Recursive Systematic Convolutional Research Articles

Improved Frey Chaotic Digital Encoder for Trellis-Coded Modulation

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this brief, nonlinear digital filters with finite precision are analyzed as recursive systematic convolutional (RSC) encoders. An infinite-impulse-response (IIR) digital filter with finite precision (wordlength of <formula formulatype="inline"><tex Notation="TeX">$N$</tex></formula> bits) is a rate-1 RSC encoder over a Galois field <formula formulatype="inline"><tex Notation="TeX">$GF(2^{N})$</tex></formula>. The Frey chaotic filter is analyzed for different wordlengths <formula formulatype="inline"><tex Notation="TeX">$N$</tex></formula>, and it is demonstrated that the trellis performances can be enhanced by proper filter design. Therefore, a modified definition for the encoding rate is provided, and a trellis design method is proposed for the Frey filter, which consists of reducing the encoding rate from 1 to 1/2. This trellis optimization partially follows Ungerboeck's rules, i.e., increasing the performances of the encoded chaotic transmission in the presence of noise. In fact, it is demonstrated that for the same spectral efficiency, the modified Frey encoder outperforms the original Frey encoder only for <formula formulatype="inline"> <tex Notation="TeX">$N = 2$</tex></formula>. To show the potential of these nonlinear encoders, it is demonstrated that a particular nonlinear digital filter over <formula formulatype="inline"><tex Notation="TeX">$GF(4)$</tex></formula> is equivalent to a <formula formulatype="inline"><tex Notation="TeX">$GF(2)$</tex></formula> conventional optimum RSC encoder. The symbol error rate (SER) is estimated for all the proposed schemes, and the results show the expected coding gains as compared to their equivalent nonencoded and linear versions. </para>

Iterative AMR-WB Source and Channel Decoding Using Differential Space–Time Spreading-Assisted Sphere-Packing Modulation

In this paper, we present a novel system that invokes jointly optimized iterative source and channel decoding for enhancing the error resilience of the adaptive multirate wideband (AMR-WB) speech codec. The resultant AMR-WB-coded speech signal is protected by a recursive systematic convolutional (RSC) code and transmitted using a noncoherently detected multiple-input-multiple-output (MIMO) differential space-time spreading (DSTS) scheme. To further enhance the attainable system performance and to maximize the coding advantage of the proposed transmission scheme, the system is also combined with multidimensional sphere-packing (SP) modulation. Furthermore, the convergence behavior of the proposed scheme is evaluated with the aid of extrinsic information transfer (EXIT) charts. The proposed system exhibits an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Eb</i> / <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> gain of about 1 dB, as compared with the benchmark scheme carrying out joint channel decoding and DSTS-aided SP demodulation in conjunction with separate AMR-WB decoding, when using only <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">system</sub> = 2 system iterations and when communicating over narrow-band correlated Rayleigh fading channels.

Turbo codes with symmetric and asymmetric component codes defined over finite fields of integers

Binary asymmetric turbo codes and non-binary turbo codes have been proposed to improve the bit error rate (BER) performance of parallel concatenated coding schemes. Both strategies have certain advantages that can be exploited when they are put together. This paper investigates turbo codes based on two component recursive systematic convolutional (RSC) codes defined over a finite field of integers. Symmetric and asymmetric non-binary turbo codes are obtained and their BER performance in both the ‘waterfall’ and the ‘error-floor’ regions is analysed. The results show good performance improvements when compared to binary and quaternary turbo codes with same throughput.

Message-passing iterative decoding between detector and RSC code decoder for PMR channel

For higher density perpendicular magnetic recording (PMR) systems, it is difficult to expect an improvement in performance when only partial response maximum likelihood (PRML) or noise predictive maximum likelihood (NPML) detection is used. Hence, we consider a system that performs better than either PRML or NPML. We concatenate a channel detector using a recursive systematic convolutional (RSC) code decoder and a message-passing algorithm. The RSC code itself is simple but its performance is inferior to other iterative decoding methods. Therefore, we used channel iteration between the channel detector and the RSC decoder in order to improve performance. We also used a precoding and random interleaver. We evaluated the performance on the PMR channel with jitter noise.

