statistical characterization of the music null spectrum statistical characterization of the music sample null spectrum is presented for an arbitrary number of possibly correlated emitters observed by a general array a remarkably simple expression is developed for the spectral covariance matrix in the vicinity of the emitters the covariance result together with prior mean spectral results support a gaussian characterization in the neighborhood of the emitters for many scenarios it is shown that the sample spectrum is exactly wishart distributed in the emitter directions the results can be utilized to quantify a variety of performance metrics as an illustration models based on physical parameters are constructed for the curve of probability of resolution versus signal to noise ratio the models predict the curve with a high degree of accuracy for example scenarios

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

performance analysis of music type high resolution estimators for direction finding in correlated and coherent scenes an asymptotic analysis is presented of a class of high resolution estimators for resolving correlated and coherent plane waves in noise these estimators are in turn constructed from certain eigenvectors associated with spatially smoothed or unsmoothed covariance matrices generated from a uniform array the analysis is first carried out for the smoothed case and from this the conventional i e unsmoothed multiple signal classification music scheme follows as a special case independent of the total number of sources present in the scene the variance of the conventional music estimator along the true arrival angles is shown to be zero within a first order approximation the bias expressions in the smoothed case are used to obtain a resolution threshold for two coherent equipowered plane waves in white noise and the result is compared to the one derived by kaveh et al 1986 for two uncorrelated equipowered plane waves

effects of amplitude and phase errors on linear predictive array processors the effects of amplitude and phase errors are determined for symmetric and asymmetric linear predictive array processors the maximum entropy method mem corresponds to linear prediction in the asymmetric case of a line array of equispaced elements with an end reference a comparison to conventional beamforming and maximum likelihood processing shows that they have the same peak location as the symmetric linear predictive array processor

investigation of difference of fundamental diagrams in pedestrian flow abstract in this paper the difference of fundamental diagrams in pedestrian flow is investigated a new measurement method in which the moving back is considered as negative contribution to pedestrian flow is proposed based on previous study firstly the comparison of different measurement methods is made and small discrepancy in the fundamental diagrams of the same experiment is observed in order to remove the effect of different measurement methods on final results we compare the fundamental diagrams of chinese and german experiments by using the same measurement method and surprisingly large difference is found from the analysis of experimental video it is observed the motivation and competitiveness of participants in two experiments are quite different which plays a dominant role on the large difference between fundamental diagrams to make it more tenable we further analyze two german experiments hermes experiment and basigo experiment in which the participants have the same average age the free velocity is adopted to measure pedestrian s motivation in the experiment and the analysis results confirm that the discrepancy of motivation and competitiveness of participants in different experiments leads to the large difference in the fundamental diagrams the study may be helpful to understand the effect of pedestrian behavior on global flow and microscopic dynamics

karush kuhn tucker conditions for interval and fuzzy optimization in several variables under total and directional generalized differentiability abstract we extend the interval and fuzzy gh differentiability to consider interval and fuzzy valued functions of several variables and to include directional gh differentiability the proposed setting is more general than the existing definitions in the literature and allows a unified view of total and direction gh differentiability and for the computation of partial gh derivatives directional gh derivative and level wise gh differentiability in the fuzzy valued case a concept of gh differential is then deduced and its values are used to define an abstract local tangency property for a gh differentiable function similar to the well known tangency between a differentiable function and its tangent plane the proposed new setting allows an analysis of conditions for local optimality dominance with respect to interval and level wise partial orders well known in the literature in terms of directional gh derivatives including concepts of local convexity and to formulate kkt like conditions for non dominated solutions in constrained optimization problems

measuring the steady state of pedestrian flow in bottleneck experiments abstract experiments with pedestrians could depend strongly on initial conditions comparisons of the results of such experiments require to distinguish carefully between transient state and steady state thus a modified version of the cumulative sum control chart algorithm is proposed to robustly detect steady states from density and speed time series of bottleneck experiments the threshold of the detection parameter in the algorithm is calibrated using an autoregressive model comparing the detected steady states with manually selected ones the modified algorithm gives robust and reproducible results for the applications three groups of bottleneck experiments are analysed and the steady states are detected the results reconfirm that the specific flow is constant as bottleneck width changes moreover we proposed a criterion to judge the difference between the flows in all states and in steady states which is the ratio of pedestrian number to bottleneck width the critical value of the ratio is found to be approximately 115 persons m this conclusion applies not only for the analysis of existing bottleneck experiments but also for the design of new bottleneck experiments and the validation of evacuation models furthermore the range of steady state in time series of pedestrian characteristics could be effectively controlled by adjusting the value of the ratio

