Applied and Computational Mathematics 2018; 7(4): 188-196 http://www.sciencepublishinggroup.com/j/acm doi: 10.11648/j.acm.20180704.12 ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online)
Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation
Tatiana Yakovleva
Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russia Email address:
To cite this article: Tatiana Yakovleva. Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation. Applied and Computational Mathematics . Vol. 7, No. 4, 2018, pp. 188-196. doi: 10.11648/j.acm.20180704.12
Received : June 7, 2018; Accepted : August 15, 2018; Published : September 18, 2018
Abstract: The Rice statistical distribution has recently become a subject of increasing scientific interest due to its wide applicability in various fields of science and technology, such as the magnetic-resonance visualization and ultrasound diagnostics in medicine, the radio and radar signals’ analysis and processing, the optical measurements, etc. The common feature of the tasks that are adequately described by the Rician statistical model consists in the fact that the value to be measured and analyzed is an amplitude, or an envelope of the output signal which is composed as a sum of the initially determined informative component and a random noise component being formed by many independent normally-distributed summands. The efficient reconstruction of the Rician signal’s informative component against the noise background is shown to be achieved only on the basis of joint determination of both a priori unknown Rician parameters. The Rice statistical distribution possesses some peculiarities that allow solving rather complicated tasks connected with the stochastic data processing. The paper considers the issues of the strict mathematical substantiations of the Rice distribution properties that are meaningful for its efficient application, namely: the Rician likelihood function features and the stable character of the Rice distribution. There are provided the rigorous proofs of the mentioned properties. Keywords: Rice Distribution, Probability Density, Likelihood Function, Data Processing
1. Introduction Modern science is characterized by dynamic development mathematically described by the Rice distribution includes the of new mathematical techniques that provide enhancing the tasks of magnetic-resonance visualization [7, 8], the radio efficiency of data analyzing and processing. As a rule the data signals reception and processing, the tasks of the radar signals being analyzed are of random, stochastic character. That’s analysis [9], as well as the analysis of the sonar signals, the why the techniques being used at solving the tasks of data measurements of the optical medium’s properties by analysis should take into account the peculiar features of the analyzing the reflected wave [10, 11], etc. statistical distribution of the data to be analyzed. Taking into account the physical meaning of the parameters A recently increasing interest to the problem of the Rician of Rice statistical distribution one can conclude that the task of signals analysis [1-5] is connected with a wide circle of the useful, informative signal reconstruction against the noise scientific and technological tasks which are adequately background at Rician data analysis is equivalent to the task of described by the Rician statistical model. The common feature the both Rician parameters estimation. of these tasks consists in the following: the value to be The paper presents the theoretical study of the Rice analyzed is an amplitude, or an envelope of the output signal statistical distribution’s peculiarities that allow efficient that is composed by a sum of initially determined signal and a processing of Rician data by means of the so-called random noise while the noise being formed by many two-parameter analysis. Such an analysis provides the independent normally-distributed summands with zero mean, possibility of joint signal and noise calculation by measuring [6]. Such a random variable is known to obey to the Rice the sum signal. statistical distribution. The range of the problems that are 189 Tatiana Yakovleva: Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation
2. Relation to Prior Art The real xRe and imaginary xIm parts of the complex signal with amplitude x are random Gaussian values with The Rice distribution was first formulated by Stephan Rice [1] as an extension of the Rayleigh distribution well known mathematical expectations xRe and xIm , satisfying the from the classical probability theory on the case of nonzero 2 2 2 condition x+ x = A 2 , and dispersion σ . The amplitude of the initial signal. The Rice statistical model [1] Re Im x adequately describes the processes of various physical nature, random value obeys the Rice distribution with parameters ν = A σ x when the noise is formed by summing a big number of and , [1]. Obviously, the value of belongs to the ∈ ∞ normally-distributed components while the analyzed value is subset of the not-negative real numbers: x (0, ) . The ratio just an amplitude of the signal. of the Rician parameters SNR =ν/ σ characterises the A significant interest to solving a task of joint estimation of signal-to-noise ratio. both parameters of the Rice distribution has appeared in 60-th So, the Rician random variable x represents the amplitude years of the 20th century due to understanding that in of the signal with the Gaussian real and imaginary parts. The conditions of Rice distribution only the knowledge of both Rician probability density function is given by the following Rician parameters allows efficient reconstructing the initial formula: required signal against the noise background. In paper [2] the two-parameter task was considered applicably to radar signals’ x x2+ν 2 x ν P() xν, σ 2 =⋅− exp ⋅ I , 2 2 0 2 (1) (1) analysis. However, this task solving has appeared to be σ2 σ σ conjugated with considerable difficulties of both the theoretical and the computational character. Later the where I is the modified Bessel function of the first kind of simplified methods of the Rician data analysis have been 0 order zero, [16]. Here and below the following denotations are elaborated in the conditions of the so-called one-parameter ( ) approximation consisting in estimating only one of the two used: Iα z is the modified Bessel function of the first kind α unknown parameters – the signal value, in supposition that the (or the Infeld function) of the order ; xi is the signal's second parameter – the noise dispersion – is known a priori, [3] value measured as the i -th element of a sample; n is the and [4]. quantity of elements in a sample, called also a sample’s length. However, in practice the condition when the Gaussian noise The final purpose of the Rician data processing is evidently dispersion is known a priori never takes place and such a the evaluation of value A that characterizes the process supposition is a severe restriction of the one-parameter under the study and coincides with parameter ν of the Rice approach’s applicability. distribution. Therefore, the theoretical problem of joint estimation of The Rician value’s mathematical expectation and both Rician parameters without any a priori conditions has dispersion are known to be expressed by the following remained unsolved for a few decades, since the 60-th years of formulas, [17]: the 20 th century. The strict mathematical grounds for the =⋅σπ ⋅ − νσ2 2 theory of the two-parameter approach to solving the task of x/2 L 1/2 ( /2) , (2) Rician data analysis have been first provided in [5, 12-15]. The present paper is directed onto the detailed study of the π σ2=⋅2 σνσ 222 +− ⋅⋅L 2() − νσ 22 /2, (3) Rice statistical distribution’s peculiarities and provides their x 2 1/2 mathematical substantiation. where function L1/2 ( z ) is the Laguerre polynomial. σ 2 3. Theoretical Basics of the Rician Data As one can see the values x and x do not coincide with Analysis the Rician parameters ν and σ 2 , correspondingly. Nevertheless the both Rician parameters have a certain The Rician distribution is known to describe an amplitude physical sense, namely: σ 2 is s dispersion of the Gaussian of the random variable, formed by summing an initially noise distorting the initial signal, while parameter ν determined complex signal and the Gaussian noise distorting coincides with the value of the initially determined signal this signal. ν = A . Let A be a determined value that characterizes the physical From the mathematical peculiarities of the Rician process to be considered. This value is inevitably distorted by distribution noticed above it follows that the Rician signal’s the Gaussian noise created by a great number of independent mean value (2) and its dispersion (3) depend on the both noise components, while the measured and analyzed value is ν σ 2 an amplitude, or the envelope of the resulting signal. The Rician parameters: and . This means that the efficient Gaussian noise distorting the initial determined signal is reconstruction of the Rician signal against the noise characterized by a zero mean value and a dispersion σ 2 . background demands solving the task of joint evaluation of a ν σ 2 Thus the task consists in the analysis of the signal’s amplitude priori unknown parameters and . The solution of the formulated two-parameter task allows =2 + 2 . x xRe x Im reconstructing the sought for values of the initial, not-noised Applied and Computational Mathematics 2018; 7(4): 188-196 190
