Trellis Termination Research Articles

Low-Power VLSI Implementation of Novel Hybrid Adaptive Variable-Rate and Recursive Systematic Convolutional Encoder for Resource Constrained Wireless Communication Systems

In the modern wireless communication system, digital technology has tremendous growth, and all the communication channels are slowly moving towards digital form. Wireless communication has to provide the reliable and efficient transfer of information between transmitter and receiver over a wireless channel. The channel coding technique is the best practical approach to delivering reliable communication for the end-users. Many conventional encoder and decoder units are used as error detection and correction codes in the digital communication system to overcome the multiple transient errors. The proposed convolutional encoder consists of both Recursive Systematic Convolutional (RSC) Encoder and Adaptive Variable-Rate Convolutional (AVRC) encoder. Adaptive Variable-Rate Convolutional encoder improves the bit error rate performance and is more suitable for a power-constrained wireless system to transfer the data. Recursive Systematic Convolutional encoder also reduces the bit error rate and improves the throughput by employing the trellis termination strategy. Here, AVRC encoder ultimately acquires the channel state information and feeds the data into a fixed rate convolutional encoder and rate adaptor followed by a buffer device. A hybrid encoder combines the AVRC encoder and RSC encoder output serially and parallel, producing the solid encoded data for the modulator in the communication system. A modified turbo code is also obtained by placing interleaver between the two encoder units and building the stronger code word for the system. Finally, the conventional encoder system is compared and analyzed with the proposed method regarding the number of LUT’s, gates, clock cycle, slices, area, power, bit error rate, and throughput.

Turbo codes and turbo algorithms

In the first part of this paper, several basic ideas that prompted the coming of turbo codes are commented on. We then present some personal points of view on the main advances obtained in past years on turbo coding and decoding such as the circular trellis termination of recursive systematic convolutional codes and double-binary turbo codes associated with Max-Log-MAP decoding. A novel evaluation method, called genieinitialised iterative processing (GIIP), is introduced to assess the error performance of iterative processing. We show that using GIIP produces a result that can be viewed as a lower bound of the maximum likelihood iterative decoding and detection performance. Finally, two wireless communication systems are presented to illustrate recent applications of the turbo principle, the first one being multiple-input/multiple-output channel iterative detection and the second one multi-carrier modulation with linear precoding.

Multi-non-binary turbo codes

Abstract This paper presents a new family of turbo codes called multi-non-binary turbo codes (MNBTCs) that generalizes the concept of turbo codes to multi-non-binary (MNB) parallel concatenated convolutional codes (PCCC). An MNBTC incorporates, as component encoders, recursive and systematic multi-non-binary convolutional encoders. The more compact data structure for these encoders confers some advantages on MNBTCs over other types of turbo codes, such as better asymptotic behavior, better convergence, and reduced latency. This paper presents in detail the structure and operation of an MNBTC: MNB encoding, trellis termination, Max-Log-MAP decoding adapted to the MNB case. It also shows an example of MNBTC whose performance is compared with the state-of-the-art turbo code adopted in the DVB-RCS2 standard.

Linear Diophantine Constrained Intra-Fix for Frequency Domain Equalization of Multi-h CPM

This paper considers the problem of constructing a symbol block to generate a cyclic multi-h continuous phase modulated (CPM) signal so as to permit frequency domain equalization (FDE) at the receiver. Like single-h CPM, the memory inherent in a multi-h CPM waveform requires the use of an additional segment, called an intra-fix, in each transmitted block. The intra-fix forces the transmitter state machine to regress to the state it had at the start of the block. The constraints on the symbols of the intra-fix, which ensure above condition, are shown to be a set of Linear Diophantine Equations (LDEs). Three different methods are proposed for finding the symbols of the intra-fix, such that the length of the intra-fix is minimized. The last two methods connect the problem of intra-fix insertion to the problem of trellis termination of the tilted phase. This connection enables us to state and justify that existing bounds established on the length of the terminating sequence are effective for the length of the intra-fix. We observe that multi-h CPM schemes typically require a shorter intra-fix, as compared to single-h CPM schemes having similar modulation parameters. We apply the proposed procedure to a dual-h CPM scheme.

Trellis Termination of Multi-h CPM and the Diophantine Frobenius Problem

This paper presents a method for terminating the phase trellis in multi-h continuous phase modulation (CPM). It is shown that tilted phase trellis for multi-h CPM signals is periodic and enables the representation of the multi-h CPM modulator as a cascade of a periodic recursive continuous phase encoder (CPE) and a phase modulator (PM). We then pose the problem of finding the minimum number of symbols that will terminate the phase trellis in a known state of the modulator and establish a connection of its solution to the Diophantine Frobenius Problem (FP). Analytical results are established for the existence of a terminating sequence and upper and lower bounds are derived on its length for arbitrary modulation parameters. We show that the trellis terminating symbols and their quantity depends upon the initial state of the encoder, the set of modulation indices, the CPE memory order, and the number of bits per modulated symbol. It is shown that required overhead for trellis termination can be significant for schemes with integer modulation indices.

