6), we find (E6.15) (E6.16) jeeven Channel Coding and Decoding: Part 1-Block Coding and Decoding (E6.17) jeeven (E6.18) jeeven (E6.19) 6.10-4 (E6.20) For the linear block decoder, we turn to equation (6.20), which leads us to (E6.21) (E6.22) =1-PU,10) (E6.23) (E6.24) (E6.25) = 0.000114 (E6.26) Chapter Six 6.4 Benefits and Costs of Block Coders So far, all we've talked about are the wonderful benefits of block coders. By mapping k bits to n bits (n > k), they are able to detect bit errors that occur in transmission, and, even better, they can correct such errors. But, alas, as is the case in all things engi- neering (big and small), there is always a tradeoff. We know what we gain from block codes-how about what we lose? Take a look at Figure 6.12(a). There you see three bits, mapped by a baseband modulator to a non-return-to-zero format. Each bit sent is of duration T. Now look at Figure 6.12(b). There you'll see that a block coder was introduced, and it took three bits and mapped it to six bits. To operate in real time, the channel coder had to smoosh (squeeze) all six bits in the time interval that the original three bits fit into. So, the same modulator maps the six bits into a non-return-to-zero format, where you can clearly see that each bit is of duration T/2. Baseband Channel 011101 Baseband modulator coder modulator Figure 6.12 Illustrating the cost of channel coding We're at a place where, without the channel coder, the bits sent across the chan- nel are of duration T, and with the channel coding, the bits sent across the channel are of duration T/2. It is easily shown (and I've shown it in Chapter 5), that the bandwidth (frequency range) occupied by a sent bit, BW, varies inversely with bit duration. So, before (or without) the channel coding, we have a bandwidth of 1/T, and with channel coding we have a bandwidth of 2/T. This means that channel coding comes at the cost of significantly increased bandwidth. Channel Coding and Decoding: Part 1-Block Coding and Decoding Specifically, the tradeoff is this.