BCD e Gray Code

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

Binary-Coded Decimal (BCD)

What Is BCD?

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

Key Difference

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

Perche Usa BCD?

Easy Decimal Display

• Each 4-bit group → one digit on 7-segment display
• No complex binary-un-decimal conversione needed
• Calculators e metri usare this

Exact Decimal Values

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

Human-Readable Storage

• Timestamps often stored in BCD
• Easy un read in hex dumps
• BIOS real-time clocks usare BCD

BCD Arithmetic

Addition Challenge

BCD addition needs adjustment quando result > 9:

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

Perche Add 6?

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

BCD Variants

Packed BCD

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

Unpacked BCD

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

Excess-3 BCD

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

Gray Code

What Is Gray Code?

Gray code e un binary sequence dove only one bit changes tra consecutive values:

Perche Gray Code Matters

Il Problem Gray Code Solves

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

• If sensors don't switch at esattamente il same instant
• Momentary false readings occur
• Could read 0000, 0001, 0011, 0111, o 1111 briefly

Gray Code Solution

• Only one bit changes at un time
• No ambiguous intermediate states
• Essential per position encoders

Gray Code Applicazioni

Rotary Encoders

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

Error Minimization

• Analog-un-digital conversione
• Genetic algorithms (crossover)
• Karnaugh maps (logic minimization)

Communication

• Some error-correcting codes
• Certain modulation schemes

Convertendo Binary un Gray Code

Method

1. Keep il most significant bit (MSB) il same
2. XOR each bit con il bit un its left

Esempio: Converti 1011 (binary) un Gray

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

Formula

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

Gray[MSB] = Binary[MSB]

Convertendo Gray Code un Binary

Method

1. Keep il MSB il same
2. XOR each Gray bit con il previous Binary bit

Esempio: Converti 1110 (Gray) un Binary

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

Conclusione

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

Articoli Correlati

• Binary in Computing: How Computers Think in 1s e 0s
• Programming Number Formats: How Languages Handle Numbers