signal A , coinciding with parameter ν : A =ν . of the sought for useful signal ν is depicted by the straight line outgoing from the origin of coordinates. The plots in Figure 1 correspond to the constant value of 4. Parameters of the Rice Distribution: parameter σ :σ = 1, so that the points of the abscissa axes Physical Meaning and Peculiarities of correspond to the value of signal-to-noise ratio SNR =ν/ σ . Estimating Therefore, if one applies to the Rician data processing the traditional technique of the filtration by means of the data The foregoing allows substantiating the choice of the Rice averaging, then within the range of small values of the statistical model for the joint signal and noise estimation for signal-to-noise ratio the result will be just a leveling of the real solving the task in which the analyzed value is an amplitude signal values. So, the peculiarities of the Rician distribution and the noise obeys to the Gaussian statistics. The Gauss and are first of all determined by the fact that the average value of the Rayleigh distributions can be considered as its special the Rician signal does not coincide with the initial, un-noised cases at the limits of very high and very low signal-to-noise signal. Similarly, the dispersion of the Rician signal does not ratio. coincide with the dispersion of Gaussian noise that forms the By virtue of the specifics of the Rice distribution the Rician signal from the initially determined value. The Rician analysis of the Rician data demands the development of noise non-linearity is also a peculiar feature of the Rice special methods and the corresponding mathematical statistical model. These properties are inherent to the Rice apparatus. It is well known that at the Gaussian data analysis a distribution and, in contrast to the Gaussian statistical model traditional and efficient filtration tool is the data averaging. do not allow analyzing data just by simple averaging. Instead However, in contrast to the case of the Gaussian distribution the indicated peculiarities of the Rician random variable the average value of the Rician signal x does not coincide ν predetermine the necessity of the development of special with the value of the sought for useful signal . This is theoretical approach for the Rician data analysis and illustrated by Figure 1 where the average value of the processing. Riciansignal x , depending upon the Rician parameters according to (2), is depicted by the curve line, while the value
Figure 1. Illustration of the non-coincidence of the Rician signal’s averaged value x and Rician parameter ν as dependent on the signal-to-noise ratio SNR.
π In a limiting case of very high signal-to-noise ratio the σ2→ σ 2 ⋅ − x 2 , what coincides with the known formula Rician signal’s dispersion coincides with the Gaussian noise ν/ σ ≪ 1 2 σ2→ σ 2 x . dispersion: ν/ σ →∞ , what is natural taking into account the for the random variable with the Rayleigh distribution, what is interconnection of the Rician and the Gaussian distributions: not unexpected as at ν/ σ ≪ 1 the Rice distribution is being at ν/ σ ≫ 1 the Rician distribution is being transformed into transformed into the Rayleigh distribution. the Gaussian distribution with the corresponding values of Due to above indicated peculiarities of the Rice distribution parameters ν and σ . the reconstruction of the initial, un-noised signal against the In the opposite limiting case ν/ σ ≪ 1 , when the value of noise background is possible only by means of estimation of the useful signal is much less than the noise value, one can get the both Rician parameters. 191 Tatiana Yakovleva: Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation
5. Substantiation of the Maximum mathematical treatment one can get the following system of ν 2 Likelihood Technique Applicability for equations for the sought for parameters and σ , [12, 13, 15]: the Rician Data Analysis n xν As mentioned above at the Rician data analysis the ν −1 ⋅ɶ i = ∑ xi I 0, n σ 2 =2 + 2 i=1 magnitude x xRe x Im to be measured is the modulus of nν n ν (6) 21 2 2 ɶ xi the complex value with the independent real xRe and σ−()x ++ ν x ⋅ I = 0. ⋅ ∑i ∑ i 2 2 n= n = σ imaginary xIm parts being distorted by the Gaussian noise. i1 i 1 Let us consider a sample of n measurements of the signal’s amplitude x . The various types of the signal’s averaging are ( ) n ɶ d I1 z k1 k Iz() =ln Iz() = . denoted by the angular brackets x= x while the where 0 : ∑ i , dz Iz0 () n i=1 So the two-parameter task implies solving the system (6) of average value at infinite large sample’s length is denoted by two essentially nonlinear equation with two unknown the over-bar: variables ν and σ 2 . The numerical solution of this system n in its initial view is connected with the necessity of the k= 1 k xlim ∑ x i optimization by two parameters simultaneously, what n→∞ n i=1 significantly complicates the computing algorithm elaboration and increases the volume of calculations. The mathematical task of the Rician data two-parameter Obviously, the properties of the above introduced function analysis consists in computing the two mentioned parameters Iɶ( z ) determine the existence of the maximum likelihood ν and σ 2 on the basis of the sampled measurements’ data. equation's solutions, their quantity and the peculiarities. The In the present paper the maximum likelihood technique is properties of this function have been studied in detail strictly to be substantiated as an appropriate tool for solving this proved. Here some of them are presented: problem. This means the necessity to prove that the ɶ( ) maximum likelihood solution exists and is a unique one. 1. Function I z satisfies to the following differential The likelihood function L(ν, σ 2 ) is expressed as a product d 1 equation: Izɶ() =−⋅1 IzIz ɶ()() − ɶ 2 . of the probability density functions for each measurement x dz z i ɶ ( i= 1,..., n ) in the sample: 2. Function I( z ) is monotonically increasing on the (0, +∞ ) n interval . L()νσ,2= ∏ P() x νσ , 2 . (4) ɶ i 3. Function I( z ) is an upward-convex function. i=1 Taking into account these properties of function Iɶ( z ) ν σ 2 where function P( x i , ) is defined by (1). At the known and having implemented some mathematical treatment allow measured sampled data function (4) is the function of to get the following simplified system (6): ν 2 unknown statistical parameters and σ . The maximum n ⋅ ⋅ ν likelihood method consists in calculating those values of ν =1 ⋅ ɶ2 xi ∑ xi I parameters ν and σ 2 , that maximize the likelihood function n 2−ν 2 i=1 x , (7) ν σ 2 L( , ) , or, what is equivalent, maximize its logarithm. 1 σ2= ⋅()x 2 − ν 2 . Taking into account (1) and (4) results in the following: 2
n It is important that the first equation of system (7) – the νσ2= νσ 2 = lnL() ,∑ ln P() x i , equation for ν – is the one variable equation. Hence the task i=1 (5) of solving the system of two nonlinear equations (6) for two n x2+ν 2 x ν variables ν and σ has been reduced to the task of solving lnx− lnσ 2 −i + ln I i . ∑ i ⋅σ20 σ 2 just one equation (7) for only one variable ν . i=1 2 Let us consider now the issue on the existence and The maximum likelihood equations’ system for calculating uniqueness of the solution of the maximum likelihood ν unknown parameters ν and σ 2 can be obtained by equations’ system (7) for the sough-for parameters and 2 differentiating the logarithmic likelihood function and setting σ . equal to zero its partial derivatives by the both sought-for Theorem parameters. Having implemented the differentiation and some Solution of the maximum likelihood equations’ system (7) Applied and Computational Mathematics 2018; 7(4): 188-196 192
for parameters ν and σ exists and is a unique. function ν= ξ ( g ) . Having proved the existence and the Proof uniqueness of the non-zero solution of equation (10) for From (7) it obviously follows that at existing the unique variable g means thereby proving the existence and the ν solution for it is imperative that solution for the second uniqueness of the non-zero solution for parameter ν . parameter σ also exists and is a unique. That’s why just the Based on (9) one can obtain an analytical expression for the first equation of system (7) (equation for ν ) is to be reciprocal function ν= ξ(g ( ν )) , taking into account the considered here. The left-hand side of this equation is depicted positivity of the physically meaningful value of parameter ν : ν= ν by a straight like yl ( ) , while the right-hand side is a linear combinations of the functions Iɶ = I/ I : x x 2 1 0 