Hamming Distance (HD)

Hamming distance (Hamming metric) In the theory of block codes intended for error detection or error correction, the Hamming distance d(u, v) between two words u and v, of the same length, is equal to the number of symbol places in which the words differ from one another. If u and v are of finite length n then their Hamming distance is finite since d(u, v) ← n.
It can be called a distance since it is non-negative, nil-reflexive, symmetric, and triangular:
0 ← d(u, v)
d(u, v) = 0 iff u = v
d(u, v) = d(v, u)
d(u, w) ← d(u, v) + d(v, w)
The Hamming distance is important in the theory of error-correcting codes and error-detecting codes: if, in a block code, the codewords are at a minimum Hamming distance d from one another, then
(a) if d is even, the code can detect d – 1 symbols in error and correct ½d – 1 symbols in error;
(b) if d is odd, the code can detect d – 1 symbols in error and correct ½(d – 1) symbols in error.

How to Calculate Hamming Distance ?
- Ensure the two strings are of equal length. The Hamming distance can only be calculated between two strings of equal length.
String 1: "1001 0010 1101"
String 2: "1010 0010 0010"
- Compare the first two bits in each string. If they are the same, record a "0" for that bit. If they are different, record a "1" for that bit. In this case, the first bit of both strings is "1," so record a "0" for the first bit.
- Compare each bit in succession and record either "1" or "0" as appropriate.
String 1: "1001 0010 1101"
String 2: "1010 0010 0010"
Record: "0011 0000 1111"
- Add all the ones and zeros in the record together to obtain the Hamming distance.
Hamming distance = 0+0+1+1+0+0+0+0+1+1+1+1 = 6

Data Formats

The data bits in a serial transmission may represent anything, including commands, sensor readings, status information, error codes, or text messages. The information may be encoded as binary or text data.

- Binary Data : The receiver interprets a received byte as a binary number with a value from 0 to 255. The bits are numbered 0 through 7, with each bit representing the bit's value (0 or 1) multiplied by a power of 2. A byte of 1111 1111 translates to 255 or FFh and 0001 0001 translates to 17 or 11h. In asynchronous mode, bit 0, the least-significant bit arrives first. Binary data works fine for many links but some links need to send messages or files containing text.

- Text Data : To send text, the program uses a code that assigns a numeric value to each text character. There are several coding conventions :
* ASCII : It consists of 128 codes and requires only seven data bits. An eighth bit, if used, may be 0 or a parity bit.
* ANSI : It consists of 256 codes with the higher codes representing special and accented characters.
Other formats use 16 bits per character, which allows 65,536 different characters.

One can also use text to transfer binary data by expressing the data in ASCII Hex format. Each byte is represented by a pair of ASCII codes that represent the byte's two hexadecimal characters. This format can represent any value using only the ASCII codes 30h through 39h (from 0 through 9) and 41h to 46h (for A through F).