ALGORITHMS OF ESTIMATING THE COMPLETE GROUP OF POLARIZATION INVARIANTS OF THE SCATTERING MATRIX (SM) BY MEASURING ALL SM ELEMENTS

V. Karnychev ^(1,2), L. Ligthart ⁽²⁾, G. Sharygin ^(1,2)

⁽¹⁾ TUCSR, Lenin Avenue, 40, Tomsk, 634050, Russia, guest scientists at IRCTR

⁽²⁾ International Research Centre of Telecommunications-transmission and Radar (IRCTR), TU-Delft, Mekelweg 4, 2628CD Delft, The Netherlands

Abstract - The estimation procedure of polarization invariants of the backscattering matrix in HV-basis is considered for radar observation of arbitrary non-reciprocal object case. Two additional polarization invariants are introduced further to the well known six Huynen-Euler invariants. These new invariants (non-reciprocity angle and difference of absolute phases of symmetrical and antisymmetric parts of the scattering matrix) describe non-reciprocal properties of object itself. Supposing the simultaneous measurement of all 8 quadratures of the scattering matrix's elements in monostatic radar case, the closed form expressions for calculation of 8 polarization invariants are found. The derived expressions are the starting point for complete estimation of the polarization properties of radar objects with arbitrary properties. The given approach can be used for research of various polarization effects in remote radar sensing of artificial and natural objects, and also for simulation of polarization measurement processes and errors' estimation caused by scattering matrix elements' measurements in different instants.

Index Terms - radar polarimetry, polarization invariants, asymmetric backscattering matrix, non-reciprocal object, Huynen-Euler parameters, non-reciprocity parameters, HV-basis, scattering matrix elements' quadratures

I. Introduction

There is a number of theoretical works published in the West, in which principles of the radar polarimetry and optimization procedures [11], the optimum polarizations finding [8, 9, 10], research of null polarizations of radar objects [12], are extended on bistatic case or monostatic radar of non-reciprocal objects. In several papers of Russian polarimetrists, the results of investigations in the field of monostatic radar of non-reciprocal objects were also published. For example, the different aspects of optimum parametrization of asymmetrical scattering matrices for monostatic radar case, an analysis of such polarization characteristic of arbitrary radar object (medium) as "non-reciprocity factor" were considered in [13, 14]. In more details the physical aspects of backscattering mechanism by partly non-reciprocal objects were analyzed in [15, 16]. In particular, an experimental test-bed, that simulates the radar channel with arbitrary polarization properties, was described in [15]. In this work, application capabilities of a controllable radar reflector with partly non-reciprocal properties were also estimated.

Evidently, the paper [1] should be considered as one of the first mentions of the scattering matrix (SM) measurement of radar object with non-reciprocal polarization properties. The inequality of the measured off-axis scattering matrix elements for some regions in electric storm has been explained by presence of formation in which very strong electrification prevailed. As another experimental confirmation of non-reciprocal objects' existence can be considered [15], in which an absolutely non-reciprocal reflector's design was described at the first time. Moreover, this work includes an analysis of possibility to remotely detect spatial regions or ground surface areas, where magnetic field and/or para- or ferromagnetics are present.

It is known that polarimetric analysis of radar objects having arbitrary polarization properties can be performed with the use of coherent decomposition theorems. The Pauli spin matrices approach for decomposition of asymmetrical scattering matrix is widely used (see [11]). In this case, the backscattering matrix can be presented in the form:        

,

 

where a, b, c and d are all complex values.

The investigation of radar objects with relation to possible backscattering mechanisms allows to connect these values with four elementary deterministic point targets. In such a case, a, b, c and d correspond to sphere or plane surface, diplane, diplane rotated on 45°, and scatterer that transforms every incident polarization into its orthogonal state, correspondingly.

In the Cameron approach [18], the scattering matrix is also decomposed with using the Pauli matrices. In this case, the matrix is firstly decomposed into reciprocal and non-reciprocal components, and then the reciprocal component is decomposed into two further components, both of which have linear eigenpolarizations.