ν()()g≡ ξ g =− + x 2 + . (12) g g 2 n ⋅ ⋅ ν 1 ɶ 2 xi y()ν = x ⋅ I , (8) r∑ i 2 2 Having analyzed the second derivative of function n = x −ν i 1 ν= ξ ( g ) it is not difficult to prove that this function is an upward-convex function. while the argument of function Iɶ in each summand of the The existence and the uniqueness of the solution of linear combination is a nonlinear function of ν . equation (10) are obviously determined by the presence and 2 2 One can see that the value (x −ν )/2 corresponds to the quantity of the points of intersection of the curves the magnitude of the signal’s dispersion and consequently depicting the right-hand side and the left-hand side of (10). changes insignificantly as a function of parameter ν because For the convenience of the further analysis let us denote the the dispersion is determined first of all by the physical nature right-hand and the left-hand sides of equation (10) of the noise. correspondingly: Let us consider the behavior of the argument of function Iɶ 1 n x in (8) in dependence of parameter ν . For this a denotation is ψ () = ɶi rg∑ xI i g , (13) introduced: ni=1 x
2 x ν ψ= ξ l ( g) ( g ). (14) g ()ν = . (9) x2−ν 2 ψ The right-hand side r ( g) of equation (10), as mentioned Then the right-hand and the left-hand sides of the first above, is a liner combination of monotonically increasing and equation of (7) can be presented as functions of a new variable ɶ xi upward-convex function I g and consequently also is a g . This variable can be shown to be a single valued function x g (ν ) of parameter ν in the interval of its physically monotonically increasing and upward-convex function that in meaningful positive values. virtue of the properties of function Iɶ asymptotically n Then the first equation of (7) can be re-written as follows: ν = = 1 approaches to value rx∑ x i . n = n i 1 1 ɶ xi ξ ()g= xI g , (10) The left-hand side of equation (10) also presents a n∑ i x ψ i=1 monotonous and upward-convex function l ( g) ,), that is analytically defined by formula (12) and asymptotically where the left-hand side is a magnitude of ν as a reciprocal approaches to value ν = x2 . function of argument g : ν= ξ( g ( ν )). l Thus the both sides of equation (10) are monotonically ν To analyze the properties of function g ( ) one can get increasing convex function outgoing from the origin of from (9): coordinates. These functions asymptotically approach to = = horizontal lines v v r and v v l correspondingly. For any 2+ν 2 x 2 2 g′ ()ν =2 ⋅ x > 0. stochastic process the condition x> () x always takes 2 (11) ()x2−ν 2 ν place. This condition means that the asymptotic value r of ψ the right-hand side r (g ) of equation (10) is less that the ′ ()ν ν ψ ( ) The positivity of derivative g means the asymptotic value l of its left-hand side l g . monotonically increasing character of function g (ν ) and So, the right-hand and the left-hand sides of equation (10) are depicted by the upward-convex curves outgoing from the ν thus the unambiguity of function g ( ) and the reciprocal origin of coordinates and having different asymptotes. It is obvious that these two monotonically increasing convex 193 Tatiana Yakovleva: Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation
curves may intersect only in the case if in the vicinity of zero and is a unique. The proved existence and uniqueness of the point g → 0 the derivative of the equation’s right-hand side solution for parameter ν obviously means the existence and the σ dψ ( g ) dψ ( g ) uniqueness of the solution for the second parameter . r exceeds the derivative of its left-hand side l . dg dg Thus, the solution of equations’ system (7) exists and is a Having conducted the non-complicated transformations unique. The theorem is proved. The solution of the equations system (7), obtained by with account of above formulas one can obtain setting the derivative of the likelihood function equal to zero, defines the extremum point of the likelihood function that may ψ () x2 2 dl g ν correspond to both the maximum and the minimum. To lim = ⋅−⋅ 1 3 . (15) → 2 g 0 dg 2⋅ x x determine the character of extremum the sign of the second derivative of the logarithmic likelihood function ∂2 Similarly for the derivative of the right-hand side of (10) in lnL()ν , σ 2 has been investigated. As its rather lengthy ∂ν 2 the vicinity of zero point one can obtain mathematical analysis cannot be presented within the frameworks of the present paper, just the resultant conclusion ψ () x2 x 4 is provided here: at the