Alphabet Size Research Articles

Computing MEMs and Relatives on Repetitive Text Collections

We consider the problem of computing the Maximal Exact Matches (MEMs) of a given pattern \(P[1\mathinner{.. }m]\) on a large repetitive text collection \(T[1\mathinner{.. }n]\) over an alphabet of size \(\sigma\) , which is represented as a (hopefully much smaller) run-length context-free grammar of size \(g_{rl}\) . We show that the problem can be solved in time \(O(m^{2}\log^{\epsilon}n)\) , for any constant \(\epsilon\gt0\) , on a data structure of size \(O(g_{rl})\) . Further, on a locally consistent grammar of size \(O(\delta\log\frac{n\log\sigma}{\delta\log n})\) , the time decreases to \(O(m\log m(\log m+\log^{\epsilon}n))\) . The value \(\delta\) is a function of the substring complexity of \(T\) and \(\Omega(\delta\log\frac{n\log\sigma}{\delta\log n})\) is a tight lower bound on the compressibility of repetitive texts \(T\) , so our structure has optimal size in terms of \(n\) , \(\sigma\) , and \(\delta\) . We extend our results to several related problems, such as finding \(k\) -MEMs, MUMs, rare MEMs, and applications. Categories and Subject Descriptors: E.1 [Data structures] ; E.2 [Data storage representations] ; E.4 [Coding and information theory]: Data compaction and compression; F.2.2 [Analysis of algorithms and problem complexity] : Nonnumerical algorithms and problems— Pattern matching, Computations on discrete structures, Sorting and searching

Capacity of an infinite family of networks related to the diamond network for fixed alphabet sizes

Capacity of an infinite family of networks related to the diamond network for fixed alphabet sizes

Matrix hypercontractivity, streaming algorithms and LDCs: the large alphabet case

We prove a hypercontractive inequality for matrix-valued functions defined over large alphabets. In order to do so, we prove a generalization of the powerful 2-uniform convexity inequality for trace norms of Ball, Carlen, Lieb (Inventiones Mathematicae’94). Using our hypercontractive inequality, we present upper and lower bounds for the communication complexity of the Hidden Hypermatching problem defined over large alphabets. We then consider streaming algorithms for approximating the value of Unique Games on a hypergraph with t -size hyperedges. By using our communication lower bound, we show that every streaming algorithm in the adversarial model achieving an ( r − ε)-approximation of this value requires Ω ( n 1 − 2/ t ) quantum space, where r is the alphabet size. We next present a lower bound for locally decodable codes ( \(\mathsf {LDC} \) ) \(\mathbb {Z}_r^n\rightarrow \mathbb {Z}_r^N \) over large alphabets with recoverability probability at least 1/ r + ε. Using hypercontractivity, we give an exponential lower bound \(N= 2^{\Omega (\varepsilon ^4 n/r^4)} \) for 2-query (possibly non-linear) \(\mathsf {LDC} \) s over \(\mathbb {Z}_r \) and using the non-commutative Khintchine inequality we prove an improved lower bound of \(N= 2^{\Omega (\varepsilon ^2 n/r^2)} \) .

Slowing down top trees for better worst-case compression

We consider the top tree compression scheme introduced by Bille et al. [ICALP 2013] and construct an infinite family of trees on n nodes labeled from an alphabet of size σ, for which the size of the top DAG is Θ(nlogσ⁡nlog⁡logσ⁡n). Our construction matches a previously known upper bound and exhibits a weakness of this scheme, as the information-theoretic lower bound is Ω(nlogσ⁡n). Lohrey et al. [IPL 2019] designed a more involved version of the original algorithm achieving the lower bound. We show that this can be also guaranteed by a very minor modification of the original scheme: informally, one only needs to ensure that different parts of the tree are not compressed too quickly. Arguably, our version is more uniform, and in particular, the compression procedure is oblivious to the value of σ.

