Discussion of digital band pass filter and its application to random water wave data

Abstract

In this report the design of a digital band pass filter is discussed. It consists of an approximation to the ideal rectangular frequency filter. The data are convoluted with a truncated Sine function. Theoretically, the Fourier transform (Frequency response} of the Sine function is the ideal low pass filter in the frequency domain. The difference of two such low pass filters of different cut-off frequencies gives a band pass filter with a suitably chosen narrow bandwidth. A band pass filter consisting of a truncated Sine function does not have ideal shape, but by keeping the truncation limits 6 to 8 times wider than the main lobe of the Sine function, this filter has a sharper cut-off than that used by Blackman-Tukey. The filter is applied to the determination of properties of particular frequency components of wind water waves. These components are filtered from the original record of water surface displacement with the aid of a digital computer. In this manner the statistical properties of the water surface, which include wave spectra, auto-correlation, and space-time correlation were studied. The variance calculated from a filtered time series of particular frequency component represents the wave energy at that filtered frequency band. Spectra constructed by this technique are in good agreement with those obtained by the Blackman-Tukey method. Filtered space-time correlations showed that the propagation speed of a water wave is independent of frequency component, and is governed by the dominant frequency of original time series only.