Turbo Detection of Precoded Sphere Packing Modulation Using Four Transmit Antennas for Differential Space-Time Spreading

This paper presents a novel turbo-detected multidimensional sphere packing (SP) modulation scheme using four transmit antennas combined with differentially encoded space-time spreading (DSTS). The DSTS scheme can be readily combined with PSK, QAM as well as SP modulation schemes. The SP-aided DSTS system advocated has low-complexity encoding and decoding algorithms that require no channel knowledge and it is capable of outperforming the DSTS system dispensing with sphere packing. Further system performance improvements can be attained by serially concatenated convolutional encoding combined with a unity-rate code (URC) referred to as a precoder. Then, at the receiver side, iterative decoding is invoked by exchanging extrinsic information between the precoder's decoder as well as the outer recursive systematic convolutional (RSC) code's decoder. Moreover, the convergence behaviour of the proposed system is evaluated with the aid of extrinsic information transfer (EXIT) charts. Explicitly, the turbo-detected precoded DSTS-SP system performs within 1.6 dB of the achievable multiple-input multiple output (MIMO) capacity. Finally, in contrast to an equivalent 0.5 bit/symbol throughput turbo- detected DSTS-SP scheme using no preceding, the turbo-detected precoded DSTS-SP scheme exhibits no error floor.

Joint Iterative Multiuser Detection and Channel Estimation for Differentially Coded Asynchronous CDMA Systems

In this paper, we propose a receiver structure that performs joint iterative multiuser detection and channel estimation for the uplink of differentially coded asynchronous direct-sequence code-division multiple-access systems in unknown multipath channels. The overall transmission scheme for each user is the serial concatenation of a recursive systematic convolutional (RSC) encoder and a differential encoder separated by an interleaver. The proposed receiver consists of two stages. The first stage performs channel estimation, soft interference cancellation, and soft-input soft-output multiuser filtering, followed by the second stage, which consists of a bank of serially concatenated single-user channel decoders and differential decoders. The single-user iterative decoder for each user consists of a powerful combination of an RSC decoder and a differential decoder, which work together in an iterative fashion. In terms of channel estimation, at the first iteration, the multipath channel is blindly estimated by exploiting the orthogonality property between the signal and noise subspaces, whereas, for the next iterations, the channel is estimated using the soft estimates of coded bits of each user provided by the single-user iterative decoders in conjunction with the output of the multiuser detector. By exchanging soft information between the different stages, the receiver performance is improved through iteration. Simulation results demonstrate that the performance of the proposed iterative multiuser detector with the integrated channel estimator approaches the single-user bound at high signal-to-noise ratio.

Correlated data gathering in wireless sensor networks based on distributed source coding

We propose in this paper a novel scheme for correlated data gathering in energy- and bandwidth-limited wireless sensor networks based on Distributed Source Coding (DSC). We develop a special Viterbi Algorithm, denoted as VA-DSC, for decoding of the sensor data encoded by DSC. DSC principles have recently been applied to sensor data gathering by constructing practical DSC schemes using channel coding approach. However, existing schemes have not yet taken into account the inherent difference between source coding and channel coding. In this proposed algorithm, we take advantage of the known parity bits at the decoder when the data is encoded by DSC. When the proposed algorithm is applied to Recursive Systematic Convolutional (RSC) and Turbo codes, we demonstrate that VA-DSC is able to reduce both decoding error probability and computational complexity. When the proposed algorithm is applied to correlated data gathering in wireless sensor networks, we demonstrate that VA-DSC is also capable of receiving all data correctly, while, at the same time, reducing the energy consumption in the networks. Our simulation results show that the proposed scheme results in superior performance in terms of data reception accuracy and energy consumption efficiency.

Performance analysis of turbo codes in quasi-static fading channels

The performance of turbo codes in quasi- static fading channels both with and without antenna diversity is investigated. In particular, simple analytic techniques that relate the frame error rate of a turbo code to both its average distance spectrum as well as the iterative decoder convergence characteristics are developed. Both by analysis and simulation, the impact of the constituent recursive systematic convolutional ( RSC) codes, the interleaver size and the number of decoding iterations on the performance of turbo codes are also investigated. In particular, it is shown that in systems with limited antenna diversity different constituent RSC codes or interleaver sizes do not affect the performance of turbo codes. In contrast, in systems with significant antenna diversity, particular constituent RSC codes and interleaver sizes have the potential to significantly enhance the performance of turbo codes. These results are attributed to the fact that in single transmit - single receive antenna systems, the performance primarily depends on the decoder convergence characteristics for E-b/N-o values of practical interest. However, in multiple transmit - multiple receive antenna systems, the performance depends on the code characteristics.