In [14], the author decomposes an arbitrary scattering matrix with the use of the Pauli matrices, reducing it to the sum of symmetrical matrix and antisymmetric matrix weighted by complex factor. With this approach, a radar object is described by a set of invariant parameters (, , q, t), which characterize the symmetrical component, and also by polarization non-reciprocity factor that describes the objects' non-reciprocal properties. The module is considered to be the ratio between radar cross-section (RCS) of non-reciprocal part and full object's RCS, and is interpreted as a spatial diversity of the reciprocal and non-reciprocal parts of the scattering matrix.

From available results of SM's theoretical studies and experimental measurements, it is possible to conclude that the SM's elements depend on many factors that are not connected with the scattering properties of observed radar objects. For example, such factors are

• choice of polarization basis (linear, circular, elliptical) realized in radar system;

• mutual orientation of a radar and object;

• spatial diversity between radar system and object, etc.

For this reason, the measurement and investigation of polarization invariants of the scattering matrix formed a fundament in the theory and practice of radar polarimetry. These invariants are measurable values, which characterise polarization properties of radar objects themselves and do not depend on the polarization basis implemented in the radar system. Some polarization invariants measured in Russia and in the West have proved to consist a high information contents concerning various radar targets.

In this paper we consider an estimation of polarization invariants for the general case of an asymmetric scattering matrix supposing that all quadratures of its SM's elements are known. In Section II we make a short review of the traditional polarization invariants. Then we analyse the asymmetric matrix case in Section III. In Section IV we present the non-reciprocity parameters supplementing the Huynen-Euler invariants group. Sections V and VI are devoted to derivation of the closed form expressions for the complete group of 8 invariants of the scattering matrix.

II. Huynen-Euler Polarization Invariants

Describing the polarization properties of reciprocal radar objects and media (within the framework of the Sinclair scattering matrix concept) the Huynen-Euler group invariants (see, for example [6]) are widely used. Their main advantage is that the given invariants group may be used not only for estimation of the polarization properties of time stable targets, but also for time-fluctuating random objects. In the latter case, it is quite acceptable to assume that the time correlation of the SM's elements of the fluctuating objects is much longer than the carrier period of the radiated signal. This allows us to speak about an opportunity of «instant» reduction of the scattering matrix to a diagonal form [3].

Without loss of generality, let us assume that the measuring polarization basis of the radar system is linear. Besides, let the radar's coordinate system coincides with Cartesian coordinates XOY. The scattering matrix of a radar object can then be written in the radar's polarization basis as:

 

,

(1)

where

(2)

 

is the radar object's scattering matrix in the eigen polarization basis;

, are complex eigen values of the scattering matrix;

 

,

(3)

 

are transformation matrices from the object's eigen basis in the linear basis of the radar [4];

superscript denotes the transposition of a matrix.

The scattering matrix can be written as [6]

         

(4)

         

so that the complex eigen values take the form

 

,

(5)

 

Therefore, the symmetric scattering matrix can be described by 6 independent Huynen-Euler parameters:

m	the maximum polarization; this value is the largest possible response from the radar target. This response is received under transmission of the optimum polarization connected with the largest eigen value, i.e. the "m" value is equal to module of the 1st eigen value;
f	the absolute phase of the scattering matrix (-180°£f<180°);
q	the orientation angle of the object's eigen basis relatively the radar's coordinates (-90°£q<90°);
e	the ellipticity angle of the object's eigen basis (-45°£e<45°);
n	the skip angle (-45°£n<45°); . If the response from the object is caused by scattering mechanisms with an even number of reflections (bounces) the parameter n is equal to 45°. In Russian polarimetry, the "phase shift" value (equal to the argument difference of the complex eigen values) is used instead of the n parameter;
g	the characteristic angle (0°£g£45°) or the "polarizability" angle; radar targets with g=45° do not change the polarization of the transmitted signal whereas targets with g=0° will completely determine the polarization state of the reflected signal.

