Finite-length Sequences Research Articles

Sampling theory of jointly bandlimited time-vertex graph signals

Time-vertex graph signal (TVGS) models describe time-varying data with irregular structures. The bandlimitedness in the joint time-vertex Fourier spectral domain reflects smoothness in both temporal and graph topology. In this paper, we study the critical sampling of three types of TVGS including continuous-time signals, infinite-length sequences, and finite-length sequences in the time domain for each vertex on the graph. For a jointly bandlimited TVGS, we prove a lower bound on sampling density or sampling ratio, which depends on the measure of the spectral support in the joint time-vertex Fourier spectral domain. We also provide a lower bound on the sampling density or sampling ratio of each vertex on sampling sets for perfect recovery. To demonstrate that critical sampling is achievable, we propose the sampling and reconstruction procedures for the different types of TVGS. Finally, we show how the proposed sampling schemes can be applied to numerical as well as real datasets.

On Run-Length-Constrained Binary Sequences

A class of binary sequences, constrained with respect to the length of zero runs, is considered. For such sequences, termed (d, k)-sequences, new combinatorial and computational results are established. Explicit expressions for enumerating (d, k)-sequences of finite length are obtained. Efficient computational procedures for calculating the capacity of a (d, k)-code are given. A simple method for constructing a near-optimal (d, k)-code is proposed. Illustrative numerical examples demonstrate further the theoretical results.

Predictive control of linear discrete-time Markovian jump systems by learning recurrent patterns

Incorporating pattern-learning for prediction (PLP) in many discrete-time or discrete-event systems allows for computation-efficient controller design by memorizing patterns to schedule control policies based on their future occurrences. In this paper, we demonstrate the effect of PLP by designing a controller architecture for a class of linear Markovian jump systems (MJSs) where the aforementioned “patterns” correspond to finite-length sequences of modes. In our analysis of recurrent patterns, we use martingale theory to derive closed-form solutions to quantities pertaining to the occurrence of patterns: (1) the expected minimum occurrence time of any pattern from some predefined collection, (2) the probability of a pattern being the first to occur among the collection. To make our method applicable to real-world dynamics, we make two extensions to common assumptions in prior pattern-occurrence literature. First, the distribution of the mode process is unknown, and second, the true realization of the mode process is not observable. As demonstration, we consider fault-tolerant control of a dynamic topology-switching network, and empirically compare PLP to two controllers without PLP: a baseline based on the novel System Level Synthesis (SLS) approach and a topology-robust extension of the SLS baseline. We show that PLP is able to reject disturbances just as effectively as the topology-robust controller at reduced computation time and control effort. We discuss several important tradeoffs, such as the size of the pattern collection and the system scale versus the accuracy of the mode predictions, which show how different PLP implementations affect stabilization and runtime performance.

The Lossless Adaptive Binomial Data Compression Method

In this paper, we propose a new method for the binomial adaptive compression of binary sequences of finite length without loss of information. The advantage of the proposed binomial adaptive compression method compared with the binomial compression method previously developed by the authors is an increase in the compression rate. This speed is accompanied in the method by the appearance of a new quality—noise immunity of compression. The novelty of the proposed method, which makes it possible to achieve these positive results, is manifested in the adaptation of the compression ratio of compressible sequences to the required time, which is carried out by dividing the initial set of binary sequences into compressible and incompressible sequences. The method is based on the theorem proved by the authors on the decomposition of a stationary Bernoulli source of information into the combinatorial and probabilistic source. The last of them is the source of the number of units. It acquires an entropy close to zero and practically does not affect the compression ratio at considerable lengths of binary sequences. Therefore, for the proposed compression method, a combinatorial source generating equiprobable sequences is paramount since it does not require a set of statistical data and is implemented by numerical coding methods. As one of these methods, we choose a technique that uses binomial numbers based on the developed binomial number system. The corresponding compression procedure consists of three steps. The first is the transformation of the compressible sequence into an equilibrium combination, the second is its transformation into a binomial number, and the third is the transformation of a binomial number into a binary number. The restoration of the compressed sequence occurs in reverse order. In terms of the degree of compression and universalization, the method is similar to statistical methods of compression. The proposed method is convenient for hardware implementation using noise-immune binomial circuits. It also enables a potential opportunity to build effective systems for protecting information from unauthorized access.