bias analysis of the music location estimator the authors present a rigorous bias analysis of the music location estimator and they derive an accurate and concise bias expression the analysis is based on the second order taylor series expansion of the derivative of the null spectrum properties of the null spectrum and statistics of the estimated signal eigenvectors it is proven that in the derivation the remainder term in the second order taylor series can be dropped but the second order terms cannot be simulations verify that the bias expression is valid over a wide range of signal to noise ratios snrs extending down into the resolution threshold region of music although asymptotic this expression can be accurately applied to a limited number of snapshot cases the utility of the expression is shown by using it in a study of music location estimator characteristics estimate bias and standard deviation are compared for variations in snr numbers of sensors and snapshots and source correlation music resolvability and estimator performance bounds are addressed accounting for bias

a resolution measure for the music algorithm and its application to plane wave arrivals contaminated by coherent interference the authors present a deterministic approach to the resolution analysis of the music algorithm for resolving plane waved contaminated by coherent interference when ensemble average covariance matrices are used a resolution measure to describe the resolution potential of music in resolving two plane waves in noise is introduced a closed form solution for the resolution measure is given and the result is compared to the resolution threshold derived by m kaveh and a j barabell 1986 when the covariance matrices are estimated from a finite number of snapshots an approximate expression for the threshold signal to interference ratio sir is derived for the cases when sirs are large and the coherent interference and direct arrivals are well separated in spatial frequency illustrations are given using an 11 element uniform array for the cases of plane wave reflection from an infinite plane and scattering from an infinite conducting circular cylinder

characterizations of generalized differentiable fuzzy functions this article deals with generalized differentiable fuzzy functions specifically we give some characterizations of generalized hukuhara differentiable fuzzy functions through the differentiability of their endpoint functions then we introduce a differentiability concept that is more general than the generalized hukuhara differentiability and extend the above characterization to this new type of differentiability as well as to g differentiable fuzzy functions these characterizations are useful tools for the calculus of derivatives of fuzzy functions

statistical resolution limits and the complexified crame spl acute r rao bound array resolution limits and accuracy bounds on the multitude of signal parameters e g azimuth elevation doppler range cross range depth frequency chirp polarization amplitude phase etc estimated by array processing algorithms are essential tools in the evaluation of system performance the case in which the complex amplitudes of the signals are unknown is of particular practical interest a computationally efficient formulation of these bounds from the perspective of derivations and analysis is presented for the case of deterministic and unknown signal amplitudes a new derivation is given using the unknown complex signal parameters and their complex conjugates the new formula is readily applicable to obtaining either symbolic or numerical solutions to estimation bounds for a very wide class of problems encountered in adaptive sensor array processing this formula is shown to yield several of the standard crame spl acute r rao results for array processing along with new results of fundamental interest specifically a new closed form expression for the statistical resolution limit of an aperture for any asymptotically unbiased superresolution algorithm e g music esprit is provided the statistical resolution limit is defined as the source separation that equals its own crame spl acute r rao bound providing an algorithm independent bound on the resolution of any high resolution method it is shown that the statistical resolution limit of an array or coherent integration window is about 1 2 spl middot snr sup 1 4 relative to the fourier resolution limit of 2 spl pi n radians large number n of array elements that is the highest achievable resolution is proportional to the reciprocal of the fourth root of the signal to noise ratio snr in contrast to the square root snr sup 1 2 dependence of standard accuracy bounds these theoretical results are consistent with previously published bounds for specific superresolution algorithms derived by other methods it is also shown that the potential resolution improvement obtained by separating two collinear arrays synthetic ultra wideband each with a fixed aperture b wavelengths by m wavelengths assumed large is approximately m b sup 1 2 in contrast to the resolution improvement of m b for a full aperture exact closed form results for these problems with their asymptotic approximations are presented

the statistical performance of the music and the minimum norm algorithms in resolving plane waves in noise this paper presents an asymptotic statistical analysis of the null spectra of two eigen assisted methods music 1 and minimum norm 2 for resolving independent closely spaced plane waves in noise particular attention is paid to the average deviation of the null spectra from zero at the true angles of arrival for the plane waves these deviations are expressed as functions of signal to noise ratios number of array elements angular separation of emitters and the number of snapshots in the case of music an approximate expression is derived for the resolution threshold of two plane waves with equal power in noise this result is validated by monte carlo simulations

