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Polarimetry with Phased Array Antennas: Theoretical Framework and Definitions
Karl F. Warnick Senior Member, IEEE, Marianna V. Ivashina Member, IEEE, Stefan J. Wijnholds Member, IEEE, and Rob Maaskant Member, IEEE

Abstract--For phased array receivers, the accuracy with which the polarization state of a received signal can be measured depends on the antenna configuration, array calibration process, and beamforming algorithms. A signal and noise model for a dual-polarized array is developed and related to standard polarimetric antenna figures of merit, and the ideal polarimetrically calibrated, maximum-sensitivity beamforming solution for a dualpolarized phased array feed is derived. A practical polarimetric beamformer solution that does not require exact knowledge of the array polarimetric response is shown to be equivalent to the optimal solution in the sense that when the practical beamformers are calibrated, the optimal solution is obtained. To provide a rough initial polarimetric calibration for the practical beamformer solution, an approximate single-source polarimetric calibration method is developed. The modeled instrumental polarization error for a dipole phased array feed with the practical beamformer solution and single-source polarimetric calibration was -10 dB or lower over the array field of view for elements with alignments perturbed by random rotations with 5 degree standard deviation. Index Terms--Phased array antennas, polarimetry, array signal processing.

polarimetric properties of each formed beam or image pixel can be adjusted on the fly by changing beamformer coefficients. If array output correlations are computed and stored, one set of observation data can be processed with multiple sets of beamformer coefficients tuned to optimize sensitivity, sidelobe level, or polarimetric accuracy. Exploiting this flexibility and achieving best possible system performance requires the development of a theory for polarimetric phased arrays, including figures of merit, optimal beamformer solutions, and practical calibration strategies. Key questions that must be answered by this theory include the following:
·

· ·

·

I . I N T RO D U C T I O N The radio astronomy community is currently developing polarimetric aperture arrays and phased array feeds (PAFs) for large reflectors [1­4]. Accurate polarization state measurements for observed sources is critical to the science goals for current and planned phased array instruments. With a traditional waveguide feed, the polarization properties of the receiver are fixed at the time of manufacture and unwanted instrumental polarization can be calibrated by observing sources with known polarization parameters. For a phased array receiver, the
Karl F. Warnick is with the Department of Electrical and Computer Engineering, Brigham Young University, 459 Clyde Building, Provo, UT 84602, USA. E-mail: warnick@ee.byu.edu. Marianna V. Ivashina is with the Department of Earth and Space Sciences, Chalmers University of Technology, S-41296 Gothenburg, Sweden. E-mail: marianna.ivashina@chalmers.se Stefan J. Wijnholds is with the Netherlands Institute for Radio Astronomy (ASTRON), Oude Hoogeveensedijk 4, NL-7991 PD, Dwingeloo,The Netherlands. E-mail: wijnholds@astron.nl Rob Maaskant is with the Department of Signals and Systems, Chalmers University of Technology, S-41296 Gothenburg, Sweden. Email: rob.maaskant@chalmers.se

How do astronomical performance criteria relate to the standard IEEE definitions for polarimetric antennas? What beamforming algorithm will simultaneously optimize for high SNR and polarimetric accuracy? Which requirements should be set on the antenna array and beamformer design to achieve optimal performance? How can a polarimetric array be accurately and efficiently calibrated?

This paper will consider the first two questions in detail and addresses the third empirically through a numerical study. An approximate single-source calibration scheme is presented to address the fourth issue. A full treatment of polarimetric calibration is beyond the scope of this paper and will be addressed in future work. The first question arises because antenna engineers assess the polarimetric performance of antenna systems in terms of the axial ratio, cross-polarization discrimination (XPD), and cross-polarization isolation (XPI), while astronomers judge instrument performance and express system requirements in terms of Stokes parameters [5, 6]. Another challenge is that the standard IEEE definitions of the axial ratio, XPD and XPI have been established for single port systems, and these figures of merit must be extended to phased array systems that are capable of forming multiple dual-polarized beams simultaneously. Astronomical antenna applications also have unique constraints because radiation in terrestrial communication systems is typically highly polarized,


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whereas astronomical sources have a small but important polarized component of a few percent or less relative to the total power flux density. The starting point for answering these questions is the development of a signal and noise model for a polarimetric phased array receiver. This is accomplished in Sec. II. The treatment leads to the general problem of beamforming for rank-two signals and the concept of a polarimetric array beam pair. In Sec. III, the standard IEEE definitions for polarimetric figures of merit are related to the beam pair Jones matrix. In Sec. IV, the optimal beam pair solution is derived for a perfectly known instrument, and a practical beamforming method based on signal correlation matrix eigenvectors is considered for partially characterized polarimeters. An important result is that if sample estimation error (the error incurred by computing the array output correlation matrix from a finite number of voltage samples) is neglected, the practical eigenvector max-SNR method is equivalent to the optimal solution when polarimetrically calibrated. The performance of practical beamformers are compared to the ideal solution using a numerical model in Sec. V. Voltage and field quantities are phasors with the ej t convention. An overbar (E ) is used to denote threedimensional field and position vectors, whereas vectors of voltages are typeset in boldface (v). The superscript designates the complex conjugate and H the conjugate transpose. · denotes expectation over time. I I . P O L A R I M E T R I C P H A S E D A R R AY M O D E L The purpose of a radio polarimeter is to measure the polarization properties of an incident electromagnetic wave as a function of the angle of arrival. Figure 1 illustrates an N -element polarimetric beamforming array comprised of two groups of antenna elements with nominally orthogonal polarization. The antenna system is assumed to be illuminated by a point source radiating partially-polarized fields. The electric field intensity vector at the point r radiated by such a source can be approximated in the neighborhood of the receiver by the incident plane wave E (r, t) = [Eu (t) u + Ev (t) v ] ej ^ ^
k·r

E (r, t) = Eu(r, t)u + Ev (r, t)v ^ ^

x-oriented antenna elements

y -oriented antenna elements

1

2

N

···

···

v

w

1

w

2

v 1 = wH v 1

H v 2 = w2 v

Fig. 1. A radio polarimeter comprised of a dual-polarized actively beamformed receiving antenna array.

