Maximum Likelihood Decoding Research Articles

Low Complexity Transmitter Architecture for SC-FDMA Systems With SFBC Code

In this letter, a novel low complexity transmitter architecture is proposed for a space frequency block coding (SFBC) single carrier frequency division multiple access (SC-FMDA) system, where the transmitted signals for each antenna are generated using a time-domain linear sequence combination (TDLSC) scheme. In the TDLSC method, a specific set of various transmitted signals is designed to reduce the peak-to-average power ratio (PAPR) without destroying the SFBC orthogonality and requiring no side information. Additionally, a low-complexity side information detection method is also presented. Computer simulations show that the PAPR reduction performance of the TDLSC outperforms the existing simplified selective mapping scheme and has a slight performance loss of less than 0.2 dB compared to an ordinary selective mapping scheme. The proposed side information detection method has a significantly lower complexity while achieving the same bit error rate performance compared to the selected mapping-based method with a maximum likelihood decoder.

The Decoding Error Probability of Linear Codes Over the Erasure Channel

In this paper, we study the decoding error probability of linear codes over the erasure channel under the list decoding. The notion of the $q^\ell $ -incorrigible sets of linear codes is introduced to characterize its decoding error probability under the list decoding or the maximum likelihood decoding. By calculating the $q^\ell $ -incorrigible set distributions, the decoding error probability of a linear code over the erasure channel under the list decoding or the maximum likelihood decoding is expressed by its support weight distributions. For the ensemble of all $[n,k]$ linear codes, the average decoding error probability under the maximum likelihood decoding and the average unsuccessful decoding probability under unambiguous decoding are determined. Furthermore, the error exponent of the average unsuccessful decoding probability under the unambiguous decoding is determined for the ensemble of all $[n,nR]$ linear codes.

Massively Parallel Tree Search for High-Dimensional Sphere Decoders

The recent paradigm shift towards the transmission of large numbers of mutually interfering information streams, as in the case of aggressive spatial multiplexing, combined with requirements towards very low processing latency despite the frequency plateauing of traditional processors, initiates a need to revisit the fundamental maximum-likelihood (ML) and, consequently, the sphere-decoding (SD) detection problem. This work presents the design and VLSI architecture of MultiSphere; the first method to massively parallelize the tree search of large sphere decoders in a nearly-concurrent manner, without compromising their maximum-likelihood performance, and by keeping the overall processing complexity comparable to that of highly-optimized sequential sphere decoders. For a $10\times 10$10×10 MIMO spatially multiplexed system with 16-QAM modulation and 32 processing elements, our MultiSphere architecture can reduce latency by $29\times$29× against well-known sequential SDs, approaching the processing latency of linear detection methods, without compromising ML optimality. In MIMO multicarrier systems targeting exact ML decoding, MultiSphere achieves processing latency and hardware efficiency that are orders of magnitude improved compared to approaches employing one SD per subcarrier. In addition, for $16\!\times \!16$16×16 both “hard”- and “soft”-output MIMO systems, approximate MultiSphere versions are shown to achieve similar error rate performance with state-of-the art approximate SDs having akin parallelization properties, by using only one tenth of the processing elements, and to achieve up to approximately $9\!\times$9× increased energy efficiency.

Fast and Scalable Soft Decision Decoding of Linear Block Codes

Ordered statistics-based decoding (OSD), which exhibits a near maximum likelihood decoding performance, suffers from huge computational complexity as the order increases. In this letter, we propose a fast and scalable OSD by considering the OSD as a fast searching problem. In the searching process, if the up-to-date minimum cost value is less than a predicted threshold value, then we can safely skip the search for remaining higher orders. The computational complexity of the proposed algorithm converges quickly to that of order one, regardless of the maximum order. Compared with the probabilistic necessary conditions-based OSD, the proposed algorithm exhibits speed-up gains of a factor of approximately 2,740 (at 3.0dB) for (127,64) BCH codes, with an indistinguishable decoding performance.

