Repeated convolutional codes for high-error-rate channels

An error-correction scheme for an M-ary symmetric channel (MSC) characterized by a large error probability p_e is considered. The value of p_e can be near, but smaller than, 1-1/M, for which the channel capacity is zero, such as may occur in a jamming environment. The coding scheme consists of an outer convolutional code and an inner repetition code of length m that is used for each convolutional code symbol. At the receiving end, the m inner code symbols are used to form a soft-decision metric, which is passed to a soft-decision decoder for the convolutional code. The effect of finite quantization and methods to generate binary metrics for M>2 are investigated. Monte Carlo simulation results are presented. For the binary symmetric channel (BSC), it is shown that the overall code rate is larger than 0.6R₀, where R₀ is the cutoff rate of the channel. New union bounds on the bit error probability for systems with a binary convolutional code on 4-ary and 8-ary orthogonal channels are presented. For a BSC and a large m, a method is presented for BER approximation based on the central limit theorem     

A fast algorithm for computing distance spectrum of convolutionalcodes

A fast algorithm for searching a tree (FAST) is presented for computing the distance spectrum of convolutional codes. The distance profile of a code is used to limit substantially the error patterns that have to be searched. The algorithm can easily be modified to determine the number of nonzero information bits of an incorrect path as well as the length of an error event. For testing systematic codes, a faster version of the algorithm is given. FAST is much faster than the standard bidirectional search. On a microVAX, d_∞=27 was verified for a rate R=1/2, memory M=25 code in 37 s of CPU time. Extensive tables of rate R=1/2 encoders are given. Several of the listed encoders have distance spectra superior to those of any previously known codes of the same rate and memory. A conjecture than an R=1/2 systematic convolutional code of memory 2M will perform as well as a nonsystematic convolutional code of memory M is given strong support     

Asymptotic performance of M-ary orthogonal modulation ingeneralized fading channels

The asymptotic (M→∞) probability of symbol error P_e,_m for M-ary orthogonal modulation in a Nakagami-m fading channel is given by the incomplete gamma function P(m, mx) where x=In 2/(E_b/N₀) and E_b is the average energy per bit. For large signal-to-noise ratio this leads to a channel where the probability of symbol error varies as the inverse mth power of E_b/N₀. These channels exist for all m⩾1/2. The special case of m=1 corresponds to Rayleigh fading, an inverse linear channel     

High-rate punctured convolutional codes for Viterbi and sequentialdecoding

An investigation is conducted of the high-rate punctured convolutional codes suitable for Viterbi and sequential decoding. Results on known short-memory codes M⩽8 discovered by others are extended. Weight spectra and upper bounds on the bit error probability of the best known punctured codes having memory 2⩽M ⩽8, and coding rates 2/3⩽R⩽7/8 are provided. Newly discovered rate-2/3 and -3/4 long-memory punctured convolutional codes with 9⩽M⩽23 are provided together with the leading terms of their weight spectra and their bit error performance bounds. Some results of simulation with sequential decoding are given     

More on the decoder error probability for Reed-Solomon codes

A combinatorial technique similar to the principle of inclusion and exclusion is used to obtain an exact formula for P_E (u), the decoder error probability for Reed-Solomon codes. The P_E(u) for the (255, 223) Reed-Solomon code used by NASA and for the (31, 15) Reed-Solomon code (JTIDS code) are calculated using the exact formula and are observed to approach the Qs of the codes rapidly as u gets large. An upper bound for the expression |P_E(u)/ Q-1| is derived and shown to decrease nearly exponentially as u increases     

Bounds for abnormal binary codes with covering radius one

The normality of binary codes is studied. The minimum cardinality of a binary code of length n with covering radius R is denoted by K(n,R). It is assumed that C is an (n,M)R code, that is, a binary code of length n with M codewords and covering radius R. It is shown that if C is an (n,M)1 code, then it is easy to find a normal (n ,M)1 code by changing C in a suitable way, and that all the optimal (n,M)1 codes (i.e. those for which M=K(n,1)) are normal and their every coordinate is acceptable. It is shown that if C is an abnormal (n,M) code, then n⩾9, and an abnormal (9118)1 code which is the smallest abnormal code known at present, is constructed. Lower bounds on the minimum cardinality of a binary abnormal code of length n with covering radius 1 are derived, and it is shown that if an (n,M)1 code is abnormal, then M⩾96     

Hybrid ARQ schemes for point-to-multipoint communication overnonstationary broadcast channels

