A robustness test for the MVD filter and MLD algorithm

J.P. Todoeschuck and O.G. Jensen (Geophysics, vol.53, no.11, p.1410-14, 1988) recently reported that some reflectivity sequences denoted μ(k), calculated from sonic logs, are not white and have a power spectral density approximately proportional to frequency, which is called a Joseph spectrum. A robustness test is now presented for the case of μ(k) having a Joseph spectrum for the minimum-variance deconvolution (MVD) filter and the maximum-likelihood deconvolution (MLD) algorithm, which were developed based on the whiteness assumption about μ(k). From the simulations performed, it is concluded that the possible Joseph spectrum of μ(k) is not a concern when applying the MVD filter and MLD algorithm towards estimating μ(k) from seismic data     

Maximum-rank array codes and their application to crisscross errorcorrection

A μ-[n×n,k] array code C over a field F is a k-dimensional linear space of n×n matrices over F such that every nonzero matrix in C has rank ⩾μ. It is first shown that the dimension of such array codes must satisfy the Singleton-like bound k⩽n(n-μ+1). A family of so-called maximum-rank μ-[n×n,k=n ( n-μ+1)] array codes is then constructed over every finite field F and for every n and μ, 1⩽μ⩽n . A decoding algorithm is presented for retrieving every Γ∈C, given a received array Γ+E, where rank (E)+1⩽(μ-1)/2. Maximum-rank array codes can be used for decoding crisscross errors in n×n bit arrays, where the erroneous bits are confined to a number t of rows or columns (or both). This construction proves to be optimal also for this model of errors. It is shown that the behavior of linear spaces of matrices is quite unique compared with the more general case of linear spaces of n×n. . .×n hyper-arrays     

Minimum-variance deconvolution and maximum-likelihood deconvolutionfor nonwhite Bernoulli-Gaussian processes with a Joseph spectrum

Todoeschuck and Jensen (1988) recently reported that the reflectivity sequences, denoted p(k), calculated from some sonic logs are not white and have a power spectral density approximately proportional to frequency, called a Joseph spectrum. It is shown here how to compute the minimum-variance estimate and maximum-likelihood estimate for a μ(k) modeled as a nonwhite Bernoulli-Gaussian (B-G) process with a Joseph spectrum. Also presented are the corresponding estimates for a statistically equivalent white B-G process μ*(k) which mimics μ(k). Some conclusions regarding the acceptability of these estimates are drawn     

Efficient algorithm for reliability of a circular consecutive-k-out-of-n:F system

The time complexities of previously published algorithms for circular consecutive-k-out-of-n:F systems are O (nk²) and O(nk). The authors propose a method to improve the original O(nk² ) algorithm, and hence derive an O(nk) algorithm     

Reliability of linear and circular consecutively-connected systems

J.G. Shanthikumar (IEEE Trans. Reliability, vol.R-36, p.546-50, Dec. 1987) proposed a new system structure called the consecutively connected system, which is a generalization of the well-studied consecutive-k-out-of-n:F system. An O(n ²) algorithm was developed for its reliability evaluation. This work studies the structure of the consecutively connected system and provides more efficient algorithms for its reliability evaluation. The complexities of the algorithms are of order O(kn) for linear systems and O(k³n) for circular systems (k <n). The concepts of transmitting capability and receiving capability are introduced     

An efficient non-recursive algorithm for computing the reliabilityof k-out-of-n systems

An algorithm for computing the reliability of k-out-of- n systems is proposed. It is simple, easy to implement on a computer, time and memory efficient, and good for numerical computation. The memory complexity is O(n-k), and for a given value of n-k the computation time is proportional to n . Its FORTRAN implementation is presented     