(1)

where u and v are orthogonal unit vectors according to ^ ^ one of Ludwig's polarization definitions relative to the coordinate system of the array [7], and k is the wave vector corresponding to the angle of arrival. Since u and ^ v are defined with respect to the coordinate system of the ^ array, we are neglecting in this treatment a rotation of the polarization state from the astronomical coordinate system on the sky to the coordinate system of the array. In the following theoretical development, we will consider the polarimetric calibration problem for one beam-

steering direction. The calibration process must be applied to every formed phased array beam for each desired beam direction, as the numerical results make clear. Array calibration is accomplished with point sources, rather than extended sources, so all signal response quantities are assumed to arise from a point source. For an imaging array, each pixel represents a different beam steering direction and set of array beamforming coefficients. The figures of merit and beamforming procedures developed in this paper apply independently to each image pixel. The source of interest is assumed to be a point source, and all results for figures of merit are calculated at the beam center. Since extended astronomical sources are common, beam polarization patterns are important, but this aspect of phased array polarimetry will be considered in future work. The antenna output signals are amplified to form the N -element output voltage vector v, which is subsequently combined into the output voltages v1 and v2 using the beamformer weight vectors w1 and w2 respectively, each of which is a column vector of size N â 1. Together, w1 and w2 constitute a polarimetric beam pair for a given sky pointing direction1 . The electric field components Eu (r, t) and Ev (r, t) are complex random processes in the phasor or complex baseband representation. The polarization state of the plane wave is determined by the covariance matrix of
1 In the less general bi-scalar method, w and w are weight vectors 1 2 of size (N/2 â 1) and are only associated to the x- and y -oriented antenna elements, respectively.


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the two field components, which is the 2 â 2 Hermitian matrix 2 |Eu | Eu Ev RE = . (2) 2 Eu Ev |E v | The covariance matrix has two real and one complex degrees of freedom, or four real degrees of freedom. The time-average power flux density of the incident wave is Ssig = |Eu |
2

The N â N array output signal voltage covariance matrix is
H H H Rs = vs vs = vu vu |Eu |2 + vu vv Eu Ev H H 2 + vv vu Eu Ev + vv vv |Ev |

(7)

+ |Ev | 20

2

which is of rank one for a fully polarized wave and of rank two for a partially polarized or unpolarized incident wave. Upon introducing the N â 2 matrix V = [vu vv ] we can write (7) in the more compact form Rs = VRE V
H

=

1 tr[RE ] 20

(3)

(8)

where 0 is the characteristic impedance of free space and tr is the matrix trace operation. Some authors rearrange the covariance matrix to form the coherency vector [6] |Eu |2 E E (4) c = vec R = u v E Eu Ev 2 |Ev | where vec(R ) is the vector obtained by stacking the E columns of R . The relative magnitudes and phases of E these quantities determine the degree of polarization and the polarization state of the polarized part of the wave. A. Array Signal Response With reference to Fig. 1, we will model the antenna array output signals {v1 , v2 } in terms of the voltage responses of the array to the unit intensity, time-harmonic, linearly polarized waves E u = ue ^ E v = ve ^
j k·r

(9)

Assuming that the phased array system noise can be characterized by the noise covariance matrix Rn , the complete array output voltage covariance matrix can be described as Rv = Rs + Rn (10) This expression provides a signal and noise model for a polarimetric array. The precise form of the noise response Rn for an array receiver is not important here, but is considered in detail in [8­10]. B. Beam Pair Signal and Noise Response After beamforming, the two output voltages obtained with the beamformer pair are (cf. Fig. 1)
H v1 = w1 v H v2 = w2 v.

(11a) (11b)

The covariance matrix Ro of the two beam outputs is = v = v
u

(5a) (5b) Ro =

j k ·r

v

where vu and vv are vectors of the respective voltages induced by the two waves at the array receiver outputs H before beamforming. Even though E u E v = 0, vu and vv may not be orthogonal due to possible non-ideal polarimetric response of the receiver. The voltage responses vu and vv are not exactly known in practice, but are useful in developing a system model for a beamforming phased array antenna. We will show in Section IV-A that if vu and vv were known, the beam pair w1 , w2 can be exactly calibrated polarimetrically. Experimental procedures for calibrating the array in the case that vu and vv are not known are discussed in Section IV-B. Since u and v are orthogonal, it follows by linearity ^ ^ that for an arbitrarily polarized wave, the array signal voltage response vector can be written using vu and vv as vs = Eu vu + Ev vv (6)

(12) In terms of the array signal and noise covariance matrices, the beam pair output covariance matrix is Ro =
H w1 (Rs + Rn ) w1 H w2 (Rs + Rn ) w1 H w1 (Rs + Rn ) w2 H w2 (Rs + Rn ) w2

v v

1 2

v1

v

2

=

|v1 |2 v2 v1

v1 v2 |v2 |2

.

. (13) (14)

Introducing the N â 2 vector W = [w1 w2 ] leads to the more compact expression Ro = Ro,s + Ro where Ro,s = WH VRE VH W Ro
,n ,n

(15)

(16a) (16b)

= W Rn W.

H

This result provides a signal and noise model for the polarimetric beam pair W.


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I I I . P O L A R I M E T R I C D E FI N I T I O N S In this section, we consider the relationship between IEEE definitions for polarimetric figures of merit and the Jones and Mueller matrix formulations that are common in radio astronomy and remote sensing. We also consider the problem of quantifying the joint sensitivity of a polarimetric beam pair. A. IEEE Definitions for Single Port Antennas The IEEE definitions for polarization terms for single port antennas are as follows [5]: Cross-polarization discrimination (XPD) is the ratio of the power level at the output of a receiving antenna, nominally co-polarized with the transmitting antenna, to the output of a receiving antenna of the same gain but nominally orthogonally polarized to the transmitting antenna. Cross-polarization isolation (XPI) is the ratio of the wanted power to the unwanted power in the same receiver channel when the transmitting antenna is radiating nominally orthogonally polarized signals at the same frequency and power level. Other figures of merit such as axial ratio and intrinsic cross-polarization ratio [11] can be applied to polarimetric phased arrays but will not be considered further in this paper. B. Definitions for Phased Array Antennas With reference to the polarimetric phased array model developed in Sec. II, the signal response of an array antenna to co- and cross-polarized incident fields Eu and Ev is described by the two beamformer output voltages v v and
1u 2u

may not hold due to polarization leakage caused by imperfections in the element geometry or mutual coupling between the array elements, in particular when the antennas are placed in a finite array exhibiting strong truncation effects. In the latter case, the polarization characteristics of the embedded element patterns differ significantly from each other. The degree of polarization leakage is dependent not only on the array geometry and mechanical construction, but also on the values of the beamformer coefficients. To quantify the beam-dependent polarization leakage, we extend the standard definitions in Sec. III-A as follows: Cross-polarization discrimination is defined as the ratio of powers received at the beamformer outputs 1 and 2 due to the same incident field, i.e. E u or E v : XPDu = XPDv = |v1u | |v2u |
2 2 2 2

= =

H H w1 vu vu w1 wH Ruu w =1 H v vH w H w2 u u 2 w2 Ruu w

1 2

(19)

|v2v |

|v1v |

H H w2 vv vv w2 wH Rvv w2 =2 (20) H v vH w H w1 v v 1 w1 Rvv w1

H H where Ruu = vu vu and Rvv = vv vv are the signal covariance matrices in response to pure E u and E v signals of unit intensity. Cross-polarization isolation is defined as the ratio of powers received at the same beamformer output, 1 or 2, due to orthogonally-polarized incident fields E u and Ev:

