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Probability density functions of instantaneous values of a signal under influence of additive and multiplicative noise

DOI 10.18127/j20700784-201903-08

Keywords:

V.I. Volovach – Dr.Sc. (Eng.), Associate Professor, Head of Department «Information and Electronic Service», Volga Region State University of Service (Tolyatti)

E-mail: volovach.vi@mail.ru

V.М. Artyushenko – Dr.Sc. (Eng.), Professor, Head of Department «Information Technology and Management Systems», Technological University («TU» GBOU VO MO) (Korolev)

E-mail: artuschenko@mail.ru


It is shown that for the successful synthesis and analysis of information-measuring systems it is necessary to consider the impact of additive-multiplicative noise on the processed signal.

Expressions are obtained to determine the probability density distribution of the instantaneous values of the useful signal, as well as a mixture of the signal and additive noise under the action of multiplicative (modulating) noise.

It is shown that the obtained mathematical expressions make it possible to simulate the probability density distribution of the said mixture at an arbitrary distribution law of the envelope signal.

Probability density functions of instantaneous values
of a signal under influence of additive and multiplicative noise

© Authors, 2019
© Publishing House Radiotekhnika, 2019

V.I. Volovach – Dr.Sc. (Eng.), Associate Professor, Head of Department «Information and Electronic Service»,

Volga Region State University of Service (Tolyatti)

E-mail: volovach.vi@mail.ru

V.М. Artyushenko – Dr.Sc. (Eng.), Professor, Head of Department «Information Technology and Management Systems», Technological University («TU» GBOU VO MO) (Korolev)

E-mail: artuschenko@mail.ru

 


Abstract

It is shown that for the successful synthesis and analysis of information-measuring systems it is necessary to consider the impact of additive-multiplicative noise on the processed signal.

Expressions are obtained to determine the probability density distribution of the instantaneous values of the useful signal, as well as a mixture of the signal and additive noise under the action of multiplicative (modulating) noise.

It is shown that the obtained mathematical expressions make it possible to simulate the probability density distribution of the said mixture at an arbitrary distribution law of the envelope signal.

Keywords

Probability density function, additive noise, multiplicative noise, amplitude and phase distortion.

DOI: 10.18127/j20700784-201903-08

Introduction

The applied theory of random processes is widespread in statistical radio engineering, radar and radio navigation as well as in the theory of communication and automatic control. As a rule, models where the received signals are described as an additive mixture of the useful signal and Gaussian noise are considered [1–4]. However, in reality, useful signals are often exposed to multiplicative (modulating) noise [5–9]. For the synthesis and analysis of information-measuring systems processing such signals, mathematical models are needed, which would allow to simulate the probability density distribution (PDF) of these mixtures close to reality [10–12].

The purpose of this work is to obtain mathematical expressions that would allow simulating the PDF of
instantaneous values of the sum of additive noise and the signal exposed to modulating (multiplicative) noise.

The PDF of instantaneous values of the signal exposed to multiplicative noise

We will consider the problem of estimating the information parameters of the useful signal under the
simultaneous influence of multiplicative and additive non-Gaussian noise. The problem of estimation of information parameters will be solved in discrete observation time.

Taken, that during the time [0,T] samples of the random process Yh Y(th) (h = 1, …, H) are observed. This process is a mixture of the useful signal  containing information about the measured parameter, the additive noise n(th) and the multiplicative noise h(th) both of them being non-Gaussian:

,

where  is a dimensionless multiplier, characterizing changes of the envelope of the signal caused by modulating noise (amplitude distortions); U(t) is the envelope of the signal determined by the law of the signal's amplitude modulation; wс is a carrier frequency of the signal; Ф(t) is the law of phase modulation of the signal (if the frequency modulation is , where W(t) is the frequency modulation law); φ0 is the initial phase of the signal; φ(t) are phase changes of the signal caused by modulating noise (phase distortions).

We present the signal exposed to modulating noise over the interval t Î (0, T), as

                                                 (1)

where h(t), φ(t) are, generally, functionally related random functions characterizing, respectively, the amplitude and phase distortions; U(t), Ф(t) are the known laws of phase and amplitude changing defining transmitted information; Y(t) is a complete instantaneous phase of the signal s(t).

The joint PDF of instantaneous values of the signal s(t), the envelope h(t) and the phase φ(t) with the account for the ratio (1) and the properties of the δ-function as well as the periodic nature of the function cosY can be described as [1,13]

                                                     (2)

The expression (2) allows to obtain the PDF of instantaneous values of the signal W(s) in the presence of amplitude and phase distortions with the PDF W(h, φ).

