2 Spectral analysis and unevenly spaced data

2.1 Introduction

In what follows, we assume x to be a physical variable measured at discrete times t_i. x(t_i) can be written as the sum of the signal $x_{\rm s}$ and random errors R: $x_{i}=x(t_{i})=x_{\rm s}(t_{i})+R(t_{i})$. The problem we are dealing with is how to estimate fundamental frequencies which may be present in the signal $x_{\rm s}(t_{i})$ (Deeming 1975; Kay 1988; Marple 1987).

If x is measured at uniform time steps (even sampling) (Horne & Baliunas 1986; Scargle 1982) there are a lot of tools to effectively solve the problem which are based on Fourier analysis (Kay 1988; Marple 1987; Oppenheim & Schafer 1965). These methods, however, are usually unreliable for unevenly sampled data. For instance, the typical approach of resampling the data into an evenly sampled sequence, through interpolation, introduces a strong amplification of the noise which affects the effectiveness of all Fourier based techniques which are strongly dependent on the noise level (Horowitz 1974).

There are other techniques used in specific areas (Ferraz-Mello 1981; Lomb 1976): however, none of them faces directly the problem, so that they are not truly reliable. The most used tool for periodicity analysis of evenly or unevenly sampled signals is the Periodogram (Lomb 1976; Scargle 1982); therefore we will refer to it to evaluate our system.

2.2 Periodogram and its variations

The Periodogram (P), is an estimator of the signal energy in the frequency domain (Deeming 1975; Kay 1988; Marple 1987; Oppenheim & Schafer 1965). It has been extensively applied to pulsating star light curves, unevenly spaced, but there are difficulties in its use, specially concerning with aliasing effects.

2.2.1 Scargle's periodogram

This tool is a variation of the classical P. It was introduced by J.D. Scargle (Scargle 1982) for these reasons: 1) data from instrumental sampling are often not equally spaced; 2) due to P inconsistency (Kay 1988; Marple 1987; Oppenheim & Schafer 1965), we must introduce a selection criterion for signal components.

In fact, in the case of even sampling, the classical P has a simple statistic distribution: it is exponentially distributed for Gaussian noise. In the uneven sampling case the distribution becomes very complex. However, Scargle's P has the same distribution of the even case (Scargle 1982). Its definition is:

$$\begin{eqnarray} P_x(f) & = & \frac{1}{2} \frac{[\sum_{n=0}^{N-1} x(n)\cos 2\pi ... ...sin 2\pi f(t_n-\tau)]^2}{\sum_{n=0}^{N-1} \sin^2 2\pi f(t_n-\tau)}\end{eqnarray}$$ (1)

where

$$\tau=\frac{1}{4\pi f}\frac{\sum_{n=0}^{N-1} \sin 4\pi ft_n}{\sum_{n=0}^{N-1} \cos 4\pi ft_n}$$

and $\tau$ is a shift variable on the time axis, f is the frequency, $\{x\left( n\right) ,t_{n}\}$ is the observation series.

2.2.2 Lomb's periodogram

This tool is another variation of the classical P and is similar to the Scargle's P. It was introduced by Lomb (Lomb 1976) and we used the Numerical Recipes in C release (Numerical Recipes in C 1988-1992).

Let us suppose to have N points x(n) and to compute mean and variance:

 

$$\bar{x}=\frac{1}{N}\sum_{n=1}^{N}x(n)\qquad \qquad \sigma ^{2}=\frac{1}{N-1} \sum_{n=1}^{N}\left( x(n)-\bar{x}\right) ^{2}.$$ (2)

Therefore, the normalised Lomb's P (power spectra as function of an angular frequency $\omega \equiv 2\pi f\gt$) is defined as follows

$$\begin{eqnarray} P_{N}(\omega ) & = & \frac{1}{2\sigma^{2}}\left[ \frac{[\sum_{n... ...au )]^{2}}{\sum_{n=0}^{N-1}\sin ^{2}\omega (t_{n}-\tau )} \right]\end{eqnarray}$$ (3)

where $\tau$ is defined by the equation

$$\tan \left( 2\omega \tau \right) =\frac{\sum_{n=0}^{N-1}\sin 2\omega t_{n}} {\sum_{n=0}^{N-1}\cos 2\omega t_{n}}$$

and $\tau$ is an offset, $\omega$ is the frequency, $\{x\left( n\right) ,t_{n}\}$ is the observation series. The horizontal lines in the Figs. 19, 22, 25, 27, 32 and 34 correspond to the practical significance levels, as indicated in (Numerical Recipes in C 1988-1992).

2.3 Modern spectral analysis

Frequency estimation of narrow band signals in Gaussian noise is a problem related to many fields (Kay 1988; Marple 1987). Since the classical methods of Fourier analysis suffer from statistic and resolution problems, then newer techniques based on the analysis of the signal autocorrelation matrix eigenvectors were introduced (Kay 1988; Marple 1987).

2.3.1 Spectral analysis with eigenvectors

Let us assume to have a signal with p sinusoidal components (narrow band). The p sinusoids are modelled as a stationary ergodic signal, and this is possible only if the phases are assumed to be indipendent random variables uniformly distributed in $[ 0,2\pi )$ (Kay 1988; Marple 1987). To estimate the frequencies we exploit the properties of the signal autocorrelation matrix (a.m.) (Kay 1988; Marple 1987). The a.m. is the sum of the signal and the noise matrices; the p principal eigenvectors of the signal matrix allow the estimate of frequencies; the p principal eigenvectors of the signal matrix are the same of the total matrix.