XPI1 = XPI2 =

| v 1u | |v1v | |v2v |

2

2

= =

H H wH Ruu w1 w1 vu vu w1 =1 H v vH w H w1 v v 1 w1 Rvv w1 H H H w2 vv vv w2 w2 Rvv w2 =H H v vH w w2 u u 2 w2 Ruu w2

(21) (22)

2 2

=

H w1 H w2

vu vu

| v 2u |

(17)

v1 v2

v v

=

H w1 vv H w2 vv

(18)

where vu and vv are the output voltage vectors of the receiving elements in response to pure E u and E v signals. The weight vectors w1 and w2 of the two beamformers are assumed to be defined such that they optimally receive or at least approximate the optimal reception of E u and E v , respectively, according to some specified criterion to be discussed in Sec. IV. This implies that the polarizations of the two formed beams are nominally aligned with the u and v directions. While ^ ^ u and v are defined as real vectors in (1), the definitions ^ ^ in this section hold for any pair of waves E u and E v with orthogonal polarizations, whether linear, circular, or elliptical. Ideally, one expects the two output voltage vectors to be directly proportional to E u and E v , but this

C. Jones Matrix Formulation Unlike radio communications, for which antennas are fabricated to meet a fixed polarization purity requirement, applications such as radio astronomy and remote sensing that rely on accurate measurement of wave polarization states also require operational polarimetric calibration. When modeling system effects that contribute to the measured wave polarization state, it is convenient to use the Jones matrix formulation [12]. The Jones formulation allows matrix terms for various system components to be chained together into an overall Jones matrix that must be modeled or measured in order to infer the incident wave polarization state. For a phased array, each polarimetric beam pair has an associated Jones matrix. The general relationship between the beam outputs and the incident electric field intensity vector is v v
1 2

=

J11 J21

J12 J22

Eu Ev

(23)


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where the two by two matrix on the right side of this equation is the beam pair Jones matrix, which we will denote in the following as J. Using the notation developed above, for an array this relationship becomes v v
1 2

In terms of the elements of the Jones matrix, the crosspolarization figures of merit (19)­(22) are [11] XPDu = XPI1 = |J11 | |J21 | |J11 | |J12 |
2 2

XPDv = XPI2 =

|J22 | |J12 | |J22 | |J21 |

2 2 2 2

(30a) (30b)

= =

wH (vu Eu + vv Ev ) 1 wH (vu Eu + vv Ev ) 2 wv wv
H 1 H 2 u u

2 2

wv wv

H 1 H 2

v v

Eu Ev

(24)

The Jones matrix can be identified as J= where W= w
1

wH vu 1 wH vu 2

wH vv 1 = WH V wH vv 2

These expressions show that the standard antenna polarization figures of merit are measures of the magnitudes of the off-diagonal elements J12 and J21 relative to the diagonal elements J11 and J22 . D. Mueller Matrix Formulation While Jones matrices operate in the voltage phasor domain, 4 â 4 Mueller matrices represent transformations on wave polarization states in the correlation or Stokes parameter domain. Since the Mueller matrix formulation is commonly used in the astronomical literature, the treatment will be rehearsed here and placed into the mathematical framework developed in earlier sections. For a signal characterized by the field coveriance matrix (2), the Stokes vector containing the four Stokes parameters is defined as [14, pp. 97­98] I |Eu |2 + |Ev |2 Q |E |2 - |Ev |2 (31) S= = u U 2 Re [ Eu Ev ] V 2 Im [ Eu Ev ] Some authors define the Stokes parameters with a factor of 1/0 , where 0 is the intrinsic impedance of space, so that the Stokes parameters have units of power density (W/m2 ). It can be shown that [15, p. 29] Q2 + U 2 + V 2 I 2 . The degree of polarization is Q2 + U 2 + V 2 . (33) I For an unpolarized wave, Q = U = V = 0. A Stokes polarimeter can be represented by the relationship So = MSi (34) m= where Si is a vector of the Stokes parameters of the incident wave given by (31) and M is the Mueller matrix of the system. The vector of Stokes parameters referred to the beam output voltages is |v1 |2 + |v2 |2 |v |2 - |v2 |2 So = 1 (35) 2 Re v1 v2 2 Im v1 v2 (32)

(25)

w

2

, and V = vu

vv .

(26)

Using this result, the beam pair output signal voltage covariance matrix (16a) can be written as Ro,s = JRE JH (27)

The goal of polarimetric calibration is to transform a given beam pair W to a new beam pair W for which the radiation patterns are steered to the same angle of arrival as the original beam pair but having Jones matrix J as close as possible to the identity matrix. If J I, the cross-polarization figures of merit are large in value. If the initial Jones matrix J = WH V is known, it can be readily seen that the beam pair W = WJH
-1

(28)

is ideally polarimetrically calibrated and the realized Jones matrix is J = W H V = I. If the beam pair output signal coveriance matrix Ro,s is measured for one signal with a known polarization state RE , the relationship (27) does not uniquely fix the beam pair Jones matrix. Given known input and output polarization parameters, by substitution it can be shown that the general solution to (27) has the form J = R1/s2 URE o,
- 1 /2

(29)

where U is an arbitrary unitary matrix. This shows that with only a single calibrator source, there remains a 2â2 unitary degree of freedom in the Jones matrix. The physical interpretation of the unitary degree of freedom is discussed in [6, 13]. The ambiguity can be removed using knowledge of the nominal element polarizations (see Section IV-C), measurements of additional calibrator sources, or correlation with a second calibrated polarimetric antenna.


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The beam pair is said to be polarimetrically calibrated if the Stokes parameters of the source, (31), and the measured Stokes parameters, (35), are identical, apart from a constant gain factor. In the following, we will include the gain calibration with polarimetric calibration, and consider the system to be polarimetrically calibrated when M = I. The Mueller and Jones matrix formulations are connected by known formulas, which we will review here. Stokes parameters are related to the coherency vector (4) by Si = Tc (36) where 1 00 1 00 T= 0 11 0 -j j 1 -1 0 0 matrix is 0 0 1 j 1 -j 0 0

Polarimetric accuracy can be quantified by the error in the measured Stokes parameters for a given source or by the deviation of the Mueller matrix from the identity. The relative RMS Stokes error is (I )2 + (Q)2 + (U )2 + (V )2 I (44) This metric can be applied to computed Stokes parameters before the instrument is polarimetrically calibrated, to determine the raw or uncalibrated Stokes error, or it can be applied after calibration, to assess the performance of both the calibration procedure and the native instrumental polarization properties. As with the Jones formulation, source-independent measures of polarimetric accuracy are also desirable. By the definition of the induced operator norm, we have the bound S (M - I)Si = M-I (45) Si Si ES = S = I For weakly polarized sources, Si approximately ES M - I The quantity EM = M - I (47) therefore is an approximate upper bound on the relative RMS Stokes error. EM might be referred to as the Stokes instrumental polarization bound. This bound is derived for the case of a noise-free system. Since estimation error due to noise can be reduced to arbitrarily low levels by integration, (47) is adequate for many purposes, including antenna design optimization, but when analyzing astronomical observation data in practice, the additional effect of noise on polarimetric accuracy should be considered. E. Beam Pair Sensitivity The above treatment has considered traditional polarimetric formulations for the case of a phased array antenna. Another important aspect of beamforming for polarimetric arrays is that the beamformer coefficients control the beam output SNR and antenna sensitivity. For one beam, it is straightforward to determine the beamformer coefficients that maximize sensitivity given a knowledge of the array signal and noise responses. For a beam pair, the sensitivity of both beams should ideally be as high as possible. In particular, the beam sensitivities of an uncalibrated beam pair W and the polarimetrically calibrated beam pair W are in general different, and there are many possible polarimetrically calibrated beam pairs that have lower sensitivity than the classical maximum-SNR beamformer solution. Therefore, it is of interest to characterize the sensitivity of both I , and we have (46)

(37)

The inverse of this transformation 1 1 0 -1 1 0 T = 2 0 0 1 -1

(38)

Using this relationship for the incident wave and the beam output voltages, co = T-1 MTci (39)

Using the properties of the Kronecker product, (27) can be rearranged in the form J J vec (RE ) = vec (Ro,s ) In view of (4), this becomes J J ci = co (41) (40)

Combining these results shows that the system Mueller matrix is related to the Jones matrix by (see also [16]) M = T(J J )T
-1

(42)

In the present context, the Jones matrix J(w1 , w2 ) is weight dependent, so that, by using (25) along with the properties of the Kronecker product, M = T(WH V) (WT V )T-
H T 1 -1

= T(W W )(V V )T

.

(43)

If the polarimeter is ideally calibrated, then J = I2â2 , M = I4â4 , and So = Si . For a realistic system configured as a Stokes polarimeter, polarimetric calibration means measuring the beam pair Mueller matrix M and using that information to transform the weight pair W, or in general adapt the beam pair covariance matrix Ro , such that the effective Mueller matrix becomes proportional to the identity matrix.


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outputs of a beam pair with a metric that is independent of the beam pair polarimetric calibration. For one beam w, the sensitivity is Ae kb B wH Rs w = Tsys Ssig wH Rn w (48)

where B is the system noise equivalent bandwidth, kb is Boltzmann's constant, and Ssig is the power flux density of the incident wave in W/m2 . As required by the definition of effective area, the incident wave is polarization matched with the polarization of the beam. The signal correlation matrix for a time-harmonic incident wave can be expressed as Rs = VEEH V
H

(49)

where E = [Eu Ev ]T . The polarization of the incident wave is matched to the beam when wH Rs w is maximized, which occurs for the incident field state 20 Ssig VH w (50) w VVH w where the scale factor follows from (3). Using these results in (48) leads to E=
H

Ae wH VVH w = 20 kb B (51) Tsys wH Rn w This same result can be obtained from (48) for an unpolarized wave (RE = 0 Ssig I) with Ssig in the leading scale factor replaced by half the power flux density of the unpolarized wave. To define the beam pair sensitivity, we seek a figure of merit that is independent of the beam pair polarimetric calibration. This means finding bounds on the beam sensitivity (48) with w an arbitrary linear combination of w1 and w2 . Using (51), the sensitivity is Ae aH W H V V H W a = 20 kb B H H Tsys a W Rn Wa (52)

where min and max are the eigenvalues of C. By substituting W = WA, where A is an arbitrary invertible 2 â 2 matrix, it can be seen that the matrix C is independent of linear transformation of the beam subspace and hence of the beam pair polarimetric calibration. Since the response V to orthogonal polarized waves is not directly available for a phased array in practice, it is of interest to express the sensitivity bound in terms of measurable array output quantities. In the Appendix it is shown that the sensitivity bound can be expressed in the form kb B Ae kb B min tr[R-1 Ro,s ] max tr[R-1 Ro,s ] o,n o,n Sin Ssig Sin Ssig Tsys (56) The calibration-independence of this form of the bound follows from the properties of the matrix trace. We will refer to two beam pairs W and W as sensitivity equivalent if they lead to the same upper and lower sensitivity bounds in (55) or (56). By the above derivation, if the beam pairs are related by a linear transformation, they are sensitivity equivalent, and a beam pair is sensitivity equivalent to its polarimetrically -1 calibrated counterpart WJH . Finally, we observe that a polarimetrically calibrated beam pair does not necessarily achieve the upper bound in (55). The degree to which polarimetric calibration reduces beam sensitivity will be studied empirically in Section V. I V. P O L A R I M E T R I C B E A M F O R M I N G For a polarimetric phased array, calibration has two aspects: 1) determining the beamformer pair W so that sensitivity and other beam figures of merit are optimal, and; 2) determining the beam pair Jones matrix or the Mueller matrix so that the incident wave polarization state can be inferred from the beam pair outputs. Combining 1) and 2) enables high sensitivity polarimetry. Beamformer coefficients applied to a phased array can be updated dynamically to accomplish various goals, such as sensitivity maximization as the noise environment changes, interference mitigation, or beam pattern sidelobe control. Consequently, the polarization properties of the array beam outputs are not necessarily fixed. This introduces the possibility of combining the beamformer optimization and polarimetric calibration steps, so that the beam pair satisfies a specified set optimality criteria and also is polarimetrically calibrated with both the Jones and Mueller matrices approximately equal to identity matrices. We will consider the ideal case of a perfectly known system, in order to understand the ultimate performance

where a = [a1 a2 ]T is an arbitrary vector. With a = C-1/2 VH Wa, where C = VH W(WH Rn W)-1 WH V (52) becomes a H Ca Ae = 20 kb B H (54) Tsys aa This shows that the ratio of quadratic forms in (52) lies within the field of values of the matrix C [17]. Since C is Hermitian, the eigenvalues are real and the field of values is an interval on the real line. It follows that for any beamformer w in the subspace spanned by the beam pair the sensitivity is bounded by 20 kb B
min

(53)



Ae 20 kb B Tsys

max

(55)