Euclidean-Geometry-Based Space-Time Block Coded Spatial Modulation

In this paper, we propose Euclidean geometric schemes to construct the codebooks for the space-time block coded spatial modulation system. The new codebook-constructing schemes are more efficient than the previous exhaustive searching scheme. The result system is referred to as Euclidean-geometry-based space-time block coded spatial modulation (EG-STBC-SM) system. To achieve transmit diversity, two different schemes are introduced to construct sub-codebooks for the EG-STBC-SM system. In the first scheme, active transmits are selected based on the points and $\mu$ -flats of $d$ -dimensional Euclidean geometry over the Galois field $\text{GF}(p^s)$ . Furthermore, the extended orthogonal space-time block codes are used to support more than two active transmit antennas in the EG-STBC-SM system. In the second scheme, the flats of a Euclidean geometry are grouped into a cyclic structure, which makes the encoding of the EG-STBC-SM scheme very easy. In addition, a simple maximum-likelihood decoder is derived in this paper. Bit error rate performance of the EG-STBC-SM system is evaluated via computer simulation and theoretical upper bound. It is shown that the proposed EG-STBC-SM system outperforms various existing multiple-input multiple-output transmission counterparts of the same or lower spectral efficiencies.

New space-time block codes from spectral norm.

Current research proposes a natural environment for space-time codes and a new design criterion is obtained for space-time block codes in multi-antenna communication channels. The objective of this criterion is to minimize the pairwise error probability of the maximum likelihood decoder, endowed with the matrix spectral norm. The random matrix theory is used and an approximation function for the probability density function for the largest eigenvalue of a Wishart Matrix is obtained.

Union Bound Analysis and Code Design for Multilevel Flash Memory Channels

Multilevel flash memories enable multiple bits to be stored in a single memory cell and hence a significant increase of the storage capacity. The multiple bits that are used for labeling the threshold voltage level of a memory cell belong to different pages. A multilevel flash memory channel resembles a multi-user channel with asymmetric noise, while the data of different pages, which are encoded independently, resembles data of multiple users. In this paper, performance analyses are proposed for this channel by using the union bound technique. In particular, we investigated two different binary labeling schemes of a cell level, the Gray labeling and non-Gray labeling with three maximum-likelihood (ML)-based decoding schemes, which are the joint multi-page ML decoding, page-separate ML decoding, and default setting ML decoding. Our analysis reveals an asymptotic diminishing rate of decoding errors as the channel noise approaches zero, based on which the code design criteria are proposed. It is shown theoretically that the Gray mapping has no joint (i.e., multi-page) decoding gain. The corresponding diminishing decoding error rate is dominated by the weakest code in each page and hence a separate decoding scheme is adequate. On the other hand, the non-Gray mapping has a joint decoding gain which means the weak code can exploit the decoding of the strong code and the diminishing rate of its decoding errors is not subject to the cask effect. Therefore, for Gray mapping, a symmetric coding scheme using equal-strength code for each page achieves better error performance, while for non-Gray mapping with joint decoding, the symmetric coding is not necessary. Moreover, by using the asymmetric coding scheme through assigning different code rates to different pages, the non-Gray mapping can achieve higher overall sum rate than Gray mapping with a similar decoding error rate performance.

Deep Learning-Based Sphere Decoding

In this paper, a deep learning (DL)-based sphere decoding algorithm is proposed, where the radius of the decoding hypersphere is learned by a deep neural network (DNN). The performance achieved by the proposed algorithm is very close to the optimal maximum likelihood decoding (MLD) over a wide range of signal-to-noise ratios (SNRs), while the computational complexity, compared to existing sphere decoding variants, is significantly reduced. This improvement is attributed to the DNN’s ability of intelligently learning the radius of the hypersphere used in decoding. The expected complexity of the proposed DL-based algorithm is analytically derived and compared with existing ones. It is shown that the number of lattice points inside the decoding hypersphere drastically reduces in the DL-based algorithm in both the average and worst-case senses. The effectiveness of the proposed algorithm is shown through the simulation for high-dimensional multiple-input multiple-output (MIMO) systems, using high-order modulations.

