Solved Examples Problems [Number Systems]

SOLVED EXAMPLES

Problem 4.28

Determine which bit, if any, is in error in the even parity. Hamming coded character 1100111. Decode the message.

Solution :

Step 1: Construct the bit location table.

Step 2: Check for parity bits

For P1 : P1 checks locations 1, 3, 5 and 7.

There are three 1 s in the group.

⸫ Parity check for even parity is wrong → 1 (LSB)

For P1 : P2 checks locations 2, 3, 6 and 7.

There are four 1 s in the group.

⸫ Parity check for even parity is correct → 0

For P4 : P4 checks locations 4, 5, 6 and 7.

There are two 1s in the group.

⸫ Parity check for even parity is correct → 0 (MSB)

The resultant word is 0 0 1. This says that the bit in the number 1 location is in error. It is 1 and should be á 0. Therefore, the correct code is 1 1.0 0 1 1 0 and the message is 1 1 0 1.

Problem 4.29

Construct a seven-bit error-correcting code to represent the decimal digits by augmenting the excess-3 code and by using odd-1 parity check.

Solution :

Excess 3 code for decimal digits are as follows:

Using odd parity code for 1 0 1 1 is constructed as shown below. The remaining code can be constructed using similar logic. The column 3 of above table gives the hamming codes for all excess-3 codes.

Problem 4.30

The message below has been coded in the even parity Hamming code and transmitted through a noisy channel. Decode the message assuming that at most a single error has occured in each word code.

(i) 1001001

(ii) 0111001

(iii) 1110110

(iv) 0011011

Solution :

(i)

Error in bit position 2. Therefore, hamming code should be 1001011 and message is 1000.

(ii)

Error in bit position 6. Therefore, hamming code should be 0011001 and message is 0010.

(iii)

Error in bit position 3. Therefore, hamming code should be 1100110 and message is 1101.

(iv)

Error in bit position 2. Therefore, hamming code should be 0011001 and message is 0010.