Finite-length Sequences Research Articles

An Input-Output Construction of Finite State <inline-formula> <tex-math notation="TeX">$\rho/\mu$</tex-math></inline-formula> Approximations for Control Design

We consider discrete-time plants that interact with their controllers via fixed discrete alphabets. For this class of systems, and in the absence of exogenous inputs, we propose a general, methodical approach for constructing a sequence of finite state approximate models starting from finite length sequences of input and output signal pairs. We explicitly derive conditions under which the proposed construct, used in conjunction with a particular generalized structure, satisfies desirable properties of ρ/μ approximations thereby leading to nominal deterministic finite state machine models that can be used in certified-by-design controller synthesis. We also show that the cardinality of the minimal disturbance alphabet that can be used in this setting equals that of the sensor output alphabet. Finally, we show that the proposed construct satisfies a relevant semi-completeness property.

Data Requirement for Phylogenetic Inference from Multiple Loci: A New Distance Method.

We consider the problem of estimating the evolutionary history of a set of species (phylogeny or species tree) from several genes. It is known that the evolutionary history of individual genes (gene trees) might be topologically distinct from each other and from the underlying species tree, possibly confounding phylogenetic analysis. A further complication in practice is that one has to estimate gene trees from molecular sequences of finite length. We provide the first full data-requirement analysis of a species tree reconstruction method that takes into account estimation errors at the gene level. Under that criterion, we also devise a novel reconstruction algorithm that provably improves over all previous methods in a regime of interest.

Enforcing Meter in Finite-Length Markov Sequences

Markov processes are increasingly used to generate finite-length sequences that imitate a given style. However, Markov processes are notoriously difficult to control. Recently, Markov constraints have been introduced to give users some control on generated sequences. Markov constraints reformulate finite-length Markov sequence generation in the framework of constraint satisfaction (CSP). However, in practice, this approach is limited to local constraints and its performance is low for global constraints, such as cardinality or arithmetic constraints. This limitation prevents generated sequences to follow structural properties which are independent of the style, but inherent to the domain, such as meter. In this article, we introduce meter, a constraint that ensures a sequence is 1) Markovian with regards to a given corpus and 2) follows metrical rules expressed as cumulative cost functions. Additionally, meter can simultaneously enforce cardinality constraints. We propose a domain consistency algorithm whose complexity is pseudo-polynomial. This result is obtained thanks to a theorem on the growth of sumsets by Khovanskii. We illustrate our constraint on meter-constrained music generation problems that were so far not solvable by any other technique.

Hilbert space formulation of symbolic systems for signal representation and analysis

Probabilistic finite state automata (PFSA) have been widely used as an analysis tool for signal representation and modeling of physical systems. This paper presents a new method to address these issues by bringing in the notion of vector-space formulation of symbolic systems in the setting of PFSA. In this context, a link is established between the formal language theory and functional analysis by defining an inner product space over a class of stochastic regular languages, represented by PFSA models that are constructed from finite-length symbol sequences. The norm induced by the inner product is interpreted as a measure of the information contained in the respective PFSA. Numerical examples are presented to illustrate the computational steps in the proposed method and to demonstrate model order reduction via orthogonal projection from a general Hilbert space of PFSA onto a (closed) Markov subspace that belongs to a class of shifts of finite type. These concepts are validated by analyzing time series of ultrasonic signals, collected from an experimental apparatus, for fatigue damage detection in polycrystalline alloys.

Parallel Hardware Implementation of Convolution using Vedic Mathematics

Convolution is fundamental operation of most of the signal processing systems. It is necessity of time to speed up convolution process at very appreciable extent. Here Direct method of computing the discrete linear convolution of finite length sequences is used. The approach is easy to learn because of the similarities to computing the multiplication of two numbers by a pencil and paper calculation. Multipliers are basic building blocks of convolver. Since it dominates most of the execution time, for optimizing the speed, 4×4 bit Vedic multipliers based on Urdhva Tiryagbhyam sutra are used. Convolver has delay of 17.996 ns when implemented on 90 nm process technology FPGA. It also provides necessary modularity, expandability, and regularity to form different convolutions for any number of bits. The coding is done in VHDL (Very High Speed Integrated Circuits Hardware Description Language) for the FPGA , as it is being increasingly used for variety of computationally intensive applications. Simulation and synthesis is done using Xilinx 9.2i.

