- Open Access
Minimum decoding trellis length and truncation depth of wrap-around Viterbi algorithm for TBCC in mobile WiMAX
© Liu and Tsai; licensee Springer. 2011
Received: 21 June 2011
Accepted: 25 September 2011
Published: 25 September 2011
The performance of the wrap-around Viterbi decoding algorithm with finite truncation depth and fixed decoding trellis length is investigated for tail-biting convolutional codes in the mobile WiMAX standard. Upper bounds on the error probabilities induced by finite truncation depth and the uncertainty of the initial state are derived for the AWGN channel. The truncation depth and the decoding trellis length that yield negligible performance loss are obtained for all transmission rates over the Rayleigh channel using computer simulations. The results show that the circular decoding algorithm with an appropriately chosen truncation depth and a decoding trellis just a fraction longer than the original received code words can achieve almost the same performance as the optimal maximum likelihood decoding algorithm in mobile WiMAX. A rule of thumb for the values of the truncation depth and the trellis tail length is also proposed.
The IEEE 802.16 defines the wireless metropolitan area network (MAN) technology that is commonly referred to as WiMAX. The IEEE 802.16 includes two sets of standards, IEEE 802.16-2004 (802.16d)  for fixed WiMAX and IEEE 802.16-2005 (802.16e)  for mobile WiMAX. In mobile WiMAX, tail-biting convolutional codes (TBCCs)  are designated as the mandatory error-correcting codes. In the WiMAX transmitters, data bursts are divided into data blocks, and each data block is separately encoded by a TBCC encoder. The circular decoding algorithm [4–6], in which the wrap-around Viterbi algorithm traverses on the circular code trellis, has been shown to be a simple and effective decoding method for TBCCs. Its performance depends on both the truncation depth of the Viterbi algorithm  and the length of the circular decoding trellis . The larger the truncation depth or the longer the decoding trellis, the better the performance, but also more computational overhead and longer delay.
The goal of this paper is to investigate how to choose truncation depth and decoding trellis length in mobile WiMAX. The rule of thumb for truncation depth has been studied in the literature [9, 10], but never for higher order modulations on the Rayleigh channel. Several circular decoding algorithms with adaptive decoding trellis length were proposed in [11–14]. These methods do not guarantee fixed number of computations. However, for DSP/ASIC implementation, fixed decoding trellis length with fixed number of computations and delay is preferable. In this paper, we examine the performance of the circular decoding algorithm with finite truncation depth and fixed trellis length for all transmission rates in mobile WiMAX. We first derive upper bounds on the error probabilities induced by finite truncation depth and finite trellis length. We show that the circular decoding algorithm with an appropriately chosen truncation depth and a fixed-length decoding trellis just a fraction longer than the original one can achieve almost the same performance as the maximum likelihood (ML) decoding algorithm in mobile WiMAX. Moreover, the truncation depths and trellis lengths that yield losses of 0.05 dB relative to ML decoding algorithm are obtained for all transmission rates on the Rayleigh channel. Finally, we also obtain a rule of thumb for the relative values of truncation depth and trellis tail length.
2 Circular decoding algorithm
In mobile WiMAX systems, data bursts are divided into data blocks. Each data block is separately encoded by the binary (171, 133) convolutional encoder with memory m = 6. Before encoding, the convolutional encoder memory is initialized with the last 6 bits of the data block being encoded. Thus, the initial state of the code trellis is the same as the end state. After encoding, the TBCC code word is then punctured to realize the designated code rate r, where r can be one of the three possible code rates 1/2, 2/3, or 3/4. Let L denote the length of a data block, and let (d0, d1,⋯, dL-1) denote the data block. It follows that the resulting TBCC code word with length n = L/r can be viewed as one period of the periodic convolutional code word generated by periodic data bits with period L. The circular decoding algorithm (similar to the one in ) with truncation depth W and trellis tail length U discussed in this paper is described as follows:
Step 1: For each received codeword metric sequence , lengthen the sequence by copying the first U/r entries of the sequence and appending them to the end of the sequence.
Step 3: Replace the first U - W + 1 decoded data bits by the last U - W + 1 decoded bits to obtain the final data sequence of length L. Since the initial state of the TBCC encoder is unknown to the Viterbi decoder, the bit error rates (BERs) of the first few decoded data bits are much larger than those of the rest. Thus, the first few unreliable decoded bits are replaced by those decoded bits obtained in the second traverse of the circular decoding trellis.
3 Upper bounds on error probabilities
The chosen path at decoding depth k + W - 1 diverges from the correct path at state , 0 ≤ t 1 ≤ k and merges into the correct path for the first time at state , k < t 2 ≤ k + W and the decoded data .
The chosen path at decoding depth k + W - 1 diverges from the correct path at state , 0 ≤ t 1 ≤ k, never merges with the correct path, and reaches state , .
The chosen path has an initial state and merges into the correct path for the first time at state with k + m < t 2 ≤ k + W. (This is because if a path merges into correct path at state , the last m data bits must be correct.)
The chosen path has an initial state , never merges with the correct path and reaches state .
where d4 is the minimum weight of error paths in the fourth error event. This error probability is caused by both finite truncation depth and the uncertainty of the encoder's initial state. The bit error probability of the k th decoded bit is upper bounded by the sum of four upper bounds in (2), (7), (8), and (9).
The values of W*, k*, and U* for TBCCs
From Table 1 we observe that even though the three codes in mobile WiMAX do not have symmetric generator polynomials, W* -m -1 is still a good estimation for k*.
4 Simulation results
The values of , , and for all transmission rates.
E b /N0(dB)
We have investigated the error probabilities of TBCCs caused by memory truncation and the uncertainty of the initial state. From the upper bounds on the error probabilities, we found that if the same criterion is used to choose the truncation depth W and the first reliable decoded bit k, then k = W - m - 1 for symmetric convolutional codes. The truncation depth, the index of the first reliable bit, and the trellis tail length with 0.05 dB losses on the Rayleigh channel were obtained by simulation for each transmission rate in the mobile WiMAX standard. From the results, we obtain a rule of thumb for the truncation depth W and trellis tail length U . The rate-1/2 code requires a truncation depth of six to seven times the memory m, and the rate-2/3 and rate 3/4 codes require a truncation depth of ten to eleven times m. Moreover, W - m - 1 is an appropriate rule of thumb for the first reliable decoded bit k. Thus, the rule of thumb for trellis tail length is U = 2W - m - 3. The results show that the circular decoding algorithm with an appropriately chosen truncation depth and a circular trellis just a fraction longer than the original trellis can achieve almost the same performance as the optimal ML decoding algorithm in mobile WiMAX. Moreover, it is observed that high-rate TBCCs require larger truncation depths and longer trellis length than low-rate ones, and high-order modulations require larger truncation depths and longer trellis length than low-order ones.
This work was supported by the National Science Council, R.O.C., under the contract NSC 97-2218-E-027-010.
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