Linear time online algorithms for constructing linear-size suffix trie

The suffix trees are fundamental data structures for various kinds of string processing. The suffix tree of a text string T of length n has O(n) nodes and edges, and the string label of each edge is encoded by a pair of positions in T. Thus, even after the tree is built, the input string T needs to be kept stored and random access to T is still needed. The linear-size suffix tries (LSTs), proposed by Crochemore et al. [Linear-size suffix tries, TCS 638:171-178, 2016], are a “stand-alone” alternative to the suffix trees. Namely, the LST of an input text string T of length n occupies O(n) total space, and supports pattern matching and other tasks with the same efficiency as the suffix tree without the need to store the input text string T. Crochemore et al. proposed an offline algorithm which transforms the suffix tree of T into the LST of T in O(nlog⁡σ) time and O(n) space, where σ is the alphabet size. In this paper, we present two types of online algorithms which “directly” construct the LST, from right to left, and from left to right, without constructing the suffix tree as an intermediate structure. Both algorithms construct the LST incrementally when a new symbol is read, and do not access the previously read symbols. Both of the right-to-left construction algorithm and the left-to-right construction algorithm work in O(nlog⁡σ) time and O(n) space. The main feature of our algorithms is that the input text string does not need to be stored.

Limitations of the Impagliazzo–Nisan–Wigderson Pseudorandom Generator Against Permutation Branching Programs

AbstractThe classic Impagliazzo–Nisan–Wigderson (INW) pseudorandom generator (PRG) (STOC ‘94) for space-bounded computation uses a seed of length $$O(\log n \cdot \log (nw/\varepsilon )+\log d)$$ O ( log n · log ( n w / ε ) + log d ) to fool ordered branching programs of length n, width w, and alphabet size d to within error $$\varepsilon $$ ε . A series of works have shown that the analysis of the INW generator can be improved for the class of permutation branching programs or the more general regular branching programs, improving the $$O(\log ^2 n)$$ O ( log 2 n ) dependence on the length n to $$O(\log n)$$ O ( log n ) or $${\tilde{O}}(\log n)$$ O ~ ( log n ) . However, when also considering the dependence on the other parameters, these analyses still fall short of the optimal PRG seed length $$O(\log (nwd/\varepsilon ))$$ O ( log ( n w d / ε ) ) . In this paper, we prove that any “spectral analysis” of the INW generator requires seed length $$\begin{aligned} \Omega \left( \log n\cdot \log \log \left( \min \{n,d\}\right) +\log n\cdot \log \left( w/\varepsilon \right) +\log d\right) \end{aligned}$$ Ω log n · log log min { n , d } + log n · log w / ε + log d to fool ordered permutation branching programs of length n, width w, and alphabet size d to within error $$\varepsilon $$ ε . By “spectral analysis” we mean an analysis of the INW generator that relies only on the spectral expansion of the graphs used to construct the generator; this encompasses all prior analyses of the INW generator. Our lower bound matches the upper bound of Braverman–Rao–Raz–Yehudayoff (FOCS 2010, SICOMP 2014) for regular branching programs of alphabet size $$d=2$$ d = 2 except for a gap between their $$O\left( \log n \cdot \log \log n\right) $$ O log n · log log n term and our $$\Omega \left( \log n \cdot \log \log \min \{n,d\}\right) $$ Ω log n · log log min { n , d } term. It also matches the upper bounds of Koucký–Nimbhorkar–Pudlák (STOC 2011), De (CCC 2011), and Steinke (ECCC 2012) for constant-width ($$w=O(1)$$ w = O ( 1 ) ) permutation branching programs of alphabet size $$d=2$$ d = 2 to within a constant factor. To fool permutation branching programs in the measure of spectral norm, we prove that any spectral analysis of the INW generator requires a seed of length $$\Omega \left( \log n\cdot \log \log n+\log n\cdot \log (1/\varepsilon )\right) $$ Ω log n · log log n + log n · log ( 1 / ε ) when the width is at least polynomial in n ($$w=n^{\Omega (1)}$$ w = n Ω ( 1 ) ), matching the recent upper bound of Hoza–Pyne–Vadhan (ITCS 2021) to within a constant factor.

