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Fast Sparse Period Estimation
Abstract
The problem of estimating the period of a point process from observations that are both sparse and noisy is considered. By sparse it is meant that only a potentially small unknown subset of the process is observed. By noisy it is meant that the subset that is observed, is observed with error, or noise. Existing accurate algorithms for estimating the period require operations where N is the number of observations. By quantizing the observations we produce an estimator that requires only operations by use of the chirp z-transform or the fast Fourier transform. The quantization has the adverse effect of decreasing the accuracy of the estimator. This is investigated by Monte-Carlo simulation. The simulations indicate that significant computational savings are possible with negligible loss in statistical accuracy.
... The analyses of time-of-arrival (TOA) of received pulse train is most widely used in the pulse train deinterleaving The existing TOA-based pulse train deinterleaving methods, such as Spectrum or Periodogram based method [3,4], cumulative difference histogram (CDIF) [5], sequential difference histogram (SDIF) [6], PRI transform [7], SAPD algorithm [8], improved version of SDIF [9,10], the periodic dictionary based method [11] and the correlation matching method [12], have been applied to ELINT. In recent year, the deinterleaving methods with recurrent neural networks method [13], with the finite-state automaton [14] and with denoising autoencoder [15] were also proposed. ...
... For the interleaved pulse train described by (6), the spectrum of x s (t) is defined as [3] M(q) N n 1 exp(i2π t n /q) (7) or the periodogram of x s (t) is defined as [4] ...
... However, SPBM-CMM has less CPU times than FMA. Furthermore, the computation complexity of deinterleaving methods mentioned in Table 2 has been involved in references [4,7,12]. ...
For the classification of radar emitters, it is important to be able to separate pulses in the interleaved pulse train in terms of their source. In practical scenarios, the received pulse train may miss a number of pulses. The missing pulses can degrade the performance of the existing TOA-based pulse train deinterleaving methods. However, this degradation is not identical for different methods. If multiple pulse train deinterleaving methods, which work with different principle, are used in fusion, it may be possible to have better performance than performance of any of pulse train deinterleaving methods used alone. In this paper, a framework to fuse multiple pulse train deinterleaved methods is proposed for separating individual pulse train included in the received pulse train. Under this framework, a Fusion Machine Approach (FMA) is proposed and the implementing steps of FMA are discussed in detail. Two types of FMA are employed in simulation. The results of simulation show that the proposed FMA has better performance than existing pulse train deinterleaving methods. The FMA has the best robustness to missing pulse and PRI jitter, and can effectively deinterleave the received pulse train with PRI stagger and jittered PRI.
... If in the measurement sequence some measurements are missing, it is called sparse. We use a point process model to describe such a measurement sequence [10]. Most highly accurate frequency estimation methods operate on a frequency domain representation of signals, e.g. ...
... In the special case of identifying the fundamental frequency, one of the most common methods is the periodogram estimation [11]- [13]. Current research [10] shows a variance of the estimation error according to O(N −3 ) at a computational complexity of O(N log(N )), where N is the number of samples used in the estimation. As we focus on periodic signals, recent research on frequency estimation of cyclostationary signals is relevant. ...
... , and, thus, a simple product N µ d can be used in (10) and therefore, ...
... Usually, high accuracy is linked to high complexity which is a contradiction to low power sensor nodes [1]. Nevertheless, energy efficiency and synchronicity is a vital demand for many systems [2], [3] and, therefore, we present an O(N ) complexity algorithm, albeit state-of-the-art algorithms with the same clock frequency estimation accuracy feature at least a complexity of O(N log N ) [4]. The algorithm can be used for frequency estimation in sparse and non-sparse processes. ...
... Actual research [4] show a variance of the estimation error according to O(N −3 ) at a computational complexity of O(N log(N )). The spectral range of ∆f is used to guarantee identifiability [4], [8] and N is the number of samples considered. ...
... Actual research [4] show a variance of the estimation error according to O(N −3 ) at a computational complexity of O(N log(N )). The spectral range of ∆f is used to guarantee identifiability [4], [8] and N is the number of samples considered. As we focus on pulse signals, recent research on frequency estimation of cyclostationary signals is relevant. ...
... One of the common methods is the periodogram estimation [10,11,12] by considering stationary processes. Its computational complexity is in the order of O(N log(N )) [13]. In [14,15], fundamental frequency estimation of cyclo-stationary processes was introduced, leading to similar results. ...
