000 1110001 1110010 1110011 1110100 1110101 1110110 1110111 1111000 1111001 1111010 1111011 1111100 1111101 1111110 1111111 Figure 6.10 All possible 7-bit words, with valid Hamming code (7,4) codewords shown in bold Figure 6.11 Representation of a Hamming decoder in bold in Figure 6.10. The minimum code distance can be verified to be d min =3: this code is capable of detecting two and correcting a onebit error. Another way to explain the (7,4) Hamming code's mathematical process is shown in Figure 6.11. Here three intersecting circles are drawn and the 7-bit received codeword of the Hamming encoder is written into the seven sections of the circles. For example the codeword '1010001' is shown here. The inner numbers '1010' (shown in bold) are the data bits and the outer numbers '001' (shown in black)are the parity bits. Each individual circle contains 4 bits. The receiver rule is that each circle must include an even number of 1's. This means that if the receiver detects that one or more circles have an odd number of 1's, it will change one bit (and one bit only) to restore the rule. If only one circle has an odd number of ones, it changes the outer bit (which is not shared with other circles.) If two circles have an odd number of 1's then it changes the bit that is shared by the two circles. Similarly if all three circles contain an odd number of 1's, then the bit in the centre is changed to restore the rule. This example further illustrates the point that error control coding techniques create mathematical relationships bet- ween data bits and parity bits to enable the receiver to detect/correct reception errors. The performance of the Hamming (7,4) coding technique may be studied by comparing the BER with and without coding. As it can be seen from Figure 6.12 the coded system per- forms better by some 0.5 dB at BER of 10-4. This demonstrates the value of a mathematical inter-relationship between the transmitted bits, and signal processing at the receiver to reduce the required energy for a desired telecommunica