We assume that amplitude and phase fluctuations are independent at least at coinciding moments of time. Then, the PDF of instantaneous values has the form:

                                                                                                                                                 (3)

where  is a one-dimensional characteristic function of phase distortions of the signal s(t) (expressions for some one-dimensional characteristic functions  are given in [5]);  is the mathematical expectation Y(t); Ф0 is the average value of phase distortions φ(t).

If there are only phase distortions , where h0 is the mathematical expectation h(t), the PDF of instantaneous values of the signal (3) can be written as:

         (4)

The first four initial moments m1(s) – m4(s) for the PDF (4) given in [14].

Note that in the relations (3) and (4) the first terms correspond to the PDF of the signal s(t) with uniformly distributed phase within the interval (0, 2p). The weight of the other components is determined by the characteristic function of phase distortions, and .

It is also important to note that the PDF (3) and (4) for nonuniform phase distribution over the interval (0, 2p) depend on time, whereas the signal is a non-stationary function.

If the amplitude and phase of the signal s(t) are independent random functions, then, by introducing a integration variable , into (3), after some transformations, we obtain:

,

where  is the PDF of instantaneous values of the signal s(t) with independent amplitude-phase distortions when phase distortions are distributed uniformly within the interval (0, 2p) or distributed arbitrarily if the variance of the phase is .

Using the results of [15], omitting complex mathematical transformations to determine WF(s), we write the final expression to determine the PDF of the instantaneous values of the signal (3), which will have the form

                   (5)

The obtained expression (5) shows that if amplitude and phase distortions are independent, when  whereas , then little does the PDF of the radio signal W(s) depends on the PDF of the phase WF(s) and it is determined mainly by the PDF W0(s) only, meaning, that practically it depends only on the PDF of the envelope W(h).

If there is a functional relation between the amplitude and phase fluctuations φ = f(h), then the two-dimensional PDF W(h, φ) can be presented as

.                                                                                             (6)

Substituting (6) into (2), after integration and all necessary mathematical transformations, we obtain

                                         (7)

It is important to note that in the presence of a functional relationship between the amplitude and phase of the PDF of the transmitted signal, the W(s) can be expressed through the statistical characteristics of the envelope.

The initial moment mk(s) of the k-th order of the distribution W(s) (7) given in [14].

Analysis of the ratio initial moment mk(s) shows that the average value of the signal s(t) (1) largely depends on the nature of amplitude distortions. In addition to the component determined by the PDF through the uniform law of phase distribution over the interval (0, 2p), it is also determined by the component that depends on the derivative characteristic of the function of amplitude distortions.

The PDF of the sum of additive noise and the signal in the presence of modulating (multiplicative) noise

In practice, together with the signal exposed to modulating noise (1), as a rule, there is additive Gaussian noise n(t), whose PDF can be described as

,

where sn2 is the variance of the additive noise.

Using the obtained relations for the PDF of instantaneous values of the signal W0(s), we determine the PDF of the mixture of the signal (1) and additive noise

                                                                                              (8)

where from (1) follows .

Substituting (8) in (2), taking into account (1) and having replaced the integration variable , after all necessary transformations, we will get:

,           (9)

where In(a) is a modified Bessel function.

Then the PDF of the mixture of the signal and additive noise can be found from the ratio

where ;

We introduce a notation  that plays the role of the signal-to-noise ratio (SNR). We consider two cases  and , which are of significant interest when solving practical problems.

We assume that amplitude and phase distortions are independent , while SNR is . Assuming that  if the law of phase distribution is arbitrary and amplitude distortions are absent :

                   (10)

For phase distortions uniformly distributed over the interval (0, 2p) or deep phase distortions :

.

If the law of the envelope distribution is arbitrary, the PDF of the sum signal will be:

,

where  is an integral transformation of the PDF of the envelope W(h) [13].

Taking it into account, the expression (10) will take the following form

When the SNR is ОСП  after simple transformations for the PDF (9) we get

When the laws of amplitude distribution W(h) and phase distribution W(φ) are arbitrary:

        (11)

Here Гc(yU) and Гs(yU) are functions expressed through the initial moments of the signal envelope (1), where ,  is the initial moment of the (2k – 1)-th, 2k-th order of amplitude fluctuation [14].

When the law of phase distribution is uniform, or when the variance of phase fluctuations is , the ratio (11) will take the form

.

We will find the PDF of the mixture of the signal and additive noise (8), when amplitude-phase distortions of the signal uм(t) are functionally related, that is, when  and the ratio (6) is satisfied.

When the SNR is , having integrated the expression (11) over h and φ, we finally write

where .