 

The angles q and e determine the ellipticity and orientation of the larger axis of the polarization ellipse of the electromagnetic wave. If a radar object is irradiated by the wave with such parameters, then the signal power received in a single-channel radar will be maximum. This means that the reflection factors for the transmission case of two orthogonal waves with the ellipticity angles e and -e and orientation angles q and q±p/2, correspondingly, will be proportional (to a precision of phase factor) to the eigen values , of the object’s scattering matrix.

III. Asymmetric Matrix Case

For the monostatic radar, the backscattering matrix is considered to be symmetric, and gives equality of its off-diagonal elements . As a rule, the asymmetry of the scattering matrix becomes apparent in bistatic configurations. However, there are experimental data which prove that the scattering matrix may be asymmetric () in the monostatic case too [1]. In particular, this may occur in cases in which strong magnetic or electric field strengths caused by exterior energy sources are present in a bounded spatial volume.

The possibility of the existence of real objects, which polarization properties are described by asymmetric () scattering matrices in the monostatic radar sensing case, issues the challenge of widening the mentioned group of the Huynen-Euler parameters by additional independent polarization invariants.

It is known that an asymmetric scattering matrix cannot be diagonalized by the following congruent transformation        

(6)

 

which is typical for symmetric SMs. Here the matrix

 

(7)

 

is the unitary unimodular transformation matrix [5] according to Takagi factorization.

In order to set up the complete group of polarization invariants we start from the consideration that the scattering matrix of an arbitrary radar object in the Cartesian basis is known and presented by four complex values           

.

(8)

 

In this case, the inequality is considered as manifestation of the object's non-reciprocal properties.

From elementary matrix theory it is known that any asymmetric matrix can be decomposed into a symmetric (S^(s)) and a skew-symmetric (S^(a)) component. Thus the initial scattering matrix (8) is written as follows:

 

,

(9)

where 

,	(9a)
.	(9b)

 

It is easily seen that these matrices take the form

 

,

(10)

and 

,

(11)

where Δ is the complex weighting coefficient

 

.

(12)

 

Let us find the scattering matrix (8) in the polarization basis with parameters e¢, q¢

 

,

(13)

 

where the transformation matrix is the following

 

.

(14)

 

Substituting (9) in (13), we obtain

 

,

(15)

 

where the second term becomes

 

.

 

One can show that for any unitary transformation matrix (14) the following equality exists:

 

.

(16)

 

Therefore, expression (15) may be rewritten in the general form as

 

.

(17)

 

Since the ellipticity and orientation angles are chosen arbitrarily, it is possible to conclude that the second item in (17) with the proportional factor will not depend on e¢, q¢ parameters in (14) for transformation matrix U¢ . In other words, the difference of the off-diagonal elements of the scattering matrix will be invariant value to the radar's polarization basis. Thus, the parameter will be only determined by the non-reciprocal properties of radar object and can be considered as an objective characteristic of this object.

4. The Non-Reciprocity Parameters

Suppose that the unitary matrix U¢=U₀ and the given values of the ellipticity e¢ and orientation q¢ angles coincide with the eigen polarization basis (e₀, q₀) parameters of the symmetric part S^(s) of the scattering matrix. In this case, the matrix S^(s) is diagonalized into

      

,

(18)

 and the initial (asymmetric) scattering matrix in this basis can be presented as

 

.

(19)

        

Let unit vectors and denote the polarization states of receiving and transmitting antennas, correspondingly, with . Then, the received signal voltage at the receiving antenna terminals is written as

 

.

(20)

 

As we consider the monostatic radar configuration, in which the same antenna is used for radiation and reception of radar signals, then expression (20) can be rewritten in the following form:

 

.

(21)

 

In this case, the normalized power transfer equation in single-channel monostatic system becomes

 

.

(22)

         

The vector is represented as         

,

(23)

 

where transformation matrix is the following

 

.

 

The signal scattered by a non-reciprocal object with scattering matrix (8) and received in the single-channel system becomes

 

.

 

By taking (11) into consideration, we derive

 

.

(24)

         

For the last term in (24) it is possible to show that for any e², q² values the following equality will take place

 

.