presence of a non-trivial solution dr g 3 2 lim = 1 − ν . (16) ν ≠ 0 the second derivative of the logarithmic likelihood g→0 dg 2⋅ x 2 3 ()x2 function is positive at ν → 0 . This means that at existence of the non-trivial solution of system (7) for parameter ν the trivial solution ν = 0 corresponds to just a minimum, not a Expressions (15) and (16) are written with the accuracy of ν 2 maximum of the likelihood function. By the other words the the series expansion terms of order . Calculating the trivial solution in this case is not a sought for solution of the difference of magnitudes (15) and (16) results in maximum likelihood method. So, just the non-trivial solution ν ≠ 0 of the equations system (7) always corresponds to the dψ( g) d ψ ( g ) lim r− l = likelihood function’s maximum. g→0 dg dg The situation, when the useful signal is absent (ν = 0 ), corresponds to the Rayleigh distribution as a particular case of 2 4 (17) x3ν 2 x the Rice distribution, and in this case the only solution of the = ⋅1 − . equation for parameter ν of system (7) is a trivial solution 2 x 2 2 x 2()x2 ɶ ν = 0 . With account of the series expansion of function I( z ) at small values of the argument, one can get the following Thus, the presence of the point of corresponding lines expression for the second derivative of the logarithmic intersection, i.e. the existence of the solution of (10), depends 2 likelihood function lnL (ν , σ ) for the Rayleigh distribution: upon the sign of the following magnitude: R
4 x4 ∂2 3 x lnL ()ν , σ2=−⋅ ⋅ ν 2 < 0. β =1 − , 2R 2 (20) 2 (18) ∂ν 8 2 2⋅()x2 x
The negative value of the second derivative (20) means that n n 41 4 21 2 the single extremum of the likelihood function, corresponding where x= x , x= x . ∑ i ∑ i to the trivial solution ν = 0 is a maximum of this function. n i=1 n i=1 Taking into account the known formulas for the moments of the Rician variable, [17], leads to the following expression: 6. The Stable Character of the Rice
8σ4+ 8 σν 22 + ν 4 Statistical Distribution limβ = 1 − > 0. (19) n→∞ 2⋅() 4σ4 + 4 σν 22 + ν 4 If a sum of two independent random variables obeying to the same distribution (probably, of different parameters) also This means that with the growth sample length n there exists obeys to the same distribution then such a distribution is said such a sample length at exceeding of which condition β > 0 to be a stable one. This section considers the issue of the Rice always takes place. This means the inevitable intersection of the distribution’s stable character. To prove it let us present the curves ψ ( g) and ψ ( g) . The considered curves’ intersection Rician signal as a sum of the determined signal and the l r Gaussian noise, as illustrated by Figure 2. point, in virtue of the above substantiated smoothness, Taking into account that the noise amplitude r obeys to the monotonous character and upward-convexity of both curves Rayleigh distribution while the noise phase ψ is distributed and with account of their radii of curvature, may be only a π single one, i.e. the non-zero solution of equation (10) and exists uniformly in interval (0,2 ) , one can obtain for the joint Applied and Computational Mathematics 2018; 7(4): 188-196 194
ϕ probability density function uR ( R , ) for the resulting vector’s amplitude R and phase ϕ the following formula:
ϕ ϕ = uR () R, dRd R2+ A 2 −2 RA cos ()ϕ − ϕ RdRd ϕ (21) exp − 0 2 2 2σ 2 πσ
To prove the stable character of the Rice distribution let us consider a distribution of a signal having been formed by summing of two Rician signals. In practice usually the situation takes place when the signals being summed are characterized by an amplitude and a phase. So, these signals are represented as vectors while the case of the scalar adding is just a particular case of the equal phases of the signals to be added. Figure 3. Illustrationto the proof of the Rice distribution’s stable character. The normal distribution is known to possess a quality of unlimited decomposability what means that any normally distributed random variable can be presented as a sum of an arbitrary number of normal random variables. Due to this fact it is not difficult to see that amplitude R of signal = + R R1 R 2 obeys to the Rice distribution (as well as amplitudes R and R ), because the components of its noise 1 2 = + vector r r1 r 2 are the normally distributed random variables. The next task being solved below consists in the determination of the Rician parameters for amplitude R of = + the resultant vector R R1 R 2 .