When the SNR is  and ,  the expression (11) will be written as

Having carried out transformations similar to those when , we obtain

                       (12)

where Сk(y) [14].

Analysis of the expression (12) shows that in contrast to (11), if amplitude-phase distortions are functionally related when expanding the PDF of the signal-additive noise mixture according to the normal law, the weight multipliers are determined from the derivative of the characteristic function of  amplitude distortions (envelope fluctuations). The only difference between non-stationary terms in the expressions (12) and (11) is the presence of the initial phase in (11), determined by the connecting function φ = f(h)at the zero point.

n     Thus, mathematical expressions are considered and analyzed that allow to simulate the probability density distribution of instantaneous values of the signal mixture exposed to multiplicative (modulating) and additive noise, with  the envelope of the signal distributed arbitrarily.

It is shown that the average value of the signal s(t) under the influence of multiplicative noise is largely determined by the nature of the amplitude distortions. Called the signal value includes two components. The first component is determined by PDF with a uniform law of phase distribution at a given interval (0, 2p). The second component depends on the derivative characteristic of the amplitude distortion function.

Also it is shown that in the presence of a functional relationship between the amplitude and phase fluctuations of the PDF signal can be expressed through the statistical characteristics of the envelope. In addition, for functionally coupled amplitude-phase distortions in the decomposition of the PDF signal mixture and additive noise according to the normal law, the weight factors are determined by the derivative of the characteristic function of the amplitude distortions.

References

1.      Levin B.R. Teoreticheskie osnovy statisticheskoj radiotehniki. M.: Radio i svyaz'. 1989.

2.      Tihonov V.I. Nelinejnoe preobrazovanie sluchajnyh processov. M.: Radio i svyaz'. 1986.

3.      Tihonov V.I. Statisticheskaya radiotehnika. Izd. 2–e, pererab. i dop. M.: Sov. radio. 1982.

4.      Trifonov A.P., Shinakov Yu.S. Sovmestnoe razlichenie signalov i ocenka ih parametrov na fone pomeh. M.: Radio i svyaz'. 1986.

5.      Kremer I.Ya., Vladimirov V.I., Karpuhin V.I. Moduliruyuschie (mul'tiplikativnye) pomehi i priem radiosignalov / Pod red. I.Ya. Kremera. M.: Sov. radio. 1972.

6.      Vasil'ev K.K. Priem signalov pri mul'tiplikativnyh pomehah. Saratov: Izd-vo Saratovskogo un-ta. 1983.

7.      Akimov P.S., Bakut P.A., Bogdanovich V.A. i dr. Teoriya obnaruzheniya signalov / Pod red. P.A. Bakuta. M.: Radio i svyaz'. 1984.

8.      Artyushenko V.M., Volovach V.I. Kvazioptimal'naya diskretnaya demodulyaciya signalov na fone korrelirovannyh negaussovskih flyuktuacionnyh mul'tiplikativnyh pomeh // Radiotehnika. 2016. № 6. C. 106–112.

9.      Artyushenko V.M., Volovach V.I. Ocenka pogreshnosti vektornogo informacionnogo parametra signala na fone mul'tiplikativnyh pomeh // Radiotehnika. 2016. № 2. S. 72–82.

10.    Artyushenko V.M., Volovach V.I. Kvazioptimal'naya obrabotka signalov na fone additivnoj i mul'tiplikativnoj negaussovskih pomeh // Radiotehnika. 2016. № 1. S. 124–130.

11.    Artyushenko V.M., Volovach V.I. Modelirovanie plotnosti veroyatnosti signala i additivnogo shuma pri vozdejstvii mul'tiplikativnyh pomeh // Radiotehnika. 2016. № 12. S. 28–36.

12.    Artyushenko V.M., Volovach V.I., Shakurskiy M.V. The Demodulation Signal under the Influence of Additive and Multiplicative non-Gaussian Noise // Proceedings of IEEE East-West Design & Test Symposium (EWDTS'2016). Yerevan, Armenia, October 14–17, 2016. Kharkov: KNURE. 2016. P. 591–594.

13.    Gradshtejn I.S., Ryzhik I.M. Tablicy integralov, summ, ryadov i proizvedenij. M.: Fizmatgiz. 1963.

14.    Artyushenko V.M., Volovach V.I. Modelirovanie plotnosti raspredeleniya veroyatnosti smesi signala, podverzhennogo vozdejstviyu amplitudnyh iskazhenij i additivnoj pomehi // Radiotehnika. 2017. № 1. S. 103–110.

15.    Smirnov V.I. Kurs vysshej matematiki / 21–e izd., stereotip. T. 2. M.: Nauka. 1974.

Поступила 18 февраля 2019 г.