(25)

         

It means that the signal scattered by a non-reciprocal object and received in a single-channel system will depend only on the «symmetric» part of the object's scattering matrix

 

.

(26)

 

The value will take maximum (in power) values only when the polarization state of the transmitting-receiving antenna coincides with the eigen polarizations of the symmetric part S^(s): where

 

,

(27)

 

and e₀, q₀ are the ellipticity and orientation angles of the eigen basis of the matrix S^(s).

        In other words, the polarization invariants e₀, q₀, , (i.e. 6 Huynen-Euler invariants) will play the same role in describing a non-reciprocal object as in the general reciprocal case. For description of non-reciprocal properties of radar objects with asymmetric scattering matrix, V. Khlusov [14] introduced into practice a complex non-reciprocity factor        

,

(28)

 

where ||S|| is Euclidean norm of the scattering matrix (8)

 

.

 

Since the difference of the off-diagonal elements () and do not depend on the polarization basis, the value is also polarization invariant and will be only determined by the non-reciprocal properties of radar object.

It is not difficult to show that the following equality follows from expression (16)

         

,

(29)

 

Where 

,
,
.

 

We now define the squared norm ratio of the antisymmetric part and the same value for the matrix S itself, as

 

.

(30)

 

With

,

 

expression (30) can be rewritten in the form

 

,

 

and we learn that 

.

(31)

Thus, the physical sense of the non-reciprocity factor is that the squared module of this value contains information on the ratio between the radar cross-section (RCS) of the non-reciprocal part and full RCS of the radar object. It is obvious that for all reciprocal objects , while for partially non-reciprocal objects Î (0; 1).

In the general case, the non-reciprocity factor value is complex. One of the possible interpretation of is given in [14]. The author considers this value as the difference of the absolute phases of the symmetric and antisymmetric parts of the scattering matrix. This value can be represented as a spatial diversity of "reciprocal" and "non-reciprocal" parts of the scattering matrix. Just as of the symmetric scattering matrix can be treated as spatial diversity of orthogonal dipoles along the line of sight in so called "two-dipole model" [2] of a radar object.

Since the complex factor contains full information concerning the non-reciprocal properties of arbitrary radar objects, the group of 6 Huynen-Euler invariants may be supplemented by two additional polarization invariants having angular dimension:

z	the non-reciprocity angle (0°£z£45°) equal to arctangent of the module; the value z=0° describes the case of radar sensing of reciprocal objects with symmetric scattering matrix, whereas objects with z=45° will be completely non-reciprocal;
h	the difference of absolute phases of the symmetric and antisymmetric parts of the scattering matrix (-180°£h<180°).

       

Thus, the non-reciprocity factor can be represented as:

         

.

(32)

         

From the point of view of an irredundant number of polarization invariants and their clear physical interpretation, it should be noted that the suggested complete group of 8 polarization invariants describes the polarization properties of arbitrary radar objects in an optimum way

        

m, f, q, e, n, g, z, h.

(33)

V. Derivation of the Complete Invariants Group

Let the radar's polarization basis be linear one and its coordinates system coincides with Cartesian system XOY. In this basis the scattering matrix of a non-reciprocal radar object is written as

 

,

(34)

where .

We assume that all 8 quadratures of the SM's elements were simultaneously measured by the radar, meaning:

 

(35)

 

where I_k is in-phase and Q_k are the quadrature components (k = 1, ...,4) of the corresponding elements. In such a case, we may find the analytic forms for the complete invariants group (33). Let us present the initial scattering matrix S_xy as the sum of the symmetric and antisymmetric matrices:

 

,

(36)

 

where the corresponding elements of and are written as follows:

 

, ,	(37)
, .	(38)

         

We first focus our attention to the analytic forms for the polarization invariants describing the properties of the symmetric part of the scattering matrix. Toward this end, we denote the matrix quadratures as

         

(39)

 

To find the solution, we need to present the elements of the SM's symmetric part as functions of the Huynen-Euler invariants. In

 

(40)

 

we, therefore, substitute the expression for the symmetric part of the scattering matrix in the object's eigen basis,

 

,

use the transformation matrices

 

, ,

 

and the expressions for the complex eigen values

 

, ,

 

where , .

