Viterbi Decoding of Convolutional Codes

At the receiver, we have a sequence of voltage samples corresponding to the parity bits that the transmitter has sent. For simplicity, and without loss of generality, we will assume 1 sample per bit. In the previous lecture, we assumed that these voltages have been digitized to form a received bit sequence. If we decode this received bit sequence, the decoding process is termed hard decision decoding (aka “hard decoding”). If we decode the voltage samples directly before digitizing them, we term the process soft decision decoding (aka “soft decoding”). The Viterbi decoder can be used in either case. Intuitively, because hard decision decoding makes an early decision regarding whether a bit is 0 or 1, it throws away information in the digitizing process. It might make a wrong decision, especially for voltages near the threshold, introducing a greater number of bit errors in the received bit sequence. Although it still produces the most likely transmitted sequence given the received sequence, by introducing additional errors in the early digitization, the overall reduction in the probability of bit error will be smaller than with soft decision decoding. But it is conceptually a bit easier to understand hard decoding, so we will start with that, before going on to soft decision decoding. As mentioned in the previous lecture, the trellis provides a good framework for understanding decoding. Suppose we have the entire trellis in front of us for a code, and now receive a sequence of digitized bits (or voltage samples). If there are no errors (i.e., the noise is low), then there will be some path through the states of the trellis that would exactly match up with the received sequence. That path (specifically, the concatenation of the encoding of each state along the path) corresponds to the transmitted parity bits. From