CC BY
22
ApmwMeuKo B. M. Artyushenko V. M.
Dr. Sci. Tech., Professor, Head of Department «Information Technology and Control Systems», SBEI HE MR «Technological University», Korolev, Russian Federation
Bonoean B. H. Volovach V. I.
Dr. Sci. Tech., Associate Professor, Head of Department «Information and Electronic Service», FSBEIHE «VolgaRegion State University of Service», Togliatti, Russian Federation
UDC 621.391.26:519.2
МОДЕЛИ ПЛОТНОСТИ РАСПРЕДЕЛЕНИЯ ВЕРОЯТНОСТИ И СТАТИСТИЧЕСКИЕ ХАРАКТЕРИСТИКИ СИГНАЛА, ОТРАЖЕННОГО ОТ ПРОТЯЖЕННОГО ОБЪЕКТА
Показано, что для аппроксимации аналитических выражений для ПРВА и и-х начальных моментов огибающей может быть использована ПРВ Накагами и начальные моменты; приведены соответствующие выражения. Приведены выражения, используемые для связи параметров ПРВ огибающих отраженного сигнала с параметрами ПРВ Накагами для наиболее распространенных распределений. Показано, что при использовании ПРВ Накагами распределение фаз отдельных компонент отраженного сигнала независимо от их огибающих, и они распределены либо равномерно, либо по нормальному закону.
Приведены обобщенная модель ПРВ и ее частные случаи для широкого класса негаус-совских сигналов. Приведены графики ПРВ сигнала, аппроксимируемого распределением Накагами, и проанализировано влияние на них различных параметров последнего.
Проведенные экспериментальные исследования показывают хорошее совпадение результатов с предложенными математическими моделями ПРВ; в частности, подтверждается многолучевость отраженного сигнала, и для большинства из них характерен вид амплитуд-но-модулированных (АМ) колебаний. Для проверки полученных в результате обработки ПРВ гипотез использовался критерий /-квадрат. Показано, что в этом случае сигнал подвержен воздействию мультипликативной помехи.
Осуществлен анализ статистических характеристик огибающей сигнала при воздействии мультипликативных (модулирующих) помех. Показано, что огибающие принимаемых сигналов в одних случаях хорошо аппроксимируются ПРВ Накагами, а в других случаях — ПРВ Вейбулла. Определены пределы изменения ключевых статистических характеристик для каждого из двух случаев. Показано, что ПРВ огибающей зависит не только от вида протяженного объекта, но и изменяется в процессе его движения в зоне действия радиолокационного измерителя (РИ), при этом преобладающим является ПРВ Накагами.
Произведен анализ полученных экспериментально статистических характеристик мгновенных значений сигнала, полученных при обработке временных реализаций сигналов, который показывает, что ПРВ мгновенных значений преимущественно носит ярко выраженный бимодальный характер. Показано, что увеличение глубины АМ сигнала приводит к расширению его ПРВ мгновенных значений, изменению ее статистических параметров.
Отмечается, что среди статистических характеристик сигнала, помимо рассмотренных, значительный интерес представляют оценка длительности статистических характеристик выбросов обрабатываемых процессов, анализ параметров спектра сигнала, отраженного от
протяженного объекта, а также нахождение статистических характеристик воздействующих на сигнал помех.
Показано, что для решения задач, связанных работой РИ в условиях ближнего действия, необходимо учитывать явно выраженный негауссовский характер как обрабатываемого сигнала, так и воздействующих на него в общем случае аддитивно-мультипликативных помех.
Ключевые слова: плотность распределения вероятности огибающей сигнала, негаус-совская помеха, многолучевость, протяженный объект, распределение Накагами.
MODELS OF PROBABILITY DENSITY AND STATISTICAL CHARACTERISTICS OF THE SIGNAL REFLECTED FROM AN EXTENDED OBJECT
It is shown that the Nakagami probability density function (PDF) and its initial moments can be used to approximate the analytical expression for PDFA and »-th initial moments of the envelope; the corresponding expressions are given. The expressions, which are used to relate parameters of the PDF of the envelope of the reflected signal to the parameters of the Nakagami PDF for the most common distributions, are given. It is shown that when the Nakagami PDF is used, phases of separate components of the reflected signal are distributed independently of their envelopes, besides they are distributed either uniformly or according to the normal law.
The generalized PDF model and its specific cases for a wide class of non-Gaussian signals are presented. Graphs of the PDF of the signal approximated by the Nakagami distribution are given, and it is analyzed how they are influenced by different parameters of the Nakagami distribution.
