- Main

## An Analytical Packet Error Rate Prediction for Punctured Convolutional Codes and an Application to CRC Code Design

- Author(s): Lou, Chung-Yu
- Advisor(s): Daneshrad, Babak
- et al.

## Abstract

The performance of a packet-based communication system is determined by its packet error rate (PER) rather than its bit error rate (BER). Although the PER and the BER are correlated, it is not straightforward to calculate the PER using its BER due to the bit errors' correlation brought about by the channel coding. Therefore, the existing PER predictions are mostly heuristic methods.

The heuristic PER prediction methods, such as exponential effective SNR mapping (EESM) and mean mutual information per bit (MMIB), follow two steps: 1) compute an averaged single metric from the channel coefficients, and 2) use a simulated curve to map from the single metric to the PER. These methods require off-line simulations, curve-fitting, and parameter calibrations for every combination of modulation, code rate, and packet length. These requirements lower their applicability as modern systems support numerous modes (combinations of the above) thus needing an impractical number of off-line simulations. In this dissertation, we develop an analytical PER prediction method that requires no off-line simulations but delivers high accuracy for a wide range of transmission schemes and environments.

First, we assume that the system uses a uniform random interleaver and treat the bits' log-likelihood ratios (LLRs) as i.i.d. random variables. The LLR distribution is modeled as a mixture of folded normal distributions and approximated by a mixture of normal distributions. The PER is then calculated using a Gaussian Q-function approximation and the transfer function of the punctured convolutional code. Second, we focus on a realistic multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) system that uses a repeated-pattern interleaver. Therefore, the LLRs are dependent and their correlation is governed by their locations in the packet. In general, there are three types of correlations: a) correlations across subcarriers, b) correlations across spatial streams, and c) correlations within a subcarrier. All the three types of correlations are essential and their effect is characterized in our PER prediction. The simulation result shows that the analytical method achieves higher prediction accuracy than all the existing heuristic methods.

The analytical PER prediction can be applied to the design of cyclic redundancy check (CRC) code, which is used for error detection. We consider a system employing a CRC code concatenated with a convolutional code. The undetected error probability of this system can be calculated by two methods: the exclusion method and the construction method. The exclusion method enumerates the error patterns of the convolutional code and tests if each of them is detectable. The construction method reduces complexity significantly by exploiting the equivalence of the undetected error probability to the frame error rate of an equivalent catastrophic convolutional code. We further propose a design of the CRC code for a specific convolutional code and codeword length such that the undetected error probability is minimized. This probability can be minimized when the CRC code detects the most probable residual error patterns at the output of the Viterbi decoder. In our example, the designed CRC codes have significant reduction in undetected error probability compared to the existing CRC codes.