The Current Spectrum Formation of a Non ‐ Periodic Signal: A Differential Approach

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Introduction
The analysis of the spectra of current physical processes in real time is one of the most important tasks of signal processing [1][2][3]. This problem is relevant, not only in the analysis of various control and communication systems, but it also finds wide application in navigation systems, seismology, astronomy, and others.
Presently, this problem is solved either by calculating the Fourier transform for function f(t) with a variable upper limit of integration [4][5][6], or the classical Fourier-series expansion [7][8][9] on a time interval Ti (Ti = Ti+1 + ΔT; i = 1,2,…) constantly updated with a given sampling interval ΔT: 2 2 2 , In the first case, the spectrum that formed depends on time as a parameter, thus obtaining the signal spectrum in its classical definition (that is, independent of time) is possible only at a fixed t. In the second case, there is a large amount of computational costs [9] to reduce which methods of discrete transformation, such as Fast Fourier Transform (FFT) [2,10,11], are widely used. The amount of FFT calculations that require ~Nlog2N multiplication operations [10,12] is somewhat reduced as compared to traditional conversion. This does not radically change the situation, since, when calculating the current spectrum, this number of multiplication operations must be performed at each new time step with a constantly increasing time interval. Such shortcomings of both approaches create significant difficulties in their practical use.

Task Definition
In this regard, there is a need to develop such an algorithm for calculating the spectrum of nonperiodic functions on a time interval that was updated with a given sampling interval, which, on one hand, does not require large computational costs and, on the other hand, provides the formation of the spectrum in its original-classical, definition (1).

Task Solution
For the construction of this algorithm, the variations and of spectral coefficients ak and bk that are caused by a change in the time interval Ti by the value ΔT are preliminarily found. Using the equation of generalized differentiation by variable Ti (Leibniz formula), we have for the corresponding spectral components: Considering further that 2 2 2 , equations (2) are written as Since k changes discretely in increments of one, an approximation is possible , , which allows us to finally write the differential equations that describe the dynamics of changes in spectral coefficients, as follows: The search for the solution of differential equations (5) under known initial conditions leads to the formulation of the Cauchy problem, where the initial conditions are: is the end time of the previous time interval, and are the coefficients that are calculated on the previous interval according to the equation (1).
In practice, the most common method for solving ordinary differential equations , is the fourth-order Runge-Kutta method [13][14][15]. In this case, each step calculates the value of the function , of the right side four times; each value 1 is obtained based on the previous one, according to the equation: 1 , where ℎ 6 ⁄ 2 2 ; , ; h is the size of the grid step according to the argument x. When calculation spectral coefficients ak and bk in equation (5) 1 1 .
The accuracy of the solution of which is largely influenced by the value of the step h of the grid, a numerical estimate of the accuracy of determining the dynamics of changes in the spectral coefficients, given below, was made to analyse the computational efficiency of the obtained equations.

Experimental Design
The numerical simulation of the developed algorithm was carried out using the MATLAB (The MathWorks, Inc., USA) mathematical modeling package, which allows for additionally estimating the time costs for the solution. To assess the accuracy of the proposed algorithms, three essentially nonlinear functions were considered as a test function: The spectra of this function were constructed for time intervals T0 = 50.1 s and T0 = 51.0 s both using classical integral transformations (1) to form the reference spectra and while using the proposed algorithm for the sampling interval ΔT = 0.001 s. The grid step h for the Runge-Kutta algorithm was assumed to be equal h = 0.0001.
The   The time signals for the f1(t) function that was reconstructed from the received spectra are shown in Figure 2. Here, a solid line shows the restored signal for 50.1 s, as calculated from the spectrum that was obtained using the integral transformation (1), and dotted line from the spectrum obtained using the proposed algorithm.   The time signals for the f1(t) function reconstructed from the obtained spectra are shown in Figure 4. Here, a solid line shows the recovered signal for 51.0 s, calculated from the spectrum obtained using the integral transformation (1), and dotted line from the spectrum that was obtained using the proposed algorithm.

) ( 1 t f
At this time interval T0 = 51.0 s (for 10 3 time samples of the current spectrum formation), the coincidence of the spectral coefficients that were calculated by the given algorithm with the reference ones, as well as the reconstructed signals, has somewhat deteriorated when compared to the previous time interval, which is due to the accumulation of step-by-step calculation errors in the proposed recurrent algorithm. However, nevertheless, the time spent on the process of calculating the current spectrum in the proposed algorithm was still more than 80% less as compared to the classical one.
Since the proposed algorithm shows fairly stable results for small time intervals, the results of numerical simulation for the f2(t) and f3(t) functions are immediately presented for the time interval T0 = 51.0 s, after which a generalized evaluation of the results for all three functions for the time intervals T0 = 50.1 s and T0 = 51.0 s. Figure 5 shows   Figure 6 shows the time signals for the f2(t) function that was reconstructed from the obtained spectra. Here, a solid line shows the recovered signal for 51.0 s, which was calculated from the spectrum obtained using the integral transformation (1), and the dotted line from the spectrum that was obtained using the proposed algorithm.    Figure 8 shows the time signals for the f3(t) function that was reconstructed from the obtained spectra. Here, a solid line shows the recovered signal for 51.0 s, as calculated from the spectrum that was obtained using the integral transformation (1), and dotted line from the spectrum obtained using the proposed algorithm.  Table 1 provides a generalized estimate of the results of numerical simulation for functions f1(t), f2(t), and f3(t) at time intervals T0 = 50.1 s and T0 = 51.0 s. As follows from the results, for all three functions, the performance estimates have approximately the same values-the relative performance of the process of calculating the current spectrum for the time interval T0 = 50.1 s is more than 96%, for T0 = 51.0 s about 80%.

Evaluation of Numerical Modelling
In order to assess the effect of the value of the time interval on the growth of errors that were caused by the accumulation of incremental error calculation in the proposed algorithm, for the considered nonlinear functions are constructed based on changes ( From the analysis of these dependencies, we can conclude that the temporal nature of the mismatch of spectral coefficients depends on the type of function being modeled and the nature of its nonlinearity. Accordingly, for the function f1(t) deviation of the recovered spectrum above the value of 0.005 is observed at 52 s, for the function f2(t) at 59 s, for the function f3(t) at 51 s. In this case, for 51 s (Figure 8), the reсovered signal generally shows a slight deviation, but fluctuations that occur at the spectrum boundaries lead to an increase in the standard deviations of the spectral coefficients.

Conclusions
The results of numerical modeling allow for drawing a conclusion regarding the possibility of effective practical use of the proposed approach to the calculation of the current spectrum of nonperiodic functions under the requirement of small sampling steps (i.e., when calculating the spectrum in real time). In comparison with the most effective method of spectral characteristics formation-FFT, the developed algorithms have the following advantages. Firstly, their number of multiplication and addition operations at each step is constant and it does not depend on the increasing number N of time samples of the signal, being determined by the step of its sampling and the current value T of the increasing time interval and, secondly, this number of operations is incommensurably small when compared to the constantly increasing number of FFT operations at the current interval.
The implementation of the developed algorithm is particularly relevant in systems of measurement and the spectral processing of broadband signals in real time-telecommunications, navigation, and others.