RU2133041C1 - Method determining spectrum of electric signals - Google Patents

Abstract

FIELD: measurement technology, spectral analysis of electric signals. SUBSTANCE: method determining spectrum of electric signal y(t) lies in generation of digital reading y[k] of signal after equal time intervals Δt, in formation of interpolating functions between neighboring readings of signal, in determination of approximation function approximating signal and in finding of Fourier image of this function, in making function of signal y(t) of type of

in which interpolation time function presents polynomial 0≤t<Δt in first discretization interval

, in next interval it is presented by difference of polynomials interpolating analyzed and preceding sections of function of signal as

,

,

, where a_k,i is sum of weight coefficients at tⁱ. In last discretization interval linear polynomial is used after which spectral coefficients are computed by formula

Description

 The present invention relates to measuring equipment and can be used in electrical and radio measuring devices, information and computing and other (eg, navigation) systems for determining the spectrum of electrical signals.

 There is a method of determining the spectrum of an electrical signal, which consists in sampling the signal at regular intervals, with a sampling frequency of at least twice the upper frequency of the signal, and directly processing the samples obtained after sampling using the fast Fourier transform (FFT) method [1].

 However, this method has low accuracy in determining the spectral components of the analyzed signal, especially at frequencies close to the upper frequency of the signal, which significantly narrows the range of operating frequencies.

 In addition, there is a known method for determining the spectrum of an electric signal [2], which is a prototype of the present invention, which consists in obtaining N + 1 digital samples y [k] (samples) of an electric signal y (t) at an equal time interval Δt, forming N piecewise linear interpolating functions between adjacent samples of the signal, obtaining the approximate function of the signal and finding the Fourier transform of this function.

 However, this method also has low accuracy in determining the spectrum of the analyzed signal at frequencies close to, equal to or greater than half the sampling frequency of the signal.

 The objective of the invention is to provide a method for determining the spectrum of electrical signals with higher accuracy of measuring the spectral components of the analyzed signal, as well as expanding the working frequency range for the sampling frequency of the signal.

(1)
в которой на первом интервале дискретизации 0 ≤ t < Δt, т.е. при k = 0) интерполяционную временную функцию представляют полиномом L - степени (L≥3)

на последующих интервалах - разностью полиномов, интерполирующих анализируемый и предшествующий участки функции сигнала

где

a_k,i - сумма весовых коэффициентов при tⁱ,

(2)
на последнем интервале дискретизации ((N-1)Δt, NΔt) используют линейный полином (L = 1)

после чего спектральные коэффициенты рассчитывают по формуле

(3)
где n = 0, N-1 - порядковые номера дискретных частотных отсчетов.This is achieved by the fact that in the method consisting in obtaining digital samples y [k] of the signal at equal time intervals Δt and generating interpolating functions between adjacent samples of the signal, obtaining an approximate function approximating the signal, and finding the Fourier transform of this function, an estimate of the function (approximating function) of the signal y (t) of the form

(1)
in which, in the first sampling interval, 0 ≤ t <Δt, i.e. for k = 0) the interpolation time function is represented by a polynomial L - degrees (L≥3)

at subsequent intervals, the difference in polynomials interpolating the analyzed and previous sections of the signal function

Where

a _{k, i} is the sum of the weights at t ⁱ ,

(2)
in the last sampling interval ((N-1) Δt, NΔt) use a linear polynomial (L = 1)

after which the spectral coefficients are calculated by the formula

(3)
where n = 0, N-1 - serial numbers of discrete frequency samples.

 The method is illustrated by drawings, where: in FIG. 1 shows one of the possible structural diagrams of the device, explaining a method for determining the spectrum of an electrical signal; figure 2 - graphs of the relative errors of the methods for determining the spectral components of an electrical signal; a - for the prototype, b - for the proposed method (L = 3), c - for the proposed method (L = 4).

 The device (Fig. 1) contains a series-connected sensor of an electric signal 1 (DES), analog-to-digital converter 2 (ADC), communication interface 3 (IC), electronic computer 4 (computer).

 The method can be understood by considering the operation of the device. The analog electrical signal y (t), generated by the signal sensor 1, enters the analog-to-digital converter 2, the output of which is the code of each k-th discrete sample of the signal y (k) (where k = 0, ..., N-1) enters through the communication interface 3 to the electronic computer 4.

