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By inserting interleavers between intermediate stages of the polar encoder, a new class of polar codes, termed interleaved polar (i-polar) codes, is proposed. By the uniform interleaver assumption, we derive the weight enumerating function (WEF) and input-output weight enumerating function (IOWEF) averaged over the ensemble of i-polar codes. The average WEF can be used to calculate the upper bound on the average block error rate (BLER) of a code selected at random from the ensemble of i-polar codes. Also, we propose a concatenated coding scheme that employs P high rate codes as the outer code and Q i-polar codes as the inner code with an interleaver in between. The average WEF of the concatenated code is derived based on the uniform interleaver assumption. Simulation results show that BLER upper bounds can well predict BLER performance levels of the concatenated codes. The results show that the performance of the proposed concatenated code with P = Q = 2 is better than that of the CRC-aided i-polar code with P = Q = 1 of the same length and code rate at high signalto-noise ratios (SNRs). Moreover, the proposed concatenated code allows multiple decoders to operate in parallel, which can reduce the decoding latency and hence is suitable for ultra-reliable low-latency communications (URLLC).

This letter proposes an alternative expression of polar codes using polynomial representations. Polynomial representations may help to explore further algebraic properties of polar codes, same as those for convolutional codes. The relationship between message and codeword polynomials of polar codes indicates that polar codes are highly related to convolutional codes with generator polynomials 1 + D and 1/(1 + D). By using polynomial representations, we show the input and output bit shift properties of polar codes. This property is then employed to construct redundant trellises for overcomplete representations. Simulation results show that belief propagation (BP) decoding over the proposed overcomplete representation can achieve a significant performance gain as compared with BP decoding over the overcomplete representation using trellis permutations.

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In this paper, an impulsive noise estimation algorithm for generating bit log‐likelihood ratios (LLRs) for channel coded systems in impulsive noise environments is proposed. This approach is to design the LLR detector in the maximum‐likelihood (ML) sense, which requires the parameters of the impulsive noise. The expectation‐maximisation (EM) algorithm is utilised to estimate the parameters of the Bernoulli–Gaussian (B–G) impulsive noise model. The estimated parameters is then used to generate the bit LLRs for the soft‐input channel decoder. Simulation results show that over a wide range of impulsive noise power, the proposed algorithm approaches the optimal performance (with ideal estimation) even under Middleton class‐A (M‐CA) impulsive noise models.