Code Converters - Binary to/from Gray Code

In this article, we will go through Code Converters - Binary to/from Gray Code, we will start our article by defining Code converters, Binary code and Gray code, and then we will go through the conversion of binary code to gray code and vice versa.

What are Code Converters?

Code Converters are the digital circuits or algorithms that are designed to translate data representation from one format to the other format. In this article, we will go through binary to gray code converters. The binary-to-Gray code converter takes binary input and translates it to its corresponding gray code representation.

For more about conversions please go through Number System and base conversions.

What is a Binary Code?

The Binary code is the numerical system used in digital electronics. It only consists of two Symbols which are 0 and 1. In binary code each digit is represented by power of 2 with the starting bit (right most bit) as 2⁰ and bit as 2¹ and so on. The binary code is serves as the basis for encoding text, number and various other types of data in the digital devices.

What is a Gray Code?

Gray Code system is a binary number system in which every successive pair of numbers differs by only one bit. It is used in applications in which the normal sequence of binary numbers generated by the hardware, is read while the system changes from the initial state to the final state. This could have serious consequences for the machine using the information. The Gray code eliminates this problem since only one bit changes its value during any transition between two numbers.

Another Name of Gray Code

• Unity Hamming Distance Code
• Cyclic Code
• Reflecting Code

Converting Binary to Gray Code

Let b₀, b₁, b₂ and b₃ be the bits representing the binary numbers, where b₀ is the LSB and b₃ is the MSB, and Let g₀, g₁, g₂, and g₃ be the bits representing the gray code of the binary numbers, where g₀ is the LSB and g₃ is the MSB.

The truth table for the conversion is

To find the corresponding digital circuit, we will use the K-Map technique for each of the gray code bits as output with all of the binary bits as input.

K-map for g₀:

K-map for g₁:

K-map for g₂:

K-map for g₃:

Corresponding Minimized Boolean Expressions and Digital Circuit for Gray Code Bits

g₀ ​= b₀​b₁′ ​+ b₁​b₀′ ​= b₀ ​⊕ b₁​

g₁ ​= b₂​b₁′ ​+ b₁​b₂′ ​= b₁ ​⊕ b₂​

g₂ ​= b₂​b₃′ ​+ b₃​b₂′ ​= b₂​ ⊕ b₃​

g₃ ​= b₃

Generalized Boolean Expression for conversion of Binary to Gray Code

Boolean expression for conversion of binary to gray code for n-bit :

g_n = b_n

g_n-1 = b_n XOR b_n-1 : :

g₁ = b₂ XOR b₁

Converting Gray Code to Binary

Converting gray code back to binary can be done in a similar manner. Let b₀, b₁, b₂ and b₃ be the bits representing the binary numbers, where b₀ is the LSB and b₃ is the MSB, and Let g₀, g₁, g₂, and g₃ be the bits representing the gray code of the binary numbers, where g₀ is the LSB and g₃ is the MSB.

Truth table:

Using K-map to get back the binary bits from the gray code.

K-map for b₀:

K-map for b₁:

K-map for b₂:

K-map for b₃:

Corresponding Minimized Boolean Expressions

b₃ = g₃

b₂ = g₃ ⊕ g₂

b₁ = g₂ ⊕ g₁

b₀ = g₁ ⊕ g₀

Corresponding Digital Circuit

Generalized Boolean Expression for conversion of Gray to Binary Code

Boolean expression for conversion of gray to binary code for n-bit :

b_n = g_n

b_n-1 = b_n XOR g_n-1 = g_n XOR g_n-1 : :

b₁ = b₂ XOR g₁ = g_n XOR ............ XOR g₁

Application of Code Converters - Binary to/from Gray Code

Given Below are the Applications of the Code Converters - Binary to/from Gray Code

• Rotary Encoders: The Rotary encoders are device which is used to convey rotary motions into digital signals. The code converters use to convert the gray code output of rotary encodes into binary.
• Control Systems: The Gray code converters are used in the Control system to integrate between system using binary and gray code.
• Signal Processing: The Code converters are used in the signal processing to translate between binary and gray coded signals.
• Analog-to-Digital Conversion: The code converters are used in ADC to interface between gray coded ADC outputs and binary based systems.