Chapter X: Binary Encoding Scheme

Section X.2 Non-Weighted Codes

Binary Encoding

Although the binary number system has many practical advantages and is widely used in digital computers, in many cases it is convenient to work with the decimal number system, especially when the communication between man and the machine is extensive since most numerical data generated by man are in terms of decimal numbers. To simplify the communication problem between man and machine, a number of codes have been devised so that the decimal digits are represented by sequences of binary digits.

Classification of binary codes

X.1 Weighted Codes

*Binary to Decimal
*BCD Code
*Self-Complementing Code
*Two's complement

X.2 Non-Weighted Codes

*Excess-3 Code
*Gray Code

Alphanumeric Codes

*American Standard Code for Information Interchange (ASCII)
*Extended Binary Coded Decimal Interchange Code (EBCDIC)

X.3 Error Detecting Codes

*Parity Bit

X.4 Error Correcting Codes

*7-Bit Hamming Code

Nonweighted Codes

In this type of binary code, the positional weights are not assigned. The examples of non-weighted codes are Excess-3 code and Gray code.

Excess-3 code

The Excess-3 code is also called as XS-3 code. It is a non-weighted code used to express decimal numbers. The Excess-3 code words are derived from the 8421 BCD code words adding \( (0011)_2 \)or \( (3)_{10} \) to each code word in 8421. The excess-3 codes are obtained as follows :

The Excess-3 code is a binary-coded decimal (BCD) representation where each decimal digit is represented by its 4-bit binary equivalent plus 3. It is a self-complementary code used in digital systems for error detection and arithmetic operations. The 1's complement of the Excess-3 code aligns with the Excess-3 code of \(9-N\), which is why it's self-complementing.

Decimal Digit (N)	Excess-3 Code (N+3)	1's Complement of Excess-3	Decimal 9's Complement (9-N)	Excess-3 Code of 9-N
0	0011	1100	9	1100
1	0100	1011	8	1011
2	0101	1010	7	1010
3	0110	1001	6	1001
4	0111	1000	5	1000
5	1000	0111	4	0111
6	1001	0110	3	0110
7	1010	0101	2	0101
8	1011	0100	1	0100
9	1100	0011	0	0011

Excess-3 code Application

1. Error Detection Example

Scenario: Decimal Value Transmission in a Digital System

Error Scenario:

Error Detection Process:

1. Check the received code 1111

2. Action: Since 1111 is invalid, the system flags it as an error.

Gray Code

Gray code, also known as reflected binary code, is a type of binary numbering system where two successive values differ by only one bit.
It is a non-weighted code and it is not an arithmetic code. That means there are no specific weights assigned to the bit position. It has a very special feature that, only one bit will change each time the decimal number is incremented as shown in Fig X.1. As only one bit changes at a time, the gray code is called as a unit distance code.

The gray code is a cyclic code . The term "cyclic" in the context of Gray code means that the Gray code sequence forms a closed loop. The last value in the sequence transitions back to the first value, and this transition involves only a single-bit change, just like any other consecutive values in Gray code. Gray code cannot be used for arithmetic operations. In this code, two consecutive values are differed by one bit of binary digits.

Binary to Gray conversion :

The Most Significant Bit (MSB) of the gray code is always equal to the MSB of the given binary code.
Other bits of the output gray code can be obtained by XOR binary code bit at that index and previous index.

Note: XOR, short for "exclusive or", is a logical operation that returns true if the inputs are different, and false (or 0) if they are the same.

Gray to binary conversion :

The Most Significant Bit (MSB) of the binary code is always equal to the MSB of the given gray code.
Other bits of the output binary code can be obtained by checking the gray code bit at that index. If the current gray code bit is 0, then copy the previous binary code bit, else copy the invert of the previous binary code bit.

n-bit Gray Code Generating

The n-bit Gray code is a member of a class called reflected codes. The term ″reflected″ is used to designate codes which have the property that the n-bit code can be generated by reflecting the (n-1)st-bit code, as illustrated here (left to right is the 1 bit, the two bit, the three bit.

Steps:

1. Start with 1-bit Gray Code:
- 0
- 1
2. Reflect the list
3. Prefix 0 to each code in the original list and 1 to each in the reflected list.
4. Concatenate the two lists to form the next level Gray Code.
5. Repeat steps 2–4 until you reach the desired n-bit Gray Code.

Figure X.1

A four-bit Gray code is shown in the following table:

Decimal	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

Key Characteristics:

1. Single-Bit Change:
2. Cyclic:
3. Reflection:

Gray code Application

Rotary Encoders: Gray code is commonly employed in rotary encoders, which are used in various devices like robotics, industrial control systems, and navigation systems. It ensures accurate tracking of rotational position by encoding the position values in Gray code, where only one bit changes at a time. This property helps to prevent errors or glitches during the position tracking process.

Example: 3-Bit Gray Code Rotary Encoder

Imagine a rotary encoder divided into \(2^3=8\) positions. These positions are encoded using 3-bit Gray code:

Position	Gray Code	Binary Code (for comparison)
0	000	000
1	001	001
2	011	010
3	010	011
4	110	100
5	111	101
6	101	110
7	100	111

Explanation:

1. Gray Code Property:

last bit changes

middle bit changes

2. Binary Code Issue:

Reference

BCD Code
Binary Codes
Gray to Binary and Binary to Gray conversion
Gray Code