On the design and implementation of digital filters to process meteorological signals

Separation of different frequency bands in the complex and combined signals related to meteorological variables and also climatic indices requires the use of digital filtering methods. In this way, the information on different frequency bands can be organized and used. Given that these signals generally exhibit complex and nonlinear behavior, the use of mathematical filtering methods to identify their stochastic and periodic components leads to a better understanding of their behavior and helps modeling them as well. Therefore, the use of digital filters in order to recognize regular variabilities and facilitate statistical forecasting is one of the main goals in this field.
The design and implementation of these filters are possible both in time and frequency domains. In the frequency space, this process is performed based on the Fourier transform of the signals on the basis of the Fast Fourier Transform Algorithm (FFT), in which the variances of the desired signal can be extracted based on spectral analysis in different frequencies. By employing different types of non-recursive and recursive digital filters, which they can be implemented as low-pass, high-pass, band-pass, and band-stop, the related signal inthe  time domain for each state can be constructed, and the corresponding spectrum can be studied. The isolated spectrum can be related to the effect of a special phenomenon that influences the main signal. In addition, it is possible to remove high frequency components from the original signal, which include noises and may not contain important information. Moreover, the process of optimal smoothing the original signal can also be carried out.
In this study, different digital filters have been designed and then applied to meteorological data such as monthly surface temperature and precipitation. Two synoptic stations over Iran are selected and the related discrete monthly signals are constructed for 504 months during 1979-2021. Then, the moving average (MA) filter is used as a main filter, because it is the most common filter in digital signal processing (DSP), and also it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average filter is optimal for a common task such as reducing random noise while retaining a sharp step response. This makes it the premier filter for time domain encoded signals. The filtering process in this study is conducted to denoise the original signals, and also to examine seasonal, annual, and inter-annual components of the original signals. Since employed filters are digital, they must be applied to the initial discrete signal in the form of convolution with the finite impulse response (FIR) of the filter in the time domain, or they can be applied in the form of multiplication in the frequency domain based on discrete Fourier transform and then using of the inverse Fourier transform to recover the desired signal.
The results of this study show the importance of using digital filters in analyzing the spectral contents of meteorological signals. Furthermore, the Hamming filter, which is defined based on the cosine truncation and windowing, shows better performance in attenuating Gibbs oscillations in the lateral sidelobes of the filter frequency response than the simple moving average (MA) filter. In addition, the correlation analysis is carried out separately to indicate the linear relationships between different frequency components of the signals. The higher correlations are observed in annual frequency bands of the temperature and precipitation signals for the selected stations. It shows the effect of external climate forcing on both temperature and precipitation that is stemmed from the earth’s motion around the sun during a year. Obviously, choosing more weights in the design of a filter can improve the filtering performance, but it should be avoided to use more weights than necessary.