... Within B, p t [n] is white and therefore it implies an infinite number of frequency components θ with individual MSE θ [N ]. To emphasize this we write MSE[N, θ] using(13). With N θ 1, its frequency dependence is proportional to θ 4 and for N θ > 1 the MSE is limited withp 2 T 2 .The interpolation error is proportional to θ 2 and thereforeMSE[N, θ] ≤ MSE[N, B] if θ ≤ B ∧ B < ∞ . ...
... This is called the closest lattice point problem (or closest vector problem) and a solution is called a closest lattice point (or simply closest point) to y [35,42,43]. The problem has found numerous applications in computer science, engineering, and statistics [44][45][46][47][48][49][50][51][52]. The closest lattice point problem and the Voronoi cell are related in that x ∈ Λ is a closest lattice point to y if and only if y − x ∈ Vor Λ. ...
We consider the problem of estimating the distance, or range, between two locations by measuring the phase of multiple sinusoidal signals transmitted between the locations. Traditional estimators developed for optical interferometry include the beat wavelength and excess fractions methods. More recently, estimators based on the Chinese remainder theorem (CRT) and least squares have appeared. Recent research suggests the least squares estimator to be most accurate in many cases. The accuracy of all of these range estimators depends upon the wavelengths chosen. This leads to the problem of selecting wavelengths that maximise accuracy. Procedures for selecting wavelengths for the beat wavelength and excess fractions methods have previously been described, but procedures for the CRT and least squares estimators are yet to be developed. In this paper we develop an algorithm to automatically select wavelengths for use with the least square range estimator. The algorithm minimises an optimisation criterion connected with the mean square error. Interesting properties of a particular class of lattices simplify the criterion allowing minimisation by depth first search. Monte-Carlo simulations indicate that wavelengths that minimise the criterion can result is considerably more accurate range estimates than wavelengths selected by ad hoc means.
... Moreover, the tests are performed when few pulses are missing (generated by a Bernoulli process of parameter p = 0.5). A fast log-linear estimator is derived in [12], but at the expense of some accuracy by quantizing the observations. A linear estimator of comparable precision is presented in [13]. ...
We consider the frequency estimation of periodic signals using noisy time-of-arrival (TOA) information with missing (sparse) data contaminated with outliers. We tackle the problem from a mathematical optimization standpoint, formulating it as a linear regression with an unknown increasing integer independent variable and outliers. Assuming an upper bound on the variance of the noise, we derive an online, parallelizable, near-CRLB optimization-based algorithm amortized to a linear complexity. We demonstrate the outstanding robustness of our algorithm to noise and outliers by testing it against diverse randomly generated signals. Our algorithm handles outliers by design and yields precise estimations even with up to 20% of contaminated data.
... Finding the periodicity of a signal is a well-studied problem in signal processing research (Schuster 1898;Berberidis et al. 2002;Vlachos et al. 2005;Li 2012;McKilliam et al. 2014;Malode et al. 2015;Unnikrishnan and Jothiprakash 2018;Puech et al. 2019;Gubner 2006). Periodogram (Schuster 1898) and circular autocorrelation (Gubner 2006) are among the widely used methods to find a plausible set of periods for a signal. ...
With the growth in complexity of real-time embedded systems, there is an increasing need for tools and techniques to understand and compare the observed runtime behavior of a system with the expected one. Since many real-time applications require periodic interactions with the environment, one of the fundamental problems in guaranteeing their temporal correctness is to be able to infer the periodicity of certain events in the system. The practicability of a period inference tool, however, depends on both its accuracy and robustness (also its resilience) against noise in the output trace of the system, e.g., when the system trace is impacted by the presence of aperiodic tasks, release jitters, and runtime variations in the execution time of the tasks. This work (i) presents the first period inference framework that uses regression-based machine-learning (RBML) methods, and (ii) thoroughly investigates the accuracy and robustness of different families of RBML methods in the presence of uncertainties in the system parameters. We show, on both synthetically generated traces and traces from actual systems, that our solutions can reduce the error of period estimation by two to three orders of magnitudes w.r.t. the state of the art.
... One of the common methods for that is the periodogram estimation [9,10,11] considering stationary processes. Its computational complexity is in the order of O(N log(N )) [12]. In [13,14] fundamental frequency estimation of cyclostationary processes was introduced, which lead to similar results. ...
... This is called the closest lattice point problem (or closest vector problem) and a solution is called a closest lattice point (or simply closest point) to y [35,42,43]. The problem has found numerous applications in computer science, engineering, and statistics [44][45][46][47][48][49][50][51][52]. The closest lattice point problem and the Voronoi cell are related in that x ∈ Λ is a closest lattice point to y if and only if y − x ∈ Vor Λ. ...
