Instantaneous Spectrum Estimation from Event-Based Densities
- PDF / 759,047 Bytes
- 5 Pages / 600 x 792 pts Page_size
- 72 Downloads / 211 Views
nstantaneous Spectrum Estimation of Event-Based Densities Lorenzo Galleani Dipartimento di Elettronica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy Email: [email protected]
Leon Cohen City University of New York, 695 Park Avenue, New York, NY 10021, USA Email: [email protected]
Douglas Nelson US Department of Defense, Fort Meade, MD 20755, USA Email: [email protected]
Jeffrey D. Scargle Space Science Division, NASA Ames Research Center, Moffett Field, CA 94035, USA Email: [email protected] Received 31 July 2001 and in revised form 30 September 2001 We present a method for obtaining a time-varying spectrum that is particularly suited when the data are in event-based form. This form arises in many areas of science and engineering, and especially in astronomy, where one has photon counting detectors. The method presented consists of three procedures. First, estimating the density using the kernel method; second, highpass filtering the manifestly positive density; finally, obtaining the time-frequency distribution with a modified Welch’s method. For the sake of validation, event-based data are generated from a given distribution and the proposed method is used to construct the time-frequency spectrum and is compared to the original density. The results demonstrate the effectiveness of the method. Keywords and phrases: time-frequency analysis, density estimation, kernel method.
1.
INTRODUCTION
An important problem in many fields is the study of densities and spectra where the data are event-based. We illustrate with a specific example of the study of a signal coming from the X-ray binary system XTE-J1550. The signal is made by a sequence ti of the photon arrival times, measured by an Xray detector orbiting the Earth [1, 2]. The more concentrated are the photons in a given time interval, the higher will be the intensity of the radiation (in the X-ray region) for that particular time interval. Hence the density f (t) of the sequence ti is here intended as the normalized intensity of the radiation underlying the arriving photons. Since a fundamental property of such a density is its instantaneous spectrum [3, 4], because it is believed to carry important physical information on the system that generated the events, we present here a method that enables us to perform this estimation in an effective way. We want to build an
estimation algorithm that performs the following operation ti −→ P (t, ω),
(1)
where P (t, ω) is the estimated instantaneous spectrum of the (normalized) density f (t) underlying the sequence of events ti . The astronomy case mentioned earlier and described in detail in [1, 2] can be considered as an excellent paradigm for instantaneous spectrum estimation of event-based densities. Here the main physical quantity is time, but the identical method can be applied to any quantity x (with associated density f (x)). The method consists of three steps. First, a density estimation is accomplished by using the kernel method, namely sliding a window that makes a we
Data Loading...