An improved inverse filtering method for parametric spectralestimation

For a wide-sense stationary process x(k), it is well known that its power spectrum P_xx(f) can be estimated by whitening the data with the inverse filter, V (z)=1/H(z), of the assumed minimum-phase rational model H(z) associated with x(k ). However, the initial conditions for computing the output e (k) of the recursive filter V(z) are unknown and must be preassigned. An improved inverse filtering method which simultaneously estimates the coefficients of V(z) as well as the initial conditions is proposed. The resultant power spectral estimator, with the initial conditions being estimated, outperforms that with the initial conditions wrongly set to zero as the time constant of V(z) is comparable to the number of data. Some simulation results which support the superior performance of the former are presented     

Multistate consecutively-connected systems

G. Shanthikumar (ibid., vol.R-36, p.546-550, Dec. 1987) has proposed a system called the consecutively connected system, which is a generalization of the consecutive-k-out-of-n:F system. He gave an O(n²) algorithm for computing the reliability of the consecutively connected system. In the present work, the authors further generalize it by assuming that the components are multistate. They give an O(Kn) algorithm for computing the reliability of the multistate system where K(K<n) is the maximum number of states for a component. They discuss the difficulties of obtaining corresponding algorithms for circular systems and two-way communication, although for two-state components they are able to extend the Shanthikumar algorithm to circular systems     

An O(kn)-time algorithm for computing thereliability of a circular consecutive-k-out-of-n:Fsystem

I. Antonopoulou and S. Papastavridis (1987) published an algorithm for computing the reliability of a circular consecutive-k-out-of-n:F system which claimed O (kn) time. J.S. Wu and R.J. Chen (1993) correctly pointed out that the algorithm achieved only O(kn²) time. The present study shows that the algorithm can be implemented for O(kn) time     

New minimum distance bounds for certain binary linear codes

Let an [n, k, d]-code denote a binary linear code of length n, dimension k, and minimum distance at least d. Define d(n, k) as the maximum value of d for which there exists a binary linear [n, k, d]-code. T. Verhoeff (1989) has provided an updated table of bounds on d(n, k) for 1⩽k⩽n⩽127. The authors improve on some of the upper bounds given in that table by proving the nonexistence of codes with certain parameters     

Recursive updating the eigenvalue decomposition of a covariancematrix

The author addresses the problem of computing the eigensystem of the modified Hermitian matrix, given the prior knowledge of the eigensystem of the original Hermitian matrix. Specifically, an additive rank-k modification corresponding to adding and deleting blocks of data to and from the covariance matrix is considered. An efficient and parallel algorithm which makes use of a generalized spectrum-slicing theorem is derived for computing the eigenvalues. The eigenvector can be computed explicitly in terms of the solution of a much-reduced (k ×k) homogeneous Hermitian system. The overall computational complexity is shown to be improved by an order of magnitude from O(N³) to O(N ²k), where N×N is the size of the covariance matrix. It is pointed out that these ideas can be applied to adaptive signal processing applications, such as eigen-based techniques for frequency or angle-of-arrival estimation and tracking. Specifically, adaptive versions of the principal eigenvector method and the total least squares method are derived     

Calculation of the binomial survivor function [reliabilityapplications]

A method is presented for calculating the binomial SF (cumulative binomial distribution), binfc(k;p,n), especially for a large n, beyond the range of existing tables, where conventional computer programs fail because of underflow and overflow, and Gaussian or Poisson approximations yield insufficient accuracy for the purpose at hand. This method is used to calculate and sum the individual binomial terms while using multiplication factors to avoid underflow; the factors are then divided out of the partial sum whenever it has the potential to overflow. A computer program uses this technique to calculate the binomial SF for arbitrary inputs of k, p, and n. Two other algorithms are presented to determine the value of p needed to yield a specified SF for given values of k and n and calculate the value where p=SF for a given k and n. Reliability applications of each algorithm/program are given, e.g. the value of p needed to achieve a stated k-out-of-n :G system reliability and the value of p for which k -out-of-n:G system reliability equals p     