The components Rx and Ry can be presented as follows:
= + β Rx R x 0 r cos (23) = + β Ry R y 0 r sin Figure 2. Illustration of formingthe Ricisn signal R= R as a result of the impact of Gaussian noise r onto the initially determined signal of amplitude A . where =ϕ +() ϕ +∆ ϕ Let vectors R and R denote two initial signals with the Rx0 A 1cos 10 A 2 cos 10 1 2 (24) R= Asinϕ + A sin () ϕ +∆ ϕ amplitudes R1 and R2 obeying to the Rice distribution with y0 1 10 2 10 parameters A ,σ , and A ,σ , correspondingly. Let β 1 1 2 2 1 On the other hand one can obviously put down R and and β be the phases of the corresponding noise components x 2 R ϕ β β y through amplitude R and phase of the resulting vector of two signals to be added. So, phases 1 and 2 are R : uniformly distributed in interval (0,2π ) . This is illustrated by Figure 3. = ϕ Rx R cos The components of the sum vector R= R + R are as = ϕ (25) 1 2 Ry R sin follows: From (23) – (25) it follows: =ϕ + ϕ = Rx R1cos 12 R cos 2 ϕ+ β + ϕϕ +∆+ β rcosβ= R cos ϕ − ArA1cos 101 cos 12 cos() 10 r 2 cos 2 , (22) −ϕ +() ϕ +∆ ϕ R= Rsinϕ + R sin ϕ = A1cos 10 A 2 cos 10 y 1 12 2 (26) ϕ+ β + ϕϕ +∆+ β rsinβ= R sin ϕ − ArA1sin 101 sin 12 sin() 10 r 2 sin. 2 −ϕ +() ϕ +∆ ϕ A1sin 10 A 2 sin 10 ϕ ϕ= ϕ + ∆ ϕ where 10 and 20 10 - the phases of vectors R1 The values in the square brackets are the predetermined, and R , correspondingly. 2 not random ones. 195 Tatiana Yakovleva: Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation
Taking into account the Rayleigh distribution of amplitude background is shown to be achieved only on the basis of joint r of the noise vector r and the uniform distribution of its determination of both a priori unknown Rician parameters. phase β one can get: The Rice distribution properties, such as the Rician likelihood function features and the stable character of the Rice − 2σ 2 e r /2 distribution, are mathematically investigated. The uniqueness u dr= rdr (27) r σ 2 of the likelihood function maximum is strictly proved, what substantiates the applicability of the maximum likelihood where σ2= σ 2 + σ 2 . Then, from (26) as a result of technique for the Rican data analysis. Besides, the Rice 1 2 distribution is shown to be a stable one, what makes it possible non-complicated mathematical transformation it follows: to apply the Rice model in many tasks when the signal being 2=+++ 2 2 2 ∆−ϕ analyzed is formed as a sum of the Rician signals. The r R A1 A 22 AA 12 cos (28) investigated features of the Rice statistical distribution allow −()()ϕϕ −+ ϕϕ − 2R A1 cos 10 A 2 cos 20 efficient solving the task of calculation of the both Rician parameters which is of special importance at data processing Taking (28) into account and having conducted some as it is directly connected with solving the problem of the mathematical treatment one can put down the joint separation of the informative and the noise components of the probability density function u( R ,ϕ) foramplitude R and analyzed data. R phase ϕ of the summand signal as follows: Acknowledgements ɶ ϕ ϕ = uR ( R, ) dRd The work has been supported by RFBR, project 2 2 R+ A −2 RA cos ()ϕ − δ RdRd ϕ (29) N17-07-00064 within the Fundamental Research program. =exp − 0 0 2σ2 2 πσ 2 where References [1] S. O. Rice, “Mathematical Analysis of Random Noise”, Bell =2 ++ 2 ∆ ϕ A0 A 1 A 22 AA 12 cos Syst. Tech. Journal, 1944. Vol. 23. p. 282-322. ()A− A ∆ϕ (30) δ= ϕ + 2 1 [2] T. R. Benedict, T. T. Soong, “The joint estimation of signal and 0 arctg tg ()A+ A 2 noise from the sum envelope”, IEEE Trans. Inf. Theory, 1967. 2 1 Vol. IT-13. No. 3. p. 447-454.
