Two types of filters are widely used to remove semidirunal and diurnal tidal signals and other high frequency noises in oceanography. The first type of filters uses moving average with weights in time domain, and can be easily operated. Some data will be lost at each end of the time series, especially for the low low-pass filters. The second type of filters uses the discrete Fourier transform filter (DFTF) which operates in the frequency domain, and there are no data loss at the ends for the forward transform. However, owing to the Gibbs phenomenon and the discrete sampling (Nyquist effect), ringing appears in the inverse transformed data, which is especially serious at each end. Thus some data at the ends are also discarded. The present study tries to find out what causes the ringing and then to seek for methods to overcome the ringing. We have found that there are two kinds of ringings, one is the Gibbs phenomenon, as defined before. The other is the "Nyquist" ringing due to sampling Nyquist critical frequency. The former is due to the abrupt transition in frequency band. The Gibbs and Nyquist effects show the ringing at each end of the filtered time series. Thus, the use of a cosine taper or a linear taper on the window in the frequency domain makes the transition band smooth, so that the Gibbs phenomenon will be minimized. Before applying the Fast Fourier Transform (FFT),the original time series at each end is properly tapered by a split cosine bell that reduces significant ringing since this method limits the energy transfer from outside of the Nyquist frequency. Thus, the DFTF can be a powerful tool to suppress the signals in which we are not interested, with sharp peaks in low frequency variation and less data loss at each end.
DownLoad: