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

 

From theoretical and experimental radar scattering matrix (SM) research in the past, it is possible to draw the conclusion that the measurement results of the scattering matrix elements depend on many factors which are not connected with scattering properties of the radar object itself. For example, such factors are: a) the choice of measuring polarization basis of the radar system (linear, circular, elliptical), b) the mutual orientation of radar and object, c) relative spatial diversity between radar 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 high information contents concerning various radar targets.

In this paper we consider an estimation of polarization invariants for the general case of an a-symmetrical scattering matrix supposing that the quadratures of its SM elements are known.

Describing the polarization properties of reciprocal radar objects and media (within the framework of the Sinclair scattering matrix concept) the Huynen-Euler group invariants is widely used (see, for example [6]). Their main advantage is that the given invariants group may be used not only for estimation of the polarization properties of stable (in time) targets, but also for fluctuating 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].

The scattering matrix can be written as [6]

 

(4)

 

so that the complex eigen values take the form

 

,

(5)

 

The symmetric scattering matrix can, therefore, 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.

For the monostatic radar case, the backscattering matrix (BSM) 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 asymmetrical () in the monostatic case too [1]. In particular, this may occur in cases in which in a bounded spatial volume strong magnetic or electric field strengths caused by exterior energy sources are present,.

The possibility of the existence of real objects, which polarization properties are described by asymmetrical () 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 asymmetrical 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 as the result of the object's non-reciprocal properties manifestation. We decompose this scattering matrix with the use of an orthogonal system of Pauli spin matrices supplemented by the unitary matrix,

 

, , , ,

(9)

 

so that 

(10)

 

In such a case, the decomposition factors are written as follows

 

(11)

 

where is the sum of the diagonal elements of matrix A.

As a result of the given decomposition, the scattering matrix S is written as

 

(12)

 

It is easily seen that the first three members of the decomposition (12) form a symmetric matrix

 

(13)

 

and the fourth member of the decomposition

 

,

(14)

 

is an antisymmetric matrix with the weighting coefficient

 

(15)

 

The initial scattering matrix can thus be represented as the sum of symmetric S^(s) and antisymmetric S^(a) matrices

 

(16)

 

We now want to find the scattering matrix (8) in the polarization basis with parameters e¢, q¢

 

,

(17)

where 

.

(18)

 

Substituting (16) in (17), we obtain

 

.

(19)

 

The second term becomes 

,

 

One can show that for any unitary transformation matrix (18) the following equality exists:

 

(20)

 

Therefore, the expression (19) may be written as

 

.

(21)

 

Since the ellipticity and orientation angles are chosen arbitrarily, it is possible to draw the conclusion that the second item in (21) with the proportional factor will not depend on parameters e¢, q¢ in the expression for the transformation matrix U¢. In other words, the difference of the off-diagonal elements of the scattering matrix will be value invariant to the polarization basis of the radar. Thus, the parameter will be determined by the non-reciprocal properties of the radar object only and can be considered as an objective characteristic of this object.

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

 

,

(22)

 

and the initial (asymmetrical) scattering matrix in this basis will be written as

 

.

(23)

 

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

.

(24)

 

As we consider the monostatic radar case, in which the same antenna is used for radiation and reception of radar signals, then expression (24) can be re-written in the following form:

 

.

(25)

 

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

.

(26)

The vector is represented as

,

(27)

where

.

 

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

.

By taking (14) into consideration, we derive

 

.

(28)

 

For the last term in (28) it is possible to show that for any e², q² values the following equality will take place

 

.

(29)

 

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

 

.

(30)

 

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

,

(31)

 

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

In other words, in describing a non-reciprocal object the polarization invariants e₀, q₀, , , and, therefore the complete group of 6 Huynen-Euler invariants, will play the same role as in the reciprocal object's case with symmetric scattering matrix. To describe the non-reciprocal properties of radar objects with asymmetrical scattering matrix, it was suggested by V. Khlusov [7] to use the complex non-reciprocity factor

 

,

(32)

where is Euclidean norm of the scattering matrix (8). Knowing that the difference of the off-diagonal elements and the norm of the scattering matrix do not depend on the polarization basis choice of radar system, the value is also polarization invariant and will be determined by the non-reciprocal properties of the radar object only.

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

 

(33)

where

,

,

.

 

We now define the ratio of the squared norms of the antisymmetric and symmetric parts of the matrix S, as

.

(34)

With

,

expression (34) can be written in the form

and we learn that

.

(35)

 

Thus, the physical sense of the non-reciprocity factor is that the squared module of this value contains information on the ratio between the RCS of the non-reciprocal part of the radar object and its full RCS. It is obvious that for all reciprocal radar objects the factor will be equal to zero, and for objects with arbitrary polarization properties (partially non-reciprocal objects) the module of is in the interval (0; 1).

In the general case the non-reciprocity factor value is complex and the following interpretation of this value is given in [7]. The author [7] considers as the difference of the absolute phases of the symmetric and anti-symmetric parts of the scattering matrix. This value can be represented, for example, as a spatial diversity of “reciprocal” and “non-reciprocal” parts of the scattering matrix, just as the arguments difference of the eigen values , of the symmetric scattering matrix can be treated as spatial diversity of the elements of radar object's physical model (for example, orthogonal dipoles in socalled “two-dipole” models [2]) along the line of sight.

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 anti-symmetric parts of the scattering matrix (-180° £ h< 180°).

 

Thus, the non-reciprocity factor can be represented as

 

.

(36)

 

In summary it should be noted that the suggested system of representation of arbitrary scattering matrix with the use of 8 invariants

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

(37)

 

describes the polarization properties of radar objects in an optimum way, from the point of view of an irredundant number and also clear physical interpretation of these parameters.

Let the radar's polarization basis be linear and its coordinates system coincides with Cartesian system XOY, the scattering matrix of a radar object can now be written in the radar's basis as

 

,

(38)

 

with . We assume that all 8 quadratures (4 in-phase and 4 quadrature components) of the scattering matrix can be simultaneously measured by the radar, meaning:

 

(39)

 

where I_k is in-phase and Q_k is quadrature components (k = 1, ...,4) of the corresponding elements of the scattering matrix.

When the quadratures values of all SM's elements of non-reciprocal objects are known, we may find the analytic forms for the complete invariants group

 

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

 

In this case, the initial scattering matrix S_xy can be presented as the sum of the symmetric and antisymmetric matrices:

 

(40)

 

where the corresponding matrices elements are written as

 

, ,

(41)

 

, .

(42)

 

We first focus our attention to the analytic forms for the polarization invariants describing the properties of the symmetric part of the scattering matrix. For this purpose we write for the quadratures of the matrix

(43)

 

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

 

(44)

 

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 substitution 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 (43):

 

,

,

,

,

,

.

 

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:

 

(45)

 

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

The orientation angle “q₀”

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

 

.

(46)

Accordingly, from the equation for we find the expression

 

(47)

 

which we substitute in equation (46) and we obtain

 

.

(48)

From the equation for we write

 

.

(49)

The equation for learns that

.

(50)

We obtain the expression for sin(2q₀)

 

.

(51)

With considering the equality

,

(52)

 

following from the first two equations of (45), the expression for the ratio of sin(2q₀) (51) and cos(2q₀) (48) can be represented as: 

.

This expression is written using the quadrature elements of the SM's symmetric part directly. Taking into consideration the expressions (43) connecting these quadratures with the elements' quadratures of the initial matrix, we can re-write the last equation in its final form, namely

 

.

(53)

 

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

,

(54)

 

It is easy to see that the orientation angle is a function of all 8 quadratures of the SM's elements, i.e. .

The ellipticity angle “e₀”

From the first equation of (45) the expression

 

,

(55)

is substituted into the 4^th and 6^th equation 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

 

,

(56)

.

(57)

 

Substitution in (56)

,

(58)

 

as derived from (57), and making a simple transformation, we get

 

.

(59)

 

Considering (43), the expression (59) well take its final form:

 

.

(60)

 

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

 

,

(61)

 

This functions contains 6 quadratures of the initial scattering matrix and the trigonometric function of the doubled orientation angle 2q₀, i.e.

The maximum polarization (the eigen value module) “m”

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

 

(62)

 

and the determinant module

 

(63)

 

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

Using these expressions, we find the maximum polarization value as

 

,

(64)

 

where

.

(65)

 

It is obvious that the “span” (62) and squared module of the determinant (63), which are used in the definition of “m”, are elements quadrature functions of the initial scattering matrix

 

,

(66)

 

.

(67)

 

Thus, is a function of all 8 quadratures of the BSM elements.