After such substitutions the resultant expressions for the matrix elements have a quite complicated form. To simplify the expressions, we introduce:

 

, ,
, .

 

With using these shortenings, we can write for the in-phase and quadrature components (39):

 

,
,
,
,
,
.

  

This set of 6 equations can be reduced into a more convenient form, in order to derive expressions for the invariants. For this purpose it is necessary to find sums and differences of the real and imaginary parts of the diagonal elements, and also the doubled real and imaginary parts of the off-diagonal element of the symmetric matrix. After simple transformations this new set of equations takes the form:

 

.

(41)

 

The solution for the invariants of the symmetric scattering matrix with using this set of equations is given in the following sub-sections.

V.1. Orientation Angle

From the equation for the difference of the in-phase quadratures we write an expression for cosine of the doubled orientation angle in the eigen polarization basis

          

.

(42)

Accordingly, from the equation for we find the expression

 

,

(43)

and its substituting in (42) gives the result

 

.

(44)

From the equation for we write

         

.

(45)

The equation for learns that

 

.

(46)

We obtain the expression for sin(2q₀)

 

.

(47)

With considering the equality         

,

(48)

 

following from the first two equations of (41), the expression for the ratio of sin(2q₀) (47) and cos(2q₀) (44) can be represented as: 

.

 

This expression is written using the quadrature elements of the SM's symmetric part directly. Taking into consideration the expressions (39), which connect these quadratures with the elements' quadratures of the initial matrix, we can re-write the last equation in its final form, namely

 

.

(49)

     

Therefore, the orientation angle of the eigen polarization basis of an arbitrary radar object is written as

         

.

(50)

 

It is easy to see that the orientation angle depends on all 8 quadratures of the SM's elements, i.e. q₀=f_q(I₁, Q₁, I₂, Q₂, I₃, Q₃, I₄, Q₄).

V.2. Ellipticity Angle

From the first equation of (45) the expression         

(51)

 

is substituted into the fourth and sixth equations of this set of equations, namely, in the equation for the difference of the imaginary parts of the diagonal elements of the SM's symmetric part and in the equation for the doubled imaginary part of the off-diagonal elements. As a result, these equations can be written as

 

,

(52)

.

(53)

Substitution in (52) 

,

(54)

 

as derived from (53), and making a simple transformation, we get

 

.

(55)

         

Considering (43), the expression (55) will take its final form:

         

.

(56)

 

The ellipticity angle of the eigen polarization basis of an arbitrary radar object becomes

 

.

(57)

 

This function contains 6 quadratures of the initial scattering matrix and trigonometric functions of the doubled orientation angle 2q₀, i.e. e₀=f_e(I₁, Q₁, I₂, Q₂, Q₃, Q₄, 2q₀).

V.3. Maximum Polarization

It is known (see, for example, [3]) that the sum of squared modules of the symmetric matrix elements

 

(58)

 

and the determinant's module

 

(59)

 

are polarization invariants, which do not depend on the radar basis.

Using these expressions, we find the maximum polarization value as

        

,

(60)

where 

.

(61)

 

It is obvious that the “span” (58) and squared module of the determinant (63), which are used in the definition of “m”, are functions of the initial SM elements' quadratures

 

,

(62)

 

.

(63)

 

Thus, the maximum polarization depends on all 8 quadratures of the SM's elements, i.e. m₀=f_m(I₁, Q₁, I₂, Q₂, I₃, Q₃, I₄, Q₄,).

V.4. Characteristic Angle

Since the characteristic angle value is connected with the eigenvalues' modules via

 

,

and the module of the 2^nd complex eigenvalue can be written as

 

,

 

then the given invariant is easily found from the equation

 

(64)

 

and is, therefore, also a function of all 8 quadratures of the initial scattering matrix, i.e. g₀=f_m(I₁, Q₁, I₂, Q₂, I₃, Q₃, I₄, Q₄,).