The conducted experimental studies show that their results coincide well with the proposed mathematical models of the PDF; in particular, it is confirmed that the reflected signal is multipath and the form of amplitude-modulated (AM) fluctuations is characteristic for most of them. X-square criterion was used to check hypothesis obtained as a result of the PDF processing. It is shown that in this case the signal is exposed to multiplicative noise.
The statistical characteristics of the envelope of the signal under the influence of multiplicative (modulating) noise are analyzed. It is shown that the envelopes of received signals in some cases are well approximated by the Nakagami PDF, while in other cases they are approximated by the Weibull PDF. Bounds of variations in main statistical characteristics for each of the two cases are defined. It is shown that the PDF of the envelope depends not only on the type of an extended object, but it also varies in the course of its motion within the range of the radar meter (RM), whilst the Nakagami PDF is predominant.
The analysis of the experimentally obtained statistical characteristics of instantaneous values of the signal is carried out. These characteristics were obtained during the processing of time realizations of signals. The analysis shows that the PDF of instantaneous values in most of the cases is of a pronounced bimodal nature. It is shown that increasing the depth of AM signal leads to the expansion of its PDF of instantaneous values and changing its statistical parameters.
It is noted that among the statistical characteristics of the signal in addition to the considered above, there are other issues of interest such as estimating the length of the statistical characteristics of the emissions of the processed processes, the analysis of parameters of the spectrum of the signal reflected from an extended object and finding the statistical characteristics of noise affecting the signal.
It is shown that to solve the problems related to the operation of RM in short-range conditions, a distinctive non-Gaussian nature of both the processed signal and additive multiplicative noise affecting the signal should be taken into account.
Key words: probability density distribution of signal envelope, non-Gaussian noise, multi-path, extended object, Nakagami distribution.
Introduction
For the synthesis and analysis of radio systems and devices that detect and measure parameters of motion of extended objects, an effective
model of the probability density (PDF) of the signal reflected from extended objects, as well as of noise affecting it, is needed. There is quite a large number of mathematical models describ-
ing the signal reflected from spatially distributed radar targets: aircrafts, ships, etc. [1-5]. However, many aspects related to the determination of PDF models of the signal and its statistical characteristics for so-called short-range conditions are still insufficiently studied and have a number of specific features [6-11].
The purpose of this work is to select and justify such models of the PDF of the signal reflected from an extended object, which will be as close to the real models as possible; a number of statistical characteristics of the signal is supposed to be described.
1. Selection and justification of PDF models for the signal reflectedfrom an extended object
As it is known [12-16], during radio detecting and ranging in short range conditions, the detected object is usually regarded as complex, extended, consisting of a set of N reflecting elements.
The resulting signal at the input of a receiving device, reflected from an extended object can be written as:
= (1) where st(t,X) = Re{a;.(t)U(t -Tt)exp j[(a0 -- Afd i)t- co0Ti - ©,.]} is the signal received from
an arbitrary i-th point of the object; «,■(*) i(t) is the attenuation coefficient of the «amplitude» of the received signal (as compared to the emitted signal at the time t from the i-th point; U(t) = /(i)exp[ j(p{t)] is a complex envelope of the signal; f(t) and y(t) are functions describing the laws of amplitude and phase (frequency) modulation; Ti is the time of signal delay from the i-th «brilliant» point; is a carrier frequency; Afdi is Doppler frequency shift from the i-th «brilliant» point; 0. is a phase of the signal reflected from the i-th «brilliant» point usually evenly distributed in the interval [- n; n]; A a vector parameter characterizing the set of the parameters rn0, f(t), y(t), a(t), t and Afdi.
Values a(t) and 0,- are considered to be random and mutually independent.
A wide variety of frequent types of signal models is possible (1). Thus, to describe a mul-tipath nature of the signal reflected from an
extended object in [13] the following model is considered:
s(i,X) = Re{[/(i)exp y[®0i + ®(i)]} =
= Re{£itf(0«py[fflo' + ®(0]}» (2)
which explicitly introduces the envelope of the received signal U(t) and the resulting phase 0,. The density of probability distribution of instantaneous values, the envelope (amplitude) (PDFA) and the phase (PDFP) of the received signal is of the greatest interest.
