CC BY
20
Artyushenko V. M. ApmwrneuKO В. М.
Doc. Sci. Tech., Professor, Head of «Information Technology and Control Systems» Chair, FSBEI HE «Technological University», Korolev MR, Russian Federation
UDC 621.396.96
Volovach V. I. BoMoeau B. H.
Doc. Sci. Tech., Associate Professor, Head of Information and Electronic Service Chair, FSBEI HE «Volga Region State University of Service», Togliatti, Russian Federation
APPLYING OF DOPPLER SIGNAL AMPLITUDE SELECTION FOR MEASURING OF EXTENDED OBJECTS MOTION SPEED
The article considers the applying of amplitude selection of the Doppler signal used to determine the speed of motion of extended objects by short-range radiolocation devices. The task of using the channel of speed measurement in extended objects detectors is justified. It is shown that to solve the above-mentioned task the threshold method of processing the Doppler signal can be used, where only the part of the signal that has the amplitude exceeding some specified value is subjected to demodulation. The main statistical characteristics of the instantaneous frequency of a signal depending on the threshold of processing are considered and the gain in measurement accuracy achieved by that is estimated.
The existence of statistical relationships between random deviations of frequency of the processed signal and values of its envelope is shown. The expression for conditional density of probability distribution of frequency at which the values of the envelope of the signal exceed the specified threshold is given; the graphs of corresponding dependencies are presented. It is shown that the average value of the absolute deviation of signal frequency is inversely proportional to the relative level of its envelope.
The error in measurement of frequency of the processed signal is estimated. It is shown that the relative RMS value of the error for instantaneous frequency is determined as a function of threshold voltage. The correlation between the duration of the processed signal and the total duration of implementation is obtained. The rational value of threshold voltage is found. The expression determining the RMS relative error in determination of instantaneous speed is given.
The spectral density of the frequency of the processed signal in the threshold mode of demodulation is analyzed. It is shown that the variance of the frequency of the Doppler signal in this mode is finite and decreases with the threshold increase. The values of its correlation function at zero, in contrast to non-threshold demodulation, are finite and will decrease with the increase of threshold voltage.
It is shown that the spectral density of instantaneous frequency fluctuations at zero, which is the integral of the correlation function, also will decrease with the threshold increase. The obtained expression of correlation function of the instantaneous frequency through the use of mathematical modeling is analyzed.
It is shown that the estimate of the spectral density of a random component of the instantaneous frequency of the signal can be found in two ways: using Fourier-transform of the correlation function and using the method of Cooley-Tuckey. The graphical dependencies of the families of correlation functions of a random frequency and a spectral density of frequency for different threshold values are presented. It is shown that the increase of the threshold leads to a narrowing of the spectrum and also to the decrease of its absolute values at the respective frequencies.
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Electrical and data processing facilities and systems. № 1, v. 13, 2017
As a result of the theoretical and numerical analysis, the estimates of a potentially achievable accuracy of measurement of motion parameters of extended objects by Doppler short-range detection devices in the threshold mode of demodulation are given. It is proved that the introduction of amplitude threshold in contrast to the non-threshold demodulation of the Doppler signal allows us not only to obtain the final values of variance of the estimation of motion speed of an extended object in a broad band of frequencies, but also to reduce spectral components of phase noise.
Key words: amplitude selection, threshold method, Doppler signal, extended object, statistical characteristics of instantaneous frequency, frequency measurement accuracy, spectral density of frequency.
ИСПОЛЬЗОВАНИЕ АМПЛИТУДНОЙ СЕЛЕКЦИИ ДОПЛЕРОВСКОГО СИГНАЛА ПРИ ИЗМЕРЕНИИ СКОРОСТИ ДВИЖЕНИЯ ПРОТЯЖЕННЫХ ОБЪЕКТОВ
В статье рассмотрено использование амплитудной селекции доплеровского сигнала, используемой при определении скорости движения протяженных объектов радиотехническими устройствами ближнего действия. Обоснована задача применения канала измерения скорости в обнаружителях объектов. Показано, что для решения названной задачи может быть использован пороговый способ обработки доплеровского сигнала, при котором демодуляции подвергается только часть сигнала, имеющая амплитуду, превышающую некоторое заданное значение. В статье рассмотрены основные статистические характеристики мгновенной частоты обрабатываемого сигнала в зависимости от порога и определена оценка достигаемого при этом выигрыша в точности измерения.
