Search for: All records

With a sufficiently large list size, the serial list Viterbi algorithm (S-LVA) provides maximum likelihood (ML) decoding of a concatenated convolutional code (CC) and an expurgating linear function (ELF), which is similar in function to a cyclic redundancy check (CRC), but doesn't enforce that the code be cyclic. However, S-LVA with a large list size requires considerable complexity. This paper exploits linearity to reduce decoding complexity for tail-biting CCs (TBCCs) concatenated with ELFs. 

Convolutional codes have been widely studied and used in many systems. As the number of memory elements increases, frame error rate (FER) improves but computational complexity increases exponentially. Recently, decoders that achieve reduced average complexity through list decoding have been demonstrated when the convolutional encoder polynomials share a common factor that can be understood as a CRC or more generally an expurgating linear function (ELF). However, classical convolutional codes avoid such common factors because they result in a catastrophic encoder. This paper provides a way to access the complexity reduction possible with list decoding even when the convolutional encoder polynomials do not share a common factor. Decomposing the original code into component encoders that fully exclude some polynomials can allow an ELF to be factored from each component. Dual list decoding of the component encoders can often find the ML codeword. Including a fallback to regular Viterbi decoding yields excellent FER performance while requiring less average complexity than always performing Viterbi on the original trellis. A best effort dual list decoder that avoids Viterbi has performance similar to the ML decoder. Component encoders that have a shared polynomial allow for even greater complexity reduction. 

Maximum-likelihood (ML) decoding of tail-biting convolutional codes (TBCCs) with S=2v states traditionally requires a separate S-state trellis for each of the S possible starting/ending states, resulting in complexity proportional to S2. Lower-complexity ML decoders for TBCCs have complexity proportional to S log S. This high complexity motivates the use of the wrap-around Viterbi algorithm, which sacrifices ML performance for complexity proportional to S.This paper presents an ML decoder for TBCCs that uses list decoding to achieve an average complexity proportional to S at operational signal-to-noise ratios where the expected list size is close to one. The new decoder uses parallel list Viterbi decoding with a progressively growing list size operating on a single S-state trellis. Decoding does not terminate until the most likely tailbiting codeword has been identified. This approach is extended to ML decoding of tail-biting convolutional codes concatenated with a cyclic redundancy check code as explored recently by Yang et al. and King et al. Constraining the maximum list size further reduces complexity but sacrifices guaranteed ML performance, increasing errors and introducing erasures. 

List Viterbi decoders are a very effective way to improve the performance of block codes in combination with an error detection outer code. In this work, we combine an efficient serial list Viterbi decoder design with an existing serially concatenated, convolutionally-encoded, pulse position modulated code (SCPPM) used in space communication, that exhibits poor performance because of an error floor. The SCPPM code features a 32-bit CRC that provides powerful error detection capability and an outer four-state convolutional code that makes it suitable for a list Viterbi decoder. The system’s code is very long, consisting of 15, 120 bits, which renders a high complexity decoder impractical, while the high error detection allows for a list decoder with very low undetected error probability. We use a very efficient list Viterbi decoder algorithm to avoid most of the redundant operations to produce low complexity serial list Viterbi decoder. The combined system reduces the error floor, moderately for the original version of the system, and completely suppresses it when the code length is increased to four times longer. 

Maximum-likelihood (ML) decoding of tail-biting convolutional codes (TBCCs) with S = 2^v states traditionally requires a separate S-state trellis for each of the S possible starting/ending states, resulting in complexity proportional to S^2. Lower-complexity ML decoders for TBCCs have complexity proportional to S log S. This high complexity motivates the use of the wrap-around Viterbi algorithm, which sacrifices ML performance for complexity proportional to S. This paper presents an ML decoder for TBCCs that uses list decoding to achieve an average complexity proportional to S at operational signal-to-noise ratios where the expected list size is close to one. The new decoder uses parallel list Viterbi decoding with a progressively growing list size operating on a single S-state trellis. Decoding does not terminate until the most likely tailbiting codeword has been identified. This approach is extended to ML decoding of tail-biting convolutional codes concatenated with a cyclic redundancy check code as explored recently by Yang et al. and King et al. Constraining the maximum list size further reduces complexity but sacrifices guaranteed ML performance, increasing errors and introducing erasures. 