УДК 621.391.26:519.2

Плотность распределения вероятности
мгновенных значений сигнала при воздействии
аддитивной и мультипликативной помех

© Авторы, 2019
© ООО «Издательство «Радиотехника», 2019

В.И. Воловач – д.т.н., доцент, зав. кафедрой «Информационный и электронный сервис»,

Поволжский государственный университет сервиса (г. Тольятти)

E-mail: volovach.vi@mail.ru

В.М. Артюшенко – д.т.н., профессор, зав. кафедрой «Информационные технологии и управляющие системы», Технологический университет (ГБОУ ВО МО «ТУ») (г. Королев)

E-mail: artuschenko@mail.ru

 

Аннотация

Рассмотрены вопросы, связанные с воздействием мультипликативной (модулирующей) (иначе – амплитудных искажений) и аддитивной помех на обрабатываемый сигнал; статистические характеристики плотности распределения вероятностей (ПРВ) мгновенных значений сигнала при воздействии на него мультипликативных (модулирующих) помех. Получено выражение для совместной ПРВ мгновенных значений сигнала, огибающей и фазы, а также для ряда практически значимых случаев: при
независимости флюктуации амплитуды и фазы в совпадающие моменты времени и при наличии только фазовых искажений.

Приведено выражение для ПРВ мгновенных значений сигнала в случае, когда амплитуда и фаза сигнала являются независимыми случайными функциями. Отмечено, что при независимости амплитудных и фазовых искажения ПРВ радиосигнала мало зависит от ПРВ фазы и определяется в основном ПРВ мгновенных значений сигнала. Даны выражения для ПРВ радиосигнала при воздействии модулирующей помехи для равномерно распределенных на интервале (0, 2π) и для глубоких фазовых искажений в наиболее часто встречающихся ПРВ огибающей.

Приведено выражение для ПРВ мгновенных значений сигнала в случае, когда между флуктуациями амплитуды и фазы имеется функциональная связь. При этом показано, что при наличии такой связи ПРВ сигнала может быть выражена через статистические характеристики огибающей. Найдены выражения для определения начальных моментов k-го порядка ПРВ сигнала для названного случая. Установлено, что среднее значение сигнала в значительной степени определяется характером амплитудных искажений.

Рассмотрены статистические характеристики ПРВ суммы аддитивной помехи и сигнала при наличии модулирующих (мультипликативных) помех. Получены выражения для определения ПРВ смеси сигнала и аддитивной помехи, включая при произвольных законах распределения амплитуды и фазы, а также при произвольном законе распределения огибающей сигнала. Получены ПРВ смеси сигнала и аддитивной помехи при функциональной связи амплитудно-фазовых искажений сигнала между собой. Выявлено, что при функционально связанных амплитудно-фазовых искажениях при разложении ПРВ смеси сигнала и аддитивной помехи по нормальному закону весовые множители определяются производной от характеристической функции амплитудных искажений (флуктуаций огибающей).

Ключевые слова

Плотность распределения вероятности, аддитивная помеха, мультипликативная помеха, амплитудные и фазовые искажения.

DOI: 10.18127/j20700784-201903-08

ЛИТЕРАТУРА

1.     Левин Б.Р. Теоретические основы статистической радиотехники. М.: Радио и связь. 1989.

2.     Тихонов В.И. Нелинейное преобразование случайных процессов. М.: Радио и связь. 1986.

3.     Тихонов В.И. Статистическая радиотехника. Изд. 2-е, перераб. и доп. М.: Сов. радио. 1982.

4.     Трифонов А.П., Шинаков Ю.С. Совместное различение сигналов и оценка их параметров на фоне помех. М.: Радио и связь. 1986.

5.     Кремер И.Я., Владимиров В.И., Карпухин В.И. Модулирующие (мультипликативные) помехи и прием радиосигналов / Под ред. И.Я. Кремера. М.: Сов. радио. 1972.

6.     Васильев К.К. Прием сигналов при мультипликативных помехах. Саратов: Изд-во Саратовского ун-та. 1983.

7.     Акимов П.С., Бакут П.А., Богданович В.А. и др. Теория обнаружения сигналов / Под ред. П.А. Бакута. М.: Радио и связь. 1984.

8.     Артюшенко В.М., Воловач В.И. Квазиоптимальная дискретная демодуляция сигналов на фоне коррелированных негауссовских флюктуационных мультипликативных помех // Радиотехника. 2016. № 6. C. 106–112.