Streaming approximation resistance of every ordering CSP

An ordering constraint satisfaction problem (OCSP) is defined by a family F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal F$$\\end{document} of predicates mapping permutations on {1,…,k}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{1,\\ldots,k\\}$$\\end{document} to {0,1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{0,1\\}$$\\end{document}. An instance of Max-OCSP(F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal F$$\\end{document}) on n variables consists of a list of constraints, each consisting of a predicate from F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal F$$\\end{document} applied on k distinct variables. The goal is to find an ordering of the n variables that maximizes the number of constraints for which the induced ordering on the k variables satisfies the predicate. OCSPs capture well-studied problems including ‘maximum acyclic subgraph’ (MAS) and “maximum betweenness”. In this work, we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, when an instance is presented as a stream of constraints. We show that for every F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal F$$\\end{document}, Max-OCSP(F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal F$$\\end{document}) is approximation-resistant to o(n)-space streaming algorithms, i.e., algorithms using o(n) space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of MAS, our result shows that for every ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon>0$$\\end{document}, MAS is not (1/2+ϵ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1/2+\\epsilon)$$\\end{document}-approximable in o(n) space. The previous best inapproximability result, due to Guruswami & Tao (2019), only ruled out 3/4-approximations in o(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$o(\\sqrt n)$$\\end{document} space. Our results build on recent works of Chou et al. (2022b, 2024) who provide a tight, linear-space inapproximability theorem for a broad class of “standard” (i.e., non-ordering) constraint satisfaction problems (CSPs) over arbitrary (finite) alphabets. Our results are obtained by building a family of appropriate standard CSPs (one for every alphabet size q) from any given OCSP and applying their theorem to this family of CSPs. To convert the resulting hardness results for standard CSPs back to our OCSP, we show that the hard instances from this earlier theorem have the following “partition expansion” property with high probability: For every partition of the n variables into small blocks, for most of the constraints, all variables are in distinct blocks.

Perfect Codes over Non-Prime Power Alphabets: An Approach Based on Diophantine Equations

Perfect error-correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of e-error correcting perfect codes over q-ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect e-error-correcting codes were found if e≥3, and it was conjectured that no perfect 2-error-correcting codes exist over any q-ary alphabet, where q>3. In the 1970s, this was proved for q a prime power, for q=2r3s and for only seven other values of q. Almost 50 years later, it is surprising to note that there have been no new results in this regard and the classification of 2-error-correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of the generalised Ramanujan–Nagell equation and from modern computational number theory to show that perfect 2-error-correcting codes do not exist for 172 new values of q which are not prime powers, substantially increasing the values of q which are now classified. In addition, we prove that, for any fixed value of q, there can be at most finitely many perfect 2-error-correcting codes over an alphabet of size q.

Pattern matching algorithms in blockchain for network fees reduction

Blockchain received a vast amount of attention in recent years and is still growing. The second generation of blockchain, such as Ethereum, allows execution of almost any program in EVM, making it a global protocol for distributed applications. The code deployment and each operation performed in EVM cost the network fee called gas, whose price varies and can be significant. That is why code optimization and well-chosen algorithms are crucial in programming on the blockchain. This paper evaluates the gas usage of several exact pattern matching algorithms on the EVM. We also propose an efficient implementation of the algorithms in the Solidity/YUL language. We evaluate the gas fees of all the algorithms for different parameters (such as pattern length, alphabet size, and text size). We show a significant gas fee and execution time reduction with up to 22-fold lower gas usage and 55-fold speed-up compared to StringUtils (a popular Solidity string library).

On the parameterized complexity of freezing dynamics

In this paper we establish how alphabet size, treewidth and maximum degree of the underlying graph are key parameters which influence the overall computational complexity of finite freezing automata networks. First, we define a general decision problem, called Specification Checking Problem, that captures many classical decision problems such as prediction, nilpotency, predecessor, asynchronous reachability.Then, we present a fast-parallel algorithm that solves the general model checking problem when the three parameters are bounded, hence showing that the problem is in NC. Moreover, we show that the problem is in XP on the parameters tree-width and maximum degree.Finally, we show that these problems are hard from two different perspectives. First, the general problem is W[2]-hard when taking either treewidth or alphabet as single parameter and fixing the others. Second, the classical problems are hard in their respective classes when restricted to families of graph with sufficiently large treewidth.

On the Practical Power of Automata in Pattern Matching

Many papers in the intersection of theoretical and applied algorithms show that the simple, asymptotically less efficient algorithm, performs better than the bestcomplex theoretical algorithms on random data or in specialized “real world” applications. This paper considers the Knuth–Morris–Pratt automaton, and shows a counter-intuitive practical result. The classical pattern matching paradigm is that of seeking occurrences of one string—the pattern, in another—the text, where both strings are drawn from an alphabet set Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma$$\\end{document}. Assuming the text length is n and the pattern length is m, this problem can naively be solved in time O(nm). In Knuth, Morris and Pratt’s seminal paper of 1977, an automaton, was developed that allows solving this problem in time O(n) for any alphabet. This automaton, which we will refer to as the KMP-automaton, has proven useful in solving many other problems. A notable example is the parameterized pattern matching model. In this model, a consistent renaming of symbols from Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma$$\\end{document} is allowed in a match. The parameterized matching paradigm has proven useful in problems in software engineering, computer vision, and other applications. It has long been believed that for texts where the symbols are uniformly random, the naive algorithm will perform as well as the KMP algorithm. In this paper, we examine the practical efficiency of the KMP algorithm versus the naive algorithm on a randomly generated text. We analyze the time under various parameters, such as alphabet size, pattern length, and the distribution of pattern occurrences in the text. We do this for both the original exact matching problem and parameterized matching. While the folklore wisdom is vindicated by these findings for the exact matching case, surprisingly, the KMP algorithm always works significantly faster than the naive in the parameterized matching case. We check this hypothesis for DNA texts and image data and observe a similar behavior as in the random text. We also show a very structured exact matching case where the automaton is much more efficient.