Показано существование статистической связи между случайными отклонениями частоты обрабатываемого сигнала и значениями его огибающей. Получено выражение для условной плотности вероятности частоты, при которой значения огибающей сигнала превышают некоторый заданный порог; приведены соответствующие графики зависимостей. Показано, что среднее значение модуля отклонений частоты сигнала обратно пропорционально относительному уровню его огибающей.
Произведена оценка погрешности измерения частоты сигнала. Показано, что относительное среднеквадратичное значение ошибки для мгновенной частоты определяется как функция порогового напряжения. Получена зависимость между длительностью полезного сигнала и общей длительностью реализации. Установлено рациональное значение уровня порогового напряжения. Приведено выражение для определения среднеквадратичной относительной ошибки определения мгновенной скорости.
Осуществлен анализ спектральной плотности частоты обрабатываемого сигнала в пороговом режиме демодуляции. Показано, что дисперсия частоты доплеровского сигнала в этом режиме конечна и падает с ростом порога. При этом, значения ее корреляционной функции в нуле, в отличие от беспороговой демодуляции, также конечны и будут падать с ростом порогового напряжения.
Показано, что спектральная плотность флуктуаций мгновенной частоты в нуле, представляющая собой интеграл от корреляционной функции, будет падать в функции порога. Осуществлен анализ полученного выражения корреляционной функции мгновенной частоты посредством использования машинного моделирования.
Показано, что оценка спектральной плотности случайной компоненты мгновенной частоты сигнала может быть найдена двумя способами: через Фурье-преобразование корреляционной функции и с помощью метода Кули и Тьюки. В качестве результатов проведенного численного анализа представлены графические зависимости семейств корреляционных функций случайной частоты и спектральной плотности частоты при различных величинах порога. Увеличение порога приводит как к сужению спектра, так и к уменьшению всех его абсолютных значений на соответствующих частотах.
В результате проведенных теоретического и численного анализов даны оценки потенциально достижимой точности измерения параметров движения протяженных объектов доплеровскими устройствами обнаружения ближнего действия в пороговом режиме демодуляции. Доказано, что введение амплитудного порога в отличие от беспороговой демодуляции доплеровского сигнала позволяет получить конечные значения дисперсии оценок
скорости движения протяженного объекта в широкой полосе частот и уменьшает все спектральные компоненты фазового шума.
Ключевые слова: амплитудная селекция, пороговый способ, доплеровский сигнал, протяженный объект, статистические характеристики мгновенной частоты, погрешность измерения частоты, спектральная плотность частоты.
Introduction
In [1] was noted that short-range radio location devices (SRRLD) may be optimized primarily by adaptation. One of the ways of adaptation is possible through speed measuring of detected object if the detection is fulfilled on the basis of Doppler signal processing. As a result, in addition to detection SRRLD will perform the function of speed meter. Speed measurement (in a wide sense the parameters of movement of the object such as acceleration, the geometric sizes of the object, location) lets us solve the task of object detection of short-range radiolocation devices more accurately, for example we get some extra data about the object including spectrum of detection signal [2-5]. It also lets us predict its behavior in relation to boundaries of the detection area. In many cases, the measurement of the parameters of motion of the object is a separate task [6-13].
Next some aspects of the implementation of the speed measurement channel that can supplement regular SRRLD will be studied.
1. Problem statement
It is known [14, 15], that the presence of multiple «shiny» points in the measuring volume and their random position leads to significant fluctuations in the Doppler signal frequency, which is the measure of speed. These fluctuations, called phase noise [16] can reach significant values and significantly limit the accuracy of speed estimation.
