Component Convolutional Codes Research Articles

Upper bounds on the minimum distance for turbo codes using CPP interleavers

Analysis of error correction performance for error correcting codes is very important when using such codes in digital communication systems. At medium-to-high signal-to-noise ratios, the distance spectrum of the error correcting code represents a good indicator for the error correction performance of the code. It is desired that the minimum distance of the code is as large as possible and that the corresponding multiplicity (i.e. the number of codewords having the weight equal to the minimum distance) is as small as possible. If we know an upper bound of the minimum distance of the code, then we have a good indication about the capabilities and the limitations of the code. One of the classes of the error correcting codes with the best performance is that of turbo codes. For such codes, establishing upper bounds on the minimum distance is challenging because it depends on the interleaver component of the turbo code. In this paper we consider turbo codes with component convolutional codes as in the Long Term Evolution standard. The interleaver lengths are of the form $$16 \varPsi $$ or $$48 \varPsi $$ , with $$ \varPsi $$ a product of different prime numbers greater than three. The first achievement in the paper is that for these interleaver lengths, we show that cubic permutation polynomials (CPP), with some constraints on the coefficients, when $$3 \not \mid (p_i-1)$$ for a prime $$p_i > 3$$ , always have a true inverse CPP. The most accurate upper bounds on the minimum distance for turbo codes are achieved by identifying bit information sequences leading to a certain weight of the corresponding turbo-codeword. In this paper we have indentified such bit information sequences by means of the full range dual impulse method to estimate the weight of the turbo-codewords. For the previously mentioned turbo codes and CPP interleavers, we show that the minimum distance is upper bounded by the values of 38, 36, and 28, for three different classes of coefficients. Previously, it was shown that for the same interleaver lengths and for quadratic PP (QPP) interleavers, the upper bound of the minimum distance is equal to 38. Several examples show that $$d_{min}$$ -optimal CPP interleavers are better than $$d_{min}$$ -optimal QPP interleavers because the multiplicities corresponding to the minimum distances for CPPs are about a half of those for QPPs. A theoretical explanation in terms of nonlinearity degrees for this result is given for all considered interleaver lengths and for the class of CPPs for which the upper bound is equal to 38.

Further upper bounds on the minimum distance for turbo codes using CPP interleavers

This paper addresses turbo codes with component convolutional codes as those in the Long Term Evolution (LTE) standard. We consider the interleaver lengths of the form L=32kLΨ, with kL∈{1,3} and Ψ is a product of prime numbers greater than three. For these interleaver lengths, we have shown that a true cubic permutation polynomial (CPP) f1x+f2x2+f3x3(modL), under the constraint f3=0(modpi) when a prime pi>3 and 3 does not divide (pi−1), always has an inverse true CPP, except when f3=L∕(4kL) and f2∈{L∕16,3L∕16,5L∕16,7L∕16}. For the previously mentioned exception, a true CPP has an inverse true quadratic PP (QPP). For CPP interleavers with true inverse CPPs, we have shown that the minimum distance is upper bounded by the values of 44, 36, and 27, for three different classes of coefficients. Previously, it was shown that for the same interleaver lengths and for QPP interleavers, the upper bound on the minimum distance is equal to 50. Due to the symmetry of the classical turbo codes, the minimum distance is also upper bounded by the value of 50 for CPP interleavers with a QPP inverse. The nonlinearity degrees for all considered CPP interleavers are computed. Several examples of dmin-optimal CPP interleavers and their inverse QPPs are given.

Braided Convolutional Codes With Sliding Window Decoding

In this paper, we present a novel sliding window decoding scheme based on iterative Bahl–Cocke–Jelinek–Raviv decoding for braided convolutional codes, a class of turbo-like codes with short constraint length component convolutional codes. The tradeoff between performance and decoding latency is examined and, to reduce decoding complexity, both uniform and nonuniform message passing schedules within the decoding window, along with early stopping rules, are proposed. We also perform a density evolution analysis of sliding window decoding to guide the selection of the window size and message passing schedule. Periodic puncturing is employed to obtain rate-compatible code rates of 1/2 and 2/3 starting from a rate 1/3 mother code and a code rate of 3/4 starting from a rate 1/2 mother code. Simulation results show that, with nonuniform message passing and periodic puncturing, near capacity performance can be maintained throughout a wide range of rates with reasonable decoding complexity and no visible error floors.

