## 1 Introduction
### Filter Bank and Common Models  
Based on a Fourier series [27; 28], each signal consists of many sine waves with different pulses, phases, and amplitudes. Therefore, brain activity signals include different pulses, phases, and amplitudes over large spatial ranges. The main brain activity formula is supported by large-scale frequency bands based on EEG systems. However, based on previous research [19; 20], formulas for distinctive patterns between brain activations are distributed in the range of 4-40 Hz [23] for more than two activations and 8-30 Hz [21] for only two activations. When all frequency domains are used for processing [21], it is crucial to determine the best feature extraction methods to discover the distinctive patterns of all frequency domains for extracting features. The focus of most researchers [15; 16; 17; 18]has been on 8-30 Hz to detect the distinctions of brain activities, such as the left hand and right hand. Based on previous studies [13; 14], important distinctive patterns of brain activation are within the 4 Hz-40 Hz and 8-30 Hz domains. When researchers consider a large frequency domain for processing, it is necessary to process both the most similar and most distinctive patterns, leading to the extraction of ineffective features. When researchers consider a suitable limited frequency domain [20; 21], similar patterns are decreased while retaining most of the distinctive patterns, which leads to the extraction of effective, good features. Based on previous research [19; 20], finding small and suitable frequency bands with important distinctive patterns is necessary to increase the accuracy of classification. Many studies [15; 16; 17; 18; 19; 20; 21; 22; 23] confirm that the most distinctive patterns are distributed in different frequency domains. This study aimed to determine a near-optimal, locally optimal, or slightly optimal frequency domain to extract the most distinctive features. The frequency range can be small and limited to between 4 Hz and 40 Hz. A specific large frequency domain can arise from several small frequency domains acting as filter banks. For this purpose, combinations (two, three, four, etc.) can be made with various filter sizes of limited banks (0.1, 0.2, 0.5, \(\ldots\), 4). In some studies, 4 Hz filter banks were applied for processing, and they showed better performance. As preliminary work, we used a combination of two limited frequency domains as filter banks, which led to the discovery of the most significant distinct patterns of brain activity signals for classification. The level of classification accuracy can be expressed as a measure of the proportion of distinctive patterns in newly combined signals. Therefore, the ultimate goal of this study was to find frequency bands with different features associated with many brain activities. In this study, we considered two combinations of two filter banks as two small frequency bands. In future research, the majority of models will examine the types of combinations and sizes of filter banks. It is possible to determine the optimal maximum or close to the optimal maximum.  
This paper focuses on different combinations of pairs of 4 Hz sized filter banks to extract better feature sets for recognizing users' brain activities with high accuracy. In some studies [20; 21], 4 Hz was applied as a small frequency range for the filter bank. The new combination of filter banks was analyzed by CSP and FBCSP (CSP with mutual information) to extract and select features, respectively. Moreover, new features were extracted using the Lagrangian polynomial coefficient (LPC) method. Finally, the impact of the new combinations of signals was investigated using principal component analysis (PCA).