ε-arithmetics for real vectors and linear processing of real vector-valued signals

In this paper, we introduce a new concept, namely ε-arithmetics, for real vectors of any fixed dimension. The basic idea is to use vectors of rational values (called rational vectors) to approximate vectors of real values of the same dimension within ε range. For rational vectors of a fixed dimension m, they can form a field that is an mth order extension Q(α) of the rational field Q where α has its minimal polynomial of degree m over Q. Then, the arithmetics, such as addition, subtraction, multiplication, and division, of real vectors can be defined by using that of their approximated rational vectors within ε range. We also define complex conjugate of a real vector and then inner product and convolutions of two real vectors and two real vector sequences (signals) of finite length. With these newly defined concepts for real vectors, linear processing, such as linear filtering, ARMA modeling, and least squares fitting, can be implemented to real vector-valued signals with real vector-valued coefficients, which will broaden the existing linear processing to scalar-valued signals.

The Mirror Transform

This paper explains how to describe any finite-energy signal through a unique representation consisting of an ordered set of positions and a sparse set of signals. This is obtained by designing an iterative decomposition through a series of mirror operations around those positions. The purpose is to find at any step of the decomposition the location that provides for the maximum decoupling between the even and odd components of the signal with respect to it. By limiting such even and odd components on three separate information bearing support, the algorithm can be iterated at infinity determining a sequence of positions. The per location information determines the optimal energy decoupling strategy at each stage providing remarkable sparsity in the representation. The resulting transformation leads to a 1-to-1 mapping. The approach is easily extended to finite-energy sequences, and in particular for sequences of finite length <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>, at most <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> iterations of the decomposition are required. Thanks to the sparsity of the resulting representation, experimental simulations demonstrate superior approximation capabilities of this proposed non-linear Mirror Transform with potential application in many domains such as approximation and coding. Its implementation has been made publicly available.

Profile Hidden Markov Models Are Not Identifiable.

Profile Hidden Markov Models (HMMs) are graphical models that can be used to produce finite length sequences from a distribution. In fact, although they were only introduced for bioinformatics 25 years ago (by Haussler et al., Hawaii International Conference on Systems Science, 1993), they are arguably the most commonly used statistical model in bioinformatics, with multiple applications, including protein structure and function prediction, classifications of novel proteins into existing protein families and superfamilies, metagenomics, and multiple sequence alignment. The standard use of profile HMMs in bioinformatics has two steps: first a profile HMM is built for a collection of molecular sequences (which may not be in a multiple sequence alignment), and then the profile HMM is used in some subsequent analysis of new molecular sequences. The construction of the profile thus is itself a statistical estimation problem, since any given set of sequences might potentially fit more than one model well. Hence, a basic question about profile HMMs is whether they are statistically identifiable, which means that no two profile HMMs can produce the same distribution on finite length sequences. Indeed, statistical identifiability is a fundamental aspect of any statistical model, and yet it is not known whether profile HMMs are statistically identifiable. In this paper, we report on preliminary results towards characterizing the statistical identifiability of profile HMMs in one of the standard forms used in bioinformatics.

Non-Asymptotic Classical Data Compression With Quantum Side Information

In this paper, we analyze classical data compression with quantum side information (also known as the classical-quantum Slepian–Wolf protocol) in the so-called large and moderate deviation regimes. In the non-asymptotic setting, the protocol involves compressing classical sequences of finite length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> and decoding them with the assistance of quantum side information. In the large deviation regime, the compression rate is fixed, and we obtain bounds on the error exponent function, which characterizes the minimal probability of error as a function of the rate. Devetak and Winter showed that the asymptotic data compression limit for this protocol is given by a conditional entropy. For any protocol with a rate below this quantity, the probability of error converges to one asymptotically and its speed of convergence is given by the strong converse exponent function. We obtain finite blocklength bounds on this function, and determine exactly its asymptotic value. In the moderate deviation regime for the compression rate, the latter is no longer considered to be fixed. It is allowed to depend on the blocklength <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> , but assumed to decay slowly to the asymptotic data compression limit. Starting from a rate above this limit, we determine the speed of convergence of the error probability to zero and show that it is given in terms of the conditional information variance.

Sequences, q-multinomial Identities, Integer Partitions with Kinds, and Generalized Galois Numbers

Using sequences of finite length with positive integer entries and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their q-analog form via combinatorial proofs. Using the major index statistic on sequences, a connection between finite differences of the coefficients of generalized Galois numbers and integer partitions with kinds is established.