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limits of a given beam pair, as well as the practical case for which polarization properties of a beam pair must be calibrated empirically using observations of sources with known polarization parameters. To avoid sacrificing observation time for polarimetric calibration, one can observe with non-polarimetrically calibrated beam pairs having imperfect polarization discrimination. Therefore, it is important to find an approach for obtaining best possible polarimetric beam pairs using the single-source calibration data required for non-polarimetric array beamforming, and to characterize the polarization purity of beam pairs that are obtained without additional effort at polarimetric calibration. A. Optimal Beamforming for a Perfectly Known System Any beam pair that responds to two non-colinear incident polarizations can be polarimetrically calibrated, but the resulting beams may have poor sensitivity, which implies a low SNR and inaccurate measured Stokes parameters for a given integration time. We will show that there is a unique polarimetrically calibrated beam pair that minimizes estimation error in measured Stokes parameters for a point source at a given angle of arrival. Measured Stokes parameters are a linear combination of the elements of a sample estimate of the 2 â 2 covariance matrix Ro,s given by (16a). We will denote the sample estimated matrix for a given integration length ^ K as Ro,s . With the standard radiometric detection ^ technique, Ro,s is the difference between an on-source measurement and off-source measurement of the beam pair output coveriance matrix: ^ ^ ^ Ro,on = JRE JH + WH Rn ^ o,off = WH Rn,off W ^ R
,on

polarization parameters obtained from an uncalibrated beam pair W, the measurement error due to system noise can be analyzed as if the beam pair were replaced by a polarimetrically calibrated beam pair. The goal is to find the polarimetrically calibrated beam pair that minimizes the estimation error in (59), which we will write with (60) as ^ En = W
H

^ (Rn

,on

^ - Rn,off )W

(61)

^ The matrix Rn is described stochastically by the Wishart ^ distribution [18], which means that En is the difference of two identically distributed Wishart random matrices. ^ The mean of En is the zero matrix. In the low SNR ^ limit, the variance of the entries of En is 2 ^ var(En,kl ) = Ro P
,n,kk

Ro

,n,ll

(62)

where P is the number of voltage samples that are ^ ^ averaged to produce Ro,on and Ro,off . This result shows that in order to minimize estimation error in the Stokes parameters, the diagonal elements of the 2 â 2 beam pair output noise correlation matrix Ro,n must be minimized. This consideration reduces the polarimetric beamformer calibration problem to the joint constrained minimization problem argmin wH Rn w1 , argmin wH Rn w 1 2
w
1

w

2

(63) (64)

2

subject to J = WH V = I

W

(57a) (57b)

Since Rn is positive definite, we can add the two objective functions and recast the constrained minimization problem in the form argmin tr(WH Rn W), subject to WH V = I
W

(65)

^ where R signifies a sample-estimated coveriance matrix. The difference is ^ Ro
,diff

The constrained optimization problem can be solved by setting to zero the matrix derivative
WH

^ ^ = Ro,on - Ro,off = ^ JH + WH (Rn ^ JRE

tr WH Rn W - (WH V - I) = 0

(66)

,on

^ - Rn

,off

)W

(58)

After polarimetrically calibrating the beam pair outputs, ^ the output covariance matrix Ro,cal becomes ^ Ro
,cal -1 ^ = J-1 Ro,diff JH = ^ ^ ^ RE + J-1 WH (Rn,on - Rn ^ En

where is a matrix of Lagrange multipliers, which will be chosen to satisfy the constraint in (64). Evaluating the derivative leads to Rn W - V = 0 (67) (68)

,off

)W J

H -1

(59)

or W = R-1 V n Using the constraint WH V = I, and solving for , we find that H VH R-1 V = I (69) n from which it follows that H = = (VH R-1 V)-1 . n (70)

This is the same measurement equation that would be obtained with (58) subject to the calibrated beam pair W = WJ
H -1

, so that

J =W

H

V=I

(60)

Therefore, if Stokes parameters are computed by applying the beam pair inverse Jones matrix J-1 to the


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The solution to the optimization problem is found by substituting (70) in (68) to yield W
opt

= R-1 V(VH R-1 V)-1 . n n

(71)

This pair of beamformer weight vectors minimizes the output system noise power while constrained to be polarimetrically calibrated with respect to a source with a given angle of arrival. The optimal solution is related to a simple ranktwo generalization of the classical maximum-SNR beamformer [19]: WmaxSNR = R-1 V (72) n This beam pair realizes two maximum sensitivity beamformer weight vectors, with the Jones matrix VH R-1 V = VH WmaxSNR = JH n maxSNR . The optimal beamformer solution (71) can be written as W
opt

This is the eigenvector max-SNR beamformer algorithm. To compare this beamformer with the optimal solution presented in the previous section, we note that the voltage response vectors vu and vv must span the same subspace as the eigenvectors veig,1 and veig,2 . This implies that vu vv = A11 v = A12 v
eig,1 eig,1

+ A21 v + A22 v A12 A22

eig,2 eig,2

(78a) (78b)

which is equivalent to V = [veig
,1

v

eig,2

]

A A

11 21

=V

eig

A

(79)

Substitution in (71) gives W
opt

= R-1 V n = R-1 V n

eig eig

A VH R-1 V n VH R-1 V n

=W

H -1 maxSNR JmaxSNR

(73)

eig A -1 eig

-1

(80)

which shows that the beam pair WmaxSNR is not in general polarimetrically calibrated but is sensitivity equivalent to the optimal beam pair. B. Practical Polarimetric Array Calibration and the Eigenvector Method In practice, the voltage response matrix V is difficult to measure for a PAF on a reflector antenna, since ideal, orthogonally polarized astronomical sources are unavailable. A practical polarimetric array calibration procedure using observations of unpolarized and partially polarized sources is needed. For a dual-polarized array, neglecting estimation error, the signal covariance matrix Rs is of rank two for an unpolarized source. The latter is readily concluded from (9) by taking RE = I, so that Rs = VRE V
H RE =I H H = vu vu + vv vv

which has a similar form as (71). Upon introducing Weig = R-1 Veig and Jeig = WH V, (80) can be n eig written as -1 (81) Wopt = Weig JH eig This result shows that the eigenvector max-SNR beam pair is sensitivity equivalent to the optimal beam pair. A modification of the above approach can be obtained using the generalized eigenvalue form of the max-SNR beamformer: Rn w = Rs w (82) Since Rs is rank two, the generalized eigenvalue problem has two nonzero eigenvalues. The corresponding principle eigenvalues provide a maximum-SNR beam pair. It can be easily shown that this beam pair is also sensitivity equivalent to the optimal beam pair (71). The same beam pair can also be obtained by using the noise correlation matrix to pre-whiten the array output signals and using the eigenvector CFM method with the prewhitened signal correlation matrix. To summarize, we have so far five polarimetric beamforming algorithms: W W
opt