We consider the problem of estimating the distance, or range, between two locations by measuring the phase of multiple sinusoidal signals transmitted between the locations. Traditional estimators developed for optical interferometry include the beat wavelength and excess fractions methods. More recently, estimators based on the Chinese remainder theorem (CRT) and least squares have appeared. Recent research suggests the least squares estimator to be most accurate in many cases. The accuracy of all of these range estimators depends upon the wavelengths chosen. This leads to the problem of selecting wavelengths that maximise accuracy. Procedures for selecting wavelengths for the beat wavelength and excess fractions methods have previously been described, but procedures for the CRT and least squares estimators are yet to be developed. In this paper we develop an algorithm to automatically select wavelengths for use with the least square range estimator. The algorithm minimises an optimisation criterion connected with the mean square error. Interesting properties of a particular class of lattices simplify the criterion allowing minimisation by depth first search. Monte-Carlo simulations indicate that wavelengths that minimise the criterion can result is considerably more accurate range estimates than wavelengths selected by ad hoc means.
An efficient one-dimensional search solution is proposed for single-sensor emitter localization based on signal periodicity. By introducing an intermediate variable, this single-sensor localization problem can be solved by solving a set of equations for classic multi-sensor time difference of arrival localization, whose algebraic solutions can be obtained by the well-known two-stage weighted least squares (TSWLS) algorithm. In this manner, the two-dimensional or three-dimensional search for the emitter location can be replaced by a one-dimensional search for the intermediate variable. Simulations indicate that, at the expense of a small loss of localization accuracy, the proposed method achieves a significant advantage in computation time, compared with the conventional maximum likelihood technique.
For a moving target indicating (MTI) radar system to eliminate “blind speeds” or synthetic aperture radar (SAR) to eliminate “blind ranges,” two or more pulse repetition intervals (PRIs) are often used sequentially. This article proposes a new correlation-matching algorithm to estimate the PRIs, despite the radar receiver possibly missing some time samples. This proposed algorithm operates in the orthonormal Ramanujan subspace, leading to fewer nonzero elements therein, thus reducing the computational load. This proposed algorithm employs the
-norm (instead of the
-norm) in computing the correlation coefficients, for better robustness against outliers. Simultaneously, a new divisor rule, instead of the least common multiple (LCM) rule, and an algorithm for computing the threshold are proposed to identify the frame period of the staggered pulse train in the presence of spurious peaks. This new algorithm improves estimation accuracy and is robustness to missing pulse and PRI jitter.
Wireless sensor networks have tremendous advantages in monitoring environment, civilian, military, health, home, education and scientific, it requires degree of synchronization to best results. The operation confirming that the processors spread over a wide area possess a standard time is synchronization of clock. At each sensor node their local clocks are synchronized to a common time. Its main applications are communication, security, industry. The focus on wireless sensor network is due to extensive application. There are many difficulties in building and designing sensor network and is a challenge of providing clock synchronization. It is very important that the application in wireless sensor network choose a correct clock synchronization method. This article illustrate the existing clock synchronization approaches in wireless sensor network and estimates the clock skew and clock offset in synchronization protocol.
Traffic signal phase and timing (TSPaT) information is valuable for various applications, such as velocity advisory systems, navigation systems, collision warning systems, and so forth. In this paper, we focus on learning baseline timing cycle lengths for fixed-Time traffic signals. The cycle length is the most important parameter among all timing parameters, such as green lengths. We formulate the cycle length learning problem as a period estimation problem using a sparse set of noisy observations, and propose the most frequent approximate greatest common divisor (MFAGCD) algorithms to solve the problem. The accuracy performance of our proposed algorithms is experimentally evaluated on both simulation data and the real taxi GPS trajectory data collected in Shanghai, China. Experimental results show that the MFAGCD algorithms have better sparsity and outliers tolerant capabilities than existing cycle length estimation algorithms. Copyright © 2018 The Institute of Electronics, Information and Communication Engineers.
This paper examines the uncertainty of period measurement, using a real time segmentation and labeling algorithm, proposed for the definition of a new standard of the IEEE 1451, the family of standards for smart sensors. Firstly, an explanation of the algorithm is shown. Secondly, several simulation results are analyzed. Finally, experimental result are shown.
This paper investigates the problem of extracting the pulse repetition interval (PRI) of stationary pulsed radar emitter from noisy time of arrival(TOA) measurements with missing observations (incomplete) by the moving receiver. The relative motion between the receiver and emitter induces the large estimation bias to estimate PRI, we first discuss the conditions on which identifying the period integer numbers under the circumstance of relative movement, then propose two robust algorithms based on compensating delay of each observations on emitter's status space grids. The proposed algorithms can efficiently reduce the estimation bias introduced by model mismatch without the requirement of specific emitter location information. Simulation studies indicates that the proposed solutions can improve the PRI estimation accuracy.