Deterministic Designs with Deterministic Guarantees: Toeplitz Compressed Sensing Matrices, Sequence Design and System ID

In this paper we present a new family of discrete sequences having random like uniformly decaying auto-correlation properties. The new class of infinite length sequences are higher order chirps constructed using irrational numbers. Exploiting results from the theory of continued fractions and diophantine approximations, we show that the class of sequences so formed has the property that the worst-case auto-correlation coefficients for every finite length sequence decays at a polynomial rate. These sequences display doppler immunity as well. We also show that Toeplitz matrices formed from such sequences satisfy restricted-isometry-property (RIP), a concept that has played a central role recently in Compressed Sensing applications. Compressed sensing has conventionally dealt with sensing matrices with arbitrary components. Nevertheless, such arbitrary sensing matrices are not appropriate for linear system identification and one must employ Toeplitz structured sensing matrices. Linear system identification plays a central role in a wide variety of applications such as channel estimation for multipath wireless systems as well as control system applications. Toeplitz matrices are also desirable on account of their filtering structure, which allows for fast implementation together with reduced storage requirements.

Model reduction and composite control for overhead hoist transport system by singular perturbation technique

An innovative Overhead Hoist Transport (OHT) system is proposed and analyzed to transport fragile semi-finished products in factories. A triplet of double-link arm is used to carry the load, in replacement of the cables used conventionally. Unlike conventional OHT, the proposed OHT exhibits superior capability for high-speed transportation, flexible stiffness and is able to account for the inherent auto-sway characteristics and parameters uncertainties of the OHT system. The three-time-scale plant model of the OHT system, including the drive motors, flexible links and rigid links, is developed. By singular perturbation order-reduction technique, the highly nonlinear high-order dynamics of the OHT system can be modeled as a low-order linearized plant so that the synthesis of the feedback controller becomes simpler. The composite control, composed of sliding mode control and input shaping technique, is proposed. The sliding mode control is, as usual, employed to account for the system parameters uncertainties. On the other hand, to suppress theresidual vibration, i.e., auto-swaying, the input shaping technique is utilized by implementation of a finite-length sequence of impulses in the appropriate amplitude and time epoch. Finally, the efficacy of the proposal composite control strategy is examined and verified by intensive computer simulations.

On minimal polynomials over [formula omitted] and over [formula omitted] of a finite-length sequence over [formula omitted

In this paper we consider two problems, finding minimal polynomials over F q m and over F q of a finite-length sequence over F q m , in a lattice and its sublattice respectively using lattice theory over polynomial rings. And we deduce a relationship between them by a lattice basis transformation matrix. As a byproduct, we present a new synthesis algorithm for the first problem just using operations in F q instead of F q m as in Berlekamp–Massey algorithm.

The Line Spectral Frequency Model of a Finite-Length Sequence

The line spectral frequency (LSF) of a causal finite length sequence is a frequency at which the spectrum of the sequence annihilates or the magnitude spectrum has a spectral null. A causal finite-length sequence with (L + 1) samples having exactly L-LSFs, is referred as an Annihilating (AH) sequence. Using some spectral properties of finite-length sequences, and some model parameters, we develop spectral decomposition structures, which are used to translate any finite-length sequence to an equivalent set of AH-sequences defined by LSFs and some complex constants. This alternate representation format of any finite-length sequence is referred as its LSF-Model. For a finite-length sequence, one can obtain multiple LSF-Models by varying the model parameters. The LSF-Model, in time domain can be used to synthesize any arbitrary causal finite-length sequence in terms of its characteristic AH-sequences. In the frequency domain, the LSF-Model can be used to obtain the spectral samples of the sequence as a linear combination of spectra of its characteristic AH-sequences. We also summarize the utility of the LSF-Model in practical discrete signal processing systems.

RECURRENCE PLOTS FOR SYMBOLIC SEQUENCES

This paper introduces an extension of recurrence analysis to symbolic sequences. Heuristic arguments based on Shannon–McMillan–Breiman theorem suggest several relations between the statistical features of "symbolic" recurrence plots and the entropy per unit time of the dynamics; their practical efficiency for experimental sequences of finite length is checked numerically on two paradigmatic models, namely discretized logistic map trajectories and Markov chains, and also on experimental behavioral sequences. Specific advantages of RP analysis are presented, among which is the detection of nonstationary features.

Properties of the magnitude terms of orthogonal scaling functions

The spectrum of the convolution of two continuous functions can be determined as the continuous Fourier transform of the cross-correlation function. The same can be said about the spectrum of the convolution of two infinite discrete sequences, which can be determined as the discrete time Fourier transform of the cross-correlation function of the two sequences. In current digital signal processing, the spectrum of the contiuous Fourier transform and the discrete time Fourier transform are approximately determined by numerical integration or by densely taking the discrete Fourier transform. It has been shown that all three transforms share many analogous properties. In this paper we will show another useful property of determining the spectrum terms of the convolution of two finite length sequences by determining the discrete Fourier transform of the modified cross-correlation function. In addition, two properties of the magnitude terms of orthogonal wavelet scaling functions are developed. These properties are used as constraints for an exhaustive search to determine an robust lower bound on conjoint localization of orthogonal scaling functions.

Phylogenetic distances for neighbour dependent substitution processes

We consider models of nucleotidic substitution processes where the rate of substitution at a given site depends on the state of the neighbours of the site. We first estimate the time elapsed between an ancestral sequence at stationarity and a present sequence. Second, assuming that two sequences are issued from a common ancestral sequence at stationarity, we estimate the time since divergence. In the simplest non-trivial case of a Jukes–Cantor model with CpG influence, we provide and justify mathematically consistent estimators in these two settings. We also provide asymptotic confidence intervals, valid for nucleotidic sequences of finite length, and we compute explicit formulas for the estimators and for their confidence intervals. In the general case of an RN model with YpR influence, we extend these results under a proviso, namely that the equation defining the estimator has a unique solution.

Compact Matched Filter for Integrand Code Using a Real-Valued Shift-Orthogonal Finite-Length Sequence

In this paper, we proposed the compact construction of a matched filter for integrand code, which do not require the high-rate clock pulse in two-valued PWM (pulse width modulation) code, using a real-valued shift-orthogonal finite-length sequence, which has a sharp aperiodic autocorrelation function with zero sidelobes except at left and right shift-ends. This matched filters are implemented on a field programmable gate array (FPGA) corresponding to 400, 000 logic gates. A proposed matched filter for the sequence of length 129 can be constructed by the circuit scale of about 47% compared with conventional filter.

The Universal LZ77 Compression Algorithm Is Essentially Optimal for Individual Finite-Length $N$-Blocks

Consider the case where consecutive blocks of N letters of a semi-infinite individual sequence X over a finite alphabet are being compressed into binary sequences by some one-to-one mapping. No a priori information about X is available at the encoder, which must therefore adopt a universal data-compression algorithm. It is known that there exist a number of asymptotically optimal universal data compression algorithms (e.g., the Lempel-Ziv (LZ) algorithm, context tree algorithm and an adaptive Hufmann algorithm) such that when successively applied to N-blocks then, the best error-free compression for the particular individual sequence X is achieved as N tends to infinity. The best possible compression that may be achieved by any universal data compression algorithm for finite N-blocks is discussed. Essential optimality for the compression of finite-length sequences is defined. It is shown that the LZ77 universal compression of N-blocks is essentially optimal for finite N-blocks. Previously, it has been demonstrated that a universal context tree compression of N blocks is essentially optimal as well.

Newtonian and Schinzel sequences in a domain

A Schinzel or F sequence in a domain is such that, for every ideal I with norm q , its first q terms form a system of representatives modulo I , and a Newton or N sequence such that the first q terms serve as a test set for integer-valued polynomials of degree less than q . Strong F and strong N sequences are such that one can use any set of q consecutive terms, not only the first ones, finally a very well F ordered sequence, for short, a V.W.F sequence, is such that, for each ideal I with norm q , and each integer s , { u s q , … , u ( s + 1 ) q − 1 } is a complete set of representatives modulo I . In a quasilocal domain, V.W.F sequences and N sequences are the same, so are strong F and strong N sequences. Our main result is that a strong N sequence is a sequence which is locally a strong F sequence, and an N sequence a sequence which is locally a V.W.F. sequence. We show that, for F sequences there is a bound on the number of ideals of a given norm. In particular, a sequence is a strong F sequence if and only if it is a strong N sequence and for each prime p , there is at most one prime ideal with finite residue field of characteristic p . All results are refined to sequences of finite length.