An SDN oriented loop detection mechanism based on TTL statistics

The rise of the network has injected new impetus into the development of traditional networks. Due to the complexity of the network itself and its network programmability, there is a risk of routing loops occurring in the SDN network. This paper proposes a loop detection mechanism. According to the Time To Live (TTL) value of the loop packet, there is approximately periodicity in the same loop. We use sFlow to count the number of packets corresponding to each TTL value of a switch in the loop over a period of time, and perform discrete Fourier transform on the obtained finite-length sequence to observe its frequency domain performance and determine whether there are periodic features. By doing so, it is determined whether there is a routing loop, and the purpose of passively detecting the routing loop is achieved. Compared to existing algorithms, it has advantages in real-time, scalability and false positive rate. The experimental results show that the routing loop detection algorithm based on TTL statistics in this paper still maintains high judgment accuracy under the scenarios of lower stream sampling rate and smaller detection period.

Achievable simultaneous time and frequency domain energy concentration for finite length sequences

For numerous applications in the field of signal processing, it is desired to design a compact window that can simultaneously concentrate maximum energy in finite time interval and frequency band. Although zero-order discrete prolate spheroidal sequence (DPSS) meets this requirement, it is of infinite support. Limiting this sequence to finite support no longer guarantees the optimality property. This study aims at designing a discrete finite length window that can maximise the energy simultaneously in narrow time interval and frequency band. A multi-objective optimisation approach is adopted to obtain the upper bound of maximum achievable time and frequency domain energy concentrations for finite length sequences. The optimal sequence thus obtained is termed as the optimal window with finite support (OWFS), and its various associated properties are discussed. It is shown analytically that as the support of OWFS approaches infinity, it converges to zero-order DPSS. In order to illustrate its optimality, the compactness of the proposed OWFS is compared with those of various window functions. Extending the proposed OWFS, this study also discusses the formulation and associated properties of the zero-order periodic DPSS. Further, the closed form expression for the upper bound of achievable time and frequency domain energy concentrations is also derived.

Discrete Fourier transform based frequency characteristics of iterative learning control for linear discrete-time systems

For discrete-time iterative learning control systems, the discrete Fourier transform (DFT) is a powerful technique for frequency analysis, and Toeplitz matrices are a typical tool for the system input–output transmission. This paper first exploits z-transform and DFT-based frequency properties for iterative learning control systems and studies the convergence property of a Toeplitz matrix to the power of iteration index. The exploitation exhibits that for the finite-length discrete-time iterative learning control systems, the time-domain convolution theorem for the z-transform and DFT is no longer true, and the Toeplitz matrix to the power of iteration index converges if and only if the identical diagonal element lies in the unit circle. Then, by considering the DFT to a finite-length sequence as a linear transform, it is easy to equivalently reform the input–output equation of linear discrete time-invariant and time-varying ILC systems as an algebraic discrete-frequency equation. Thus the derivative-type (D-type) iterative learning control (ILC) converges in a discrete-frequency domain if and only if it converges in a discrete-time domain. Numerical simulations are carried out to exhibit the validity and effectiveness.

AMPLITUDE AND PHASE MODULATED PULSE TRAINS FOR CLUTTER REJECTION IN RADAR

In radar systems the very low side-lobes that can be achieved with amplitude and phase modulated pulse trains make their use particularly attractive in systems requiring a large dynamic range. Moreover, the excellent self-clutter rejection performance is obtained without sacrificing the ability for the separation of close targets (no main lobe widening). The additional expense of encoding and decoding in amplitude and phase may be justified for radars that must cope with land clutter or operate in a dense-target environment. This paper presents a method for generating finite length sequences that have high energy ratio and low sidelobe energy in their autocorrelation function. The method is computer based and generates the desired sequence of any given length in an iterative manner starting with a sequence that has relatively low sidelobes in its autocorrelation function. It is shown that energy ratio of polyphase sequences that have low sidelobe energy , is improved through the process of clipping.

Construction of finite-dimensional multiresolutions based on linear subdivision schemes: Application to the 4-point shifted Lagrange scheme

This paper is devoted to the construction of multiresolution frameworks related to general but linear subdivision schemes applied on sequences of finite length. Thanks to the flexible properties of subdivision schemes, a subdivision-based multiresolution is a promising powerful tool for compression, control and analysis of data. However, its construction is not straightforward either for non-interpolatory subdivisions, or for sequences of finite length. In this paper, given a subdivision for finite length sequences, we provide a first approach to construct compatible multiresolutions thanks to an extension of some classical wavelet results. As it will become clear that this type of approach can become computationally costly in practice, a new edge-adapted method that combines local consistent decimation is then developed to circumvent this limitation. An illustration in the case of the 4-point shifted Lagrange subdivision scheme, for which there is no available multiresolution up to now, is then provided. Finally, some properties of the new multiresolutions are analyzed and their performances are evaluated in the framework of image approximation.

