1 Introduction

Astronomical photometric observations are often obtained through different passbands (channels) simultaneously or quasi-simultaneously. As a rule, astronomers perform time series analysis for each observed channel separately. Sometimes normalization is used to put the channels on an equal footing (see for instance Jetsu & Pelt 1999) and sometimes the results from different channels are averaged (You 1999). It is quite clear that in this way part of the information in the data is lost, especially if the light curves and error estimates for separate channels are substantially different. If we assume (and this is a quite reasonable assumption for most variable stars) that the periods we are seeking are the same in different passbands then we can analyze all data together using combined statistics. The new method described in detail below is a simple and straightforward generalization of the multistage methods (Pelt 1980,1983,1993; Jetsu & Pelt 1999). It is based on the phase dispersion minimization method (PDM) at the first (pilot search) stage, a linear model fit (LM) at the second (grid search) stage and finally a nonlinear model fit (NLM) at the last (refined search) stage. The first stage allows the detection of probable periods in a wide range of possible periods using fast algorithms on the coarse grid of trial frequencies. Computed periods can then be analysed to identify spurious periods which occur due to periodicities in the observation time points (Tanner 1948). Candidates which remain in the list of probable periods are then used as input for the second stage of analysis.

The LM is an intermediate stage, where refined period values are computed using equispaced high resolution grids of trial frequencies around period estimates obtained from the first stage. The purpose of the LM is to provide best initial values and correct value bracketing (search range for periods) for the final refinement.

The third stage (NLM) is in essence a nonlinear minimization procedure where the best estimates for linear model parameters (amplitudes of harmonic functions of different orders in the model) and nonlinear model parameters (periods) are obtained. It is now assumed that the initial period values for minimization and corresponding search brackets are already so well known that the convergence to a unique minimum is guaranteed. In our case of single period analysis there is only one nonlinear parameter and we can use the classical Brent minimization algorithm (Brent 1973) combined with a linear fit of the amplitudes for every particular period. The error estimate of the final value of the period can be computed from the curvature of the $\chi^{2}$ hypersurface or by using other standard methods.

The purpose of this paper is to demonstrate the usefulness of our three stage weighted multichannel period analysis (MPA) where all the available information in the input data sets are used fully and uniformly. The paper is organized as folllows. First we describe the MPA method in full mathematical detail in Sect. 2. Then, in Sect. 3, we apply the MPA to two groups of artificially generated data and show how all the three period searching stages help to recover the correct period. The test cases reveal the principal advantages of the new method. The results are briefly summarized in Sect. 4.