VLSI architectures for turbo codes

A great interest has been gained in recent years by a new error-correcting code technique, known as “turbo coding”, which has been proven to offer performance closer to the Shannon's limit than traditional concatenated codes. In this paper, several very large scale integration (VLSI) architectures suitable for turbo decoder implementation are proposed and compared in terms of complexity and performance; the impact on the VLSI complexity of system parameters like the state number, number of iterations, and code rate are evaluated for the different solutions. The results of this architectural study have then been exploited for the design of a specific decoder, implementing a serial concatenation scheme with 2/3 and 3/4 codes; the designed circuit occupies 35 mm², supports a 2 Mb/s data rate, and for a bit error probability of 10^-6, yields a coding gain larger than 7 dB, with ten iterations     

Coded FH/SS Communications in the Presence of Combined Partial-Band Noise Jamming, Rician Nonselective Fading, and Multiuser Interference

In this paper we address the problem of combatting combined interference in spread-spectrum communication links. We consider frequency-hopped spread-spectrum systems withM-ary FSK modulation and noncoherent demodulation which employ forward-error-control coding. The interference consists of partial-band noise jamming, nonselective Rician fading, other-user interference, and thermal noise. The coding schemes which we analyze include: ReedSolomon codes (with or without diversity and error-only, erasure-only, or parallel erasure/error decoding), binary, nonbinary, and dual-kconvolutional codes with and without side information (information about the state of the channel), and concatenated schemes (Reed-Solomon outer codes with either inner detection-only block codes or inner convolutional codes). In all cases we derive 1) the minimum signal-to-jammer energy ratio required to guarantee a desirable bit error rate as a function of ρ, the fraction of the band which is jammed, when the number of interfering users is fixed; and 2) the maximum number of users that can be supported by the system as a function of ρ, when the signal-to-jammer energy ratio is fixed.     

Self-synchronizing codes derived from binary cyclic codes

The sensitivity of a binary block code to loss of synchronism (misplacement of the "commas" separating codewords) can be characterized by a pair of numbers[s, delta]such that any synchronization slip of s bits or less produces an overlap sequence differing from a legitimate codeword in at leastdeltaplaces. This definition is broader than that of comma freedom of indexdelta, which is included as the special case of s equal to the integer part of half the code block length. For codes having the slip-detecting characteristic[s, delta]there exists the possibility of implementation to restore synchronism during an interval relatively free from bit errors. It is shown that certain error-correcting binary cyclic block codes can be altered to obtain the characteristic[s, delta]by the addition of a fixed binary vector to each codeword prior to transmission. These altered cyclic codes retain the full error-correcting power of the original cyclic codes. An error-detecting/correcting data format providing protection against the acceptance of misframed data is thus obtained without the insertion of special synchronizing sequences into the bit stream.     

Power-Bandwidth Performance of Smoothed Phase Modulation Codes

Constant envelope phase varying sinusoids of the formsqrt{2E/T} cos(omega_{c}t + phi(t))are studied, in which the phase functionphi(t)follows some coded pattern in response to data. Power and bandwidth performance are studied for such patterns. The patterns depend on a phase shaping function, a modulation index (h), and a sequence ofM-ary underlying changes in phase which are chosen at random. A cutoff rate-like parameter R0is computed, which guarantees existence of codes at all ratesR < R_{0}bits/T-interval whose error performance varies as exp[-N(R_{0} - R)], whereNis the code word length inT-intervals. Plots of R0are given as a function of interval energyE, the shaping functionhandM. Extensive spectral calculations give the spectra of these phased sinusoids, and their performance is plotted in the power-bandwidth plane. The results give strong evidence that phase codes can approximate any power-bandwidth combination consistent with Shannon's Gaussian channel capacity, and that linear channels are not required for narrow-band transmission.     

A Technique for Improving the Efficiency of M-ary Signaling

The coding gain ofM-ary communication systems using orthogonal alphabets have been shown to improve by using concatenated orthogonal codes, each of lengthm = sqrt{M}. In the receiver, a two-stage MF/correlator and ML detection are utilized to provide the final decision through only2(m- 1)comparisons instead of the(M- 1)comparisons normally required. A coding gain of 8 dB forP_{e} = 10^{-5}is obtained withm = 2^{l0}only. Higher coding gain is obtained with higher order concatenation.     