Implementation of Fast Convolution using Robust Vedic Multiplier of Radix-2

Convolution is an algorithm which is mainly used in video, audio and image processing. Convolution calculation is simple in steps however it consumes a lot of memory as well as power in the computational process. It is a mathematical algorithm which is also used in the applications like filtering, edge detection, de-noising, compression etc., as it can be exploit computational power. In this paper, we implemented the speed of discrete linear convolution using robust Vedic multiplier which is one of the fastest multipliers with two finite-length sequences. By implementing convolution with Vedic multiplier power, area and delay are reduced. This implementation process can be realized by simplifying the convolution building block.

Entrance and escape dynamics for the typical set.

According to the asymptotic equipartition property, sufficiently long sequences of random variables converge to a set that is typical. While the size and probability of this set are central to information theory and statistical mechanics, they can often only be estimated accurately in the asymptotic limit due to the exponential growth in possible sequences. Here we derive a time-inhomogeneous dynamics that constructs the properties of the typical set for all finite length sequences of independent and identically distributed random variables. These dynamics link the finite properties of the typical set to asymptotic results and allow the typical set to be applied to small and transient systems. The main result is a geometric mapping-the triangle map-relating sequences of random variables of length n to those of length n+1. We show that the number of points in this map needed to quantify the properties of the typical set grows linearly with sequence length, despite the exponential growth in the number of typical sequences. We illustrate the framework for the Bernoulli process and the Schlögl model for autocatalytic chemical reactions and demonstrate both the convergence to asymptotic limits and the ability to reproduce exact calculations.

Finite length sequences with large nonlinear complexity

Finite length sequences with large nonlinear complexity over $\mathbb{Z}_{p}\, (p≥ 2)$ are investigated in this paper. We characterize all $p$-ary sequences of length $n$ having nonlinear complexity $n-j$ for $j=2, 3$, where $n$ is an integer satisfying $n≥ 2j$. For $n≥ 8$, all binary sequences of length $n$ with nonlinear complexity $n-4$ are obtained. Furthermore, the numbers and $k$-error nonlinear complexity of these sequences are completely determined, respectively.

Predictive Rate-Distortion for Infinite-Order Markov Processes

Predictive rate-distortion analysis suffers from the curse of dimensionality: clustering arbitrarily long pasts to retain information about arbitrarily long futures requires resources that typically grow exponentially with length. The challenge is compounded for infinite-order Markov processes, since conditioning on finite sequences cannot capture all of their past dependencies. Spectral arguments confirm a popular intuition: algorithms that cluster finite-length sequences fail dramatically when the underlying process has long-range temporal correlations and can fail even for processes generated by finite-memory hidden Markov models. We circumvent the curse of dimensionality in rate-distortion analysis of finite- and infinite-order processes by casting predictive rate-distortion objective functions in terms of the forward- and reverse-time causal states of computational mechanics. Examples demonstrate that the resulting algorithms yield substantial improvements.

Alternation Bias as a Consequence of Pattern Overlap: Comments on Miller and Sanjurjo (2015)

Miller and Sanjurjo (2015) suggest that the alternation bias in people’s observation of independent and identically distributed Bernoulli trials might be due to a selection bias, by which an empirical conditional probability p^(H|H) is repeatedly sampled from sequences of finite length. Here we show that the alternation bias originates from the overlap of patterns, rather than finite sample length. When measured by the pattern time statistics, the bias takes shape early on and approach an asymptote as the sample length approaches infinity. We argue that emphasizing the overlap properties that are intrinsic of patterns themselves instead of the sample length may have different implications about the nature of human statistical learning and decision-making under uncertainty. Instead resorting to the limited working memory capacity or the belief of the law of small numbers, our analysis points to the effectiveness of predictive learning and the sensitivity of the human mind to the latent structures in the learning environment.

A steady-state Kalman predictor-based filtering strategy for non-overlapping sub-band spectral estimation.

This paper focuses on suppressing spectral overlap for sub-band spectral estimation, with which we can greatly decrease the computational complexity of existing spectral estimation algorithms, such as nonlinear least squares spectral analysis and non-quadratic regularized sparse representation. Firstly, our study shows that the nominal ability of the high-order analysis filter to suppress spectral overlap is greatly weakened when filtering a finite-length sequence, because many meaningless zeros are used as samples in convolution operations. Next, an extrapolation-based filtering strategy is proposed to produce a series of estimates as the substitutions of the zeros and to recover the suppression ability. Meanwhile, a steady-state Kalman predictor is applied to perform a linearly-optimal extrapolation. Finally, several typical methods for spectral analysis are applied to demonstrate the effectiveness of the proposed strategy.