This paper applies probabilistic amplitude shaping (PAS) to cyclic redundancy check (CRC)-aided tail-biting trellis-coded modulation (TCM). CRC-TCM-PAS produces practical codes for short block lengths on the additive white Gaussian noise (AWGN) channel. In the transmitter, equally likely message bits are encoded by a distribution matcher (DM) generating amplitude symbols with a desired distribution. A CRC is appended to the sequence of amplitude symbols, and this sequence is then encoded and modulated by TCM to produce real-valued channel input signals. This paper proves that the sign values produced by the TCM are asymptotically equally likely to be positive or negative. The CRC-TCM-PAS scheme can thus generate channel input symbols with a symmetric capacity-approaching probability mass function. The paper provides an analytical upper bound on the frame error rate of the CRC-TCM-PAS system over the AWGN channel. This FER upper bound is the objective function used for jointly optimizing the CRC and convolutional code. Additionally, this paper proposes a multi-composition DM, which is a collection of multiple constant-composition DMs. The optimized CRC-TCM-PAS systems achieve frame error rates below the random coding union (RCU) bound in AWGN and outperform the short-blocklength PAS systems with various other forward error correction codes studied in Coşkun et al. (2019).

This paper derives a union bound on the frame error rate (FER) of a probabilistic amplitude shaping (PAS) system which uses a CRC-aided,$\text{ rate }-\frac{k}{k+1}$, systematic, recursive trellis-coded modulation (TCM). A tail-biting convolutional code (TBCC) provides the feed-forward error correction (FEC) code for the TCM. The system is referred as CRC-TCM-PAS [1]. In order to derive the union bound, we first prove that the concatenation of a CRC and a$\text{ rate }-\frac{k}{k+1}$convolutional code is equivalent to a new convolutional code. Then, we give the generating function of the new convolutional code using Biglieri's product-state-diagram approach. A union bound can be cal-culated using the generating function. Simulation results show that the derived union bound is tight in the high signal-to-noise ratio (SNR) regime and can be used to design the convolutional and CRC codes. Simulation results also show that the optimized CRC-TCM-PAS system exceeds the random coding union (RCU) bound and outperforms the PAS systems with various FEC codes studied in [2] for the same number of input bits and the same transmission rate.

This paper applies probabilistic amplitude shaping (PAS) to a cyclic redundancy check (CRC) aided trellis coded modulation (TCM) to achieve the short-blocklength random coding union (RCU) bound. In the transmitter, the equally likely message bits are first encoded by a distribution matcher to generate amplitude symbols with the desired distribution. The binary representations of the distribution matcher outputs are then encoded by a CRC code. Finally, the CRC-encoded bits are encoded and modulated by Ungerboeck’s TCM scheme, which consists of a systematic $\frac{{{k_0}}}{{{k_0} + 1}}$ tail-biting convolutional code and a mapping function that maps coded bits to channel signals with capacity-achieving distribution. This paper proves that, for the proposed transmitter, the CRC bits have uniform distribution and that the channel inputs have symmetric distribution. In the receiver, the serial list Viterbi decoding (S-LVD) is used to estimate the information bits. Simulation results show that, for the proposed CRC-TCM-PAS system with 87 input bits and 65-67 8-AM coded output symbols, the decoding performance under additive white Gaussian noise channel achieves the RCU bound with properly designed CRC and convolutional codes.

This paper presents a method for computing a finite-blocklength converse for the rate of fixed-length codes with feedback used on discrete memoryless channels (DMCs). The new converse is expressed in terms of a stochastic control problem whose solution can be efficiently computed using dynamic programming and Fourier methods. For channels such as the binary symmetric channel (BSC) and binary erasure channel (BEC), the accuracy of the proposed converse is similar to that of existing special-purpose converse bounds, but the new converse technique can be applied to arbitrary DMCs. We provide example applications of the new converse technique to the binary asymmetric channel (BAC) and the quantized amplitude-constrained AWGN channel.

This paper compares the accuracy and complexity of Raghavan and Baum’s Reliability Output Viterbi Algorithm (ROVA), Polyanskiy’s accumulated information density (AID), and Fricke and Hoeher’s lower complexity approximation of ROVA. It turns out that AID is far less accurate than ROVA in practice. This paper proposes codeword information density (CID), which modifies AID to improve its accuracy and leads to a lower-complexity implementation of ROVA. The paper includes an analytical expression for the random variable describing the correct decoding probability computed by ROVA and uses this expression to characterize how the probabilities of correct decoding, undetected error, and negative acknowledgement behave as a function of the selected threshold for reliable decoding. This paper examines both the complexity and the simulation time of ROVA, CID, AID, and the Fricke and Hoeher approximation to ROVA. This paper also derives an expression for the union bound on the frame error rate for zero-terminated trellis codes with punctured symbols and uses it to optimize the order of symbol transmission in an incremental retransmission scheme. This paper concludes by comparing the performance of an incremental retransmission scheme using ROVA as a stopping condition to one that uses a CRC as a stopping condition.

Previously, dynamic-assignment Blahut-Arimoto (DAB) was used to find capacity-achieving probability mass functions (PMFs) for binomial channels and molecular channels. As it turns out, DAB can efficiently identify capacity-achieving PMFs for a wide variety of channels. This paper applies DAB to power-constrained (PC) additive white Gaussian Noise (AWGN) Channels and amplitude-constrained (AC) AWGN Channels.This paper modifies DAB to include a power constraint and finds low-cardinality PMFs that approach capacity on PC-AWGN Channels. While a continuous Gaussian PDF is well-known to be capacity-achieving on the PC-AWGN channel, DAB identifies low-cardinality PMFs within 0.01 bits of the mutual information provided by a Gaussian PDF. Recall the results of Ozarow and Wyner requiring a constellation cardinality of ⌈2C+1⌉ to approach capacity C to within the asymptotic shaping loss of 1.53 dB at high SNR. PMF’s found by DAB approach capacity with essentially no shaping loss with cardinality less than 2C+1.2. As expected, DAB’s numerical approach identifies PMFs with better mutual information vs. SNR performance than the analytical approaches to finite-support constellations examined by Wu and Verdu.This paper also uses DAB to find capacity-achieving PMFs with small cardinality support sets for AC-AWGN Channels. The resulting evolution of capacity-achieving PMFs as a function of SNR is consistent with the approximate cardinality transition points of Sharma and Shamai.