Trellis Coded Line Packing: Large Dimensional Beamforming Vector Quantization and Feedback Transmission

In this paper we study the beamforming vector quantization and feedback transmission problem for a cooperative transmission environment. Feedback is the primary means of providing the transmitter with channel state information (CSI) in a frequency division duplex (FDD) system. The conventional codebook approach uses a common codebook at the transmitter and at the receiver, while the feedback takes the form of the index of the codebook. This approach has a complexity that scales exponentially with the number of transmit antennas M, making the approach less favorable when M is large. We propose to use trellis-based quantization to reduce the complexity to grow linearly in M. A codebook design criteria is derived for the trellis-based quantizer. We also present three enhancements to the base scheme, each giving 0.2-0.8 dB gain. These enhancements address issues relevant to feedback error, trellis termination, and unequal path-loss from different transmission points. Performance results are shown in numerical simulations.

Performance Analysis of Serial Concatenation Convolutional Codes Based on Amplitude Matrix

Serial Concatenation Convolutional Codes (SCCCs) encoder is built using a serial concatenation of two Recursive Systematic Convolutional (RSC) encoders, separated by an interleaver. These two RSC encoders depending on the trellis termination criteria are software implemented and the performance of each one of them is analyzed under different conditions and circumstances. The output data from the encoder are multiplied by an amplitude matrix (AM) at the transmitter side and the Inverse of Amplitude Matrix (IAM) at the receiver side. The reliability estimation, log-likelihood algebra, and soft channel outputs for Soft Output Viterbi Algorithm (SOVA) are examined. Then the modified Viterbi metric that incorporates a-priori information used for SOVA decoding is derived. A low memory implementation of the SOVA decoder is presented. The iterative SOVA for SCCCs is described with illustrative examples. The behavior of the SCCCs encoder-decoder scheme is tested under different circumstances with AM and without AM at the AWGN and Rayleigh fading channels with unlike frame sizes (FS) and constraint length (K). The results show that the performance of system with AM outperforms the other conventional system that worked without AM.

Coded modulation in the block-fading channel: coding theorems and code construction

We consider coded modulation schemes for the block-fading channel. In the setting where a codeword spans a finite number N of fading degrees of freedom, we show that coded modulations of rate R bit per complex dimension, over a finite signal set /spl chi//spl sube//spl Copf/ of size 2/sup M/, achieve the optimal rate-diversity tradeoff given by the Singleton bound /spl delta/(N,M,R)=1+/spl lfloor/N(1-R/M)/spl rfloor/, for R/spl isin/(0,M/spl rfloor/. Furthermore, we show also that the popular bit-interleaved coded modulation achieves the same optimal rate-diversity tradeoff. We present a novel coded modulation construction based on blockwise concatenation that systematically yields Singleton-bound achieving turbo-like codes defined over an arbitrary signal set /spl chi//spl sub//spl Copf/. The proposed blockwise concatenation significantly outperforms conventional serial and parallel turbo codes in the block-fading channel. We analyze the ensemble average performance under maximum-likelihood (ML) decoding of the proposed codes by means of upper bounds and tight approximations. We show that, differently from the additive white Gaussian noise (AWGN) and fully interleaved fading cases, belief-propagation iterative decoding performs very close to ML on the block-fading channel for any signal-to-noise ratio (SNR) and even for relatively short block lengths. We also show that, at constant decoding complexity per information bit, the proposed codes perform close to the information outage probability for any block length, while standard block codes (e.g., obtained by trellis termination of convolutional codes) have a gap from outage that increases with the block length: this is a different and more subtle manifestation of the so-called "interleaving gain" of turbo codes.

Simultaneous zero-tailing of parallel concatenated codes

In a parallel concatenated convolutional code, an information sequence is encoded by a convolutional encoder, and an interleaved version of the information sequence is encoded by another convolutional encoder. We discuss the situation in which we require both convolutional encoders to end in the all-zero state. To do so, we have to split an information word in two parts. One part contains the true information bits, and the second part contains the so-called tail bits, which are special bits with values computed such that both encoders end in the all-zero state. Depending on the interleaver, a different number of tail bits are needed. By using a constructive method, we give a characterization of all interleavers for a prescribed number of tail bits. We explain the method of encoding. In addition, simulations have been carried out to investigate the performance of codes resulting from simultaneous zero-tailing. This shows that simultaneous zero-tailing is similar in performance as compared to previously known zero-tailing methods (but with fewer trellis termination bits) and that it is better than zero-tailing just one of the encoders.

A self-terminated interleaver design for Turbo codes

Turbo codes can achieve excellent performance at low signal-to-noise ratio (SNR), but the performance can be severely degraded if no trellis termination is employed. This paper proved that if trellis termination bits were appended to RSC1, trellis of RSC2 could be terminated by designing the interleaver properly, consequently, derived the designing condition of such self-terminated interleaver (STI). Then we presented an algorithm of implementing a kind of STI, which could terminate RSC2 as well on condition that the RSC1 was terminated. We verified the performance of STI for turbo codes by simulation, and the simulation results showed that turbo codes with STI outperformed interleavers that could not terminate RSC2 as well.

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.