Analysis of the Block Error Probability of Concatenated Polar Code Ensembles

In this paper, we provide an analysis of the performance of concatenation of polar codes with outer cyclic redundancy check (CRC) codes, separated by an interleaver, in the short and moderate block length regimes. The analysis addresses maximum likelihood decoding as a proxy to the code performance under successive cancellation list decoding. The analysis is carried out by introducing the concatenated polar code (CPC) ensembles, whose distance properties can be analyzed (for sufficiently short block lengths) by means of the uniform interleaver approach. At moderate block lengths, we resort to the Monte Carlo simulations. Results show that if the inner polar code possesses a low minimum distance and the outer CRC code has a sufficiently large amount of redundancy, then the choice of the outer code generator polynomial and the interleaver may yield to a large variability in the performance of the resulting CPC.

Distributed Reed Muller Code with Multiple Relays for Cooperative Broadband Wireless Networks

The authors have investigated the bit-error rate (BER) performance of Reed-Muller coded cooperative single-carrier frequency domain equalization (RMCC-SC-FDE) scheme and Reed-Muller coded cooperativeOFDM(RMCC-OFDM) scheme incorporating multiple relays and multiple antennas. The maximum ratio combining (MRC) technique is utilized for OFDM/SC-FDE signal detection at the destination terminal. The joint soft maximum likelihood decoding (JSMLD) and the joint majority logic decoding (JMLD) are employed at the destination terminal. The deployment of multiple relays have invigorated the BER performance of RMCC-OFDM and RMCC-SC-FDE schemes. Numerical results demonstrate that the RMCC-SC-FDE scheme exhibits a betterBERperformance over theRMCC-OFDM scheme in identical conditions. Furthermore, the simulated results reveal that the RMCC-SC-FDE and RMCC-OFDM schemes not only yield a better BER performance gain over their corresponding non-cooperative coded counterpart schemes but also outperform the turbo coded cooperative SC-FDE and turbo coded cooperative OFDM schemes, respectively, under identical conditions.

Universal Sparse Superposition Codes With Spatial Coupling and GAMP Decoding

Sparse superposition codes, or sparse regression codes, constitute a new class of codes, which was first introduced for communication over the additive white Gaussian noise (AWGN) channel. It has been shown that such codes are capacity-achieving over the AWGN channel under optimal maximum-likelihood decoding as well as under various efficient iterative decoding schemes equipped with power allocation or spatially coupled constructions. Here, we generalize the analysis of these codes to a much broader setting that includes all memoryless channels. We show, for a large class of memoryless channels, that spatial coupling allows an efficient decoder, based on the generalized approximate message-passing (GAMP) algorithm, to reach the potential (or Bayes optimal) threshold of the underlying (or uncoupled) code ensemble. Moreover, we argue that spatially coupled sparse superposition codes universally achieve capacity under GAMP decoding by showing, through analytical computations, that the error floor vanishes and the potential threshold tends to capacity, as one of the code parameters goes to infinity. Furthermore, we provide a closed-form formula for the algorithmic threshold of the underlying code ensemble in terms of Fisher information. Relating an algorithmic threshold to a Fisher information has theoretical as well as practical importance. Our proof relies on the state evolution analysis and uses the potential method developed in the theory of low-density parity-check (LDPC) codes and compressed sensing.

An improved robust image-adaptive watermarking with two watermarks using statistical decoder

This paper presents an improved image-adaptive watermarking technique. Two image watermarks are embedded in the high entropy 8 × 8 blocks of the host image. DWT is applied on these blocks using the principle of sub band coding. This decomposes the high entropy blocks into four sub band coefficients, wherein the approximation and vertical frequency coefficients are modeled using Gaussian (or Normal) distribution. The two watermarks are inserted in the host image using Adjustable Strength Factor (ASF). It is calculated adaptively using the fourth statistical moment known as kurtosis. A limited side information is also transmitted along with the watermarked image. This side information consists of high entropy block positions and Gaussian distribution parameters. To extract both watermarks from the received watermarked image, the high entropy block positions sent in the side information help in applying DWT to calculate the approximation and vertical frequency coefficients. Gaussian (or Normal) distribution is similarly used for modeling and calculating the distribution parameters. This helps the Maximum Likelihood (ML) decoder to recover the watermarks successfully using a statistical approach. Two important contributions are presented in this paper. Firstly, adjustable kurtosis values are used which improves the capacity and robustness of the proposed technique. Secondly, the proposed work is implemented on medical applications and gives better performance as compared to the existing methods. Further, the efficiency of the proposed work is evaluated by better simulation results using PSNR, NCC, SSIM and GMSD under different attacks. The technique is highly robust as watermarks survive under different attacks. This increases security and ensures copyright protection.

