We develop an automatic smoothing procedure for an estimate of the spectral density of a random process. The procedure is based on smoothing the periodogram with variable bandwidth and a spline interpolation. Effective varying bandwidth is obtained by approximating the log periodogram with a step function whose positions of level changes are determined using a dynamic programming technique. We show that the step function can be obtained by minimizing the cost function <i>D</i>(<i>C</i>||μ<i>k</i>) for a given <i>K</i>. The number of partitions <i>K</i> also can be chosen by minimizing another cost function <i>L(K)</i>. Some numerical examples show that the resulting estimates are shown to be good representations of the true spectra.
In this paper, we propose a method for tracking of a instantaneous equivalent bandwidth (IEBW) of non-stationary random signals. IEBW is defined on the positve time-frequency distribution of a non-stationary random signal by using Renyi entropy. It is a natural extension of a equivalent bandwidth for stationary random signals. In order to obtain the positive time-frequency satisfying the marginals of a random signal, we have modified a copula-based time-frequency technique slightly. We, then, showed the results of two simple computer simulations. The results show that the method presented here can track the IEBW of the random signals properly. We also applied the method to track the change of the IEBW of the heart sound. The results suggest that tracking the IEBW could be a useful index for automatic diagnostic of heart disease.
In this paper, we present the definition of a the generalized equivalent bandwidth (EBW) of a stochastic process. The generalized EBW is defined by <i>W</i><sup>(<i>α</i>)</sup> = exp(<i>H</i><sup>(<i>α</i>)</sup>)/2, where <i>H</i><sup>(<i>α</i>)</sup> is Renyi's entropy <i>H</i><sup>(<i>α</i>)</sup> = 1/(1-<i>α</i>)log (<sub> -infinity</sub>Integral<sup>infinity </sup>) <i>p<sup>α</sup></i>(<i>f</i>)<i>df</i>, <i>p</i>(<i>f</i>) is the normlized power spectrum and <i>α</i> greater than or equal to 0 is the order of the EWB. The generalized EBW is a new class of EBW which can represent major equivalent bandwidths uniformly. We also argue an interpretation of the generalized EBW from a different perspective. In latter of this article, we examine an estimation property of the generalized EBW. When we obtain an estimated smoothed power spectrum by using the convolution of periodogram and smoothing window, we evaluate how smoothing window length, data length or the variance of an estimated spectrum affect estimation of the generalized EBW. The result indicates that if we increase the data length while keeping the variance constant, the increase rate of the generalized EBW caused by smoothing window will decrease. On the other hand, if we decrease the variance while holding the data length fixed, the generalized EBW of estimated power spectrum will increase.
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Noise and Fluctuations in Photonics, Quantum Optics, and Communications