Neural Joint Entropy Estimation.

Estimating the entropy of a discrete random variable is a fundamental problem in information theory and related fields. This problem has many applications in various domains, including machine learning, statistics, and data compression. Over the years, a variety of estimation schemes have been suggested. However, despite significant progress, most methods still struggle when the sample is small, compared to the variable's alphabet size. In this work, we introduce a practical solution to this problem, which extends the work of McAllester and Statos. The proposed scheme uses the generalization abilities of cross-entropy estimation in deep neural networks (DNNs) to introduce improved entropy estimation accuracy. Furthermore, we introduce a family of estimators for related information-theoretic measures, such as conditional entropy and mutual information (MI). We show that these estimators are strongly consistent and demonstrate their performance in a variety of use cases. First, we consider large alphabet entropy estimation. Then, we extend the scope to MI estimation. Next, we apply the proposed scheme to conditional MI estimation, as we focus on independence testing tasks. Finally, we study a transfer entropy (TE) estimation problem. The proposed estimators demonstrate improved performance compared to existing methods in all of these setups.

SARN: Script-Aware Recognition Network for scene multilingual text recognition

Recent research in script-specific text recognition has achieved remarkable performance on scene text images. However, the challenges of multilingual text in real-world images have not been effectively addressed, thereby limiting the practical application. For multilingual text recognition, the prevailing single-head scheme that uses the same recognition head for all scripts suffers from the issues of enormous alphabet size and linguistic knowledge discrepancies. Meanwhile, the multi-head scheme, which assigns distinct heads for individual scripts, encounters error accumulation within the two-step inference process. In this paper, we explore the influence of script attribute information within existing methods on achieving accurate recognition. Correspondingly, we propose a script-aware recognition network encompassing script identification and text recognition, which leverages script information to make character features more discriminative in text recognition. In particular, we design the script-aware module to utilize script information at the local level, global level, and their combination, respectively. Furthermore, our method is extended to the multi-head scheme, utilizing mutual assistance between script and character features to improve the performance in both script identification and text recognition tasks. Extensive experiments demonstrate that our method obtains superior results in multilingual text recognition, and reveal the importance of script information in alleviating the issues of single-head and multi-head recognition scheme.

Be My Guesses: The interplay between side-channel leakage metrics

In a theoretical context of side-channel attacks, optimal bounds between success rate, guessing entropy and statistical distance are derived with a simple majorization (Schur-concavity) argument. They are further theoretically refined for different versions of the classical Hamming weight leakage model, in particular assuming a priori equiprobable secret keys and additive white Gaussian measurement noise. Closed-form expressions and numerical computation are given. A study of the impact of the choice of the substitution box with respect to side-channel resistance reveals that its nonlinearity tends to homogenize the expressivity of success rate, guessing entropy and statistical distance. The intriguing approximate relation between guessing entropy and success rate GE=1/SR is observed in the case of 8-bit bytes and low noise. The exact relation between guessing entropy, statistical distance and alphabet size GE=M+12−M2SD for deterministic leakages and equiprobable keys is proved.

Symbolic Aggregate Approximation-Based Clustering of Monthly Natural Gas Consumption

Natural gas is an indispensable non-renewable energy source for many countries. It is used in many different areas such as heating and kitchen appliances in homes, and heat treatment and electricity generation in industry. Natural gas is an essential component of the transportation sector, providing a cleaner alternative to traditional fuels in vehicles and fleets. Moreover, natural gas plays a vital role in boosting energy efficiency through the development of combined heat and power systems. These systems produce electricity and useful heat concurrently. As nations move towards more sustainable energy solutions, natural gas has gained prominence as a transitional fuel. This is due to its lower carbon emissions when compared to coal and oil, thus making it an essential component of the global energy framework. In this study, monthly natural gas consumption data of 28 different European countries between 2014 and 2022 are used. Symbolic Aggregate Approximation method is used to analyse the data. Analyses are made with different numbers of segments and numbers of alphabet sizes, and alphabet vectors of each country are created. These letter vectors are used in hierarchical clustering and dendrogram graphs are created. Furthermore, the elbow method is used to determine the appropriate number of clusters. Clusters of countries are created according to the determined number of clusters. In addition, it is interpreted according to the consumption trends of the countries in the determined clusters.