This paper considers using a moving receiver to locate a stationary source that transmits periodically with an unknown period. The receiver records the time of arrivals (TOAs) of the intercepted signals for source localization. A new two-step localization algorithm is developed. Its first step ignores the signal propagation delay and identifies the signal period using existing period estimation techniques. Sufficient conditions for correctly relating the TOA measurements are established. The second step of the new algorithm jointly estimates the source position and the signal period using an iterative maximum likelihood estimator. Simulation results demonstrate the effectiveness of the newly proposed technique.
To estimate the period of a periodic point process from noisy and incomplete observations, the classical periodogram algorithm is modified. The original periodogram algorithm yields an estimate by performing grid search of the peak of a spectrum, which is equivalent to the periodogram of the periodic point process, thus its performance is found to be sensitive to the chosen grid spacing. This paper derives a novel grid spacing formula, after finding a lower bound of the width of the spectral mainlobe. By employing this formula, the proposed new estimator can determine an appropriate grid spacing adaptively, and is able to yield approximate maximum likelihood estimate (MLE) with a computational complexity of O(n2)O(n2). Experimental results prove that the proposed estimator can achieve better trade-off between statistical accuracy and complexity, as compared to existing methods. Simulations also show that the derived grid spacing formula is also applicable to other estimators that operate similarly by grid search.
Lattice reduction is a powerful concept for solving diverse problems involving point lattices. Signal processing applications where lattice reduction has been successfully used include global positioning system (GPS), frequency estimation, color space estimation in JPEG pictures, and particularly data detection and precoding in wireless communication systems. In this article, we first provide some background on point lattices and then give a tutorial-style introduction to the theoretical and practical aspects of lattice reduction. We describe the most important lattice reduction algorithms and comment on their performance and computational complexity. Finally, we discuss the application of lattice reduction in wireless communications and statistical signal processing. Throughout the article, we point out open problems and interesting questions for future research.
In this correspondence, we focus on blind estimation of the chip duration of time-hopping signals by introducing a cost function based on time-of-arrival (ToA) folding over multiple observation sets. An optimization algorithm that takes advantage of the highly oscillatory behavior in the nearby of the global minimum is proposed and a performance bound for the chip time estimate is derived. The pertinence of our approach is shown through numerical results, considering alternative methods like periodogram and separable least squares line search. The proposed technique enables a good tradeoff between statistical accuracy and computational complexity.
The Coxeter lattices are a family of lattices containing many of the important lattices in low dimensions. This includes An , E 7 , E 8 and their duals An* , E 7*, and E 8*. We consider the problem of finding a nearest point in a Coxeter lattice. We describe two new algorithms, one with worst case arithmetic complexity O ( n log n ) and the other with worst case complexity O ( n ) where n is the dimension of the lattice. We show that for the particular lattices An and An * the algorithms are equivalent to nearest point algorithms that already exist in the literature.
In our previous work [1, 2] we studied detection of anomalies in packet arrival times for computer networks, most detection of denial of-service (DoS) attacks in Internet traffic. In this paper we reformulate the detection method proposed in [1] using renewal theory, providing several useful extensions. This reformulation also leads to a method that would be applicable to numerous real life signals that exist as discrete events, e.g., biological signals. Most importantly renewal theory allows us to characterize the performance of our detector and determine theoretical bounds on the time-to-detection. Compared to alternative methods that use frequency spectra or event arrival rates for detection our method is shown to be superior in terms of time-to-detection. Further, unlike rate based techniques, our method can estimate the multiple periods when multiple periodic anomalies occur simultaneously.
In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).
Analysis of periodic pulse trains based on time of arrival is
considered, with perhaps very many missing observations and contaminated
data. A period estimator is developed based on a modified Euclidean
algorithm. This algorithm is a computationally simple, robust method for
estimating the greatest common divisor of a noisy contaminated data set.
The resulting estimate, although it is not maximum likelihood, is used
as initialization in a three-step algorithm that achieves the Cramer-Rao
bound (CRB) for moderate noise levels, as shown by comparing Monte Carlo
results with the CRBs. This approach solves linear regression problems
with missing observations and outliers. Comparisons with a periodogram
approach based on a point process model are shown. An extension using
multiple independent data records is also developed that overcomes high
levels of contamination
Modifications of the Euclidean algorithm are presented for
determining the period from a sparse set of noisy measurements. The
elements of the set are the noisy occurrence times of a periodic event
with (perhaps very many) missing measurements. This problem arises in
radar pulse repetition interval (PRI) analysis, in bit synchronization
in communications, and in other scenarios. The proposed algorithms are
computationally straightforward and converge quickly. A robust version
is developed that is stable despite the presence of arbitrary outliers.