Analysis of the instantaneous frequency of Doppler signal, received from two «shiny» points moving with the same speed, showed that its maximum values correspond to the minimum amplitude and, on the contrary, for large values of the envelope of the signal its frequency almost does not change.
Based on these results, threshold method of Doppler signal processing was proposed and implemented [17], which means that only that part is subject to demodulation, amplitude of which lies above a preassigned threshold. The
speed values in intermediate moments of time are determined by interpolation of demodulation results.
Consider the basic statistical characteristics of instantaneous frequency depending on the threshold and assessment of win in the measurement accuracy achieved in this case.
The task is solved for the Doppler signal, the model for which is narrowband normal random process of the type
£(i) = £/(i)cos (1)
where U(t) is envelope signal, w0 is carrier frequency, j(t) is function reflecting the law of phase (frequency) modulation.
2. Statistical characteristics of instantaneous frequency
First, let us find a statistical relation between random deviations of the frequency q(t) and the values of the envelope U(t). Two-dimensional probability density of values q and U can be found on the basis of the known expression [18] for four-dimensional probability density
W4 (u, U, cp, q>) = [u2 / An2 aA (~Pq )) X
*exp{-[[//2a2 (-p0»)][(-A")t/2 + U2 + uV^ (2)
where the dispersion of the process (1) is a2, the value of the second derivative of the correlation coefficient at zero is
It is considered that p"0(0) = Am2, where Dm is half-width of the Doppler signal range on the level e-1/2 of its maximum. Integrating (2) for U and f within the range -&>, +x> for U and -n, +n forf we will get
W2 (iu 4>) = (u1/^*1^g) x X exp\-U2/2ct2 } exp\-U'lq>2 ¡la2 (-p"0 )}
Moving to a new relative variable: z = u/*j2=[u2/(u2)f,
where (u2Sj = 2s2 is the average square of the
envelope, we are getting
r2(^,z) = (252/V^A®)expj-z2(Ai02 + (z>2)/A®2}. (3)
On the basis of this expression we will find the conditional density of frequency probability
W2{q>\z=zt^, when the values of the signal envelope exceed a preset threshold z^ =11^ (u2^j . By the definition [19] we can write
w2 {<p\z = Ztr) = w2 (cp;ztr )/w(Ztr). (4)
Then 00 /00 W2((p\z>ztr)=\wz(<p;z)dz/¡W(z)dz, (5)
ztr / ztr
where W(z) is one-dimensional density probability of the relative envelope. In accordance with [18]
JT(z) = 2zexp(-z2). (6)
Substituting (4) and (6) in (5) and introducing the new relative variable y = p/Arn [20], we are getting
W/y\z >zj = (1/Jx)exp(z2J(1+y)-3/2r(3\2,x). Here x=z2r (l+y2); r(3/2, x) is incomplete gamma function.
A family of probability densities of instantaneous frequency fluctuations for different levels z r is shown in Figure 1.
W2(y\z>ztr)
module of the frequency deviations from the preset value of the envelope. By definition
Figure. 1. The dependence of the probability densities of instantaneous frequency fluctuations from the value of the specified threshold level
The graphs show that with the increase of the threshold levels z tr the probability of large deviations of the frequency sharply decreases and probability of small deviations increases. Therefore, the value of root-mean-square error with the increase zt should decrease. Of interest is the finding of a regression line that establishes a dependence of the expected value of the
—00
00
= 2\yW(y\z = zl)dy,
(7)
where m is the symbol of the expected value, W^y\z = zl) is the density of frequency probability with set value z1. Substituting in (7) value W(y\z = zl>) which, in accordance with (4) is equal to
w{y\z = zl) = [zJ4Ti)^v{-^y2),
we will get
w{(H)|z = zi} = 1/(zi^). (8)
As follows from (8), the mean value of the modulus of the frequency deviations is related to the relative level of the envelope z1 with inversely proportional dependence. The obtained results are the theoretical basis for the method of reducing the phase noise through amplitude selection of the Doppler signal.