A Building Block Approach for Designing Multilevel Coding Schemes

A novel building block (BB) design approach is presented for systematically constructing multilevel codes (MLCs) under the balanced distances rule (BDR). In contrast to the traditional method of starting from an overall signal constellation and then using a mapping policy, the BB approach starts by designing a BB constellation for the first level of the MLC and then uses two copies of this BB to construct a second level constellation. This process is continued to the highest level by constructing the next level constellation using multiple copies of the current level constellation as a BB. Compared to the traditional method of construction of MLCs according to the BDR, the BB approach provides additional flexibility in selecting component codes and is able to exactly balance the distances of all levels. Compared with MLCs constructed using the traditional BDR approach, MLCs constructed using the BB approach have the same decoding complexity and perform better in the error floor region, which is typically the operating region for MLCs constructed with convolutional component codes.

A gaussian sum approach to blind carrier phase estimation and data detection in turbo coded transmissions

We present a joint phase estimation and decoding method for convolutional turbo codes in the presence of strong phase noise. In order to overcome the problem of phase ambiguity and cycle slips, a combined state-space model for the time varying phase and the component convolutional codes is introduced. The proposed algorithm uses a Gaussian sum approach to approximate the multimodal a posteriori probability density function (pdf) of the phase in a blind context. We compare our method to the well known alternative consisting in discretizing the phase.Monte-Carlo simulations for the turbo code used in the DVB-RCS standard show that the performances of the proposed scheme are close to decoding with perfect knowledge of the phase.

Blind turbo-detection in the presence of phase noise

Blind coherent detection of convolutional turbo codes is a hard problem in the presence of strong phase noise. Since the operating signal-to-noise ratio is usually very low, phase synchronisation algorithms suffer from phase ambiguities and cycle slips. A possible remedy is to perform joint phase estimation and decoding on a combined state-space model for the time-varying phase and the component convolutional codes. We demonstrate that joint phase estimation and decoding is in fact mandatory only for one component code, while ordinary BCJR decoding can be used for the second component code. Monte Carlo simulations for the turbo code used in the DVB-RCS standard show that the performances of the proposed scheme are close to decoding with perfect knowledge of the phase.

VLSI Design of a Fully-Parallel High-Throughput Decoder for Turbo Gallager Codes

The most powerful channel coding schemes, namely those based on turbo codes and low-density parity-check (LDPC) Gallager codes, have in common the principle of iterative decoding. However, the relative coding structures and decoding algorithms are substantially different. This paper presents a 2048-bit, rate-1/2 soft decision decoder for a new class of codes known as Turbo Gallager Codes. These codes are turbo codes with properly chosen component convolutional codes such that they can be successfully decoded by means of the decoding algorithm used for LDPC codes, i.e., the belief propagation algorithm working on the code Tanner graph. These coding schemes are important in practical terms for two reasons: (i) they can be encoded as classical turbo codes, giving a solution to the encoding problem of LDPC codes; (ii) they can also be decoded in a fully parallel manner, partially overcoming the routing congestion bottleneck of parallel decoder VLSI implementations thanks to the locality of the interconnections. The implemented decoder can support up to 1 Gbit/s data rate and performs up to 48 decoding iterations ensuring both high throughput and good coding gain. In order to evaluate the performance and the gate complexity of the decoder VLSI architecture, it has been synthesized in a 0.18 μm standard-cell CMOS technology.

