Sliding signal processing in telecommunication networks based on two-dimensional discrete Fourier transform

. A method of vertical sliding processing of two-dimensional discrete signals in the spatial frequency domain is proposed — a method of fast vertically sliding two-dimensional discrete Fourier transform. The mathematical representation of the two-dimensional discrete Fourier transform in algebraic and matrix form is considered. An effective method of vertically sliding two-dimensional discrete Fourier transform is proposed. The algorithm developed in the framework of the proposed method allows calculating the coefficients (bins) of this transformation in real time.


Introduction
It is difficult to overestimate the role and place of digital spectral processing of discrete one-dimensional (1-D) and two-dimensional (2-D) signals in telecommunication networks (telephone, computer, television networks and radio networks). The importance and relevance of the development and improvement of the methods of digital spectral processing of 1-D and 2-D signals is increasing due to the creation of telecommunication universal multiservice networks that can equally effectively transmit any type of information: data, sound and video.
The classical method of spatial-frequency processing of two-dimensional discrete signals is the 1-D and 2-D discrete Fourier transform (DFT), which allows to obtain a 1-D and 2-D frequency spectrum [1][2][3][4][5][6][7][8][9][10][11][12][13]. At the same time, there are a number of applications [9][10][11][12] where it is necessary to find the values of the frequency spectrum not at all 1-D and 2-D frequencies, but at a subset of them. In this case, the application of the full version of 1-D and 2-D DFT, even on the basis of the fast Fourier transform (FFT), becomes ineffective, since most of the obtained 1-D and 2-D DFT coefficients are not used. The solution to this problem for 1-D signals is given in [9].
The article discusses the solution of this problem for 2-D signals, introduces the concept of moving spatial-frequency processing based on 2-D DFT. Specifically, the vertical spatial-frequency processing of 2-D signals is considered.

Direct two-dimensional discrete Fourier transform
Suppose we are given a discrete two-dimensional signal ) , ( 2 1 n n x in the form of a twodimensional sequence of finite length ) 1 ( 0 or with a matrix of size 2 1 n n  in a rectangular reference zone (specifically on a plane). The direct two-dimensional discrete Fourier transform (2-D DFT) of a two-dimensional signal ) , ( 2 1 n n x is a special case of a direct two-dimensional z-transform: and can be specified both in algebraic and in matrix form.

Matrix form:
where 2 2 1 1 1 Hereinafter there is no loss of generality in omitting the multiplier ) (3). Due to the fact that for the product of the matrices (3) holds the associative property: and the 2-D DFT core is separable, then, according to (7), you can get the 2-D DFT , in two ways, each consists of two stages.
It is easy to find out that for obtaining is necessary to perform

Sliding spatial-frequency processing of discrete signals
Let us consider the spatial-frequency processing of two-dimensional discrete signals in a sliding spatial analysis window. In contrast to the one-dimensional case for the twodimensional case, there are 4 possible ways of sliding of the spatial analysis window on the original two-dimensional discrete signal:   . In this case, the matrix equation (3) is converted into the form: At the first stage, according to (8), we multiply the basis function of frequency 2 k and duration 2 N by a matrix of a discrete two-dimensional signal 12 ( , ) x n n . As a result, we obtain a columned matrix Note that this amount of computation needs to be performed at each shift of a twodimensional spatial analysis window using a two-dimensional signal. At the same time, it is easy to see that for any kind of shift of a two-dimensional signal a large number of values 2 1 N N X  of the complex matrix in the spatial analysis window remains unchanged.
Note that the shift of the spatial window along a two-dimensional discrete signal can be considered as a shift of a two-dimensional discrete signal in the spatial analysis window in the opposite direction to the movement of the spatial window.

The algorithm of vertical sliding processing of 2-D signals based on 2-D DFT
Let us need to find one coefficient (bin) of a two-dimensional discrete transformation

Conclusion
The effectiveness of the proposed algorithm for vertical sliding processing of twodimensional discrete signals in the spatial frequency domain in comparison with the standard method for obtaining the coefficient of two-dimensional discrete transformation