Interleaving and termination in unpunctured symmetric turbo codes

The turbo code interleaver design problem is considered for large block sizes, where the effect of trellis termination is less marked. The performance is compared for various interleavers with a similar block size, including an implementation of the uniform interleaver. An optimised interleaver design technique based on simulated annealing is proposed. The results obtained show that the error performance may be significantly improved without increasing the delay. Finally, interleaver design for small turbo codes is considered. In this case it is shown that while correct termination improves the performance for an average interleaver, its effect on turbo codes with optimised interleavers is negligible. Using the proposed simulated annealing design technique it is easier to include restrictions which make the interleaver correctly-terminating or odd-even. While the S-random algorithm serves well for specifying interleaver spread, the authors believe that their algorithm is better suited for more sophisticated design criteria

A comparison of turbo codes using different trellis terminations

This paper investigates the incidence of trellis termination on the performance of turbo codes and accounts for the performance degradation often experienced in the absence of trellis termination. Analytical upper bounds on the performance for the ensemble of turbo codes using different trellis termination strategies as well as performance results obtained by computer simulation are presented. In the case of uniform interleaving, the performance differences between various termination methods are relatively small, except when using no trellis termination at all.

On the theory and performance of trellis termination methods for turbo codes

The performance of a turbo code can be severely degraded if no trellis termination is employed. This paper investigates the implications of the choice of trellis termination method for turbo codes, and explains the origin of the performance degradation often experienced without trellis termination. An efficient method to derive the distance spectrum of turbo codes for different trellis termination methods is presented. Further, we present interleaver design rules that are tailored to each termination method. Using interleavers designed with these restrictions, we demonstrate that the performance difference between various termination methods is very small, including no trellis termination at all. For example, we demonstrate a turbo code with a 500-bit interleaver that exhibits no sign of an error floor for frame error rates as low as 10/sup -8/, even though no trellis termination is employed.

Decoder-assisted frame synchronization for packet transmission

This paper proposes a novel frame synchronization methodology for packets with convolutionally encoded data. Rather than by placing sync bits in a separate header, the sync bits are placed in a midamble and encoded as part of the data sequence, using the error correction encoder to resolve time ambiguities. The scheme is based on the principle that an error in trellis termination may result in decoding a wrong information bit sequence, while the starting portion of the sequence can be decoded as error free. This frame synchronization technique is extended to the synchronization of turbo-coded systems using a list-based approach, where the first synchronizer provides a list of potential packet starting positions, and the following synchronizer makes the final decision. The performance improvement over conventional synchronization techniques is quantified via simulation.

Sliding window and interleaver design

Terminating the trellis was considered essential to lower the bit error probability in a turbo coded system. When a sliding window algorithm is used, this issue becomes even more important. It is shown that trellis termination can be completely ignored by using an interleaver that takes into account the particular window size of the sliding window algorithm.

Application and standardization of turbo codes in third-generation high-speed wireless data services

This paper addresses the application of turbo codes for third-generation wireless services. It describes the specific characteristics of high-rate data applications in third-generation wide-band code-division multiple-access (CDMA) systems that make turbo codes superior to convolutional codes. In particular, it shows the positive effect of fast power control employed in these systems on the relative performance of turbo codes with respect to convolutional codes. It also shows how turbo and convolutional codes behave differently when the figure of merit is changed to frame error rate from bit error rate for high-speed data services. Furthermore, it describes in detail how and why the standardized turbo code has been selected in the presence of other candidates, which were also based on iterative decoding. Details of turbo interleaving and trellis termination as specified in the standards are explained. Performance of turbo codes under wide-band CDMA operating conditions are presented. The suitability of turbo codes for low-rate data applications is discussed. Finally it is shown that the performance loss as a result of internal decoder parameter quantization is negligible.

Upper bound on the frame error probability of terminated trellis codes

The frame-error rate (FER) of trellis codes with trellis termination can be approximated by considering the length of the error events negligible with respect to the frame length. In this letter we prove that this approximation is actually a true upper bound. This may be useful to orientate the design of frame based transmission systems directly in terms of FER instead of the more common but less significant bit-error rate figure.

Block Codes Formed from Catastrophic Convolutional Encoders: Applications to BCM

This article describes the construction of block codes derived from trellis termination of a binary catastrophic convolutional encoder. Two construction methods are considered: the Zero Tail (ZT) method and a modified Tail Biting (TB) method. 4-PSK block coded modulation (BCM) schemes are designed based on the constructed codes. These schemes outperform similar schemes derived from the best non-catastrupltic encoder.

Performance of continuous and blockwise decoded turbo codes

In this letter we define and evaluate the average maximum-likelihood performance of the three ways of co-decoding turbo codes. In all cases the information sequence is split into blocks of N bits (N being the length of the interleaver used by the turbo code), that are encoded by the first constituent encoder and, after interleaving, by the second encoder. In the first operation mode, both constituent encoders work in a continuous fashion, whereas in the second, at the end of each block, a suitably chosen sequence of bits is appended to the information block in order to terminate the trellises of both constituent codes. In the third mode, the operation is similar to the second, but, instead of trellis termination, both constituent encoders are simply reset.