3. Estimation of frequency measurement error
The dependence of the dispersion of deviations from the threshold frequency can be found from the ratio
00 00 / 00
°1<p=(<pI) = \ J <p2W2(z,<p)d<pdz \w{z)dz,
where values W2{z,<p) and W(z) are determined by the expressions (3) and (6).
With regard to the latter, we can write
< =(^) = ( l/2)A^2exp(4)i<i(4) (9) where El is the integral exponential function.
The relative root-mean square value of error for the instantaneous frequency as a threshold function can be easily found from (9):
°tr<pl<°D=^f)/^=(W V2«zj)exp(4/2)^i(4).
Introducing the designation A co = ^/M^, we will get:
Sa = cr^l(DD={\ISM7r)™V[zll2)^l , (10) where M is any positive integer.
Analysis of the dependence (10) shows that the mean square error at first decreases rapidly with the increase of the threshold, and then its steepness decreases.
On the basis of the expression (6) it is easy to get the ratio between the length of Tus of the useful signal and the total duration of the implementation rn:
Data processing facilities and systems
00 /00 JzQ=\w{z)dz/ejm{z)dz = (-l)
(11)
where 4=t£/(tf2).
From the analysis of the dependence (11) follows that if with z = 1 the duration of the useful
tr
signal is 0,37 % from the duration of implementation, with z^ = 2 this value decreases to 2 %. From
tr
the above we can make a conclusion about inexpediency of establishment zr > 1,5, because the duration of the useful signal will be less than 10 % of the total duration of the implementation, and we will not get a significant reduction of the error.
Measurement error can be further reduced by rational filtering [18] of the output signal of frequency discriminator provided that we know correlation function or the spectral density of the instantaneous frequency for a given value of Utr.
The spectral density of the instantaneous frequency of the multi-frequency Doppler signal with nonthreshold detection, as is known [16], is equal to the spectral density of a normal narrow-band process. Its values at zero S(0) are maximum and equal to 4,66Dro [18].
For nonthreshold detecting the root-mean-square relative error of determination of the instantaneous speed will be equal to
where df is standard deviation of frequency Doppler signal, dv is standard deviation of speed of object, fD is frequency of Doppler signal, DF is range of variation of frequency of Doppler signal.
For actual values DF/fD and M = 100 we will get afD = 0,2 %.
4. Analysis of the spectral density of frequency
Next, we proceed to analyzing the spectral density of frequency in threshold mode of demodulation. First, let us make a few preliminary remarks. As it was shown, the variance of the Doppler signal frequency in this mode is finite and decreases with the threshold increase. Therefore, values of the correlation functions at zero, in contrast to the no-threshold demodulation, also are finite and will decrease with the increase of the threshold. Finite variance implies that the integral of spectral density of phase noise is finite as well, and the spectral density is falling faster than 1/^.
Since the correlation function of the instantaneous frequency at zero threshold is monotonous and decreasing, we can assume that with the increase of the threshold its values will begin to decrease not only at zero, but also in all other points. If it is so, the spectral density of fluctuations of instantaneous frequency at zero, which is an integral of the correlation function, also will fall with the increase of threshold.
Considering that obtained analytical expression for the correlation function of the instantaneous frequency was so complex that its analysis even with the use of modern PCs are too expensive, to confirm the obtained conclusions using quantitative estimates we used the method of computer simulation.
The Doppler signal computer modeling was performed with the determining of the current values of the envelope and instantaneous frequency at each time step, with identifying the signal sections with above-threshold values of the envelope and quantitative estimations of the correlation function and the spectral density of the instantaneous frequency for the considered sections.
The calculation of desired values of the random frequency is carried out in accordance with the known [18] expression
<p{t = nAt) = (USUC - UCUS)\(uj + U2C),
where US,UC,US,UC are accordingly sinus and cosine components and their time derivatives of the signal complex amplitude U(t).
