Determination of the degree of freedom of digital filtered time series with an application to the correlation analysis between the length of day and the Southern oscillation index

After digital filtering the degree of freedom of the original observational time series will be greatly decreased. For a given filter, simulation by the Monte Carlo method can be used to obtain the critical correlation coefficients at given confidence levels and the degree of freedom of the filtered time series. If the ratio between the frequency bandwidth after filtering and the Nyquist frequency is Z, then the degree of freedom of the filtered time series will be Z and Z/2 times of the original value for perfect single-sideband and double-sideband filters. For non-perfect filters, it should, and can, be found by Monte Carlo simulation.