V.5. Skip Angle

Since the arguments of the complex eigen values (with taking the absolute phase into consideration) are equal to         

and ,

    

the parameter n₀ can be found as the difference of these arguments

 

,

or

.

(65)

 

Supposing the orientation and ellipticity angles (q₀, e₀) are known, it is possible to make the following set of 4 equations         

.

(66)

 

To solve this equation set by the Kramer's method, it is necessary to find its determinant

 

.

 

The determinants D_a, D_b, D_d and D_f, can be derived from D by replacement of the corresponding columns by the column of absolute terms. For example, the determinant D_a has the form

 

.

 

After some transformations and simplifications, we present the determinants in their final form as

 

,
, ,	(67)
, ,

 

where, by taking (39) into consideration,

 

, ,
, ,	(68)
, .

 

If the determinant D is not equal to zero, i.e.

        

,

 

the equations set (66) has the unique solution:

 

, , , .

         

Then, according to (65), the "preliminary" value of the skip angle can be written as

 

,

or

.

(69)

 

After substitution of (68) in (69) and some transformations, parameter n₀ becomes

 

(70)

where 

(71)

and

,
,
,	(72)
,
.

 

V.6. Absolute Phase

The determinant of the symmetric scattering matrix

 

(73)

 

is one of the SM's invariants, i.e. does not depend on the radar polarization basis. In this respect not only the determinant's module is an invariant value, but also its argument. With

 

, ,

 

it is easy to write for the determinant in the radar object's eigen basis as

 

.

(74)

         

Thus, the "preliminary" value of the absolute phase of the SM's symmetric part can be found from

 

(75)

(cf. [8]), or

.

(76)

 

Since the argument of a complex number can only be found unambiguously in the angular interval (0; 2p) or (-p; +p), the value (as a half of this angle) is reduced to the interval (0; p) or (-p/2; +p/2). Suppose that is determined in the interval (0; p) in accordance with (76). In such a case, expression (76) gives the true estimate of the absolute phase for positive values only. Therefore, to find the correct value of the absolute phase in the interval (-p; +p), it is necessary to calculate the additional angle () and to check which of the values is true. Such verification may be carried out as a result of substitution of both values and () in the expression for a quadrature of the SM's symmetric part, and execute a comparison of the calculated value and initial one. By choosing the in-phase quadrature , it is possible to show that the absolute phase may be found according to the rule

,

(77)

 

The values of and G in (77) equal to

 

,

(78)

and

,

(79)

where

,
.

VI. Non-Reciprocity Angle and Absolute Phases Difference

In this section we pay attention to the non-reciprocal aspects. We summarize the analytic forms for the polarization invariants describing properties of the antisymmetric part of the scattering matrix. The polarization invariant describing the non-reciprocal properties of an arbitrary radar object is written as

 

,

 

where z is the non-reciprocity angle and h is the absolute phase’s difference of the symmetric and antisymmetric parts of the scattering matrix. These invariants are found in the form:

 

and .

(80)

 

Taking into consideration that the module of the non-reciprocity factor is by definition

          

,

 

where and , then the expression for the non-reciprocity angle can be written as

 

.

(81)

         

In turn, the difference of absolute phases of the SM's symmetric and antisymmetric parts is easily found as

.

(82)

VII. Conclusions

The analysis made in the paper has shown that there exists a possibility to estimate the complete group of the polarization invariants by the results of simultaneous measuring all 8 quadrature components of the scattering matrix's elements. The suggested parameters group is based on the Huynen-Euler invariants supplemented by additional two invariants describing the non-reciprocal properties of an arbitrary radar object. The derived expressions (50), (57), (60), (64), (71), (77), (81), and (82) form the algorithmic basis for estimating the complete group of polarization invariants. It should be noted that seven of eight invariants have the angular dimension that is very convenient for analysis and comparison of radar objects data. The authors intend to use the approach for investigations of the polarization properties of radar objects with using the experimental measurement results. It is possible to apply the derived expressions of the polarization invariants in statistical simulation and also for evaluation of the errors caused by non-simultaneous measuring the SM's elements.

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