The results [17], show that the signal reflected from the extended object (2) can be well described by the generalized PDFA model A; 0,^,0,0,,) (table 1), where 1 when n = 0,I„ ( ) is a Bessel function of the i-th kind
/ , 2\0,5
of the n-th order; A = Ul\axay\ ,
a=U0j(cr2cr2) ' are normalizing values;
+yo)°'5> ©o=arctg(j;0/*0) are the
module and the argument of a deterministic component of the signal; cr2 = cr2 = cr2 is the variance of quadrature components of the signal; xq and jo are deterministic quadrature components of the signal; r^ is correlation coefficient between quadrature components of the signal
a = ~ay)/(ax +ay) is the parameter of time-varying, changing within the limits
of [1, -1].
It can be seen from the above expressions, that the PDF of the envelope generally depends on four parameters a, r^, a and 0O. When they change, the shape of the PDFA curve changes as well. Numerical characteristics of the PDFA are described by the relation (table 1) mvA (a, r^, a, 0O).
The introduction of complex analytical expressions for the PDFA and v-th initial moments of the envelope can be approximated by simpler expressions [17]. In particular, the Nakagami PDF and its initial moments give good results
iT(C/)=(2/r(m))(m/Q)'"[/2w"1exp{-wC/2/Q},
U>0, (3)
where m and Q are distribution parameters:
m = Q2/{u2-Q2f >0,5; Q = U2; (4)
r(.) is a gamma function;
ml=T(m + v/2)/r(m)(Q / rn)"v/2 (5)
The expressions which relate the parameters m and Q to the parameters a, r^, a, 0O are:
+2 a
= (l + a2)2{(l + [4(l-a2) + a l + (4(l-a2) + a2)°'5x xcos^2@0-arctgr^l-a2) ' ja Q = o-2(l + a2).
+
Using these expressions, one can define the relationship between m, Q, and a, r^, a, ©0 parameters (and vice versa) which is necessary to know when you move from one distribution to another.
Table 2 shows the results of approximations of the envelope of the signal (2) by the Nakagami distribution. Here the phase of the deterministic Table 1. Generalized model of PDFA and its numerical characteristics
component is assumed to be zero (®0 = 0) without loss of generality.
In table 2 the results of approximations of the envelope of the signal (2) by the Nakagami distribution are presented. Here the phase of the deterministic component is assumed to be zero (©0 = 0) without loss of generality.
When the Nakagami distribution is used (3), the issue of the choice of phase distribution remains open. In most studies, it is generally assumed that phases of particular components of the reflected signal are independent from their envelope and are distributed either evenly within the interval [-n, n], or by the normal law [18, 19]:
W(U,®) = Wm(U)W{®)
where Wm(U) is determined by the expression (3).
In [17] the statistical characteristics of instantaneous values of non-Gaussian signals (2) are analyzed. Their amplitude U is described by the generalized PDFA model (see table 1).
Characteristics
Analytical writing of an expression
Wg(A;a,r„,a,®0)
Mf:
-exp-
-.A2 -a2 [l-i?(a,rJcos20 -/?(«,rj
*M)(1-«T
I>J„x
/1=0
*M)Mr
xcos2«v(a,r9,,0o)
hn Aa
l-'i \
r2+E2 («,©„)
KSili2@o
Mf J
N,0,5
[C-i)(i-»!)r«p
f
*M)Mr
xM 2(a,^)j->„
n ~ N{a,rv,a,®0)~
2M2(a,rJ 2
r - + 2n + 2k + l
T(2n + 1) S R-!r(n + i + l)
2M(a,rv)
— + 2n + 2k + l:
2
4M(a,rv)
Description: B(a,r^) = {a2+ r2 (l - a2 ))°'5; fi(a,rv) = arctg
f t \«.5
= f ^fj cos2©o+[^l sin20o; v{a,rv,&o) = A(*,rv,Qo).
1 + a
A(a,^,0o) = arctg
1+a
T-a
sin0o-/^cos©o
1+a
T-a
cos0o — r sin©0
M{a,rxy) = 0,5{\-r2xy){\-a2f
rl+E2{a,®0)-
2>ysin20o (1 -a2f
Table 2. Relating the parameters of the PDF of the envelope of the reflected signal to the parameters of the Nakagami PDF
Type of distribution Distribution parameters Linking by Qc parameters Linking by mc parameters Change range m
Rayleigh distribution 2 2 2 a = <** = П= 2 a2 — m= 1
Generalized Rayleigh distribution a; 2 2 2 a = ax = ay Q= 2cr2(l + a2) (l + a2)2 m - l + 2a 1 <m< oo
Hoyt distribution a; a2= <r2+a2 X у П = <72 1 m=! (l + a ) 0,5 < m <1
Generalized Hoyt distribution a; a; П = ст2 (l + a2) (l + a2)2 m =----- l + 2a2(l + a) + a2 0,5 < m < oo
p-distribution v a2 = a2+a2 X у П = 2(т2 1 m= t Ы) 0,5 < m < oo
Generalized p-distribution W> a> _2 2 x у П= 2o-2(l + a2) (l + a2 )2 m = ——, ' -l + 2a2+r2 0,5 < m < oo
Generalized PDFA «; W'a' a2=cr2x+a2 П = ст2 (l + a2) (Ws)if®= 0 0,5 < m < oo
One-sided Gaussian PDFA a = l;rv=l П = <т2 — m = 0,5
If the values U= (x2 +y2) and 0 = arctgO / x), where y and x are quadrature components of the signal, they are independent of each other, and the phase distribution is equally probable, then the PDF of instantaneous values of the signal W(sc) is determined by the ratio presented in the table 3, where ^Q is a degenerate hypergeo-metric function.
