) to 8 (1000) changes 4 bits simultaneously.

• If sensors don't switch at exactly the same instant
• Momentary false readings occur
• Could read 0000, 0001, 0011, 0111, or 1111 briefly

Gray Code Solution

• Only one bit changes at a time
• No ambiguous intermediate states
• Essential for position encoders

Gray Code Applications

Rotary Encoders

• Motor position sensing
• Volume knobs (digital)
• Robotics joint angles
• CNC machine positioning

Error Minimization

• Analog-to-digital conversion
• Genetic algorithms (crossover)
• Karnaugh maps (logic minimization)

Communication

• Some error-correcting codes
• Certain modulation schemes

Converting Binary to Gray Code

Method

1. Keep the most significant bit (MSB) the same
2. XOR each bit with the bit to its left

Example: Convert 1011 (binary) to Gray

• MSB: 1 (keep)
• 1 XOR 0 = 1
• 0 XOR 1 = 1
• 1 XOR 1 = 0
• Result: 1110 (Gray)

Formula

Gray[i] = Binary[i] XOR Binary[i+1]

Gray[MSB] = Binary[MSB]

Converting Gray Code to Binary

Method

1. Keep the MSB the same
2. XOR each Gray bit with the previous Binary bit

Example: Convert 1110 (Gray) to Binary

• MSB: 1 (keep)
• 1 XOR 1 = 0
• 0 XOR 1 = 1
• 1 XOR 0 = 1
• Result: 1011 (binary)

まとめ

BCD and Gray code are specialized binary encodings that solve specific problems. BCD simplifies decimal display and ensures exact decimal arithmetic—important for calculators and financial systems. Gray code ensures only one bit changes between consecutive values—essential for position encoders and error prevention. While neither is as common as pure binary, understanding them reveals how different number representations serve different engineering needs.

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