The calculation of each component was carried out by the following formulas:
cos ©¿¿I Us=YjUi{t~ti}'S'{nCDDti> Uc^YPAt-ti)00*^-
In the above expressions t is current time. Its discrete values at each time step are defined as qh, h is the interval of discreteness, is the envelope of the signal of a single brilliant point, t. is random arrival moment of the «brilliant» point i to the centre of the measuring volume, a is parameter defined by the transverse dimension of the measured object. When we know duration of the single-frequency signal, expressed in Doppler periods NTD, and the average number of particles n present in the measuring volume in
the same time is given, the value of l is found as n/NTD. Sequential values of intervals Dt. and arrival moments of t. particles were obtained using a random numbers generator in accordance with expressions
where Px is a random number uniformly distributed in the range [0,1], generated by the random numbers generator. Estimation of the autocorrelation function of the frequency with qh shift (q is the number of steps, h is discreteness interval in the absence of threshold limit of the envelope) was found as [21]
kq = k-{qh) = , q=1, 2, • • •, m (12)
Here N is the number of implementation samples, m is the maximum number of calculated points of the correlation function.
With the introduction of the threshold bottom limits for the envelope, part of the signal with small amplitudes was not taken into consideration, and the values of a random frequency of these sections were assumed to be equal to zero. Therefore, the expression (12) for the correlation function of the frequency with the introduction of the threshold should be amended to exclude from the total number of summands members with zero values of product PnVn+q. Then
kg{qh) = (<pn<pn+q ) = N_q_s E PnPn+qi
where s is the number of zero products <P„<P„+q (where <pn, <pn+q or both values of q> are equal to zero).
Estimation of the spectral density of the random component of the instantaneous frequency of a signal in accordance with [22] can be found in two ways:
1) a standard way, i.e. via the Fourier transform of the correlation function;
2) the method of the direct Fourier transform of the original implementation of instantaneous frequency with using the fast Fourier transform (FFT) algorithm (the so-called Cooley and Tukey method), which is more efficient in terms of computation time.
Based on requirements to ensure the accuracy of not less than 5 % at the estimates of the spectra and the correlation functions and frequency resolution in the spectral area Acb-0,iAoj, the following values of the model parameters were selected: the number of implementations
r = 400, the length of the implementation of the T = 256 samples, discreteness step h = 4TD.
The results of numerical analysis are presented in Figures 2 and 3.
Figure 2. The family of the correlation functions of the random frequency depending on the relative parameter S
Figure 3. Graphs of spectral density of frequency S(ra/Ara) at different values of the threshold
Figure 2 shows the family of the correlation functions of the random frequency built based on the relative parameter S=x/M1/2, where x = t/ TD, M1/2 = M/2, with different levels of threshold limit of the envelope. It is significant that the values of the correlation functions for S = 0 coincide exactly with the theoretically calculated values of frequency dispersion at the appropriate levels of the threshold zt , and the curves with other values of the relative shift S coincide with the above assumptions.
Figure 3 shows the graphs of the spectral density of frequency S(w) at different values of threshold (the letters T and M mean theoretical and model results). It is shown that with the increase of the threshold narrowing of the spectrum occurs, as well as reduction of all its abso-
lute values at the relevant frequencies. Changing of the spectral density at zero S(0) from the threshold is approximated as
{ztr ) = $0 exP(_zir ).
Therefore, for z = 1,5 value SJzt) will be
? tr ? tr'
reduced by 4,5 times compared to the case where ztr = 0. It means that when we study turbulent pulsations of speed the value of the root-mean-square error of speed measurement, determined mainly by the value S0, with the introduction of the threshold, will be reduced in 4,51/2 = 2,1 times compared with the case where the threshold is absent, and the frequency band is the same.
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Conclusion
Thus, the theoretical and numerical analysis allows us to estimate in common case the potentially achievable accuracy of motion parameters of extended objects for Doppler short-range radio location devices in the threshold mode of demodulation. Applying of the amplitude threshold in contrast to the no-threshold demodulation of Doppler signal allows us to get finite variances of extended object speed estimates within detection area in wide frequency band and reduces all spectral components of phase noise.
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