Multiple relay‐based Reed–Muller network‐coded cooperation for wireless communication system

This research article proposes a multiple relay-based Reed–Muller network-coded cooperative (MR-RMNCC) scheme for wireless communication. In the MR-RMNCC scheme, Hadamard transform construction is adopted to build Reed–Muller (RM) codes that effectively achieve network coded-cooperation among multiple sources and multiple relays over slow and fast Rayleigh fading channels. In any network coded-cooperative system, a proper encoding strategy at the relay nodes is vital for achieving an optimum code construction at the destination node. Therefore, the authors have proposed an efficient message bit selection algorithm at relay-2 that optimally selects the message bits. The supremacy of the message bit selection algorithm is demonstrated by Monte–Carlo simulation. Furthermore, the performance bound for average error-probability is also derived for the MR-RMNCC scheme that validates its simulated bit-error-rate performance over a fast Rayleigh fading channel. At the destination node, the MR-RMNCC scheme is jointly decoded by two different decoding strategies, i.e. joint soft maximum likelihood decoding and joint majority logic decoding. Monte–Carlo simulations suggest that the MR-RMNCC scheme not only provides a significant bit-error-rate performance gain over its corresponding coded non-cooperative scheme but also to the existing state-of-the-art RM network-coded cooperative scheme under identical conditions.

Energy-Efficient and Low-Latency Massive SIMO Using Noncoherent ML Detection for Industrial IoT Communications

To enable ultrareliable low-latency wireless communications required in the Industrial Internet of Things, in this paper we develop an energy-based modulation [i.e., non-negative pulse amplitude modulation (PAM)] constellation design framework for noncoherent detection in massive single-input multiple-output (SIMO) systems. We consider that one single-antenna transmitter communicates to a receiver with a large number of antennas over a Rayleigh fading channel, and the receiver decodes the transmitted information at the end of every symbol. For such an SIMO system with non-negative PAM modulation, we first propose a fast noncoherent maximum-likelihood decoding algorithm and derive a closed-form expression of its symbol error probability (SEP). We then enhance the system energy efficiency by finding the optimal PAM constellation that minimizes the exact SEP subject to a total signal power constraint for such a system with an arbitrary number of receiver antennas, signal-to-noise ratio (SNR), and constellation size. Furthermore, the closed-form upper and lower bounds on the optimal SEP are derived. Based on these bounds, the exact expression for coding gain of the dominant term of the SEP is presented for such an optimal massive SIMO system. We also present an asymptotic SEP expression at a high SNR regime and the approximate diversity gain of the system. Simulation results for the proposed optimal PAM constellation validate the theoretical analysis, and show that our presented optimal constellation attains significant performance gains over the currently available minimum-distance-based constellation systems.

High performance demodulation method with less complexity for image-sensor communication.

This study presents a novel method for signal demodulation for use with visible light communication systems composed of an image sensor as a receiver and light-emitting diode (LED) transmitters. Demodulation is a central challenge in the design of such a system, as the image captured at the image-sensor receiver is deteriorated by distance and noise. We propose a demodulation method that offers performance approaching that of the maximum-likelihood decoding (MLD) method and with significantly less complexity. The proposed method first applies the minimum mean square error (MMSE) method to each LED into reliable LEDs and unreliable LEDs according to the MMSE results and it demodulates the LEDs judged as reliable directly. Then, the MLD method is applied only to the unreliable LEDs to demodulate their signals. The results of numerical simulations and lab experiments are presented to evaluate the performance of this modified demodulation method.