на втором этапе для каждого k-го интервала составляют интерполяционный полином

который путем объединения подобных членов при tⁱ приводят к виду

где a_k,i - сумма весовых коэффициентов при tⁱ для (k, i) последовательностей; и на последнем интервале дискретизации ((N-1)Δt, NΔt) используют линейный полином

На третьем этапе составляют оценку функции сигнала y(t)

в которой на первом интервале дискретизации (0 ≤ t < Δt, т.е. при k = 0) интерполяционную временную функцию представляют полиномом L-й степени (L≥3)

на последующих интервалах - разностью полиномов, интерполирующих анализируемый и предшествующий участки функций сигнала

где

Затем в электронно-вычислительной машине 4 по полученным через интерфейс связи 3 дискретным выборкам сигнала y[k], в зависимости от заданной степени L полинома, вычисляются коэффициенты b_k,i (используя выражение (2)), интерполяционной функции (1)). Так, значения коэффициентов b_ki, например, в случае интерполяции функции на интервале между отсчетами полиномом третьей степени (L = 3), вычисляют по выражениям:

При использовании при интерполяции полинома четвертой степени значения b_k,i вычисляют по выражениям

После чего по формуле (3) вычисляют спектр электрического сигнала y(t).When compiling an estimate of the signal function (1) previously on each k-th time interval (ie, [kΔt, (k + 1) Δt]), the function is interpolated by the polynomial P _k (t) of the L-th degree (here L≥3) , at the first stage, on each k-th interval for a sequence of L + 1 samples (y [k], y [k + 1], ..., y [k + L]), L + 1 time functions are constructed

at the second stage, for each kth interval, an interpolation polynomial

which by combining similar terms at t ⁱ lead to the form

where a _{k, i} is the sum of weights at t ⁱ for (k, i) sequences; and in the last sampling interval ((N-1) Δt, NΔt) use a linear polynomial

At the third stage, an estimate of the signal function y (t)

in which, on the first sampling interval (0 ≤ t <Δt, i.e., for k = 0), the interpolation time function is represented by a polynomial of degree L (L≥3)

at subsequent intervals, the difference in polynomials interpolating the analyzed and previous sections of the signal functions

Where

Then, in the electronic computer 4, using the discrete samples of the signal y [k] obtained through the communication interface 3, depending on the given degree L of the polynomial, the coefficients b _{k, i} (using expression (2)), interpolation function (1)) are calculated. So, the values of the coefficients b _ki , for example, in the case of interpolation of a function in the interval between samples by a polynomial of the third degree (L = 3), are calculated by the expressions:

When using a fourth-degree polynomial for interpolation, the values of b _{k, i} are calculated from the expressions

Then, using the formula (3), the spectrum of the electric signal y (t) is calculated.

где c - масштабный множитель, τ - постоянная времени. Так, при τ = 0,1 мкс, интервале дискретизации Δt = 0,025 мкс и числе отсчетов N = 128 относительные погрешности расчета спектра d% для способов прототипа и предложенного в предлагаемом изобретении для L, равных 3 и 4, представлены соответственно на графиках фиг. 2а; фиг. 2б; фиг. 2в. Анализ приведенных на фиг. 2 относительных погрешностей показывает, что предложенный способ обеспечивает снижение погрешности определения амплитуды и фазы спектральных составляющих более чем на порядок, а также расширение области рабочих частот не менее чем в 4 раза.A comparison of the errors in determining the spectrum of the electrical signal of the proposed method and the prototype method is performed on the example of an analytical exponentially decaying signal, which is often found in practice

where c is the scale factor, τ is the time constant. So, at τ = 0.1 μs, the sampling interval Δt = 0.025 μs, and the number of samples N = 128, the relative errors in calculating the spectrum d% for the prototype methods and proposed in the present invention for L equal to 3 and 4 are presented, respectively, in the graphs of FIG. 2a; FIG. 2b; FIG. 2c. The analysis of FIG. 2 relative errors shows that the proposed method reduces the error in determining the amplitude and phase of the spectral components by more than an order of magnitude, as well as expanding the range of operating frequencies by at least 4 times.

Literature:
1. Max J. Methods and techniques for processing signals in physical measurements. / Per. with french -T. 2. Basic principles and classical methods. -M.: Mir, 1983. -312 p.

 2. Shute. A new fast Fourier transform algorithm for the analysis of linear systems - application to molecular-beam relaxation spectroscopy // Instruments for Scientific Research, N 3, 73, (1981).

Claims (1)

The method of determining the spectrum of the electric signal y (t), which consists in obtaining digital samples y [k] of the signal at equal time intervals Δt and generating interpolating functions between adjacent samples of the signal, obtaining an approximate function approximating the signal, and finding the Fourier transform of this function, which differs in that they constitute an estimate of the signal function y (t) of the form

in which the interpolation time function is represented by the polynomial in the first sampling interval (0 ≤ t <Δt, i.e., for k = о)

at subsequent intervals, the difference in polynomials interpolating the analyzed and previous sections of the signal function

Where

a _{k, i} is the sum of the weights at t ⁱ ,

in the last sampling interval ((N-1) Δt, NΔt) use a linear polynomial (L = 1)

after which the spectral coefficients are calculated by the formula

where n = 0, ..., N-1 are the serial numbers of the discrete frequency samples.

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