Construction of partial unit memory encoders for application in capacity-approaching concatenated codes

In order to achieve capacity approaching performance, the use of partial unit memory (PUM) component codes with good distance properties in turbo codes and woven turbo codes have been suggested. PUM codes are no more complex, but perhaps less well known than traditionally used convolutional component codes. Turbo codes and woven turbo codes based on PUM codes have been shown to outperform those based on convolutional codes with no extra decoding complexity. In this paper, the convolutional properties of PUM codes are exploited and how these can be used to construct encoder structures and their trellises is described so that they are suitable for use in the aforementioned parallel concatenated structures. The encoder and trellis construction techniques are illustrated by two example PUM codes. The performance of these codes as component codes in a turbo structure is presented to show the influence of the parameters of the PUM code and how this impacts on the design of suitable component codes in a concatenated structure with capacity approaching performance.

On the use of partial unit memory codes for woven turbo codes with outer and inner warp

The encoding, decoding and performance analysis of concatenated partial unit memory (PUM) code structures for capacity approaching performance are investigated. PUM codes are known for their excellent distance properties and lower decoding complexity compared to equivalent multi-memory convolutional codes. Two capacity approaching concatenated structures are considered: turbo codes (TCs) and woven turbo codes (WTCs), both initially proposed with component convolutional codes. TCs consist of a number of parallel concatenated encoders. WTCs were originally proposed with outer warp, that is a number of outer encoders are parallel concatenated to one inner encoder. WTCs are also constructed with inner warp, i.e. one outer encoder is parallel concatenated to a number of inner encoders. An iterative max-log-maximum a posteriori decoding scheme is proposed for decoding PUM codes having multiple-input parallel branches in the trellis. The distance properties and bit error rate performance of the novel concatenated PUM codes are compared with known concatenated convolutional codes. It is observed that PUM structures, both TC and WTC with inner and outer warp, have better distance properties and hence improved performance over their equivalent convolutional counterparts. Performance as close as 1.25 dB from the Shannon limit is obtained for concatenated PUM structures with an interleaver size of only 4800 bits.

Design and Performance of Turbo Gallager Codes

The most powerful channel-coding schemes, namely, those based on turbo codes and low-density parity-check (LDPC) Gallager codes, have in common the principle of iterative decoding. However, the relative coding structures and decoding algorithms are substantially different. This paper shows that recently proposed novel coding structures bridge the gap between these two schemes. In fact, with properly chosen component convolutional codes, a turbo code can be successfully decoded by means of the decoding algorithm used for LDPC codes, i.e., the belief-propagation algorithm working on the code Tanner graph. These new turbo codes are here nicknamed "turbo Gallager codes." Besides being interesting from a conceptual viewpoint, these schemes are important on the practical side because they can be decoded in a fully parallel manner. In addition to the encoding complexity advantage of turbo codes, the low decoding complexity allows the design of very efficient channel-coding schemes.

Serially concatenated space-time codes with iterative decoding and performance limits of block-fading channels

This work considers space-time channel coding for systems with multiple-transmit and a single-receive antenna, over space uncorrelated block-fading (quasi-static) channels. Analysis of the outage probability over such channels reveals the existence of a threshold phenomenon. The outage probability can be made arbitrary small by increasing the number of transmit antennas, only if the E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /N/sub 0/ is above a threshold which depends on the coding rate. Furthermore, it is shown that when the number of transmit antennas is increased, the /spl epsi/-capacity of a block-fading Rayleigh channel tends to the Shannon capacity of an additive white Gaussian noise channel. This paper also presents space-time codes constructed as a serial concatenation of component convolutional codes separated by an interleaver. These schemes provide full transmit diversity and are suitable for iterative decoding. The rate of these schemes is less than 1 bit/s/Hz, but can be made arbitrary close to 1 bit/s/Hz by the use of Wyner-Ash codes as outer components. Comparison of these schemes with structures from literature shows that performance gains can be obtained at the expense of a small decrease in rate. Computer simulation results over block-fading Rayleigh channels show that the frame-error rate of several of these schemes is within 2-3 dB from the theoretical outage probability.

New Punctured Turbo Codes

Abstract In its simple form, a turbo encoder consists of two constituent systematic encoders joined together by means of an interleaver. Turbo codes with different code rates can be obtained by deleting certain parity check bits prior to transmission, which leads to punctured turbo codes. In this paper, new punctured turbo codes with coding rate varying from 1/3 to 1 are designed using rate 1/2 component convolutional codes with 3 and 4-memory elements, by considering systematically several factors that affect the code performance. Bit error rates of the proposed codes with interleaver size of 1000 and with 3 and 4-memory elements are evaluated by means of computer simulations. Comparisons with the corresponding reference codes designed by the methods taken from the literature are also given.