The Euclidean algorithm approach is justified by a theorem that shows
that, for a set of randomly chosen positive integers, the probability
that they do not all share a common prime factor approaches one quickly
as the cardinality of the set increases. In the noise-free case, this
implies that the algorithm produces the correct answer with only 10 data
samples, independent of the percentage of missing measurements. In the
case of noisy data, simulation results show, for example, good
estimation of the period from 100 data samples with 50% of the
measurements missing and 25% of the data samples being arbitrary
outliers
The lattice A n * is an important lattice because of its covering properties in low dimensions. Clarkson described an algorithm to compute the nearest lattice point in A n * that requires O ( n log n ) arithmetic operations. In this correspondence, we describe a new algorithm. While the complexity is still O ( n log n ), it is significantly simpler to describe and verify. In practice, we find that the new algorithm also runs faster.
A good radar warning receiver should observe a radar very soon
after it begins transmitting, so in designing our radar warning receiver
we would like to ensure that the intercept time is low or the
probability of intercept after a specified time is high. We consider a
number of problems concerning the overlaps or coincidences of two
periodic pulse trains. We show that the first intercept time of two
pulse trains started in phase is a homogeneous Diophantine approximation
problem which can be solved using the convergents of the simple
continued fraction (s.c.f.) expansion of the ratio of their pulse
repetition intervals (PRIs). We find that the intercept time for
arbitrary starting phases is an inhomogeneous Diophantine approximation
problem which can be solved in a similar manner. We give a recurrence
equation to determine the times at which subsequent coincidences occur.
We then demonstrate how the convergents of the s.c.f. expansion can be
used to determine the probability of intercept of the two pulse trains
after a specified time when one or both of the initial phases are
random. Finally, we discuss how the probability of intercept varies as a
function of the PRIs and its dependence on the Farey points
Methods for estimating the pulse repetition interval and the pulse time-of-arrival jitter variance of a periodic pulse train (PT) are given. The estimators operate on data comprising a set of pulse time-of-arrival measurements which are corrupted by additive noise (timing jitter). Parameter estimators are formulated using two jitter models common to PT applications. The estimators include a closed-form maximum likelihood approach, recursive least squares estimation, and a Kalman filter approach that incorporates both jitter models. Comparison with Cramer-Rao lower bounds shows the estimators to be statistically efficient for a correct choice of jitter model. The estimator performance penalty for using the incorrect jitter model is also presented. Finally, the utility of the estimators is demonstrated using PT measurements from an operational radar. 1. INTRODUCTION Pulse train parameter estimation associated with periodic discrete event processes arises in a number of signal processing...
. This paper investigates the signal processing aspects of constructing a cross-correlation based detector for a known periodic point process signal with event arrival times corrupted by both non-cumulative and cumulative timing jitter. The detection scheme that we employ is based on a discrete event time of arrival folding algorithm and the use of circular statistics. This problem arises in the context of the detection by a passive receiver of a specific radar pulse train embedded in interference trains. The proposed detector is shown to be very effective even in a pulse train environment that is dominated by interference. We also investigate the use of an on-line stopping rule for determining the number of frequency harmonics to be included in a particular circular detection statistic. 1. Introduction The application of discrete-time signal processing techniques to pulse train signals arises in a number of different fields, e.g. radar, biomedicine and astrophysics. In this paper we ...
The problem of estimation of the period of a pulse train from a set of sparse, noisy measurements of the times-ofarrival, and the association of pulse indices with the measurements, is considered for a pair of statistical models of the measurement process. We find that the estimation and association problem can be formulated as a simultaneous Diophantine approximation problem. We propose an algorithm for obtaining estimates and associations based on the LLL algorithm [5]. We also show a relationship with the maximisation of a certain trigonometric sum, which can be regarded as the periodogram of the measurements. We present some numerical results which indicate that the algorithm is able to make correct associations of pulse indices, and therefore accurate estimates of the period, with very high probability even for very sparse, short and noisy records. This is demonstrated in examples in which 99:9% of pulses are missing and as few as 9 pulses are recorded with average errors of up to...
The spectral analysis of stationary point processes in one dimension is developed in some detail as a statistical method of analysis. The asymptotic sampling theory previously established by the author for a class of doubly stochastic Poisson processes is shown to apply also for a class of clustering processes, the spectra of which are contrasted with those of renewal processes. The analysis is given for two illustrative examples, one an artificial Poisson process, the other of some traffic data. In addition to testing the fit of a clustering model to the latter example, the analysis of these two examples is used where possible to check the validity of the sampling theory.
