Related Concepts: 01 BCD & Weighted Digital Codes | 02 Excess-3 Code & Self-Complementing Logic | 04 Error Control, Parity Generators & Checkers

2.03 Gray Code & Code Conversions

What is Gray Code?

Gray Code {also called Reflected Binary Code} is a 4-bit (or $N$-bit) non-weighted binary coding system.

Its defining property is: “A binary coding system in which ONLY ONE BIT changes at a time between any two consecutive code words.”

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1. What is Gray Code?

[concept] Concept: Gray Code Definition Gray Code (also known as Reflected Binary Code) is a non-weighted code. Its defining characteristic is that only one bit changes state (0 to 1 or 1 to 0) when transitioning between any two consecutive values.

For example:

• Decimal 1 to 2 in Binary: 01 $\to$ 10 (2 bits change simultaneously)
• Decimal 1 to 2 in Gray: 01 $\to$ 11 (Only 1 bit changes - the MSB)

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2. Why do we use Gray Code? (Physical Systems)

Unlike numerical computations which favor binary numbers, physical interface systems use Gray code to prevent reading errors:

A. Rotary Shaft Encoders (Robotics & Position Sensing)

Imagine a rotating disk divided into angular sectors with optical sensors reading binary codes.

• If standard binary is used and the sensor rotates from decimal 3 (011) to decimal 4 (100):
  • Three bits must flip simultaneously.
  • Because of tiny physical misalignments, the sensors will not trigger at the exact same instant.
  • For a fraction of a millisecond, the controller might read an intermediate state like 111 (decimal 7) or 010 (decimal 2), causing the system to register erratic movement.
• Using Gray code, transitioning from 3 (010) to 4 (110) flips only the MSB. No intermediate state is ever read.

B. Low-Power VLSI (Digital Integrated Circuits)

In high-speed data buses, switching a bit from 0 to 1 or 1 to 0 requires charging/discharging parasitic capacitance, which consumes dynamic power ($P\propto f\cdot C\cdot V^2$). By encoding bus values in Gray code, the average number of transitions per step is minimized, significantly lowering power consumption.

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3. 4-Bit Gray Code Lookup Table

Decimal Value	Standard Binary	Gray Code
0	0000	0000
1	0001	0001
2	0010	0011
3	0011	0010
4	0100	0110
5	0101	0111
6	0110	0101
7	0111	0100
8	1000	1100
9	1001	1101
10	1010	1111
11	1011	1110
12	1100	1010
13	1101	1011
14	1110	1001
15	1111	1000

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4. Why is it Called “Reflected” Code?

The code is named “reflected” because the sequence of code words for the second half of any $N$-bit Gray code sequence is the exact mirror image {reflection} of the first half, with the exception of the MSB which is inverted from 0 to 1.

2-Bit Gray Code Reflection Example:
Decimal 0:  0 0  ---| First half (MSB = 0)
Decimal 1:  0 1  ---|
-------------------- Reflection Axis
Decimal 2:  1 1  ---| Second half (MSB = 1, LSB reflected: 1 then 0)
Decimal 3:  1 0  ---|

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5. Conversion Algorithms

Conversions between pure Binary and Gray code rely entirely on the Exclusive-OR (XOR, $\oplus$) operation.

graph TD
    subgraph Binary to Gray Algorithm
        B3[B3 MSB] --> G3[G3 = B3]
        B3 & B2[B2] -->|XOR| G2[G2 = B3 ⊕ B2]
        B2 & B1[B1] -->|XOR| G1[G1 = B2 ⊕ B1]
        B1 & B0[B0 LSB] -->|XOR| G0[G0 = B1 ⊕ B0]
    end

graph TD
    subgraph Gray to Binary Algorithm
        g3[G3 MSB] --> b3[B3 = G3]
        b3 & g2[G2] -->|XOR| b2[B2 = B3 ⊕ G2]
        b2 & g1[G1] -->|XOR| b1[B1 = B2 ⊕ G1]
        b1 & g0[G0 LSB] -->|XOR| b0[B0 = B1 ⊕ G0]
    end

A. Binary to Gray Code Conversion

Given an $N$-bit Binary number ($B_{N-1}\dots B_0$), compute Gray bits ($G_{N-1}\dots G_0$):

• MSB Rule: $G_{N-1}=B_{N-1}$
• Subsequent Bits Rule: $G_i=B_{i+1}\oplus B_i$ for $i=N-2,\dots ,0$.

B. Gray Code to Binary Conversion

Given an $N$-bit Gray code ($G_{N-1}\dots G_0$), compute Binary bits ($B_{N-1}\dots B_0$):

• MSB Rule: $B_{N-1}=G_{N-1}$
• Subsequent Bits Rule: $B_i=B_{i+1}\oplus G_i$ for $i=N-2,\dots ,0$.

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6. Design: 4-Bit Reflected Code to Binary Converter Circuit

Major Exam Design Problem (2024 - 10 Marks)

Question: Design a combinational logic circuit that converts a 4-bit reflected code (Gray code) number to a 4-bit binary number using XOR gates.

Circuit Equations & Architecture:

$B_3=G_3$ $B_2=G_3\oplus G_2$ $B_1=B_2\oplus G_1$ $B_0=B_1\oplus G_0$

The circuit consists of a cascading chain of three 2-input XOR gates:

• $G_3$ passes straight to output $B_3$.
• $G_3\oplus G_2\to B_2$.
• $B_2\oplus G_1\to B_1$.
• $B_1\oplus G_0\to B_0$.

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Past Year Questions (PYQs)

• [PYQ 2017]: Define Reflected code. (04 Marks)
• [PYQ 2018]: Convert $(605{)}_{10}$ into Gray code number. (Binary: $1001011101_2⟹1101110011_{\text{Gray}}$). (12 Marks)
• [PYQ 2024]: Design a 4-bit reflected code to binary converter circuit using XOR gates. (10 Marks)