a frequency domain approach to eigenvalue based detection with diversity reception and spectrum estimation in this paper we investigate a frequency domain approach for eigenvalue based detection of a primary user based on equal gain combining egc and spectrum estimation with bartlett s method this paper considers two techniques for eigenvalue detection which are maximum eigenvalue detection med and the maximum minimum eigenvalue mme detector we exploit the eigenvalues that are associated with the hermitian form representation of bartlett s estimate to assess the performance of the aforementioned eigenvalue techniques in the frequency domain for each case we quantify the performance based on the probabilities of false alarm and missed detection over rayleigh and rician fading a bivariate mellin transform approach is employed to obtain the probability distribution function for the ratio of the extreme eigenvalues under each hypothesis all obtained formulas are validated via monte carlo simulations and the results give a clear insight into the performance of the investigated methods in frequency domain med outperforms both the mme detector and periodogram based energy detection even in a worst case scenario of noise uncertainty while the mme detector exhibits heavy tailed statistical characteristics and thus its receiver operating characteristics tend to stay on the line of no discrimination the performance of med is further enhanced by careful choice of combinations of the total length of the sensing frame and number of sub slots

on the performance of energy detection using bartlett s estimate for spectrum sensing in cognitive radio systems energy detection can be used for spectrum sensing in cognitive radios when no prior knowledge about the primary signals is available the performance of this technique however is strongly influenced by the available decision estimate this paper presents accurate performance analysis of energy detection when using bartlett s estimate as a test statistic both independent and identically distributed rayleigh and rician fading channels are investigated for unknown signals with complex envelopes the novel contribution here is threefold first the quadratic form representation of bartlett s estimate is formulated then starting from the characteristic function the cumulative distribution function is derived for each type of channel and accurate expressions are developed for the probabilities of false alarm and missed detection finally a performance comparison with the raw periodogram is presented the accuracy of the proposed analysis is confirmed using monte carlo trials the results provide valuable insight into the performance of bartlett s based energy detection it will be seen that the additional degrees of freedom associated with bartlett s method empower it to offer excellent performance when compared with the raw periodogram bartlett s consistently leads to lower probability of miss but its probability of false alarm is only superior at relatively higher sensing thresholds

modeling and performance analysis of multitaper detection using phase type distributions over mimo fading channels this paper presents modeling and analysis of two variations of the multitaper detector namely multiple antenna detection of a single user multiple input multiple output mimo node and the multitaper method mtm combined with singular value decomposition svd which is known as the mtm svd processor motivated by the reputation of the mtm as the best nonparametric power spectral density psd estimator and after reviewing the limited previous research attempts which focus on single input single output siso multitapering we present the exact analytical models for the two considered derivatives of the multitaper method over fading channels by making use of the theory of hermitian forms and phase type distributions in addition using the neyman pearson approach npa the performance of both detectors is optimized over nakagami fading for both multitaper variations we accurately derive the eigenvalues of the hermitian form of each detector where the eigenvalues identify the phase type distribution parameters this yields generalized expressions for the probabilities of false alarm and missed detection when arbitrary multitaper weights are used finally we investigate the impact of noise uncertainty on the performance of mimo mtm the results show that performance of both detectors is dependent on the total number of discrete prolate spheriodal sequences dpsss while for the mtm svd processor the performance is also dependent on the number of cooperating users and the employed frequency resolution it is also shown that mimo mtm is robust under noise uncertainty the obtained analytical models are proven to be accurate and enables further investigations on the multitaper detector

performance analysis of the periodogram based energy detector in fading channels in this paper we provide a performance analysis of cognitive radio systems employing energy detection based on power spectral density estimation specifically we derive mathematical expressions for the probabilities of false alarm and missed detection when the periodogram estimate is used as a decision statistic first using the characteristic function of the quadratic form representation of the periodogram we study the characteristics of the product of the sample covariance matrix of the observed vector and the symmetric matrix of the quadratic form we have found that this product is rank 1 with one non zero eigenvalue and therefore a simplified expression for the cumulative density function cdf is obtained second the cdf is exploited to derive accurate mathematical expressions for the probabilities of false alarm and missed detection in both rayleigh and rician fading channels also for each case we have derived the probability distribution function pdf of the corresponding eigenvalues the results demonstrate that the probabilities of false alarm and missed detection are independent of the length of the observations when all the other parameters remain fixed the results also reveal that the considered frequency domain based energy detector is superior to its conventional time domain counterpart as it provides fewer false alarms

the generalized hukuhara differentiability of interval valued function is not fully equivalent to the one sided differentiability of its endpoint functions abstract in this paper we first show by a counterexample that the generalized hukuhara differentiability of interval valued function at a point is not fully equivalent to the one sided differentiability of its endpoint functions then present a complete characterization of the generalized hukuhara differentiability of interval valued functions the obtained results include the existing cases in the literature and the new cases that correspond to the rather odd scenario of functions being gh differentiable at a point but not continuous in any deleted neighborhood of that point