The analysis of the above table shows that the expression W(sc) is basic and can serve as a base in determining the PDF of instantaneous values of the radio signal for a wide class of probabilistic models of non-Gaussian processes sc(Y).
Figure 1 shows the PDF W(sc) calculated for the case where W(U) adheres to the Nakagami PDF.
The graphs (Figure 1, a) show that when Q = 1 and m ^ 1 the PDF curve is normalized.
At m > 1 a «dip» in the probability curve W(sc) appears at sc = 0 point and we get a twomode PDF curve with modes at symmetric points -sc and +sc. The variance increases and the mode is shifted from the axis with the increase in m. In this case, the PDF curve remains symmetrical with respect to the axis. If m < 1 (in Figure 1, b m = 2,31), the PDF curve becomes two-modal, and the increase in the parameter Q leads to the increase in the «dip» of the probability curve and its variance.
Experimental studies carried out by the radar meter (RM), measuring motion parameters of extended objects, showed a good coincidence of these mathematical models with the results of the experiments [20]. Various spatially-distributed models of motor vehicles such as road
-1,6 -0,8 0 0.8 1,6 -1,6 -0,8 0 0,8 1,6 s,
a) b)
a) m; b) Q
Figure 1. Density of probability distribution W(sc) for the model (2) with different parameters of the distribution
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Электротехнические и информационные комплексы и системы. № 2, т. 14, 2018
Table 3. Generalized PDF model and its special cases for a wide class of non-Gaussian signals
Distribution law Analytical expression of the probability density Numerical characteristics of the PDF
Generalized model of the PDF f y [2' 2' 2 where 1 M2tx V ' X > A (°,b,z) „.^(-^iT* ■R UJ k'° *!r(a + )fc)r|j + lj ( V {$)■ where v=2,4,6...
sin^i [r(l + ai-i A {l + a-b,-b-z)\ b)T(b)
r(a)r(2-Ä) J
W(U) is the Nakagami PDF a = m; fi = 2m/Cl;y= 0 i-T f 2^ mSr I i2m—1 xp--- Ls 1 x fi 1 1 V " y 'O)
x,p(0,5;/« + 0,5;/«ic2
W(U) is described by the Rayleigh PDF (m= 1; Q = 2cr2 ) 1 w(s')=~ 2 rexp 2a yiK xY(0,5;3/2;i2/2cr2 "2a2, ) klx
W(U) is described by the Gaussian PDF 1 w(s-)=~ 2 rexp 2a V7T where 'V(t) is the fu through the Hermit nction e polyn J expressed omial
tankers, heavy container trucks, prefab trailers, etc. were used as extended objects.
Statistical processing of experimental data was carried out on the basis of an automated system for experimental data processing. Testing of hypotheses obtained as a result of PDF processing was carried out according to the X2-criterion [21, 22]. The methodology of experimental studies and statistical processing is presented in detail in [20] and is not given here.
Analyzing time realizations of Doppler signals received when being reflected from various models of motor vehicles, one can come to the conclusion that the reflected signal is well described by a model of a multipath signal. It is also seen that for most of them reflected signals have the form of AM oscillation, and the type of the reflected signal depends on a number of different factors [23]. The depth of AM varies within large limits and can reach 100 %, i.e. complete signal fading.
In this case, it is assumed that the received signal is affected by multiplicative noise [17, 24-27], statistical characteristics of which are obtained by means of processing the envelope of the signal.
2. Statistical characteristics of the envelope o f the signal under the influence of multiplicative (modulating) noise
Analysis of the results of statistical processing of the envelope showed that received signals can be roughly attributed to two large groups. The first group includes signals, envelopes of which are well approximated by the Nakagami PDF (3)-(5).