Hybrid automatic-repeat-request (ARQ) error control schemes make use of both error detection and error correction in order to achieve high throughput and low undetected error probabilities on two way channels. Two hybrid ARQ schemes, termed hybrid go-back-N (HGB- N) and hybrid selective-repeat (HSR), are proposed for point-to-multipoint communications over broadcast channels. Both schemes incorporate a concatenated code for error correction and error detection. The performance study of the hybrid schemes is based on a two-state Markov model of a burst noise channel. An analytic solution is derived for the throughput efficiency of the HSR scheme, while approximations and computer simulation are used to evaluate the throughput efficiency of the HGB-N scheme. It is shown that the schemes perform considerably better than the corresponding pure ARQ schemes in which a block code is used for error detection only, especially in environments with a large number of receivers and large channel roundtrip delays, such as satellite broadcast links     

On linear structure and phase rotation invariant properties ofblock M-PSK modulation codes

Two important structural properties of block M(=2^' )-ary PSK modulation codes, linear structure and phase symmetry, are investigated. An M-ary modulation code is first represented as a code with symbols from the integer group S_M-PSK=(0,1,2,---,M-1) under modulo-M addition. Then the linear structure of block M-PSK modulation codes over S_M-PSK with respect to modulo- M vector addition is defined, and conditions are derived under which a block M-PSK modulation code is linear. Once the linear structure is developed, the phase symmetry of block M-PSK modulation codes is studied. In particular, a necessary and sufficient condition for a block M-PSK modulation code that is linear as a binary code to be invariant under 2^h/180°M phase rotation, for 1⩽h⩽l is derived. Finally, a list of short 8-PSK and 16-PSK modulation codes is given, together with their linear structure and the smallest phase rotation for which a code is invariant     

A selective ARQ protocol with a finite-length buffer

Of the automatic-repeat-request (ARQ) techniques commonly used in communication systems, selective protocols, while the most efficient, have the notable drawback of requiring large buffers at the receiver side. A selective ARQ protocol with a finite-length buffer is described. If N is the number of codewords transmittable in the round-trip delay, the protocol requires a buffer length N+N_a , N_a⩾2 being an integer. A lower bound on the throughput of the protocol is derived. It achieves higher throughputs than similar schemes giving results comparable to those for selective protocols with infinite-length buffer for high error rates in the communication channel     

Modifying real convolutional codes for protecting digital filteringsystems

A method is presented for protecting the overall realization of digital filters implemented with very dense high-speed electronic devices against both hard and soft errors at the data sample level using the error-detecting properties of real convolutional codes. The normal filter system is surrounded with parallel parity channels defined by a real systematic rate k/n convolutional code. Erroneous behavior is detected by comparing externally the calculated and regenerated parity samples. Significant complexity reductions are possible by modifying the code structure, without loss of error protection, yielding simplified parity channels with finite impulse response (FIR) structures with computational rates decimated by k . The code modification procedure is described. The code modification process has been automated in a computer algorithm. The effects of parity filter quantizations are analyzed and a bound on the mean-square error in the parity comparison is given     

Bidirectional breadth-first algorithms for the decoding ofconvolutional codes

Bidirectional multiple-path tree searching algorithms for the decoding of convolutional codes are presented. These suboptimal coding algorithms use a multiple-path breadth-first bidirectional tree exploration procedure and long-memory convolution codes. It is shown that, compared to the usual M-algorithm, the bidirectional exploration considerably reduces the bit error propagation due to correct path loss. Computer simulations using rate-1/2 codes over binary symmetric channels are used to analyze the effect of the number of path extensions, code memory, and frame length on the bit error probability. The results show that with a bit error probability of 10^-5, coding gains on the order of 2 dB over the M-algorithm and 1 dB over a Viterbi decoder of equivalent complexity can be achieved     

Asymptotic performance of M-ary orthogonal signals inworst case partial-band interference and Rayleigh fading

It is shown that for worst-case partial-band jamming, the error probability performance (for fixed E_b/N_I) becomes worse with increasing M for (M>16). The asymptotic probability-of-error is not zero for any E_b/N _I(>ln 2), but decreases inverse linearly with respect to it. In the fading case, the error-probability performance (for fixed E_b/N₀) improves with M for noncoherent detection, but worsens with M for coherent detection. For large E_b/N₀ the performance of the Rayleigh fading channel asymptotically approaches the same limit as the worst case partial-band jammed channel. However, for values of M at least up to 4096, the partial-band jammed channel does better. While it is unlikely that an M-ary orthogonal signal set with M>1024 will be used in a practical situation, these results suggest an important theoretical problem; namely, what signal set achieves reliable communication     

Asymptotically optimal classification for multiple tests withempirically observed statistics