So far we have been concerned with one lattice at a time. In this chapter we are concerned with properties of sets of lattices. We first must define what is meant by two lattices Λ and M being near to each other; and this is done by means of homogeneous linear transformations. A homogeneous linear transformation X=τx of n-dimensional euclidean space into itself is said to be near to identity transformation if the coefficients τij in
are near those of the identity transformation, that is if
and
are all small.
High-accuracy pulse repetition interval (PRI) estimation is meaningful for passive sensors to identify radar emitters. This paper considers the problem of estimating the PRIs of motionless radars in moving passive sensor systems. A modified method which based on observation calibration is proposed. This method can efficiently compensate the estimation bias induced by model mismatch, through calibrating the pulse time of arrival (TOA) measurements with emitter geolocation information. Performance analysis and simulation results show that our method can improve the PRI estimation accuracy significantly.
The problem of estimation the period of a periodic event from a sequence of noisy, incomplete time-of-arrival (TOA) observations is studied. A novel grid spacing determination strategy is proposed for a class of numerical-search estimators, whose performances is determined by grid spacing selection. A modified algorithm based on the Fogel's Periodogram estimator is proposed by employing that strategy. It is shown that our algorithm is very likely to yield the maximum likelihood estimate (MLE), with a low complexity of O(n2). Simulation results demonstrate the superior performance of our estimator comparing with the other existing estimators.
We owe to MINKOWSKI the fertile observation that certain results which can be made almost intuitive by the consideration of figures in n-dimensional euclidean space have far-reaching consequences in diverse branches of number theory. For example, he simplified the theory of units in algebraic number fields and both simplified and extended the theory of the approximation of irrational numbers by rational ones (Diophantine Approximation). This new branch of number theory, which MINKOWSKI christened “The Geometry of Numbers”, has developed into an independent branch of number-theory which, indeed, has many applications elsewhere but which is well worth studying for its own sake.
The problem discussed is that of estimating the period of a sequence of periodic events when the occurrence time measurements are noisy and sparse. The problem arises in signal processing applications such as baud estimation from zero-crossings in telecommunications and in pulse repetition interval estimation in electronic support measures. Estimation techniques have been based on periodogram maximisation [1][2], Euclidean algorithms [3][4][5], least squares line search [6], lattice line search [7], Gaussian maximum likelihood [8] and least squares [9]. Aside from [9], there has been no rigorous statistical analysis. In this paper, we show that the periodogram maximiser has excellent (theoretical) asymptotic statistical properties, illustrating them via simulation.
We discuss a computational algorithm for numerically evaluating the z-transform of a sequence of N samples. This algorithm has been named the chirp z-transform algorithm. Using this algorithm one can efficiently evaluate the z-transform at M points in the z-plane which lie on circular or spiral contours beginning at any arbitrary point in the z-plane. The angular spacing of the points is an arbitrary constant; M and N are arbitrary integers.
The algorithm is based on the fact that the values of the z-transform on a circular or spiral contour can be expressed as a discrete convolution. Thus one can use well-known high-speed convolution techniques to evaluate the transform efficiently. For M and N moderately large, the computation time is roughly proportional to (N + M) log2 (N + M) as opposed to being proportional to N·M for direct evaluation of the z-transform at M points.
Applications discussed include: enhancement of poles in spectral analysis, high resolution narrow-band frequency analysis, interpolation of band-limited waveforms, and the conversion of a base 2 fast Fourier transform program into an arbitrary radix fast Fourier transform program.
The theory of numbers has repeatedly shown itself to be both practical and beautiful. This paper gives an example of this duality. We present a very efficient (and practical) algorithm for extracting the fundamental period from a set of sparse and noisy observations of a periodic process. The procedure is computationally straightforward, stable with respect to noise, and converges quickly. Its use is justified by a theorem, which shows that for a set of randomly chosen positive integers, the probability that they do not all share a common prime factor approaches one quickly as the cardinality of the set increases. The proof of this theorem rests on a (beautiful) probabilistic interpretation of the Riemann zeta function.
The problem discussed in this paper is that of estimating the period of a sequence of periodic events when the measurements of the occurrence times are noisy and sparse. The problem is common to many signal processing applications, such as baud estimation from zero-crossings in telecommunications and in pulse repetition interval estimation in electronic support measures. Previous algorithms have been based on periodogram maximisation [1, 2], Euclidean algorithms [3-5], least-squares line search [6], lattice line search [7] and Gaussian maximum likelihood [8]. Until now, very little has been known about the asymptotic statistical properties of any such algorithm. In this paper, a new algorithm is proposed, based on a modified least-squares approach. Under very general properties, the estimators of the system parameters are shown to have excellent (theoretical) asymptotic statistical properties. These properties are illustrated using a number of simulations.
