Data Communications Networking Devices - 4th Ed.pdf

4.2 MODEMS _____________________________________________________________________ 371mapped into a signal point which is selected from a 128-point 2 7 ) signalconstellation.The key to the ability of TCM to minimize errors at high data rates is theemployment of forward error correcting FEC) in the form of convolutionalcoding. With convolutional coding, each bit in the data stream is compared withone or more bits transmitted prior to that bit. The value of each bit, which can bechanged by the convolutional encoder, is therefore dependent upon the value ofother bits. In addition, a redundant bit is added for every group of bits compared inthis manner. The following examination of the formation of a simple convolutionalcode clari®es how the convolutional encoder operates.Convolutional encoder operationAssuming that a simple convolutional code is formed by the modulo 2 sum of thetwo most recent data bits, then two output bits will be produced for each data bitÐa data bit and a parity bit. If we also assume that the ®rst output bit from theencoder is the current data bit then the second output bit is the modulo 2 sum of thecurrent bit and its immediate predecessor. Figure 4.20 illustrates the generation ofthis simple convolutional code.Because each parity bit is the modulo 2 sum of the two most recent data bits, therelationship between the parity bits and the data bits becomes:P i ˆ b i ‡ b i 1 i ˆ 1; 2; 3 ...If the composition of the ®rst four data bits entering the encoder was 1101b 4 b 3 b 2 b 1 ), the four parity bits are developed as follows:P 1 ˆ b 1 ‡ b 0 ˆ 1 ‡ 0 ˆ 1P 2 ˆ b 2 ‡ b 1 ˆ 0 ‡ 1 ˆ 1P 3 ˆ b 3 ‡ b 2 ˆ 1 ‡ 0 ˆ 1P 4 ˆ b 4 ‡ b 3 ˆ 1 ‡ 1 ˆ 0Thus, the four-bit sequence 1101 is encoded as 01111011.The preceding example also illustrates how dependencies can be constructed. Inactuality, there are several trade-offs in developing a forward error correctionscheme based upon convolutional coding. When a bit is only compared with apreviously transmitted bit the number of redundant bits required for decoding atthe receiver is very high. The complexity of the decoding process is, however,minimized. When the bit to be transmitted is compared with a large number ofpreviously transmitted bits the number of redundant bits required is minimized.The processing required at both ends, however, increases in complexity.Figure 4.20Simple convolutional code generation