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Astronomical Data Analysis Software and Systems VI
ASP Conference Series, Vol. 125, 1997
Editors: Gareth Hunt and H. E. Payne

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The Time Interferometer: Synthesis of the Correlation Function

V. V. Vityazev
Astronomy Department, St. Petersburg University, Bibliotechnaya pl.2, Petrodvorets, St. Petersburg, 198904, Russia.

Abstract:

1. Review of the Fundamentals

Suppose that a time series is a stationary stochastic process with zero mean defined by a set of realizations

To describe gaps in observations we introduce the time window function

With this notation the observed time series can be represented as

Calculate the periodogram:

where <> denotes averaging over the set of realizations. Under the conditions stated above, the relationship between the periodogram and the power spectrum is given by the convolution

where the spectral window function is the periodogram of the time window function

Now, from Eq. (43), for the correlogram and the correlation window , introduced as inverse Fourier transforms of and respectively, one has:

where is the auto-correlation function of . For further details the reader is referred to Jenkins & Watts (1968), Deeming (1975), Otnes & Enocson (1978), Marple (1978), and Terebish (1992).

In interferometry (Esepkina et al. 1973; Thompson et al. 1986), the intensity of radiation from any source on the sky can be described in terms of position , wavelength , and time (t). Each measurement is an average over a band of wavelengths and over some span of time. Here, the one-dimensional, monochromatic, and instantaneous approximation is used to simplify the discussion. The resulting specific intensity that describes the distribution of the source brightness along the arc of a circle is the angular coordinate) has a Fourier transform , which is called the spectrum of spatial frequencies. Correspondingly, when an interferometer measures the visibility data , the image, or the map, can be calculated by the Fourier transform of . Two fundamental relations are valid:

where is the beam of interferometer and is the transfer function. Eq. (47) determines an interferometer as a filter of spatial frequencies, whereas Eq. (46) explains why, due to convolution of with the beam , the resulting image is called a dirty map.

Upon close examination one can see that in both sciences the rigorous (theoretical) quantities are introduced at the first level. In the spectral-analysis case they are the power spectrum and the correlation function; in the interferometry their counterparts are the distribution of brightness and the spatial spectrum. At the second level we have estimators of the strict quantities. In spectral analysis, they are the periodogram and the correlogram, whereas in the interferometry these are the map and the visibility data, respectively. Finally, equations which connect the quantities of the two levels are identical (convolution and multiplication ) and include the characteristics of observations: the beam and the transfer function and their analogs, i.e., the spectral window and the correlation window.

In reality, due to the finite dimensions of mirrors and the finite time spans of observations we cannot get the true quantities, and all we can do is to find their optimal approximations. In optics or in radio astronomy, when filled apertures are used, the maps are produced directly in the focal plane of a telescope. Analogously, when the time series is given at all points of some interval or at time points regularly spaced within the interval, the evaluation of the periodogram can be made quite easily. When an interferometer is used, the aperture is not solid, and what we can measure is the visibility data, i.e., the estimator of the spectrum of spatial frequencies. The longer the baseline, the smaller the area of the -plane (u-domain in the one-dimensional case) filled, and the more dirty the resulting map becomes.

To overcome this, various techniques of aperture synthesis are used, and this leads to complete solution of the problem since the -plane is completely filled. If the aperture synthesis provides partial filling of the -plane, the cleaning procedures can be used with the aim of eliminating the artifacts of the ``holes'' in the -plane from the map. We have the same problems in the spectral-analysis case, when the time points are distributed irregularly or have long gaps. In this case the correlograms cannot be determined for all values of time lag , and this would give false features in the resulting periodograms. The main aim of the present paper is to answer the question: is it possible to apply the aperture synthesis method to spectral analysis of time series?

2. Synthesis of the Correlation Function

It is known that to do aperture synthesis one must have an interferometer with the changeable baseline. Of all the schemes of the aperture synthesis the one proposed by Ryle (1960) is the most suitable for us. This consists of two antennas A and B fixed at separation L. The third antenna C is moving inside the interval . At each position of the moving antenna one obtains two interferometers (CB) and (AC) with the baselines l and , and they yield the values of the visibility data at the points and where . Obviously, while antenna C sweeps all the interval , the visibility data become available at all points of the interval .

Now we apply this idea to time series analysis. Every two points spaced at the distance may be called the Time Interferometer with variable baseline , since each such pair yields the estimation of the correlation function by averaging the products over the set of realizations. The value can be obtained no matter where the points t and are located inside the interval . This follows from our assumption that is a stationary stochastic process, and that is crucial for our study.

Assume that the time series is given at two points and . This Time Interferometer allows us to get the correlation function only at the point Let us make new observations at the points + T. It is clear that each new point yields two additional values of the correlation function, namely, at the points and . Obviously, when the observations cover the interval , the values of become available at all points of the interval .

Now we see that the fixed antennas A and B in the Ryle interferometer are the counterparts of the boundary points and , while each new position of the moving antenna C is nothing else but the new point of observations. Since Ryle's interferometer makes the synthesis of the visibility data, it is a good reason to call the described procedure the synthesis of the correlation function.

Obviously, complete synthesis and the evaluation of from an even time series are the same. Thus, with elsewhere for the synthesized periodogram we get

This estimator yields the clean spectrum, i.e., free of the false peaks that come from the ``holes'' in the -domain.

To proceed further, assume that we have a gap in the observations. Let the length of the gap be l and the longest distance between the borders of the gap and the boundary points of the interval be a. If then the correlation function can be evaluated at all points , otherwise only on the subintervals and . In this case the correlation function turns out to be synthesized at all points except the ``hole'' of the length l-a, and, consequently, the synthesized periodogram calculated from Eq. (48) will not be clean. Nevertheless, it is less contaminated than the periodogram calculated directly from Eqs. (42). Thus we see that to clean the spectrum completely, one needs to perform more observations until the condition becomes true.

When the only realization is available, averaging over the ensemble should be replaced with averaging over time. For further development of the method the reader is referred to Vityazev (1996).

References:

Deeming, T. J. 1975, Ap&SS, 36, 137

Esepkina, N. A., Korolkov, D. N., & Pariysky, Yu. N. 1973, Radio telescopes and Radiometers (Moscow: Nauka)

Jenkins, G. M., & Watts, D. G. 1968, Spectral Analysis and its Applications (San Francisco: Holden-Day)

Marple, Jr., S. L. 1987, Digital Spectral Analysis with Applications (Englewood Cliffs, NJ: Prentice-Hall)

Otnes, R. K., & Enocson, L. 1978, Applied Time Series Analysis (New York: Wiley-Interscience)

Ryle, M., & Hewish, A. 1960, MNRAS, 120, 220

Terebish, V. Yu. 1992, Time Series Analysis in Astrophysics (Moscow: Nauka)

Thompson, A. R., Moran, J. M., & Swenson, G. W. Jr. 1986, Interferometry and Synthesis in Radio Astronomy (New York: Wiley)

Vityazev, V. V. 1996, Astron. and Astrophys. Tr., in press

Next: A New Stable Method for Long-Time Integration in an N-Body Problem
Previous: Time Series Analysis of Unequally Spaced Data: Intercomparison Between Estimators of the Power Spectrum
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