Viterbi Decoding of Punctured Convolutional Code

Viterbi algorithm is the most well known technique in convolutional decoding process. The operation of Viterbi algorithm can be explained easily using the trellis diagram, which is generated by the encoder with all possible inputs. As we know, the convolutional encoder consists of the memory elements, one input bit and two output bits. The output bits are decided by the suitable combinations (AND and XOR) of the past input bits. The changes of the value in the memory elements are viewed as the transition from one state to another. So we can model the encoder as a finite state machine, which is useful in the analysis of trellis diagram. An example of the finite state

machine is shown in Fig. 3.13, whereas x(n-1) and x(n-2) denote the previous input and the input prior to the previous input, respectively. When we acquire a new input bit, the state of memory elements is changed and the finite state machine generates the corresponding output bits.

Figure 3.13: State Transition Diagram Example.

The trellis diagram can be derived from the state transition diagram. First, the finite state machine output is constructed by the given input and the current state. We expand the finite state machine to a trellis diagram by introducing the concept of time. The trellis diagram is consisting of all the features of finite state machine and can be viewed as the time axis expansion of the finite state machine diagram. A simple trellis diagram is shown in Fig. 3.14 as an example. We can easily see all the state transition for any possible input for every propagation time instance. In this trellis diagram, the upper outgoing branch for each state corresponds to an input of 0, and the lower outgoing branch corresponds to an input of 1. Each state has two incoming and two outgoing branches. Each information sequence, uniquely encoded into an encoded sequence, corresponds to a unique path in the trellis. Equivalently, for a given path through the trellis, we can obtain the corresponding information sequence by reading off the input

labels on all the branches that make up the path, and the procedure is also called

“Traceback”. The Viterbi algorithm is used to find the optimal path in the trellis diagram that results in the minimum errors. Then we do the traceback procedure to retrieve the information sequence, which has been the inputs to the encoder, and the details are discussed below.

Figure 3.14: Trellis Diagram Example for a Viterbi Decoder.

The Viterbi algorithm computes the branch metric of each path at each stage of the trellis. The metric is first calculated and stored as a partial metric for each branch as the trellis traversed. Since there are two paths merge at each node, the path with a smaller metric is retained while the other is discarded. This is based on the principle that the optimum path must contain the sub-optimum survivor path just like as the one shown in Fig. 3.15 [16]. The survivor path for a given state at time instance n is the sequence of symbols closest to the received sequence up to time n. For the case of puncturing convolutional code, the metric associated with the punctured bits are simply disregarded in metric calculation stage. The overall operation discussed in the above constitutes the computational core of the Viterbi algorithm and is so-called the Add-Compare-Select (ACS) operation.

Figure 3.15: Survivor path of the Trellis Diagram.

In conclusion, the Viterbi algorithm can be divided into four major steps, the first step is the branch metric calculation and state metric loading, the second step is the ACS, the third step is the state metric storing and path recording, and the last one is the traceback. The block diagram of our Viterbi decoder program is shown in Fig. 3.16, and the structure of the Viterbi algorithm is shown in Fig. 3.17. The extend received sequence block shown in Fig. 3.16 is included for decoding the puncturing and tail-biting convolutional code and will be discussed later in this subsection.

Received

Figure 3.16: Block Diagram of the Viterbi Decoder Program.

Load Branch

Figure 3.17: Structure of the Viterbi Algorithm.

Notice that we have named our Viterbi decoder in the block diagram as an SDD Viterbi decoder, where the SDD stands for Soft-Decision-Decoding. In fact, there are two kinds of decision types used in Viterbi decoding, one is called hard-decision, and another is called soft-decision. If hard-decision is adopted, then the metric value we used for calculating branch metric and state metric is the Hamming distance, which only counts the bit errors between each trellis path and the hard-limited output of the demodulator. For the case of soft-decision, the metric we used should be the Euclidean distance between each trellis path and the soft-output of the demodulator. The major difference on performance between these two decision types is the coding gain and the computational speed. For hard-decision, the calculation of Hamming distance is a simple XOR operation, On the other hand, the soft-decision in metric calculation requires a floating-point arithmetic. The hard-decision based Viterbi decoder is much faster than the soft-decision based algorithm. However, its coding gain will lose 2 to 3 dB compared to soft-decision decoding, and cannot satisfy the requirements of IEEE 802.16a standard [17]. Hence, the soft-decision decoding is adopted to implement our Viterbi decoder.