INTRODUCTION 



The purpose of this report is to present some of the possibil- 

 ities associated with statistical filtering of periodic data 

 by smoothing and to provide sets of weights to fit some par- 

 ticular cases, for example, the smoothing of low-frequency, 

 ambient sea-noise data. A brief theoretical discussion is 

 included to explain the basic concepts involved. 



A set of data arranged chronologically is called a time series . 

 Time variations in the data may be relatively smooth or of a 

 complex nature devoid of any apparent pattern. 



Assuming the Fourier theorem, any variation with time may be 

 considered the result of superposition of a number of simple 

 sinusoidal components, the amplitudes, frequencies, and 

 phases of these components being time -dependent. 



In many time series it is assumed that high frequency oscilla- 

 tions in the data are either random noise or are of no signif- 

 icance to the particular purpose for which the data are to be 

 evaluated. Consequently, one important purpose of time 

 smoothing is to attenuate the amplitudes of high frequency 

 components and, at the same time, preserve the low frequency 

 components of immediate interest. Hence, the smoothed value 

 of an experimentally observed time series is an estimate of 

 its true value free from noise and other undesirable high 

 frequency influences originally present. 



Smoothing of a time series is a special case of the general 

 process of numerical filtering and is analogous to low -pass 

 filtering of an electrical signal. However, numerical filter- 

 ing includes band-pass and high-pass filtering as veil as 

 low-pass filtering. Thus, if smoothed values are subtracted 

 from the corresponding values in the original unsmoothed 

 time se? ±es, only .high frequency components will remain. 

 Such an iper .tion is equivalent to high-pass filtering. 

 Bandpass filtering may be achieved by subtracting well 

 smooched values of a time series from corresponding values 

 smoothed to a lesser extent; only intermediate frequencies 

 will remain, thus giving the equivalent of band-pass filter- 

 ing. 



By use of the above procedures one may separate the oscilla- 

 tions of the time series into particular bands of frequencies, 

 high, intermediate, and low. This report, nowever, is 