Continuous Phase Modulation--Part I: Full Response Signaling

The continuous phase modulation (CPM) signaling scheme has gained interest in recent years because of its attractive spectral properties. Data symbol pulse shaping has previously been studied with regard to spectra, for binary data and modulation index 0.5. In this paper these results have been extended to theM-ary case, where the pulse shaping is over a one symbol interval, the so-called full response systems. Results are given for modulation indexes of practical interest, concerning both performance and spectrum. Comparisons are made with minimum shift keying (MSK) and systems have been found which are significantly better inE_{b}/N_{0}for a large signal-to-noise ratio (SNR) without expanded bandwidth. Schemes with the same bit error probability as MSK but with considerably smaller bandwidth have also been found. Significant improvement in both power and bandwidth are obtained by increasing the number of levelsMfrom 2 to 4.     

An error-correcting code for data broadcasting and its error-correction capability

By using the vertical blanking period of television signals, it is possible to transmit coded data such as teletext, newspaper, telesoftware, music score, etc. However, the quality of ordinary TV channels is very poor for digital communications and a powerful error-correcting code is required to reliably transmit coded data. From the results of simulations using error pattern data collected in field tests and the comparison of various error-correcting codes under many conditions, it has been determined that the shortened (272, 190) majority-logic decodable difference-set cyclic code is a suitable code for NTSC TV signals. Moreover, the decoding algorithm for this code has been improved in order to increase the error-correction capability and a new LSI with this function has been developed. By using error-correcting codes proposed to date for teletext, it has been difficult to obtain a page error rate (PER) of 10⁻¹in many measurement points. However, PERs of less than 10⁻² can be obtained in this system, even when random noise, ghost interference, or waveform distortion are present and bit error rates (BERs) are 10⁻². This paper also gives PERs according to the error-data collected in field tests and shows that the error-correction capability increased equivalently up to 11 error-bits in a 272-bit by improving the decoding algorithm and the results of indoor and outdoor tests.     

Zigzag codes and concatenated zigzag codes

This paper introduces a family of error-correcting codes called zigzag codes. A zigzag code is described by a highly structured zigzag graph. Due to the structural properties of the graph, very low-complexity soft-in/soft-out decoding rules can be implemented. We present a decoding rule, based on the Max-Log-APP (MLA) formulation, which requires a total of only 20 addition-equivalent operations per information bit, per iteration. Simulation of a rate-1/2 concatenated zigzag code with four constituent encoders with interleaver length 65 536, yields a bit error rate (BER) of 10^-5 at 0.9 dB and 1.3 dB away from the Shannon limit by optimal (APP) and low-cost suboptimal (MLA) decoders, respectively. A union bound analysis of the bit error probability of the zigzag code is presented. It is shown that the union bounds for these codes can be generated very efficiently. It is also illustrated that, for a fixed interleaver size, the concatenated code has increased code potential as the number of constituent encoders increases. Finally, the analysis shows that zigzag codes with four or more constituent encoders have lower error floors than comparable turbo codes with two constituent encoders     

非同步扩频系统中空时分组码的阵列处理

针对非同步DS-CDMA系统提出了一种基于空时分组码的阵列处理干扰对消的技术。通过对发射信号进行空时分组编码,可以使发射信号在瑞利衰落信道中传输时信息不会畸变,并取得最大分集增益和码增益。利用空时分组码的性质有效抑制干扰的同时,大大简化了空时分组码的解码。仿真结果表明,这里提出的方法与传统的抑制干扰的方法相比具有更低的误比特率,并且有更快的解码速率。     

Error correcting codes for fast frequency-hopped MFSK spread-spectrum satellite communications under worst case jamming

Spread-spectrum techniques such as frequency hopping (FH) have been used to reduce the vulnerability of satellite communications to jamming. In such a system, error correcting (EC) codes play an essential role. Thus a knowledge of the performance of various EC codes is necessary in order to choose an EC code and related system parameters in the design of an anti-jam system, this paper examines the performance of various types of error correcting codes under worst case partial band noise jamming and worst case multitone jamming using fast frequency-hopped, non-coherent M-ary frequency-shift keying (NCMFSK) with optimum diversity. A comprehensive study including convolutional codes, binary and non-binary block codes and concatenated codes has been conducted. This paper presents bit error rate (BER) performances of various error correcting codes. New candidate codes with superior anti-jam performance are identified.     