This expository paper traces the development of the Newton-Raphson method for solving nonlinear algebraic equations through the extant notes, letters, and publications of Isaac Newton, Joseph Raphson, and Thomas Simpson. It is shown how Newton's formulation differed from the iterative process of Raphson, and that Simpson was the first to give a general formulation, in terms of fluxional calculus, applicable to nonpolynomial equations. Simpson's extension of the method to systems of equations is exhibited.
Measurements of a process described by
t ( i )= k i T +τ+η(
i ), ( i =1, 2, . . .), where k i are
unknown integers and {η( i )} is a white noise, are
considered. The estimation of the `period' T and `phase' τ
of such processes is often required in communications and radar. Popular
linear regression techniques for this estimation problem are shown to
fail and new algorithms are derived. Statistical properties of the
estimators are presented, under reasonable assumptions. The suggested
procedure is illustrated in the digital communications symbol timing
recovery from eye-diagram zero crossings. The new bit synchronizer
obtained is shown to be superior in many practical situations to known
maximum-likelihood-type algorithms
It has been well known for at least twenty years that computing the maximizer of the periodogram, in order to estimate the unknown frequency in a noisy sinusoid, is problematic. In particular, because the periodogram is highly nonlinear, a grid size of order o (T-1) is needed to find the maximizer reliably, where T is the sample size, and that Newton's method may fail to find the zero of the first derivative of the periodogram closest to the maximizer of the periodogram calculated, for example, using the FFT. In this paper, we show that Newton's method does, in fact, work if it is applied to an appropriately chosen monotonic function of the periodogram.
The problem of estimating the period of a periodic point process is considered when the observations are sparse and noisy. There is a class of estimators that operate by maximizing an objective function over an interval of possible periods, notably the periodogram estimator of Fogel & Gavish and the line-search algorithms of Sidiropoulos et al. and Clarkson. For numerical calculation, the interval is sampled. However, it is not known how fine the sampling must be in order to ensure statistically accurate results. In this paper, a new estimator is proposed which eliminates the need for sampling. For the proposed statistical model, it calculates a maximum- likelihood estimate. It is shown that the expected arithmetic complexity of the algorithm is O(n3 log n) where n is the number of observations. Numerical simulations demonstrate the superior statistical performance of the new estimator.
This paper develops parametric methods to detect network anomalies using only aggregate traffic statistics, in contrast to other works requiring flow separation, even when the anomaly is a small fraction of the total traffic. By adopting simple statistical models for anomalous and background traffic in the time domain, one can estimate model parameters in real time, thus obviating the need for a long training phase or manual parameter tuning. The proposed bivariate parametric detection mechanism (bPDM) uses a sequential probability ratio test, allowing for control over the false positive rate while examining the tradeoff between detection time and the strength of an anomaly. Additionally, it uses both traffic-rate and packet-size statistics, yielding a bivariate model that eliminates most false positives. The method is analyzed using the bit-rate signal-to-noise ratio (SNR) metric, which is shown to be an effective metric for anomaly detection. The performance of the bPDM is evaluated in three ways. First, synthetically generated traffic provides for a controlled comparison of detection time as a function of the anomalous level of traffic. Second, the approach is shown to be able to detect controlled artificial attacks over the University of Southern California (USC), Los Angeles, campus network in varying real traffic mixes. Third, the proposed algorithm achieves rapid detection of real denial-of-service attacks as determined by the replay of previously captured network traces. The method developed in this paper is able to detect all attacks in these scenarios in a few seconds or less.
Suppose that a neuron is firing spontaneously or that it is firing under the influence of other neurons. Suppose that the data available are the firing times of the neurons present. An "integrate several inputs and fire" model is developed and studied empirically. For the model a neuron's firing occurs when an internal state variable crosses a random threshold. This conceptual model leads to maximum likelihood estimates of internal quantities, such as the postsynaptic potentials of the measured influencing neurons, the membrane potential, the absolute threshold and also estimates of derived quantities such as the strength-duration curve and the recovery process of the threshold. The model's validity is examined via an estimate of the conditional firing probability. The approach appears useful for estimating biologically meaningful parameters, for examining hypotheses re these parameters, for understanding the connections present in neural networks and for aiding description and classification of neurons and synapses. Analyses are presented for a number of data sets collected for the sea hare, Aplysia californica, by J. P. Segundo. Both excitatory and inhibitory examples are provided. The computations were carried out via the Glim statistical package. An example of a Glim program realizing the work is presented in the Appendix.