9.     Артюшенко В.М., Воловач В.И. Оценка погрешности векторного информационного параметра сигнала на фоне мультипликативных помех // Радиотехника. 2016. № 2. С. 72–82.

10.   Артюшенко В.М., Воловач В.И. Квазиоптимальная обработка сигналов на фоне аддитивной и мультипликативной негауссовских помех // Радиотехника. 2016. № 1. С. 124–130.

11.   Артюшенко В.М., Воловач В.И. Моделирование плотности вероятности сигнала и аддитивного шума при воздействии мультипликативных помех // Радиотехника. 2016. № 12. С. 28–36.

12.   Artyushenko V.M., Volovach V.I., Shakurskiy M.V. The Demodulation Signal under the Influence of Additive and Multiplicative non-Gaussian Noise // Proceedings of IEEE East-West Design & Test Symposium (EWDTS’2016). Yerevan, Armenia, October 14–17, 2016. Kharkov: KNURE. 2016. P. 591–594.

13.   Градштейн И.С., Рыжик И.М. Таблицы интегралов, сумм, рядов и произведений. М.: Физматгиз. 1963.

14.   Артюшенко В.М., Воловач В.И. Моделирование плотности распределения вероятности смеси сигнала, подверженного воздействию амплитудных искажений и аддитивной помехи // Радиотехника. 2017. № 1. С. 103–110.

15.   Смирнов В.И. Курс высшей математики / Изд. 21-е, стереотип. Т. 2. М.: Наука. 1974.

References:
  1. Levin B.R. Teoreticheskie osnovy statisticheskoj radiotehniki. M.: Radio i svyaz'. 1989.
  2. Tihonov V.I. Nelinejnoe preobrazovanie sluchajnyh processov. M.: Radio i svyaz'. 1986.
  3. Tihonov V.I. Statisticheskaya radiotehnika. Izd. 2–e, pererab. i dop. M.: Sov. radio. 1982.
  4. Trifonov A.P., Shinakov Yu.S. Sovmestnoe razlichenie signalov i ocenka ih parametrov na fone pomeh. M.: Radio i svyaz'. 1986.
  5. Kremer I.Ya., Vladimirov V.I., Karpuhin V.I. Moduliruyuschie (mul'tiplikativnye) pomehi i priem radiosignalov / Pod red. I.Ya. Kremera. M.: Sov. radio. 1972.
  6. Vasil'ev K.K. Priem signalov pri mul'tiplikativnyh pomehah. Saratov: Izd-vo Saratovskogo un-ta. 1983.
  7. Akimov P.S., Bakut P.A., Bogdanovich V.A. i dr. Teoriya obnaruzheniya signalov / Pod red. P.A. Bakuta. M.: Radio i svyaz'. 1984.
  8. Artyushenko V.M., Volovach V.I. Kvazioptimal'naya diskretnaya demodulyaciya signalov na fone korrelirovannyh negaussovskih flyuktuacionnyh mul'tiplikativnyh pomeh // Radiotehnika. 2016. № 6. C. 106–112.
  9. Artyushenko V.M., Volovach V.I. Ocenka pogreshnosti vektornogo informacionnogo parametra signala na fone mul'tiplikativnyh pomeh // Radiotehnika. 2016. № 2. S. 72–82.
  10. Artyushenko V.M., Volovach V.I. Kvazioptimal'naya obrabotka signalov na fone additivnoj i mul'tiplikativnoj negaussovskih pomeh // Radiotehnika. 2016. № 1. S. 124–130.
  11. Artyushenko V.M., Volovach V.I. Modelirovanie plotnosti veroyatnosti signala i additivnogo shuma pri vozdejstvii mul'tiplikativnyh pomeh // Radiotehnika. 2016. № 12. S. 28–36.
  12. Artyushenko V.M., Volovach V.I., Shakurskiy M.V. The Demodulation Signal under the Influence of Additive and Multiplicative non-Gaussian Noise // Proceedings of IEEE East-West Design & Test Symposium (EWDTS'2016). Yerevan, Armenia, October 14–17, 2016. Kharkov: KNURE. 2016. P. 591–594.
  13. Gradshtejn I.S., Ryzhik I.M. Tablicy integralov, summ, ryadov i proizvedenij. M.: Fizmatgiz. 1963.
  14. Artyushenko V.M., Volovach V.I. Modelirovanie plotnosti raspredeleniya veroyatnosti smesi signala, podverzhennogo vozdejstviyu amplitudnyh iskazhenij i additivnoj pomehi // Radiotehnika. 2017. № 1. S. 103–110.
  15. Smirnov V.I. Kurs vysshej matematiki / 21–e izd., stereotip. T. 2. M.: Nauka. 1974.

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