## 1 Introduction
With the Fourier transform, filtering is accomplished by multiplying the transformed signal with a filter in the frequency domain, and then performing the inverse transform to return the processed signal or image. A special case in the field of application of this theorem is linear radio circuits formed from passive elements. The classical convolution theorem [28] allows us to move on to describing the circuit in terms of the frequency range, since the Fourier transform of the convolution is the product of the Fourier transforms of the functions under its sign. In particular, this fundamental theorem allows us to correctly justify the use of such terms as the amplitude-frequency characteristic, the frequency range of the radio signal, etc. [29]. For systems described by the digital analog of the convolution theorem, a similar theorem is valid, a visual proof of which (from the point of view of digital signal processing) was given in [30] based on the results of [31]. Therefore, it is permissible to pose the question of reducing the description of any systems whose behavior is described by the operation of digital convolution (including convolutional neural networks) to a description in terms of spectra and their transformations, similar to how this is the case for classical linear radio circuits. This issue is also relevant to the description of convolutional neural networks (CNNs). Such networks have a number of advantages that have made them indispensable in various tasks for computer vision, signal processing, and data analysis [32]. The main advantage of CNNs is their ability to automatically extract features from data, especially from images, without the need for manual feature selection. This is achieved using convolutional layers that identify local patterns in the data (e.g., edges, textures, and shapes in images) and can combine them into more complex features as the depth of the network increases. Another important advantage of CNNs is the ability to reduce the number of parameters with convolutional filters with shared weights [32; 33]. This reduces computational complexity and makes the networks more resistant to overfitting. The main applications of CNNs are related to pattern recognition, object detection and image segmentation. They are widely used in medical diagnostics (e.g., for analyzing X-ray and MRI images [34; 35]), in security systems for facial recognition [36], and in autonomous vehicles for recognizing objects on the road [37]. CNNs can also effectively be used to analyze time series and signals such as sound and text [38; 39]. For example, they are used to recognize speech and music, as well as to analyze texts in classification tasks, such as when working with the sentiment of texts or analyzing the syntactic structure. However, the digital convolution theorem in the formulation in [40] and similar ones solve the problem of transitioning to a description of a linear system in terms of a spectrum only partially. Indeed, the theorem under consideration in the formulation indicated above is valid only for the case when the function describing the digital signal in terms of the Galois field \(GF(p)\) is a periodic piecewise constant function, and the period contains exactly p-1 clock cycles. This makes the issue of constructing orthogonal bases for which this requirement is not satisfied very relevant. Note that a very wide range of orthogonal bases is currently known, used to calculate the spectra of digital functions. In particular, the Walsh basis is known, which is an orthogonal system of functions used in various applications of digital signal processing [41; 42] and in problems related to the analysis of light neural networks [43]. Unlike the Fourier basis, which is based on sinusoids and cosines, the Walsh basis consists of piecewise constant functions called Walsh functions. These functions have only values of +1 or \(-1\) and change sign at certain intervals, which makes them especially convenient for digital signal processing, where discrete levels are easier to process.