The characteristic angle “g”

Since the characteristic angle value is connected with the eigen values modules via

 

,

 

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

 

,

 

the given invariant is easily found from the equation

 

(68)

 

and is, therefore, also a function of all 8 quadratures of the initial scattering matrix, i.e.

.

 

The skip angle “n₀”

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

.

(69)

 

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

 

.

(70)

 

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 can be derived. For example, the determinant D_a can be written in the form

 

.

 

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

, , , ,

(71)

 

where, by taking (43) into consideration,

 

, , , , , .

(72)

 

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

 

,

 

the set of equations (70) has the unique solution:

 

, , , .

 

Then, according to (69), the skip angle value can be written as

 

,

or

.

(72)

 

After substitution of (71) in (72) and some transformations, parameter n₀ becomes

(73)

where

(74)

and

,
,
,	(75)
,
.

 

The absolute phase “f₀”

The determinant of the symmetric scattering matrix

 

(76)

 

is one of the SM's polarization invariants, i.e. does not depend on the radar basis choice. In this repect not only the determinant 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

 

.

(77)

 

Thus, the absolute phase value of the symmetric part of the scattering matrix can be found from

 

(78)

(cf. [8]), or

.

(79)

 

Since the argument of a complex number can only be found unambiguously in the angular interval (0; 2p) or (-p; +p), the f_o value (as a half of this angle) is reduced to the interval (0; p) or (-p/2; +p/2). Suppose that the f_o value is determined, according to (79), in the interval (0; p). In such a case, the given formula gives only the true estimate of the absolute phase for positive f₀. Therefore, to find the correct value of the absolute phase in the interval (-p; +p), it is necessary to calculate the additional angle (f_o-p) and to check which of the values is true. Such verification may be carried out as a result of substitution of both values f_c and (f_c-p) in the expression for one of the quadratures 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

 

,

(80)

where

,

(81)

 

and the value G in (80) equals

 

,

(82)

where

,

.

 

The non-reciprocity angle “z₀” and absolute phases difference “h₀”

In this section we pay attention to te 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 phases difference of the symmetric and antisymmetric parts of the scattering matrix. These invariants are found in the form:

 

and .

(83)

 

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

 

.

(84)

 

The value of the absolute phase difference of the symmetric and antisymmetric parts of the scattering matrix is easily found as

 

.

(85)

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.

• The suggested invariants group are based on the Huynen-Euler invariants supplemented by additional two invariants describing the non-reciprocal properties if an arbitrary radar object.

References

1. Boerner W.M., Yamaguchi Y. «A State-of-the-Art Review in Radar and its Applications in Remote Sensing», IEEE Aerosp. and. Electron. Syst. Mag., N 5, pp. 3-6, 1990.

2. Kanarejkin D.B., Pavlov N.F., Potekhin V.A. Polarization of Radar Signals, Moscow, Publ. ”Sovetskoe Radio”, 440 p., 1966

3. Bogorodskij V.V., Kanarejkin D.B., Kozlov A.I. Polarization of Scattered and Self Radioemission of the Earth’s Covers, Leningrad, Publ. “Gidrometeoizdat”, 280 p., 1981

4. Luneburg E. «Radar Polarimetry: A Revision of Basic Concepts». Int. Workshop on Direct and Inverse Electromagnetic Scattering, Marmara Research Center, Gebze-Turkey, 24-30 Sept., 1995.

5. Luneburg E. «Canonical Bases and Huynen Decomposition», Proc. of 3rd International Workshop on Radar Polarimetry (JIPR-3, 95), IRESTE, Univ.-Nantes, France, pp.75-83, 1995.

6. Krogager, E. and W-M. Boerner, «On the importance of utilizing complete polarimetric information in radar imaging and classification», AGARD Symposium: Remote Sensing - A Valuable Source of Information, Toulouse, France, 1996 April 22-25, AGARD Proc., (528 pp.), pp. 17.1-7.12.

7. Khlusov V.A. «Parametrisation of Backscattering Matrix of Non-Reciprocal Media», Optics of Atmosphere and Ocean, Vol. 8, No. 10, pp. 1441-1445, 1995

8. Krogager, E., Z.H. Czyz, «Properties of the Sphere, Di-plane and Helix Decomposition», Proc. of 3rd International Workshop on Radar Polarimetry, IRESTE, University of Nantes, France, pp.106-114, Apr. 1995