In this work we develop an approach to extracting information from neural spike trains. Using the expectation-maximization (EM) algorithm, interspike interval data from experiments and simulations are fitted by mixtures of distributions, including Gamma, inverse Gaussian, log-normal, and the distribution of the interspike intervals of the leaky integrate-and-fire model. In terms of the Kolmogorov-Smirnov test for goodness-of-fit, our approach is proved successful (P>0.05) in fitting benchmark data for which a classical parametric approach has been shown to fail before. In addition, we present a novel method to fit mixture models to censored data, and discuss two examples of the application of such a method, which correspond to the case of multiple-trial and multielectrode array data. A MATLAB implementation of the algorithm is available for download from .
The problem of estimating the period of a series of periodic events is considered under the condition where the measurements of the times of occurrence are noisy and sparse. The problem is common to bit synchronisation in telecommunications and pulse-train parameter estimation in electronic support, among other applications. Two new algorithms are presented which represent different compromises between computational and statistical efficiency. The first extends the separable least squares line search (SLS2) algorithms of Sidiropoulos et al., having very low computational complexity while attaining good statistical accuracy. The second is an approximate maximum-likelihood algorithm, based on a low complexity lattice search, and is found to achieve excellent accuracy.
Given a noisy sequence of (possibly shifted) integer multiples of a certain period, it is often of interest to accurately estimate the period. With known integer regressors, the problem is classical linear regression. In many applications, however, the regressors are unknown integers, and only loose bounds on the period are available. Examples include hop period and timing estimation, wherein hops may be missed at the output of the frequency discriminator or the emitter may hop out of band; Pulse Repetition Interval (PRI) analysis; and passive rotating-beam radio scanning. We study several pertinent period estimators. Our emphasis is on a Quasi-Maximum Likelihood approach developed herein and an earlier method based on the Fourier Transform of a Dirac delta train representation of the data. Surprisingly, both are capable of attaining the clairvoyant Crame´r-Rao Bound at moderate signal-to-noise ratios (SNRs), even for short (e.g., 10) samples. We carefully address parameter identifiability issues and corroborate our findings with extensive simulations.
Bit synchronization algorithms based on signal zero-crossing
analysis are investigated. Their performance is compared to conventional
maximum-likelihood-type techniques for symbol timing recovery. The
zero-crossing bit synchronizers are shown to be superior for bandlimited
pulses such as raised cosine and double-jump frequency rolloff, except
for very low signal-to-noise ratios
An instrument for computing correlograms of neuronal spike trains is described. The special purpose nature of the instrument permits sufficient computation speed to allow real-time display of compiling correlograms. Such capability greatly extends the user's observation power during the important phase of data collection from multiple neural units. Due to its modular design, other useful statistical analysis programs (PST, Interval histograms) can be easily designed and incorporated.
We describe a computer program which is able to estimate the tempo and the times of musical beats in expressively performed music. The input data may be either digital audio or a symbolic representation of music such as MIDI. The data is processed off-line to detect the salient rhythmic events and the timing of these events is analysed to generate hypotheses of the tempo at various metrical levels. Based on these tempo hypotheses, a multiple hypothesis search nds the sequence of beat times which has the best fit to the rhythmic events. We show that estimating the perceptual salience of rhythmic events significantly improves the results. No prior knowledge of the tempo, meter or musical style is assumed; all required information is derived from the data. Results are presented for a range of different musical styles, including classical, jazz, and popular works with a variety of tempi and meters. The system calculates the tempo correctly in most cases, the most common error being a doubling or halving of the tempo. The calculation of beat times is also robust. When errors are made concerning the phase of the beat, the system recovers quickly to resume correct beat tracking, despite the fact that there is no high level musical knowledge encoded in the system.
Detecting periodic patterns in internet traffic with spectral and statistical methods
- X He
X. He, " Detecting periodic patterns in internet traffic with spectral and statistical methods, " Ph.D. dissertation, Univ. Southern California, Los Angeles, CA, USA, 2006.
Approximation of linear forms by lattice points with applications to signal processing
- I V L Clarkson
I. V. L. Clarkson, " Approximation of linear forms by lattice points with applications to signal processing, " Ph.D. thesis, Australian Nat. Univ., Canberra, Australia, Jan. 1997.
An algorithm to compute the nearest point in the lattice <formula formulatype="inline"><tex Notation="TeX">$A_n^\ast$</tex> </formula>
- mckilliam