Turbo equalization of serially concatenated turbo codes using a predictive DFE-based receiver

This paper investigates the performance of various “turbo” receivers for serially concatenated turbo codes transmitted through intersymbol interference (ISI) channels. Both the inner and outer codes are assumed to be recursive systematic convolutional (RSC) codes. The optimum turbo receiver consists of an (inner) channel maximum a posteriori (MAP) decoder and a MAP decoder for the outer code. The channel MAP decoder operates on a “supertrellis” which incorporates the channel trellis and the trellis for the inner error-correcting code. This is referred to as the MAP receiver employing a SuperTrellis (STMAP). Since the complexity of the supertrellis in the STMAP receiver increases exponentially with the channel length, we propose a simpler but suboptimal receiver that employs the predictive decision feedback equalizer (PDFE). The key idea in this paper is to have the feedforward part of the PDFE outside the iterative loop and incorporate only the feedback part inside the loop. We refer to this receiver as the PDFE-STMAP. The complexity of the supertrellis in the PDFE-STMAP receiver depends on the inner code and the length of the feedback part. Investigations with Proakis B, Proakis C (both channels have spectral nulls with all zeros on the unit circle and hence cannot be converted to a minimum phase channel) and a minimum phase channel reveal that at most two feedback taps are sufficient to get the best performance. A reduced-state STMAP (RS-STMAP) receiver is also derived which employs a smaller supertrellis at the cost of performance.

Performance of JPEG Image Transmission Using Proposed Asymmetric Turbo Code

This paper gives the results of a simulation study on the performance of JPEG image transmission over AWGN and Rayleigh fading channels using typical and proposed asymmetric turbo codes for error control coding. The baseline JPEG algorithm is used to compress a QCIF ( ) "Suzie" image. The recursive systematic convolutional (RSC) encoder with generator polynomials , that is, (13/11) in decimal, and 3G interleaver are used for the typical WCDMA and CDMA2000 turbo codes. The proposed asymmetric turbo code uses generator polynomials , that is, (13/11; 13/9) in decimal, and a code-matched interleaver. The effect of interleaver in the proposed asymmetric turbo code is studied using weight distribution and simulation. The simulation results and performance bound for proposed asymmetric turbo code for the frame length , code rate with Log-MAP decoder over AWGN channel are compared with the typical system. From the simulation results, it is observed that the image transmission using proposed asymmetric turbo code performs better than that with the typical system.

Performance of Quantized Turbo Decoding for Decoders with Maximum a Postriori Algorithm

This paper investigates Bit Error Performance for quantized turbo code over Gaussian channels. The turbo encoder is built using a parallel concatenation of two recursive systematic convolution encoders and a random interleaver. The turbo decoder employs' two iterative symbol by symbol MAP based algorithm decoders for decoding the received data. For a given quantization level, the Bit Error Rates are examined by considering the influence of the code structure, the block size, and the number of decoding steps. Finally, to measure loss of power efficiency due to quantization, the results are compared to previously published results for the unquantized case under the same Bit to noise ratio and the code structure.

A new class of asymmetric turbo code for 3G systems

In this paper, we propose a new asymmetric turbo encoder for 3G systems that performs well in both “water fall” and “error floor” regions. The Recursive Systematic Convolutional (RSC) encoder with generator polynomials (1, D 3+ D 2+1/ D 3+ D+1), i.e. (13/11) in decimal and 3G interleaver are used for the typical Wideband Code-Divison Multiple Access (WCDMA) and Code-Divison Multiple Access 2000 (CDMA2000) turbo codes. The proposed asymmetric turbo code uses generator polynomials (1, D 3+ D 2+1/ D 3+ D+1; D 3+ D 2+1/ D 3+1), i.e. (13/11;13/9) and a code-matched interleaver. The effect of interleaver in the proposed asymmetric turbo code is studied using weight distribution and simulation. The simulation results and performance bound for proposed asymmetric turbo code for the frame length, N = 400 , code rate, r = 1 / 3 with Log-MAP decoder over additive white Gaussian noise (AWGN) channel are obtained and compared with the typical system. The performance tests are extended over both AWGN and Rayleigh fading channels for different frame size to verify the possibility of implementation of the proposed asymmetric turbo code. From the performance results, it is observed that the proposed asymmetric turbo code performs better than the typical system at both low and high SNR and it provides a coding gain of 0.5–0.8 dB.

Turbo Codes With Rate-&amp;lt;tex&amp;gt;$m/(m+ 1)$&amp;lt;/tex&amp;gt;Constituent Convolutional Codes

The original turbo codes (TCs), presented in 1993 by Berrou et al., consist of the parallel concatenation of two rate-1/2 binary recursive systematic convolutional (RSC) codes. This paper explains how replacing rate-1/2 binary component codes by rate-m/(m+1) binary RSC codes can lead to better global performance. The encoding scheme can be designed so that decoding can be achieved closer to the theoretical limit, while showing better performance in the region of low error rates. These results are illustrated with some examples based on double-binary (m=2) 8-state and 16-state TCs, easily adaptable to a large range of data block sizes and coding rates. The double-binary 8-state code has already been adopted in several telecommunication standards.