Comparison of convolutional and turbo codes for OFDM with antenna diversity in high-bit-rate wireless applications

This letter presents simulation results for the application of turbo coding to an OFDM system with diversity. First, results are presented for convolutional and Reed-Solomon codes. It is shown that even with short constraint lengths, convolutional codes have the potential to outperform Reed-Solomon codes, provided that sufficient precision is used in the soft decoder. We then evaluate the performance of turbo codes under slow fading conditions and study the effects of varying codeword size. Increasing codeword size theoretically provides better interleaving between the two component codes. However, this advantage is less clear when the fading rate is significantly lower than the symbol rate, which is typical of the high-data-rate systems considered here. Under such conditions, the advantage of using two component convolutional codes in turbo codes is limited. A single convolutional code with a long constraint length may be a better choice.

Improved upper bounds on the ML decoding error probability of parallel and serial concatenated turbo codes via their ensemble distance spectrum

The ensemble performance of parallel and serial concatenated turbo codes is considered, where the ensemble is generated by a uniform choice of the interleaver and of the component codes taken from the set of time-varying recursive systematic convolutional codes. Following the derivation of the input-output weight enumeration functions of the ensembles of random parallel and serial concatenated turbo codes, the tangential sphere upper bound is employed to provide improved upper bounds on the block and bit error probabilities of these ensembles of codes for the binary-input additive white Gaussian noise (AWGN) channel, based on coherent detection of equi-energy antipodal signals and maximum-likelihood decoding. The influence of the interleaver length and the memory length of the component codes is investigated. The improved bounding technique proposed here is compared to the conventional union bound and to a alternative bounding technique by Duman and Salehi (1998) which incorporates modified Gallager bounds. The advantage of the derived bounds is demonstrated for a variety of parallel and serial concatenated coding schemes with either fixed or random recursive systematic convolutional component codes, and it is especially pronounced in the region exceeding the cutoff rate, where the performance of turbo codes is most appealing. These upper bounds are also compared to simulation results of the iterative decoding algorithm.

Iterative decoding of binary block and convolutional codes

Iterative decoding of two-dimensional systematic convolutional codes has been termed "turbo" (de)coding. Using log-likelihood algebra, we show that any decoder can be used which accepts soft inputs-including a priori values-and delivers soft outputs that can be split into three terms: the soft channel and a priori inputs, and the extrinsic value. The extrinsic value is used as an a priori value for the next iteration. Decoding algorithms in the log-likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code. The iteration is controlled by a stop criterion derived from cross entropy, which results in a minimal number of iterations. Optimal and suboptimal decoders with reduced complexity are presented. Simulation results show that very simple component codes are sufficient, block codes are appropriate for high rates and convolutional codes for lower rates less than 2/3. Any combination of block and convolutional component codes is possible. Several interleaving techniques are described. At a bit error rate (BER) of 10/sup -4/ the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block/convolutional component codes, interleaver sizes less than 1000 and for three to six iterations.

Multilevel ternary line codes with trellis structure

A combination of the two coding techniques given by Imai and Hirakawa (1977) and Pottie and Taylor (1989) is applied to the ternary (+, 0, -) line-code design problem. New ternary coding systems are obtained with reduced decoding complexity and improved error performance, compared with those obtained by the classical Ungerboeck&apos;s trellis-coding approach. The decoding complexity for high coding rates is reduced by the proper choice of the punctured convolutional component codes for each partitioning level. A spectral null at zero frequency is obtained by the use of a halfrate unit-memory convolutional encoder at the last partitioning level which selects line codewords with opposite disparities in an alternated fashion so that the running-digital-sum values vary in a finite interval. The effects of the path multiplicity are evaluated by upper bounding the bit error probabilities based on the union-bound approach.