Optimal Puncturing Ratios and Energy Distribution for Multiple Parallel Concatenated Codes

Extrinsic information transfer (EXIT) charts have been used extensively for designing concatenated coding schemes with iterative decoding. The area between the two transfer curves has been shown to approximate the gap to channel capacity. A curve-fitting procedure on EXIT charts was previously suggested for designing low density parity check codes. In this paper, we develop a similar approach for shaping the EXIT charts of multiple parallel concatenated codes (MPCCs) with two or more constituent codes. Random puncturing and unequal energy distributions across parallel coding streams provide additional degrees of freedom for manipulating the EXIT functions of the constituent codes. A search over all rate-one convolutional codes of memory length four or less is performed, identifying all codes with unique EXIT functions. Another search for good combinations of constituent codes is subsequently conducted. Optimal constituent codes, puncturing ratios, and energy distributions are found in terms of minimizing the average signal-to-noise ratio threshold required for convergence, leading to simple MPCCs over a wide range of code rates. The best rate-1/2 code found has a 0.15 dB gain over the original turbo code with only half the decoding complexity. Another example shows a 0.5 dB gain obtained just by optimizing the energy distribution.