The Zak transform and some counterexamples in time-frequencyanalysis

It is shown how the Zak transform can be used to find nontrivial examples of functions f, g∈L²(R) with f×g≡0≡F×G, where F, G are the Fourier transforms of f, g, respectively. This is then used to exhibit a nontrivial pair of functions h, k∈L²(R), h≠k, such that |h|=|k|, |H |=|K|. A similar construction is used to find an abundance of nontrivial pairs of functions h, k∈L² (R), h≠k, with |A_h |=|A_k| or with |W_h|=|W _k| where A_h, A_k and W_h, W_k are the ambiguity functions and Wigner distributions of h, k, respectively. One of the examples of a pair of h, k∈L²(R), h≠k , with |A_h|=|A_k| is F.A. Grunbaum's (1981) example. In addition, nontrivial examples of functions g and signals f₁≠f₂ such that f₁ and f₂ have the same spectrogram when using g as window have been found     

On the influence of coding on the mean time to failure fordegrading memories with defects

The application of a combined test-error-correcting procedure is studied to improve the mean time to failure (MTTF) for degrading memory systems with defects. The degradation is characterized by the probability p that within a unit of time a memory cell changes from the operational state to the permanent defect state. Bounds are given on the MTTF and it is shown that, for memories with N words of k information bits, coding gives an improvement in MTTF proportional to (k/n) N(d_min^-2)/(d_min ^-1), where d_min and (k/n) are the minimum distance and the efficiency of the code used, respectively. Thus the time gain for a simple minimum-distance-3 is proportional to N^-1. A memory word test is combined with a simple defect-matching code. This yields reliable operation with one defect in a word of length k+2 at a code efficiency k/(k+2)     

A WDM cross-connected star topology for multihop lightwave networks

The authors propose a star topology for multihop lightwave networks in which the conventional N×N passive star coupler is replaced by fixed wavelength division multiplexed (WDM) cross connects. The proposed topology overcomes three major limitations of the conventional star topology. First, it reduces the number of wavelengths needed in a (p,k) ShuffleNet from kp ^k+1 wavelengths in the conventional topology to p wavelengths in the proposed one. Second, the signal power loss due to the 1/N power splitting at the star coupler no longer exists in the WDM cross connects and, therefore, the restriction on the supported number of users by the star network is alleviated. Third, it completely eliminates the need for wavelength filtering at the input to the receivers as is the case in the conventional star topology     

An approximation for largeconsecutive-k-out-of-n:F systems

The authors consider a consecutive-k-out-n:F system consisting of identically distributed and statistically independent components, where the life distribution of an individual component is Weibull distributed with scale parameter 1/λ and shape parameter B. Let T_n be the life length of the consecutive-k-out-of-n:F system. The authors prove that for large values of n, the distribution of the n ¹(ka)/T_n, is satisfactorily approximated by a Weibull distribution with the same scale parameter and shape parameter k times the original shape parameter     

On runlength codes

Several results on binary (d, k) codes are given. First, a novel derivation for the capacity of these codes based on information-theoretic principles is given. Based on this result the spectrum of a (d, k) code is computed. Finally, the problem of computing the capacity of the binary symmetric channel under the condition that the input sequences satisfy the (d, k ) constraint is considered. Lower bounds on the capacity of such a channel are derived     

Coset correlation of m-sequences

It is well-known that there exists a unique shift l of the m-sequence S(k) such that the value of S₀(k)=S(k+l) is only determined by the cyclotomic coset to which k belongs. A measure called the `coset correlation' is introduced. It is proven that the shift l can be determined by the coset correlation     

Index system and separability of constant weight Gray codes

A number system is developed for the conversion of natural numbers to the codewords of the Gray code G(n,k) of length n and weight k, and vice versa. The focus is on the subcode G(n,k) of G(n) consisting of those words of G(n) with precisely k 1-bits, 0<k<n. This code is called the constant weight Gray code of length n and weight k. As an application sharp lower and upper bounds are derived for the value of |i-j|, where i and j are indices of codewords g_i and g_j of G(n,k) such that they differ in precisely 2 m bits