(74)

which is a sum of two rank-one matrices. The eigenvalues and eigenvectors of Rs are defined by Rs v = v (75) Neglecting estimation error, two of the eigenvalues are nonzero. Veidt proposed to use the two principal eigenvectors veig,1 and veig,2 of Rs as conjugate field match (CFM) beamformer weights [3]. We will refer to the beam pair W
CFM

maxSNR

=V

eig

= [veig

,1

veig,2 ]

(76)

Weig Rn w = Rs w W WCFM = Veig

= R-1 V(VH R-1 V) n n = R-1 V n = R-1 Veig n

-1

(83a) (83b) (83c) (83d) (83e)

as the eigenvector CFM beamformer. The eigenvector CFM method can be modified to form the maximum-SNR eigenvector beamformer weight vectors w w
eig,1 eig,2

= R-1 veig n = R-1 veig n

,1 ,2

(77a) (77b)

The first and second beam pairs require knowledge of the array response V to orthogonally polarized incident waves, which is not directly available in practice. The third, fourth, and fifth are practical beamformers in the sense that they can be computed from measurable signal output correlation matrices for unpolarized or partially polarized sources. The first beam pair is exactly


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polarimetrically calibrated, but in general the others are not. In the absence of estimation error, all of these beam pairs except for CFM are sensitivity equivalent. Sample estimation error in the array output correlation matrices may affect each beamformer algorithm differently, but the effect of estimation error on the beamformer weights can be driven to low levels by integrating the array output during the calibration phase for a long period of time, so this effect is not considered here. C. Approximate Calibration with a Single Unpolarized Source The practical beamforming algorithms, (83b)­(83e), are not polarimetrically calibrated. The eigenvectors computed from (75) are arbitrary up to a scale factor, and the resulting beam pair has a unitary degree of freedom according to (29). In order to give meaning to crosspolarization figures of merit, these degrees of freedom must be fixed by some concrete, repeatable algorithm. Since each beam pair can be viewed as the output of a dual-polarized single pixel feed, any polarimetric calibration technique that is used for conventional feeds could be applied to calibrate the beam pair. Standard polarimetric calibration methods require observations of multiple sources with known Stokes parameters or tracking a polarized source over time as it rotates relative to the feed. Since these observations would have to be repeated with the telescope steered so that the source was at the center of each formed phased array beam, a straightforward implementation of standard polarimetric calibration could require many hours or even days. It is possible that efficient multiple-source calibration methods for phased arrays can be found, but a full treatment of efficient polarimetric calibration procedures for phased arrays is beyond the scope of this paper. Since determining beamformer weights for a phased array requires observations of a bright (and typically unpolarized) source over a grid of telescope pointings at the center of each beam [20], the goal here is to find the best possible polarimetric calibration procedure given this already available observation data. We will show that a single unpolarized source observation per beam together with the known nominal element polarization can be used to calibrate each beam pair approximately. Given an uncalibrated beam pair W, the first step in the calibration procedure is to use the nominal polarization of array elements to find the approximate Jones matrix. The signal correlation matrix has the block form Rs = Rxx Ryx Rxy Ryy (84)

estimation error, the matrices R
s,x

=

Rxx 0

0 , 0

Rs,y =

0 0 0 Ry

y

(85)

^ ^ are rank one and have principal eigenvectors vx and vy , respectively. These vectors are orthogonally rotated as ^ ^ needed to obtain vu and vv , where the hat indicates that these are only approximate responses to orthogonally polarized waves, whereas vu and vv in (5) are exact. These vectors can be used directly as beamformer weights (the biscalar method), but we will employ them here to obtain the approximate Jones matrix ^ ^ J = WH V (86) ^ ^^ where V = [vu vv ]. This approximate Jones matrix can be used to calibrate the beam pair to obtain ^ W = WJH
-1

(87)

^ ^ Since the eigenvectors vx and vy are only determined up to a scale factor, it remains to specify the complex scaling of the beamformer weight vectors w1 and w2 in the beam pair W . Assuming that the calibrator source is unpolarized, the magnitudes of the weight vectors can be fixed by equalizing the responses to the calibrator source. The overall phases of the weight vectors are set by dividing by the phase of the largest weight vector element, so that the largest weights are real. This leads to the approximately calibrated beam pair w1 = w w2 = w I /2
1 1 /2 H 1

w

Rs w Rs w

1 1 /2

|w1 w1 |w2 w2

,max ,max ,max ,max

|

(88a)

I /2
2

|

w

H 2

(88b)

2

where x and y represent outputs from x-polarized elements and y -polarized elements, respectively. Neglecting

In practice, the phase lengths of the receiver signal paths will be different, but this phase can be measured using injected calibration tones or other means and incorporated in the phase constraint. The magnitude scaling in ^ ^H (88) ensures that the diagonal elements of JJ are unity, which through (42) implies that the element M11 of the Mueller matrix is equal to unity. This approximate single-source polarimetric calibration procedure is essentially equivalent to a polarimeter that uses the nominally orthogonally polarized outputs of a standard dual-polarized antenna (including a rotation from the astronomical coordinate system to the coordinate system of the antenna). As with a standard antenna, coupling between phased array elements and mechanical imperfections mean that the beam pair will not be exactly calibrated, and additional observations of polarized sources may be required to remove residual instrumental polarization effects. Numerical results will be given in the following section to assess the performance of this single-source polarimetric calibration method.


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V. N U M E R I C A L R E S U LT S The polarimetric figures of merit and beamforming algorithms will be illustrated for a 19 â 2 element hexagonal array of thin, lossless, x- and y -polarized crossed dipoles spaced 0.6 apart and backed by a ground plane. The phased array feeds a 20-meter reflector with f /D = 0.43. The open-circuited element response is approximated using the analytical expression for radiation by a dipole. Physical optics is used to compute secondary fields scattering from the reflector. The array mutual impedance matrix is approximated by conservation of energy from the element pattern overlap integrals. The PAF noise model includes sky noise, spillover, and receiver noise due to low noise amplifiers with parameters opt = 0, Rn = 3.4 , and Tmin = 33 K. Ludwig's first convention [7] is used to define the u and v directions, which means that u and v are aligned with the array x- and y -coordinate frame. The array is calibrated using the approach of [20] using observations of an unpolarized calibrator source for each desired beam steering direction. Beamformer weights are computed using the eigenvector max-SNR and CFM algorithms described above, and polarimetrically calibrated using the approximate single-source method of Section IV-C. Results are given for two array configurations: (1) perfectly aligned elements and (2) perturbed element orientations with small random rotations away from the nominal x- and y -polarizations. Since only a small amount of power is received by elements more than one or two wavelengths away from the focal spot associated with a given formed beam, simulations of polarimetric accuracy for the 19 element case are expected to be representative of results for larger arrays. A. Perfectly Aligned Elements The first study is a comparison of the sensitivity of the eigenvector max-SNR and CFM beam pairs to the optimal solution (Figure 2). The single-source calibration procedure of Section IV-C is used with both beamformers. The sensitivity of the max-SNR beam pair calibrated with the single-source procedure is nearly indistinguishable from that of the optimal solution. For the polarimetrically uncalibrated case, the max-SNR beam pair sensitivity is different, but still lies within the upper and lower bounds over arbitrary linear transformations of the beam pair. The beam pair sensitivity bounds are given by (55) and indicated in the figure with gray shading. For some beam steering angles, the sensitivity of one of the uncalibrated beams is larger than that of the optimal beam pair. This illustrates that the calibration constraint in (65) leads to a beam pair with sensitivity slightly smaller than is achieved by other uncalibrated beam pairs
5

