Gray Code to Binary Converter

What is Gray Code to Binary Conversion?

A Gray Code to Binary Converter reverses the process of reflected binary encoding, taking a non-weighted Gray code sequence and decoding it back into standard weighted binary (base-2).

Standard binary has a positional weighting system where the leftmost column represents the highest power of 2 ($2^{n-1}$). To perform calculations, compare values, or read standard measurements, Gray code must first be decoded into standard binary digits.

Mathematical Formula: Gray Code to Binary

Unlike the binary-to-Gray conversion (which can be done in parallel for all bits), Gray-to-binary conversion is a sequential process where each bit depends on the state of the previously decoded binary bit.

For a Gray code string $G = g_{n-1}g_{n-2}\dots g_0$, the corresponding standard binary string $B = b_{n-1}b_{n-2}\dots b_0$ is defined recursively: $$b_{n-1} = g_{n-1} \quad (\text{The most significant bit remains identical})$$ $$b_i = b_{i+1} \oplus g_i \quad \text{for } 0 \le i < n-1$$ Where $\oplus$ is the logical Exclusive OR (XOR) operator.

Example Step-by-Step Transition

Let's decode the Gray code sequence $1110_{Gray}$ (which corresponds to decimal 11) back to standard binary:

1. Bit 3 (MSB): $b_3 = g_3 = 1$
2. Bit 2: $b_2 = b_3 \oplus g_2 = 1 \oplus 1 = 0$
3. Bit 1: $b_1 = b_2 \oplus g_1 = 0 \oplus 1 = 1$
4. Bit 0 (LSB): $b_0 = b_1 \oplus g_0 = 1 \oplus 0 = 1$

So, the standard binary of $1110_{Gray}$ is $1011_2$.