Extended tail-biting schemes for turbo codes

In this letter, we investigate the problem of using tailbiting recursive systematic convolutional (RSC) codes in turbo codes. Tailbiting is not always possible for a given RSC code with fixed length. We propose an extended tailbiting method for RSC codes and compare it with another extension method proposed by Van Stralen, et al. (IEE Electron. Lett., vol. 35, pp. 1461-1462, 1999). Several schemes using these extended tailbiting RSC codes in turbo code systems are developed and compared.

Interleaver Pruning for Construction of Variable-Length Turbo Codes

The issue of pruning (i.e., shortening) a given interleaver in a parallel concatenated convolutional code (PCCC or turbo codes) employing recursive systematic convolutional (RSC) constituent codes has been addressed. The encoder uses two RSC codes coupled with an interleaver represented by the permutation vector. Since the encoding and decoding delay associated with turbo codes are essentially dominated by the block length (interleaver length), it is of considerable practical interest to be able to modify it in order to obtain a variable length block code. Most other interleaver construction techniques do not have a recursive build nature and as a consequence, each time the interleaver length is modified, the interleaver itself must be redesigned. In the case of turbo codes, it is desirable to identify a technique that would allow any interleaver performing a given permutation to be reduced in size with gradual performance degradation of the resulting PCCC, but without the need for changing almost completely the operation of the interleaver.

Convergence analysis of iterative soft decoding in partial response channels

In this paper, we apply density evolution to investigate the convergence of iterative soft decoding in partial response (PR) channels and to predict the bit-error-rate (BER) performance of concatenated PR systems. The convergence analysis and computer simulations reveal that column weight-2 low-density parity check (LDPC) codes are similar to recursive systematic convolutional (RSC) codes in terms of extrinsic information transfer characteristics, BER performance, and block-error statistics, while the decoding complexity of column weight-2 LDPC codes is significantly smaller than that of RSC codes.

Some Results on Convolutional Coupled Codes: Distance Properties, Performance, and Design

The class of Convolutional Coupled Codes is a promising alternative to classical Turbo-Codes. A Convolutional Coupled Code consists of a cascade of ν identical recursive systematic convolutional (RSC) outer codes and k inner block codes with parameters (2ν, ν, d i). The codes are linked together such that only the systematic part of the outer codes is encoded with the inner block encoders. Only the redundancy from the inner and outer codes are transmitted. An estimation of the minimum distance is derived. The influence of number, code memory and code polynomials of the outer RSC codes on the distance properties and the convergence behavior of the iterative decoding scheme is studied. With respect to this results, we present a guideline for the optimal design of the RSC outer codes.

Search for good b /( b +1) high rate recursive systematic convolutional component codes

A simplified search for good recursive systematic convolution codes with rate b/(b+1) is proposed, to be used as component codes in very high rate, parallel concatenated turbo codes. The first optimised parameter is the effective free-distance of turbo codes. The results obtained are compared to known bounds and previously published results where available.

Improvements in Turbo Decoding

In recent years iterative concatenated decoding has regained popularity starting with the remarkable results presented in a paper by a group of French researchers. They introduced a new family of convolutional codes, nicknamed “Turbo codes” due to its resemblance with the turbo engine. A turbo code is built from parallel concatenation of two recursive systematic codes linked together by non-uniform interleaving. Two maximum-a-posteriori probability decoders do decoding iteratively, each using the decoding results from other one. For sufficiently large interleaver size, the error performance seems to be close to Shannon limit.We have extended the implementation principle of turbo codes and have used an encoder, which uses one additional recursive systematic convolutional (RSC) encoder and an interleaver. We show that this multiple turbo code performs better than the standard turbo code. We have also proposed a dynamic decoding algorithm, which uses variable number of iterations. We show that dynamic algorithm requires less time as compared to the other algorithm for achieving the same error performance and that the extended turbo code can achieve better results with small interleaver size.

Optimized turbo codes for delay constrained applications

We present the results of the optimization applied to the design of interleavers for rate-1/n parallel concatenated convolutional codes (PCCC) tailored to specific recursive systematic convolutional (RSC) constituent codes. The emphasis is on low-latency codes associated with interleavers of block length less than or equal to 160. The error floors of the optimized codes are significantly lower than those associated with the use of random interleavers. The distance spectra of the equivalent block codes resulting from trellis termination applied to PCCC are evaluated and used to obtain asymptotic bit error rate (BER) curves for the optimized codes.