Capacity-Achieving Guessing Random Additive Noise Decoding

We introduce a new algorithm for realizing Maximum Likelihood (ML) decoding in discrete channels with or without memory. In it, the receiver rank orders noise sequences from most likely to least likely. Subtracting noise from the received signal in that order, the first instance that results in a member of the code-book is the ML decoding. We name this algorithm GRAND for Guessing Random Additive Noise Decoding. We establish that GRAND is capacity-achieving when used with random code-books. For rates below capacity we identify error exponents, and for rates beyond capacity we identify success exponents. We determine the scheme's complexity in terms of the number of computations the receiver performs. For rates beyond capacity, this reveals thresholds for the number of guesses by which if a member of the code-book is identified it is likely to be the transmitted code-word. We introduce an approximate ML decoding scheme where the receiver abandons the search after a fixed number of queries, an approach we dub GRANDAB, for GRAND with ABandonment. While not an ML decoder, we establish that the algorithm GRANDAB is also capacity-achieving for an appropriate choice of abandonment threshold, and characterize its complexity, error and success exponents. Worked examples are presented for Markovian noise that indicate these decoding schemes substantially out-perform the brute force decoding approach.

LDPC Codes Over the $q$ -ary Multi-Bit Channel

In this paper, we introduce a new channel model termed as the $q$ -ary multi-bit channel. This channel models a memory device, where $q$ -ary symbols ( $q=2^{s}$ ) are stored in the form of current/voltage levels. The symbols are read in a measurement process, which provides a symbol bit in each measurement step, starting from the most significant bit. An error event occurs when not all the symbol bits are known. To deal with such error events, we use GF( $q$ ) low-density parity-check (LDPC) codes and analyze their decoding performance. We start with iterative-decoding threshold analysis and derive optimal edge-label distributions for maximizing the decoding threshold. We later move to a finite-length iterative-decoding analysis and propose an edge-labeling algorithm for the improved decoding performance. We then provide a finite-length maximum-likelihood decoding analysis for both the standard non-binary random ensemble and LDPC ensembles. Finally, we demonstrate by simulations that the proposed edge-labeling algorithm improves the finite-length decoding performance by orders of magnitude.

On Decoupled Decoding of Quasi-Orthogonal STBCs Using Quaternion Algebra

Receiver complexity is an important performance metric in space time block coding (STBC). After comprehensive study and analysis of STBCs over complex domain, quaternion orthogonal designs (QODs) have been explored that can provide less decoder complexity. However, some researchers argue that existing maximum likelihood (ML) decoding rule for QODs does not yield optimal performance for quasi-orthogonal STBC-based QODs and, therefore, the design of optimal decoder for such designs remains a big challenge of this field. To counter this observation, we modify the system model slightly. In addition to this, we present a new generalized QOD construction that can generate QODs for any number of transmit antennas. These codes are quasi-orthogonal in complex domain and provide code rates higher than their respective complex orthogonal designs (CODs). We show that the proposed ML decoder provides optimal linear decoupled decoding solution for these codes because of their distinctive construction. Simulation results verify the optimal performance of the ML decoder for the proposed codes and also show that the diversity of the proposed quaternion-based codes is same as CODs, however, they provide better code rates.

Expurgated Bounds for the Asymmetric Broadcast Channel

This work contains two main contributions concerning the expurgation of hierarchical ensembles for the asymmetric broadcast channel. The first is an analysis of the optimal maximum likelihood (ML) decoders for the weak and strong user. Two different methods of code expurgation will be used, that will provide two competing error exponents. The second is the derivation of expurgated exponents under the generalized stochastic likelihood decoder (GLD). We prove that the GLD exponents are at least as tight as the maximum between the random coding error exponents derived in an earlier work by Averbuch and Merhav (2017) and one of our ML-based expurgated exponents. By that, we actually prove the existence of hierarchical codebooks that achieve the best of the random coding exponent and the expurgated exponent simultaneously for both users.

Improved LAS detector for MIMO systems with imperfect channel state information

Likelihood ascent search (LAS) detector is a neighbourhood search algorithm and one of the simplest schemes for low-complexity near-optimal detection in massive multiple-input multiple-output systems. LAS detector design under the assumption of perfect channel state information has been an area of research for decades. However, channel estimation errors have never been taken into account by conventional LAS detectors when calculating the maximum-likelihood decoding metric. As a result, the bit error rate performance of LAS detectors can be significantly degraded. This study proposes robust LAS detectors which take channel estimation errors into account in the computation of the ML decoding metric. The proposed approach involves the computation of the equivalent noise covariance matrix inverse, which may increases the computational complexity. Therefore, the authors also propose a low complexity method to inverse the covariance matrix. Simulation results show that the proposed schemes outperform the conventional LAS detector.