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. At last, we will conclude our article with some faqs.

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

$b_0,\:b_1,\:b_2\:,\:and\:b_3$

be the bits representing the binary numbers, where

$b_0$

is the LSB and

$b_3$

is the MSB, and Let

$g_0,\:g_1,\:g_2\:,\:and\:g_3$

be the bits representing the gray code of the binary numbers, where

$g_0$

is the LSB and

$g_3$

is the MSB. The truth table for the conversion is

[Tex] \begin{tabular}{||c|c|c|c||c|c|c|c||} \hline \multicolumn{4}{||c||}{Binary} & \multicolumn{4}{|c||}{Gray Code}\\ \hline b_3 & b_2 & b_1 & b_0 & g_3 & g_2 & g_1 & g_0 \\ \hline \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ \hline 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ \hline 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 \\ \hline \hline 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ \hline 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\ \hline 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 \\ \hline 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ \hline \hline 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ \hline 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \\ \hline 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ \hline 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\ \hline \hline 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ \hline 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\ \hline 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \\ \hline 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ \hline \hline \end{tabular} [/Tex]

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

$g_0$

–

K-map for

$g_1$

–

K-map for

$g_2$

–

K-map for

$g_3$

–

Corresponding Minimized Boolean Expressions and Digital Circuit for Gray Code Bits

$g_0 = b_0b_1^\prime + b_1b_0^\prime = b_0 \oplus b_1\\ g_1 = b_2b_1^\prime + b_1b_2^\prime = b_1 \oplus b_2\\ g_2 = b_2b_3^\prime + b_3b_2^\prime = b_2 \oplus b_3\\ g_3 = b_3$

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 : :

G ₁ = B ₂ XOR B ₁

Converting Gray Code to Binary

Converting gray code back to binary can be done in a similar manner. Let

$b_0,\:b_1,\:b_2\:,\:and\:b_3$

be the bits representing the binary numbers, where

$b_0$

is the LSB and

$b_3$

is the MSB, and Let

$g_0,\:g_1,\:g_2\:,\:and\:g_3$

be the bits representing the gray code of the binary numbers, where

$g_0$

is the LSB and

$g_3$

is the MSB. Truth table

[Tex] \begin{tabular}{||c|c|c|c||c|c|c|c||} \hline \multicolumn{4}{||c||}{Gray Code} & \multicolumn{4}{|c||}{Binary}\\ \hline g_3 & g_2 & g_1 & g_0 & b_3 & b_2 & b_1 & b_0\\ \hline \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ \hline 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ \hline 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 \\ \hline \hline 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\ \hline 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 \\ \hline 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ \hline 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\ \hline \hline 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ \hline 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 \\ \hline 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ \hline 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 \\ \hline \hline 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ \hline 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 \\ \hline 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ \hline 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 \\ \hline \hline \end{tabular} [/Tex]

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

$b_0$

–

K-map for

$b_1$

–

K-map for

$b_2$

–

K-map for

$b_3$

–

Corresponding Boolean expressions

[Tex] \begin{align*} b_0 &=g_3^\prime g_2^\prime g_1^\prime g_0 + g_3^\prime g_2^\prime g_1g_0^\prime + g_3^\prime g_2g_1^\prime g_0^\prime + g_3^\prime g_2g_1g_0 +g_3g_2^\prime g_1^\prime g_0^\prime + g_3g_2^\prime g_1g_0 \\ &\:\:\:+g_3g_2g_1^\prime g_0 + g_3g_2g_1g_0^\prime \\ &= g_3^\prime g_2^\prime( g_1^\prime g_0 + g_1g_0^\prime) + g_3^\prime g_2(g_1^\prime g_0^\prime + g_1g_0) +g_3g_2^\prime(g_1^\prime g_0^\prime + g_1g_0 )\\ &\:\:\:+g_3g_2 (g_1^\prime g_0 + g_1g_0^\prime) \\ &= g_3^\prime g_2^\prime(g_0\oplus g_1) + g_3^\prime g_2(g_0\odot g_1)+g_3g_2^\prime(g_0\odot g_1) + g_3g_2 (g_0\oplus g_1) \\ &= (g_0\oplus g_1)(g_2\odot g_3) + (g_0\odot g_1)(g_2\oplus g_3)\\ &= g_3\oplus g_2\oplus g_1\oplus g_0\\ b_1 &= g_3^\prime g_2^\prime g_1 + g_3^\prime g_2g_1^\prime + g_3g_2g_1 + g_3g_2^\prime g_1^\prime \\ &= g_3^\prime(g_2^\prime g_1 + g_2g_1^\prime) + g_3(g_2g_1 + g_2^\prime g_1^\prime) \\ &= g_3^\prime(g_2\oplus g_1) + g_3(g_2\odot g_1) \\ &= g_3\oplus g_2\oplus g_1\\ b_2 &= g_3^\prime g_2 + g_3g_2^\prime\\ &= g_3\oplus g_2\\ b_3 &= g_3 \end{align*} [/Tex]

Corresponding Digital Circuit

The Circuit can be given as

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 : :

B ₁ = B ₂ XOR G ₁ = G _n XOR ………… XOR G ₁

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 DSP systems.

Conclusion

In this Article we have gone through Code Converters – Binary to/from Gray Code,we have gone through the definition of the Code converters,Binary code and Gray code,then we have gone through Code conversion between Binary to Gray code and vice versa.

Code Converters – Binary to/from Gray Code – FAQs

How do you minimize the logic gates required for a Binary-to-Gray Code Converter circuit?

To reduce the logic gates the techniques like Boolean algebra simplification, Karnaugh maps and logic gate optimization algorithms can be used.

Can Code Converters introduce latency in digital systems ?

Yes,code converters bring latency due to the processing time required for the conversion.

Can Code Converters be implemented using reversible logic gates for applications requiring energy-efficient computing?

Yes the code converters can be designed using reversible logich which will have the property of information losslessness and less minimal power.

This article is contributed by

Chirag Manwani

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