On the complexity of decoding Reed-Solomon codes (Corresp.)

Certainq-ary Reed-Solomon codes can be decoded by an algorithm requiring onlyO(q log^{2} q)additions and multiplications inGF(q).     

Performance of a powerful error-correcting and -detecting codingscheme for the North American Basic Teletext System (NABTS) for randomindependent errors: methods, equations, calculations and results

For a powerful layered, upward- and downward-compatible error-correcting and error-detecting scheme for NABTS, various bit error rate (BER) related performance measures are derived and calculated for random independent errors. The methods, equations, calculations and results are given for the least powerful one-byte suffix codes, for the two-byte suffix code, called code C, and for the double and single bundle codes formed by using code C for each data block (i.e. horizontally) and also vertically, thus forming a product code, for a specified, but variable, number of data blocks. Performance bounds and equations for probabilities of correct decoding of error and of decoding failure are given. The weight enumerators for a number of one-byte suffix codes are calculated, and those of weight four are classified into types depending on the number of ones occurring in a byte, and in other arrangements. Performance analyses and comparisons with a code for Japanese teletext are included. Analyses used in computer simulation studies are described     

The Design of a High-Performance Error-Correcting Coding Scheme for the Canadian Broadcast Telidon System Based on Reed-Solomon Codes

Error correction can greatly improve the performance and extend the range of broadcast teletext systems. In this paper, the requirements for an error-correcting scheme for broadcast teletext in North America (NABTS) are set down. An error-correction scheme which meets all these requirements is then described. The simplest case employs the one parity bit in each 8 bit byte and no suffix of parity check bits at the end of each data block. The next level also uses a single byte of parity check bits at the end of each data block. Adding a second byte of parity checks at the end of each data block results in a Reed-Solomon code, called theCcode, for each data block. Adding one data block of parity checks afterh - 1data blocks results in a set ofhdata packets being encoded into a bundle, in which verticalCcodes provide powerful interleaving. In a final alternative, two data blocks hold the check bytes for the vertical codewords, and the most powerful coding scheme, the double bundle code, results. The detailed mathematical definitions of the various codes are referred to or described, formulas for performance calculations are referred to, and performance curves are presented for the AWGN channel as well as for measured field data. These performance curves are discussed and compared to the performance of a difference set cyclic code, originally designed for the Japanese teletext system, which corrects any 8 bits in error in a packet.     

LDPC-COFDM系统在多径衰落信道下的性能分析

将正交频分复用(OFDM)技术应用于多径衰落信道下的高速数据传输是一个极富吸引力的方案,包括卷积码、RS码、Turbo码在内的多种纠错编码都曾被应用在OFDM系统中.最近,一种新的编码方案--低密度奇偶校验码(LDPC)引起了人们的注意.LDPC具有低的解码复杂度和逼近香农限的性能.文中仿真分析了LDPC-COFDM(编码正交频分复用)系统的性能,并与Turbo码系统进行了对比,结果表明该系统在多径衰落信道下显示出更为优越的性能.     

MDPSK-OFDM with highly power-efficient block codes forfrequency-selective fading channels

This paper investigates the performance of M-ary differential phase shift keying orthogonal frequency-division multiplexing (MDPSK-OFDM) systems employing peak power controlling block codes over frequency-selective Rayleigh fading channels. The block codes have a capability of both error correction and reduction of the peak-to-average power ratio (PAPR). To decode the block codes with reasonable complexity, the extended version of the ordered statistic decoding of Fossorier and Lin (see IEEE Trans. Inform. Theory, vol.41, p.1379-96, 1995) is utilized. The bit error rate performance of the block codes is evaluated over typical indoor radio channels by computer simulation and compared with that of the equivalent frequency diversity of the repetition codes. The significant coding gain and improvement of the irreducible error floor are observed under the constraint of the PAPR from 3 to 6 dB     