In this group, the expectation mh variance o2 and standard deviation (RMS) o, respectively, can vary 0,6051 < mj < 8,160; 0,0075 < o2 < < 0,027; 0,086 < o < 0,1647. The variation coefficients Kv, skewness coefficient Ka and kur-tosis Kk may vary: 0,1436 < Kv < 0,3724; -0,4721 < Ka < 0,2627; -0,7948 < Kk < 0,1816 [10].
The parameters of the Nakagami distribution may vary within: 4,0559 < m < 11,1965; 0,3995 < Q < 11,29. If the number of degrees of freedom nf varies from 7 to 15, x2 for this group varies from 13,096 to 30,323 and the significance level ysl is in the range
0,002 < ysl < 0,10.
The second group includes signals envelopes of which are well approximated by the Weibull PDF
W(U) = CaUa~l exp(-Ct/a ), U> 0,C>0,a>0, where C and a are parameters of distribution.
The initial moments of the Weibull PDF are defined mv = C~v/aT(l + v/a), if a = 1, a Weibull distribution becomes exponential, if C = 2 and C = 0,5a2, it turns into a Rayleigh distribution.
Statistical parameters of the distribution vary within: 0,6377 < m 1 < 1,082; 0,0106 < a2 < 0,0427; 0,091 < a < 0,2068; 0,1217 < Kv < 0,2651; -0,7973 < Ka < -0,0626; -0,7877 < Kk < 0,9330; 0,3916 < C < 1,8122; 3,72 < a < 7,88.
If the number of degrees of freedom nf varies from 7 to 12, x2 varies from 10,289 to 28,61 and the significance level ysl is in the range 0,002 < ysl < 0,1.
The results of the processing show that the PDF of the envelope depends not only on the type of the extended object, but also changes in the process of its motion within the range of RM and at some time points the PDFA can be well approximated by the Gaussian and logarithmic normal distribution. However, the Nakagami PDF is dominant [4, 20].
3. Statistical characteristics of instantaneous values of the signal
The analysis of experimental data obtained during processing of time realizations of signals shows that the PDF of instantaneous values is mainly of a pronounced bimodal nature. For different models of extended objects, the coefficients of kurtosis are within Kk=2,..., 4, the coefficients of skewness are close to zero Ka ~ 0, the expectation varies within 0 < M1 < 0,1.
Increasing the depth of AM of the signal leads to the expansion of the PDF of its instantaneous values, varying its parameters (the mathematical expectation M1, the RMS a, the variance a2, the third M3 and the fourth M4 moments).
It should be noted that the analysis and generalization of the obtained results were carried out on numerous fragments of the processed signal, reaching 900...1000 for each considered model of the extended object. The signal was recorded by more than 100 radar meters. In addition, in order to obtain the most complete statistical picture obtained data, experimental
work was carried out under various climatic conditions: clear, sunny weather, rain, fog, frost and snowfall.
When considering some issues with the aim to improving the efficiency of radio systems and short-range devices, not only the statistical characteristics of the signal are of considerable interest, but also estimating the duration of statistical characteristics of emissions of processed processes, in particular the density and the distribution function of duration of emissions of the processes below the threshold level of processing [28]. As shown by the previous studies [23, 29], analyzing parameters of the spectrum of the signal reflected from an extended object and finding statistical characteristics of noise affecting the signal is important [8, 10, 30].
Conclusion
As the result of theoretical and experimental studies that have been carried out the following conclusions can be drawn.
1. The Doppler signal reflected from extended objects, motor vehicles in particular, is well described by a mathematical model of a multipath signal; furthermore, the above mentioned Doppler signal is affected by multiplicative and additive noise at the same time;
2. As a rule, the PDF of multiplicative noise (of the envelope of the reflected Doppler signal) is well approximated by the Nakagami PDF;
3. The PDF of instantaneous values of the Doppler signal reflected from an extended object, depends not only on the type of an object, but also varies in the course of its motion within the range of measuring. Moreover, the PDF of instantaneous values is mainly of a pronounced bimodal nature;
4. Generally, the PDFA differs from the Rayleigh PDF and it is well approximated by the Nakagami PDF and the Weibull PDF; however the Nakagami PDF is dominant;
5. The results of the experimental processing agree with the theoretical models of the Doppler signal reflected from an extended object.
Thus, when solving problems related to the detection of extended objects and measurement of their parameters, it is necessary to take into account the pronounced non-Gaussian nature of both the useful processed signal and the additive multiplicative noise affecting it.
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