The decision problem of testing M hypotheses when the source is Kth-order Markov and there are M (or fewer) training sequences of length N and a single test sequence of length n is considered. K, M, n, N are all given. It is shown what the requirements are on M , n, N to achieve vanishing (exponential) error probabilities and how to determine or bound the exponent. A likelihood ratio test that is allowed to produce a no-match decision is shown to provide asymptotically optimal error probabilities and minimum no-match decisions. As an important serial case, the binary hypotheses problem without rejection is discussed. It is shown that, for this configuration, only one training sequence is needed to achieve an asymptotically optimal test     

Computing the probability of undetected error for shortened cycliccodes

The authors present a general technique for computing P _e for all possible shortened versions of cyclic codes generated by any given polynomial. The technique is recursive, i.e. computes P_e for a given code block length n from that of the code block length n-1. The proposed computation technique for determining P_e does not require knowledge of the code weight distributions. For a generator polynomial of degree r, and |g| nonzero coefficients, the technique yields P_e for all code block lengths up to length n in time complexity O(n|g |2^r+|g|). Channels with variable bit error probabilities can be analyzed with the same complexity. This enables the performance of the code generator polynomials to be analyzed for burst errors     

Performance analysis of RS-coded M-ary FSK forfrequency-hopping spread spectrum mobile radios

Decoding performance of Reed-Solomon (RS) coded M-ary FSK with noncoherent detection in a frequency-hopping spread spectrum mobile radio channel is theoretically analyzed. Exact formulas and an approximate one for evaluating word error rates (WERs) of error correction and error-and-erasure correction schemes on decoding the RS codes are derived. It is shown that with K symbol erasure and C symbol error detection, RS coded M-ary FSK achieves the equivalent diversity order of (K+1)(C+1)     

Error performance of a digitally processed binary ESK frequencyhopping receiver in the presence of large Doppler

Analysis is made of the effects of Doppler on the error rate performance of a low data rate binary FSK frequency hopping receiver, employing a discrete Fourier transform (DFT) technique for baseband detection. Bit detection decision is made by locating the maximum of the DFT outputs which, in the frequency domain, are assumed to be separated by 1/T where T is the bit period. Both the worst case and average error performances are obtained and presented as a function of E_b/N₀ for various values of M where E_b/N₀ is the signal bit energy-to-noise density ratio and M is the degree of freedom associated with the Doppler uncertainty window. The E _b/N₀ degradation as a function of M is also presented     

On the Hamming distance properties of group codes

Under certain mild conditions, the minimum Hamming distance D of an (N, K, D) group code C over a non-abelian group G is bounded by D⩽N -2K+2 if K⩽N/2, and is equal to 1 if K>N/2. Consequently, there exists no (N, K, N-K+1) group code C over an non-abelian group G if 1<K<N. Moreover, any normal code C with a non-abelian output space has minimum Hamming distance equal to D=1. These results follow from the fact that non-abelian groups have nontrivial commutator subgroups. Finally, if C is an (N, K, D) group code over an abelian group G that is not elementary abelian, then there exists an (N, K, D) group code over a smaller elementary abelian group G'. Thus, a group code over a general group G cannot have better parameters than a conventional linear code over a field of the same size as G     

Error-correcting codes and nonlinear diversity combining againstworst case partial-band noise jamming of frequency-hopping MFSK systems

Error probability analyses are performed for a coded M-ary frequency-shift keying system (MFSK) using L hops per M-ary word frequency-hopping spread-spectrum waveforms transmitted over a partial-band Gaussian noise jamming channel. The bit error probabilities are obtained for a square-law adaptive gain control receiver with forward-error-control coding under conditions of worst-case partial-band noise jamming. Both thermal noise and jamming noise are included in the analyses. Performance curves are obtained for both block codes and convolutional codes with both binary and M-ary channel modulations. The results show that thermal noise cannot be neglected in the analysis if correct determinations of the optimum order of diversity and the worst-case jamming fraction are to be obtained. It is shown that the combination of nonlinear combining, M -ary modulation, and forward-error-control coding is effective against worst-case partial-band noise jamming     

Limited search trellis decoding of convolutional codes

The least storage and node computation required by a breadth-first tree or trellis decoder that corrects t errors over the binary symmetric channels is calculated. Breadth-first decoders work with code paths of the same length, without backtracking. The Viterbi algorithm is an exhaustive trellis decoder of this type; other schemes look at a subset of the tree or trellis paths. For random tree codes, theorems about the asymptotic number of paths required and their depth are proved. For concrete convolutional codes, the worst case storage for t error sequences is measured. In both cases the optimal decoder storage has the same simple dependence on t. The M algorithm and algorithms proposed by G.J. Foschini (ibid., vol.IT-23, p.605-9, Sept. 1977) and by S.J. Simmons (PhD. diss., Queens Univ., Kingston, Ont., Canada) are optimal, or nearly so; they are all far more efficient than the Viterbi algorithm