Asymptotics and Error Rate Bounds for M-ary DPSK

The symbol error probabilityP_{E}(M)forM-ary DPSK is shown to be bounded in terms of a recent asymptotic approximationP_{asym}(M)by the inequalitiesP_{asym}(M) < P_{E}(M) < 1.03P_{asym}(M);M geq 4, E_{b}/N_{0} geq 1whereE_{b}/N_{0}is the bit energy-to-noise spectral density ratio. Aside from the wide range of validity and the closeness of the lower and upper bounds, this result is striking in light of the often held view that such asymptotic approximations are primarily of value only in the limitE_{b}/N_{0} rightarrow infty; thus, one of the goals of this note is to demonstrate that asymptotic methods can lead to extremely good error rate approximations in lieu of the more traditional and more widely used bounding techniques. The results are also noted to be applicable in other similar situations which commonly occur.     

Low-density parity check codes over GF(q)

Gallager's (1962) low-density binary parity check codes have been shown to have near-Shannon limit performance when decoded using a probabilistic decoding algorithm. We report the empirical results of error-correction using the analogous codes over GF(q) for q>2, with binary symmetric channels and binary Gaussian channels. We find a significant improvement over the performance of the binary codes, including a rate 1/4 code with bit error probability <10^-5 at E_b/N₀=0.2 dB     

Self-Orthogonal Convolutional Coding for the DPCM-AQB Speech Encoder

A fixed-tap differential pulse code modulation (DPCM) system with a robust backward-adaptive Jayant quantizer is investigated for speech encoding at 16-40 kbits/s using binary phase shift keying over an additive white Gaussian noise channel. The performance of this system becomes unacceptable as the channel bit error rate(P_{b})approaches 10^-2. Using high-rate, long constraint length, self-orthogonal convolutional codes, the DPCM system performance is much-improved for10^{-4} < P_{b} < 10^{-2}depending on the transmitted data rate. The use of high-rate(n - 1)/n, n = 2,3,4,, and 5 codes minimizes the number of bits allocated to channel coding, and decoding complexity is reduced by employing self-orthogonal codes which admit threshold decoding. Subjectively, while there is additional quantization noise with channel coding, the irritating popping and squeaking sounds due to channel errors are eliminated.     

Some families of zero- error block codes for the two-user binary adder channel with feedback

Families of zero-error codes for the real binary adder channel with feedback that achieve high rate pairs are introduced. Two families of zero-error block codes are given for the case in which only one of the two senders receives feedback about the channel output. In the first of these families, the uninformed sender transmits at a rate of nearly one bit per symbol and the informed sender transmits slightly less that1/2bit per symbol. The second family is designed for the case in which the informed sender sends at or near one bit per symbol and the uninformed one sends nearly1/2bit per symbol. A family of zero-error codes is introduced, based on the Fibonacci recursion; these codes are readily implemented by means of a simple square-dividing strategy. The Fibonacci codes achieveR_{1}=R_{2}=log_{2} [(1 + sqrt{5})/2]in the limit of large block length. Time-sharing between members of these three code families is used to obtain an achievable rate region, or inner bound, to the zero-error capacity region for block coding. For the case in which the feedback is available to both senders, a variant of the Fibonacci difference equation is used to generate zero-error block codes with slightly higher asymptotic rateR_{1}=R_{2}=0.717.     

基于短帧交织的Turbo码编码器研究

本文分析了Turbo码编码器及其输出码字的组成原理,基于使输出码字最小汉明重量最大化和使最小汉明重量输出码字出现的概率最小化的原则,结合结构化交织器和随机交织器的优点,提出了一种伪随机的结构化反块交织器,并进一步提出了采用双伪随机反块交织器的Turbo码编码器方案.该方案有效的增大了Turbo码输出码字的最小汉明距离,同时避免了Turbo码边缘效应且具有伪随机性,因此,可有效的提高系统的纠错性能.仿真实验结果显示,该方案在短帧传输的条件下有着最佳的综合性能.特别是在信噪比高时,采用该方案与采用其它交织器的Turbo码方案相比,在提高系统的可靠性上具有明显的优势.