4 Sensitivity (m2/K)

3

2

1

Optimal Beam Pair Eigenvector max-SNR Eigenvector max-SNR (Uncalibrated) Eigenvector CFM

0 0

0.5

1 Angle (HPBW)

1.5

2

Fig. 2. Beam pair sensitivity for dipole array with perfectly aligned elements for beam steering angles from the feed boresight to 2 half power beamwidths (HPBW) from boresight. The cut is at an azimuth angle of 30 from the x-direction in the feed coordinate system. The sensitivity bound in Equation (55) is indicated with gray shading.
150

XPD, Eigenvector max-SNR XPI, Eigenvector max-SNR XPD, Eigenvector CFM XPI, Eigenvector CFM

XPD/XPI (dB)

100

50

0 0

0.5

1 Angle (HPBW)

1.5

2

Fig. 3. Beam center cross-polarization figures of merit as a function of steering angle for dipole array with perfectly aligned elements.

in the subspace spanned by the optimal beam pair. The single-source calibrated CFM beam pair sensitivity also lies within its corresponding bound (55), but the overall sensitivity is lower than that of max-SNR. Cross-polarization figures of merit are shown in Figure 3. Results are given for each beam center as a function of beam steering angle. For the beam steered in the boresight direction, the polarimetric calibration is essentially perfect, but for beams steered off-boresight, the figures of merit decrease. Results are shown for the u polarization. Curves for the v polarized beams are nearly identical. Error in measured Stokes parameters relative to the total source intensity for the single-source calibrated


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0

5
Relative Stokes Error (dB) -10

4
-20

Sensitivity (m2/K)
Instrumental Polarization Bound, Eigenvector max-SNR RMS Relative Stokes Error, Eigenvector max-SNR Instrumental Polarization Bound, Eigenvector CFM RMS Relative Stokes Error, Eigenvector CFM

3

-30

2

-40

1

-50 0

Optimal Beam Pair Eigenvector max-SNR Eigenvector max-SNR (Uncalibrated) Eigenvector CFM

0.5

1 Angle (HPBW)

1.5

2

0 0

0.5

1 Angle (HPBW)

1.5

2

Fig. 4. Relative Stokes parameter error (44) and instrumental polarization bound (47) as a function of steering angle for dipole array with perfectly aligned elements.

Fig. 5. Beam center sensitivity for dipole array with perturbed element orientations. The cut is at an azimuth angle of 30 from the x-direction in the feed coordinate system. The sensitivity bound (55) is indicated with gray shading.
150

max-SNR and CFM beam pairs is shown in Figure 4. The source Stokes parameters are tabulated values from the NRAO C Band VLA/VLBA Polarization Calibration Database for 0371+331: I = 5.0522 Jy, Q = 0.1869, U = 0.1109, V = 0. The error closely follows the instrumental polarization bound (47) for both beam pairs. It may be surprising that the perfectly aligned array, with no receiver gain imbalances, no mechanical defects, and exact analytical formulas for the element response, is only exactly calibrated for the boresight beam. For the boresight beam, the response vectors vu and vv in (7) are orthogonal, and the eigenvectors obtained from (75) are proportional to vu and vv . Thus, at boresight the eigenvector max-SNR and CFM beam pairs can be exactly calibrated without making use of the biscalar transformation (87). For off-boresight steered beams, due to the depolarizing effect of the reflector, vu and vv are not orthogonal, and the eigenvector pair veig,1 and veig,2 for the perfectly aligned array require polarimetric calibration. The Square Kilometer Array (SKA) design target for relative instrumental polarization is 25 dB for general polarimetric imaging and 40 dB for specialized applications (see [21] and other available SKA documents). For the perfectly aligned array, the instrumental polarization is better than 40 dB only for beam steering angles near boresight. For off-boresight beams, it is apparent that an additional polarimetric calibration beyond the approximate single-source method of Section IV-C is required. B. Imperfectly Aligned Elements We now study the effect of mechanical imperfections on array feed polarimetric performance. A rotation in the

XPD, Eigenvector max-SNR XPI, Eigenvector max-SNR XPD, Eigenvector CFM XPI, Eigenvector CFM

XPD/XPI (dB)

100

50

0 0

0.5

1 Angle (HPBW)

1.5

2

Fig. 6. Beam center cross-polarization figures of merit for dipole array with perturbed element orientations.

x-y plane was applied to the orientation of each element in the array, with rotation angle chosen from a zeromean normal distribution with 5 standard deviation. Figure 5 shows that the beam sensitivity is not reduced by the perturbation, but the upper and lower limits of the sensitivity bound (55) are no longer equal for the boresight beam. The cross-polarization figures of merit (Figure 6) and Stokes instrumental polarization (Figure 7) are significantly degraded. The instrumental polarization is poorer than 25 dB for many beam steering angles. For observations that require better polarimetric accuracy, the nominal calibration approach presented in this paper is not adequate, and a further calibration step for each beam pair similar to the methods used for


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0

Relative Stokes Error (dB)

-10

-20

-30

-40

achieve lower instrumental polarization than the singlesource method used in this paper, standard polarimetric calibration techniques could be applied on a beam-bybeam basis to achieve lower instrumental polarization. To avoid time-consuming observations of multiple calibrator sources or long observations of a single polarized source for each formed beam, methods for reducing the number of required telescope pointings required to calibrate all formed beams are needed. Beam cross-polarization response patterns and the temporal stability of formed beam polarization responses are also of interest.
Instrumental Polarization Bound, Eigenvector max-SNR RMS Relative Stokes Error, Eigenvector max-SNR Instrumental Polarization Bound, Eigenvector CFM RMS Relative Stokes Error, Eigenvector CFM

APPENDIX
2

-50 0

0.5

1 Angle (HPBW)

1.5

Here we derive (56). By diagonalizing C in (53), it can be shown that tr[CRE ] is bounded by
min

Fig. 7. Relative Stokes parameter error and instrumental polarization bound for dipole array with perturbed element orientations.

tr[RE ] tr[CRE ]

max

tr[RE ]