Using Matrix Graph Grammars for the Analysis of Behavioural Specifications: Sequential and Parallel Independence

In this paper we present a new approach for the analysis of rule-based specification of system dynamics. We model system states as simple digraphs, which can be represented with boolean matrices. Rules modelling the different state changes of the system can also be represented with boolean matrices, and therefore the rewriting is expressed using boolean operations only.The conditions for sequential independence between pair of rules are well-known in the categorical approaches to graph transformation (e.g. single and double pushout). These conditions state when two rules can be applied in any order yielding the same result. In this paper, we study the concept of sequential independence in our framework, and extend it in order to consider derivations of arbitrary finite length. Instead of studying one-step rule advances, we study independence of rule permutations in sequences of arbitrary finite length. We also analyse the conditions under which a sequence is applicable to a given host graph. We introduce rule composition and give some preliminary results regarding parallel independence. Moreover, we improve our framework making explicit the elements which, if present, disable the application of a rule or a sequence.

On the Nonlinear Complexity and Lempel–Ziv Complexity of Finite Length Sequences

The nonlinear complexity of binary sequences and its connections with Lempel-Ziv complexity is studied in this paper. A new recursive algorithm is presented, which produces the minimal nonlinear feedback shift register of a given binary sequence. Moreover, it is shown that the eigenvalue profile of a sequence uniquely determines its nonlinear complexity profile, thus establishing a connection between Lempel-Ziv complexity and nonlinear complexity. Furthermore, a lower bound for the Lempel-Ziv compression ratio of a given sequence is proved that depends on its nonlinear complexity.

Tuning of High-Speed Telecommunication Filters, via I/O Cross-Correlation Evaluation

Continuous-time (CT) filters are widely used in signal processing but, owing to the large spread in the process parameters which determine their time constants, they require a tuning system to align their frequency characteristics. Several techniques have already been proposed in the literature and are commonly employed, but usually they require either a doubling of the filter circuitry or some complex self-calibration scheme, resulting in sensible area overhead or high computational complexity. Here, we present a novel tuning approach which can be applied to both master-slave and self-calibration arrangements and is based on the evaluation of a few samples of the input-output cross-correlation function of the filter, when the input signal is a finite length sequence of pseudorandom pulses. The key advantages of the proposed technique are basically the ease of generation of the input bit-stream and the simplicity of the circuitry required to sample the filter output and to perform the cross-correlation operation. As a consequence, the area overhead associated to the tuning circuitry is kept very small. Some simulation results of the application to the tuning of a low-Q telecommunication filter are given and an extension to the tuning of high-Q active filters is also explored in order to assess the effectiveness and robustness of the proposed technique

Counting All Possible Ancestral Configurations of Sample Sequences in Population Genetics

Given a set D of input sequences, a genealogy for D can be constructed backward in time using such evolutionary events as mutation, coalescent, and recombination. An ancestral configuration (AC) can be regarded as the multiset of all sequences present at a particular point in time in a possible genealogy for D. The complexity of computing the likelihood of observing D depends heavily on the total number of distinct ACs of D and, therefore, it is of interest to estimate that number. For D consisting of binary sequences of finite length, we consider the problem of enumerating exactly all distinct ACs. We assume that the root sequence type is known and that the mutation process is governed by the infinite-sites model. When there is no recombination, we construct a general method of obtaining closed-form formulas for the total number of ACs. The enumeration problem becomes much more complicated when recombination is involved. In that case, we devise a method of enumeration based on counting contingency tables and construct a dynamic programming algorithm for the approach. Last, we describe a method of counting the number of ACs that can appear in genealogies with less than or equal to a given number R of recombinations. Of particular interest is the case in which R is close to the minimum number of recombinations for D.

An application of the warped discrete Fourier transform in the perceptual speech enhancement

An application of the warped discrete Fourier transform (WDFT) in the perceptual speech enhancement is of interest. The WDFT allows nonuniform sampling the z-transform of finite length sequence. We focus on the perceptual warping which allocates frequency samples in good accordance with the Bark scale. The WDFT can replace conventional DFT based analysis/synthesis block of spectral weighting method. In the case of the perceptual warping, there is a problem with signal reconstruction because the WDFT matrix is ill-conditioned. This paper addresses the problem of signal distortions generated in WDFT based synthesis block. Spectral characteristics of the reconstructed signal are analyzed and discussed in the context of the perceptual processing. A new extension of the WDFT intended to cancellation of the synthesis error is presented. The new method is also validated in practical speech enhancement system. The results show that the new algorithm outperforms pure WDFT based system.