Confidence Intervals for Parameters of Unobserved Events

Consider a finite sample from an unknown distribution over a countable alphabet. Unobserved events are alphabet symbols which do not appear in the sample. Estimating the probabilities of unobserved events is a basic problem in statistics and related fields, which was extensively studied in the context of point estimation. In this work we introduce a novel interval estimation scheme for unobserved events. Our proposed framework applies selective inference, as we construct confidence intervals (CIs) for the desired set of parameters. Interestingly, we show that obtained CIs are dimension-free, as they do not grow with the alphabet size. Further, we show that these CIs are (almost) tight, in the sense that they cannot be further improved without violating the prescribed coverage rate. We demonstrate the performance of our proposed scheme in synthetic and real-world experiments, showing a significant improvement over the alternatives. Finally, we apply our proposed scheme to large alphabet modeling. We introduce a novel simultaneous CI scheme for large alphabet distributions which outperforms currently known methods while maintaining the prescribed coverage rate. Supplementary materials for this article are available online including a standardized description of the materials available for reproducing the work.

Good-bootstrap: simultaneous confidence intervals for large alphabet distributions

Simultaneous confidence intervals (SCI) for multinomial proportions are a corner stone in count data analysis and a key component in many applications. A variety of schemes were introduced over the years, mostly focussing on an asymptotic regime where the sample is large and the alphabet size is relatively small. In this work we introduce a new SCI framework which considers the large alphabet setup. Our proposed framework utilises bootstrap sampling with the Good-Turing probability estimator as a plug-in distribution. We demonstrate the favourable performance of our proposed method in synthetic and real-world experiments. Importantly, we provide an exact analytical expression for the bootstrapped statistic, which replaces the computationally costly sampling procedure. Our proposed framework is publicly available at the first author's Github page.

In search of maximum non-overlapping codes

AbstractNon-overlapping codes are block codes that have arisen in diverse contexts of computer science and biology. Applications typically require finding non-overlapping codes with large cardinalities, but the maximum size of non-overlapping codes has been determined only for cases where the codeword length divides the size of the alphabet, and for codes with codewords of length two or three. For all other alphabet sizes and codeword lengths no computationally feasible way to identify non-overlapping codes that attain the maximum size has been found to date. Herein we characterize maximal non-overlapping codes. We formulate the maximum non-overlapping code problem as an integer optimization problem and determine necessary conditions for optimality of a non-overlapping code. Moreover, we solve several instances of the optimization problem to show that the hitherto known constructions do not generate the optimal codes for many alphabet sizes and codeword lengths. We also evaluate the number of distinct maximum non-overlapping codes.

PEAK-TO-AVERAGE POWER RATIO OF SPECTRALLY EFFICIENT FDM SIGNALS

The change of the number of subcarriers in SEFDM signal from 5 to 1024 leads to the growth of the PAPR (peak-to-average power ratio) from 4.5 to 9 dB. PAPR of SEFDM signals stays about constant (instability is no more than 5%) while normalized subcarriers frequency spacing increase from 1/2 to 1. It indicates no additional energy loss during the transition from OFDM signals to SEFDM signals, caused by an increase in the PAPR. It is proposed to use a part of the resulting gain in the frequency band to reduce PAPR. The method, based on modification of Tone Insertion method, was developed. Developed method consists in adding additional subcarriers with randomly generated manipulation symbols to the SEFDM signal. SEFDM symbol with minimal PAPR is chosen to be send. According to this method the PAPR can be reduced in average of 1.4 dB. The degree of reduction of the PAPR depends on the size of the channel alphabet of additional subcarriers (changing of the average win is 0.6 dB for BPSK to 1.4 dB for QAM-64) and the number of combinations of symbols of additional subcarriers (1.6 dB for 1000 combinations for QAM-64). According to estimation of the FPGA resources for implementation of PAPR reduction algorithm it is required relatively small resource costs compared to the formation of SEFDM signal without reduction.

Minimizing the Alphabet Size in Codes with Restricted Error Sets

Minimizing the Alphabet Size in Codes with Restricted Error Sets