Gray Code to Decimal Converter

Convert Reflected Binary Gray Code to Decimal

The Gray Code to Decimal Converter is an online developer tool designed to translate reflected binary Gray code strings back to their standard decimal (base-10) integer equivalents. It parses the Gray code bits, reconstructs standard binary bits sequentially, and converts the resulting binary sequence into decimal, hexadecimal, or spaced binary formats.

How Gray Code to Decimal Conversion Works

Converting reflected binary Gray code back to standard binary requires performing sequential bitwise XOR operations from the most significant bit (MSB) to the least significant bit (LSB).

Let $G$ be the Gray code bit sequence $G[0], G[1], \dots, G[n-1]$ and $B$ be the standard binary sequence $B[0], B[1], \dots, B[n-1]$.

The conversion rules are:

• The most significant bit is identical:
  $$B[0] = G[0]$$
• Subsequent bits are calculated using:
  $$B[i] = B[i-1] \oplus G[i]$$

Once you have the standard binary string $B$, you convert it to decimal using standard positional base-2 addition.

Step-by-Step Conversion Example

Let's convert the Gray code 1010 back to decimal:

1. Identify Gray bits: $G = 1, 0, 1, 0$.
2. Calculate standard binary bits $B$:
   • $B[0] = G[0] = 1$
   • $B[1] = B[0] \oplus G[1] = 1 \oplus 0 = 1$
   • $B[2] = B[1] \oplus G[2] = 1 \oplus 1 = 0$
   • $B[3] = B[2] \oplus G[3] = 0 \oplus 0 = 0$
3. Standard Binary representation: $B = 1100_2$.
4. Convert $1100_2$ to decimal: $$1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 8 + 4 + 0 + 0 = 12_{10}$$
5. The decoded decimal value is $12$.

Key Use Cases

• Sensor Decoding: Reading and translating physical positions from optical absolute encoders.
• Debugging Digital Circuits: Verifying signal states in digital designs utilizing asynchronous counters.
• Academic Study: Hands-on practice for students learning electrical engineering and computer science logic circuits.