An expurgating linear function (ELF) is an outer code that disallows low-weight codewords of the inner code. ELFs can be designed either to maximize the minimum distance or to minimize the codeword error rate (CER) of the expurgated code. A list-decoding sieve can efficiently identify ELFs that maximize the minimum distance of the expurgated code. For convolutional inner codes, this paper provides analytical distance spectrum union (DSU) bounds on the CER of the concatenated code. For short codeword lengths, ELFs transform a good inner code into a great concatenated code. For a constant message size of K = 64 bits or constant codeword blocklength of N = 152 bits, an ELF can reduce the gap at CER 10−6 between the DSU and the random-coding union (RCU) bounds from over 1 dB for the inner code alone to 0.23 dB for the concatenated code. The DSU bounds can also characterize puncturing that mitigates the rate overhead of the ELF while maintaining the DSU-to-RCU gap. List Viterbi decoding guided by the ELF achieves maximum likelihood (ML) decoding of the concatenated code with a sufficiently large list size. The rate-K/(K+m) ELF outer code reduces rate and list decoding increases decoder complexity. As SNR increases, the average list size converges to 1 and average complexity is similar to Viterbi decoding on the trellis of the inner code. For rare large-magnitude noise events, which occur less often than the FER of the inner code, a deep search in the list finds the ML codeword. 

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 [2]. 

Recently, rate-1/ n zero-terminated (ZT) and tail-biting (TB) convolutional codes (CCs) with cyclic redundancy check (CRC)-aided list decoding have been shown to closely approach the random-coding union (RCU) bound for short blocklengths. This paper designs CRC polynomials for rate-( n - 1)/ n ZT and TB CCs with short blocklengths. This paper considers both standard rate-( n -1)/ n CC polynomials and rate-( n - 1)/ n designs resulting from puncturing a rate-1/2 code. The CRC polynomials are chosen to maximize the minimum distance d min and minimize the number of nearest neighbors A dmin . For the standard rate-( n - 1)/ n codes, utilization of the dual trellis proposed by Yamada et al . lowers the complexity of CRC-aided serial list Viterbi decoding (SLVD). CRC-aided SLVD of the TBCCs closely approaches the RCU bound at a blocklength of 128. This paper compares the FER performance (gap to the RCU bound) and complexity of the CRC-aided standard and punctured ZTCCs and TBCCs. This paper also explores the complexity-performance trade-off for three TBCC decoders: a single-trellis approach, a multi-trellis approach, and a modified single-trellis approach with pre-processing using the wrap around Viterbi algorithm. 

This paper derives a union bound on the frame error rate (FER) of a probabilistic amplitude shaping (PAS) system which uses a CRC-aided, rate −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 rate −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 calculated 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. 

We extend earlier work on the design of convolutional code-specific CRC codes to Q -ary alphabets, with an eye toward Q -ary orthogonal signaling. Starting with distance-spectrum optimal, zero-terminated, Q -ary convolutional codes, we design Q -ary CRC codes so that the CRC/convolutional concatenation is distance-spectrum optimal. The Q -ary code symbols are mapped to a Q -ary orthogonal signal set and sent over an AWGN channel with noncoherent reception. We focus on Q=4 , rate-1/2 convolutional codes in our designs. The random coding union bound and normal approximation are used in earlier works as benchmarks for performance for distance-spectrum-optimal convolutional codes. We derive a saddlepoint approximation of the random coding union bound for the coded noncoherent signaling channel, as well as a normal approximation for this channel, and compare the performance of our codes to these limits. Our best design is within 0.6 dB of the RCU bound at a frame error rate of 10 −4 .