performance analysis of root music the authors analyze the performance of root music a variation of the music algorithm for estimating the direction of arrival doa of plane waves in white noise in the case of a linear equispaced sensor array the performance of the method is analyzed by examining the perturbation in the roots of the polynomial formed in the intermediate step of root music in particular asymptotic results for the mean squared error in the estimates of the direction of arrival are derived simplified expressions are presented for the one and two source case and compared to those obtained for least squares esprit computer simulations are also presented and they are in close agreement with the theory an important outcome of this analysis is the fact that the error in the signal zeros has a largely radial component this provides an explanation as to why the root music is superior to the spectral music algorithm

a statistical resolution theory of the ar method of spectral analysis the autoregressive ar method of spectral analysis is widely used in diverse areas for its solid theoretical foundation interesting physical interpretation computational efficiency and more importantly high resolution capability various aspects of its statistical performance have been investigated however the resolution probability that provides the most rigorous description of the spectrum resolution capability is still not available in the literature in this paper by formulating the resolution event in the framework of statistical decision theory and directly determining its probability from its characteristic function we obtain an exact asymptotic formula for the probability of resolution on this basis we determine the limiting resolving behavior of the sample ar spectrum and develop the corresponding geometrical insight in the parametric space simulation and numerical results are also presented to confirm and illustrate the theory

a statistical resolution theory of the beamformer based spatial spectrum for determining the directions of signals in white noise due to its simplicity and ease of implementation the beamformer based spatial spectrum bss is widely used in practice to determine the angles of arrival of signals in white noise in array processing as the most important performance to this application the resolution capability of the bss is roughly assessed by a deterministic measure borrowed from optics which takes into account only the influence of the effective aperture of the linear array of sensors for a linear array operating in a stochastic environment resolving two incoming signals is essentially a random event thereby demanding a probabilistic resolution measure in the paper the probabilistic performance of the bss in resolving two incoming signals is studied the author derives a simple expression in a closed form for its failure probability in distinguishing the two signals the results reveal the way various parameters of signals noise and array of sensors affect the resolution performance computer simulations are also presented to confirm the validity of the theory

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

probability of resolution of the music algorithm the music algorithm is well known for its high resolution capability and various aspects of its statistical performance have been investigated however rigorous asymptotic analysis of one of its most important performance measures the probability of resolution is not available yet we analyze the probability of the music algorithm resolving two spatially separated signal sources in the context of array processing by formulating the resolution problem in the framework of statistical decision theory and directly determining the probability density function pdf of the indefinite and singular quadratic form that defines the resolution event we arrive at an exact asymptotic formula for the probability of resolution this is accomplished by a multistep procedure computer simulations have been performed to confirm the validity of the theory

generalized derivative and π derivative for set valued functions in this paper we study the generalized derivative and the p derivative for interval valued functions we show the connections between these derivatives some illustrative examples and applications to interval differential equations and fuzzy functions are presented

extraction and quantitative analysis of microscopic evacuation characteristics based on digital image processing abstract in this study experiments of single file pedestrian movement were conducted and the movement parameters of pedestrians were extracted with a digital image processing method based on a mean shift algorithm the microscopic characteristics of pedestrian dynamics including velocity density and lateral oscillation as well as their interrelations were obtained and analyzed firstly we studied the lateral oscillation phenomena of pedestrian movement the result indicates that the trajectory of pedestrians presents a wavy form and the amplitude of the oscillation remains about 5 5 cm when the pedestrians move with free walking velocity which is the velocity when there is no obvious interaction between sequential pedestrians but when the movement velocity decreases to 0 27 m s the amplitude of oscillation increases to 13 cm with increasing density the velocity decreases and the amplitude of oscillation presents a linear increase trend the increasing oscillation amplitude widens the occupation area of a pedestrian with low velocity so as to make the moving efficiency even worse secondly we studied the frequency of the oscillation the result indicates that the frequency remains at 2 hz when pedestrians move with a free walking velocity but it presents a similar linear decrease trend when the velocity changes to a lower value the decrease of oscillation frequency is also a negative feedback to the moving efficiency thirdly it is found that with the increase of crowd density the time interval between two sequential pedestrians increases though the space gap between them decreases the quantitative relation between time interval and crowd density is obtained the study in this paper provides fundamental data and a basic method for understanding pedestrian dynamics developing and validating evacuation models the results are also expected to be useful for evacuation design