(89)

traditional single-pixel feeds would be required. VI. CONCLUSIONS A signal and noise model for a dual-polarized beamforming array receiver has been used to develop polarization figures of merit and a theory of ideal polarimetric beamforming. An optimal polarimetrically calibrated maximum-SNR beamforming algorithm was given. For each angle of arrival, the algorithm provides a beam pair with maximum sensitivity subject to a polarimetric calibration constraint. The optimal polarimetric beamforming algorithm requires exact knowledge of the responses of the array to orthogonally polarized waves from a given angle of arrival. In practice, the array responses to orthogonally polarized waves are not readily available. A practical eigenvector-based maximum-SNR beamformer solution has been shown to optimize sensitivity in the sense that when the beams are polarimetrically calibrated, in the absence of estimation error the beams become equivalent to the optimal solution. In general, the eigenvector maximum-SNR beam pair is not polarimetrically calibrated, and additional correction through estimation of the beam pair Jones matrix is required. An approximate single-source calibration procedure based on the approximate orthogonality of the array element polarizations was developed. The instrumental polarization of the approximately calibrated beams for a perturbed dipole phased array feed was better than 10 dB over the field of view. The approximate single-source method therefore provides a rough calibration method for routine observations that do not require high polarimetric accuracy. In future work, more accurate polarimetric calibration methods for phased arrays should be studied. To

In view of (3) and (55), the sensitivity of any beam in the beam pair subspace is therefore bounded by the largest and smallest values of kb B (RE ) = tr[VH W(WH Rn W)-1 WH VRE ] Ssig (90) Using the invariance of the trace with respect to the ordering of a product of two matrices, this can be expressed as kb B tr[(WH Rn W)-1 WH VRE VH W] Ssig (91) Using (16), (RE ) becomes (RE ) = (RE ) = kb B tr[R-1 Ro,s ] o,n Ssig (92)

This leads directly to the bound (56). Because of the properties of the trace, replacing W with the calibrated -1 beam pair W = WJH does not change the value of the trace: tr[R
-1 o,n

Ro,s ] = tr[(J-1 Ro,n JH = tr[ R-1 o,n Ro,s ]

-1 -1 -1

)

J

R

o,s

JH

-1

]

(93)

which shows that the trace of R-1 Ro,s is invariant o,n with respect to polarimetric calibration of the beam pair, and the bound in (55) is a polarimetric calibrationindependent measure of the intrinsic beam pair sensitivity. REFERENCES
[1] S. G. Hay et al., "Focal plane array development for ASKAP (Australian SKA Pathfinder)," in European Conference on Antennas and Propagation (EuCAP), Edinburgh, UK, Nov. 11-16 2007. [2] O. A. Iupikov, M. V. Ivashina, and O. M. Smirnov, "Reducing the complexity of the beam calibration models of phased-array radio telescopes," in European Conf. on Antennas and Propagation (EuCAP), Rome, Italy, Apr. 11-15 2011.


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[3] B. Veidt, G. Hovey, T. Burgess, R. Smegal, R. Messing, A. Willis, A. Gray, and P. Dewdney, "Demonstration of a dual-polarized phased-array feed," IEEE Trans. Ant. Propag., vol. 59, no. 6, pp. 2047­2057, Jun. 2011. [4] K. F. Warnick, D. Carter, T. Webb, J. Landon, M. Elmer, , and B. D. Jeffs, "Design and characterization of an active impedance matched low noise phased array feed," IEEE Transactions on Antennas and Propagation, vol. 59, no. 6, pp. 1876­1885, Jun. 2011. [5] IEEE Standard Denitions of Terms for Radio Wave Propagation, IEEE Std. 211-1997, 1997. [6] J. P. Hamaker, J. D. Bregman, and R. J. Sault, "Understanding radio polarimetry - I. Mathematical foundations," Astronomy & Astrophysics Supplement Series, no. 117, pp. 137­147, May 1996. [7] A. Ludwig, "The definition of cross polarization," IEEE Transactions on Antennas and Propagation, vol. 21, no. 1, pp. 116­119, Jan. 1973. [8] K. F. Warnick and B. D. Jeffs, "Efficiencies and system temperature for a beamforming array," IEEE Antennas and Wireless Propagation Letters, vol. 7, pp. 565­568, 2008. [9] K. F. Warnick, B. Woestenburg, L. Belostotski, and P. Russer, "Minimizing the noise penalty due to mutual coupling for a receiving array," IEEE Trans. Ant. Propag., vol. 57, no. 6, pp. 1634­1644, June 2009. [10] M. V. Ivashina, R. Maaskant, and B. Woestenburg, "Equivalent system representation to model the beam sensitivity of receiving antenna arrays," IEEE Ant. Wireless Propag. Lett., vol. 7, pp. 733­737, 2008. [11] T. Carozzi, "A fundamental figure of merit for radio polarimeters," IEEE Transactions on Antennas and Propagation, vol. 59, no. 6, pp. 2058­2065, June 2011. [12] O. Smirnov, "Revisiting the radio interferometer measurement equation. I. a full-sky Jones formalism," Astronomy & Astrophysics, vol. 527, 2011. [13] R. J. Sault, J. P. Hamaker, and J. D. Bregman, "Understanding radio polarimetry - II. Instrumental calibration of an interferometer array," Astronomy & Astrophysics Supplement Series, no. 117, pp. 149­159, May 1996. [14] A. R. Thompson, J. M. Moran, and G. W. Swenson Jr., Interferometry and Synthesis in Radio Astronomy, 2nd Ed. Weinheim, Germany: WILEY-VCH Verlag GmbH & Co. KGaA, 2004. [15] H. C. v. d. Hulst, Light Scattering by Small Particles. New York: Dover Publications, 1981. [16] R. J. Sault and T. J. Cornwell, "The Hamaker-Bregman-Sault Measurement Equation," Synthesis Imaging in Radio Astronomy II ­ ASP Conference Series, vol. 180, pp. 657­669, 1999. [17] R. Horn and C. Johnson, Topics in matrix analysis. Cambridge University Press, 1994. [18] J. Wishart, "The generalised product moment distribution in samples from a normal multivariate population," Biometrika, vol. 20A, no. 1-2, pp. 32­52, 1928. [19] H. L. Van Trees, Optimum Array Processing. New York: John Wiley & Sons, 2002. [20] B. D. Jeffs, K. F. Warnick, J. Landon, J. Waldron, D. Jones, J. R. Fisher, and R. Norrod, "Signal processing for phased array feeds in radio astronomical telescopes," IEEE Journal of Selected Topics in Signal Processing, vol. 2, no. 5, pp. 635­646, July 2008. [21] Lazzio, J., SKA Science Working Group, "The square kilometre array design reference mission: SKA-mid and SKA-lo," www.skatelescope.org, 2010.