In (30) A0 is the amplitude of resulting signal as [3] K. K. Talukdar, W. D. Lawing, “Estimation of the parameters calculated at the absence of a noise, and δ - an angle value of Rice distribution”, J. Acoust. Soc. Amer., 1991. Vol. 89. No. corresponding to the un-noised resulting signal. 3. p. 1193-1197. Comparing (21) and (39) provides a strict proof of the fact [4] J. Sijbers, A. J. den Dekker, P. Scheunders, D. VanDyck, that the sum signal obeys to the same distribution for its “Maximum-Likelihood Estimation of Rician Distribution amplitude and phase as a distribution of each of the summand Parameters”, IEEE Transactions on Medical Imaging, 1998. signals. Vol. 17. No 3. p. 357-361. Integrating (29) by phase variable ϕ allows getting the [5] T. V. Yakovleva, “Conditions of Rice statistical model following expression for the probability density function for applicability and estimation of the Rician signal’s parameters the resultant signal’s amplitude R : by maximum likelihood technique”, Computer Research and Modeling, 2014, vol. 6, no. 1, pp. 13-25.
RA −(R2 + A 2)/2 σ 2 ω = 0 0 RdR [6] T. M. Cover, J. A. Thomas Elements of Information Theory. — R ()R dR I0 e (31) σ2 σ 2 John Wiley and Sons, 2006. 776p. [7] A. J. den Dekker, J. Sijbers “Data distributions in magnetic This means that the amplitude of the resultant signal obeys resonance images: A review”, Physica Medica: European to the Rice distribution with parameters A0 and Journal of Medical Physics, 2014. Vol. 30. Issue 7. p. 723-741. σ2= σ 2 + σ 2 1 2 . [8] T. V. Yakovleva, “A Theory of Signal Processing at the Rice So, the stable character of the Rice statistical distribution is Distribution”, Dorodnicyn Computing Centre, RAS, Moscow, 2015. – 268p. strictly proved. [9] C. F. M. Carobbi, M. Cati, L. M. Millanta, “A New Procedure for Evaluating the Performance of the Site for 7. Conclusion Radiation Test or Antenna Calibration”, EMC Europe 2004, International Symposium on Electromagnetic Compatibility, The paper provides a detailed theoretical study of the Symposium Record, Eindhoven, The Netherlands, 2004. Vol. peculiarities inherent to the Rice statistical model. The 2. p. 702-706. efficient reconstruction of the Rician signal against the noise Applied and Computational Mathematics 2018; 7(4): 188-196 196
[10] T. V. Yakovleva, A. V. Kniazkov, “Speckle-noise computing by [14] T. V. Yakovleva, N. S. Kulberg, “Methods of Mathematical two-parameter analysis of the reflected light’s periodic Statistics in Two-Parameter Analysis of Rician Signals”, variation”, Optical Memory & Neural Networks (Information Doklady Mathematics. 2014. Vol. 90, No. 3. P. 1–5. Optics), 2014. Vol. 23. №4. p. 233-241. [15] Tatiana V. Yakovleva, Nicolas S. Kulberg, “Noise and Signal [11] T. V. Yakovleva, A. V. Kniazkov, “Comparison of Two Estimation in MRI: Two-Parametric Analysis of Techniques of Electro-Optical Coefficient Evaluation”, Physics Rice-Distributed Data by Means of the Maximum Likelihood Procedia, 2015. Vol. 73, p. 189 – 192. Approach”, American Journal of Theoretical and Applied Statistics, 2013. Vol. 2. No. 3. p. 67-79. [12] T. V. Yakovleva, N. S. Kulberg, “Special features of the Likelihood Function of the Rice Statistical Distribution”, [16] M. Abramovits, I. Stegun, Handbook of Mathematical Doklady Mathematics, 2014. Vol. 90. No. 1. p. 472–475. Functions, М.:Nauka, 1979. 832 p. [13] T. V. Yakovleva, N. S. Kulberg, “Mathematical Statistics [17] J. H. Park, Jr., “Moments of generalized Rayleigh distribution”, Methods as a Tool of Two-Parameter Magnetic-Resonance Q. Appl. Math., 1961. Vol. 19. No. 1. p. 45-49. Image Analysis”, Informatics and its application, 2014. v. 8. is. 3. p. 79-89.