research article dual-polarized planar phased array

Research ArticleDual-Polarized Planar Phased Array Analysis forMeteorological Applications

Chen Pang1 Peter Hoogeboom2 Franccedilois Le Chevalier2 Herman W J Russchenberg2

Jian Dong1 Tao Wang1 and Xuesong Wang1

1National University of Defense Technology Deya Road 109 Changsha 410073 China2Delft University of Technology Stevinweg 1 2628 CD Delft Netherlands

Correspondence should be addressed to Chen Pang pangchen1017hotmailcom

Received 27 February 2015 Revised 19 July 2015 Accepted 22 July 2015

Academic Editor Stefano Selleri

Copyright copy 2015 Chen Pang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a theoretical analysis for the accuracy requirements of the planar polarimetric phased array radar (PPPAR)in meteorological applications Among many factors that contribute to the polarimetric biases four factors are considered andanalyzed in this study namely the polarization distortion due to the intrinsic limitation of a dual-polarized antenna elementthe antenna pattern measurement error the entire array patterns and the imperfect horizontal and vertical channels Twooperation modes the alternately transmitting and simultaneously receiving (ATSR) mode and the simultaneously transmittingand simultaneously receiving (STSR) mode are discussed For each mode the polarimetric biases are formulated As the STSRmode with orthogonal waveforms is similar to the ATSR mode the analysis is mainly focused on the ATSR mode and the impactsof the bias sources on the measurement of polarimetric variables are investigated through Monte Carlo simulations Some insightsof the accuracy requirements are obtained and summarized

1 Introduction

Recently the weather radar community has paid much atten-tion to the polarimetric phased array radar (PPAR) due to itsagile electronic beam steering capability which has the poten-tial to significantly advance weather observations [1] Varioussystem designs have been presented and studied A low costmobile X-band phased array weather radar with phase-tiltantenna array was developed in [2] Fulton and Chappell[3] designed an S-band differentially probe-fed stackedpatch antenna formultifunctional phased arrayweather radarapplications and studied the calibration method [4] Zhanget al [5] proposed a cylindrical configuration for the polari-metric phased array weather radar and illustrated the advan-tages of the cylindrical configuration over the planar configu-ration In [6] an overview of the calibration techniques toolsand challenges surrounding the development of a cylindricalpolarimetric phased array radar (CPPAR) demonstrator wasprovided The design of interleaved sparse arrays [7] for theagile polarization control was analyzed with the purpose

of meteorological applications Dong et al [8] analyzed thepolarization characteristics of two ideal orthogonal Huygenssources and evaluated their polarimetry performance

As shown in [1 9] a high-accuracy measurement ofpolarimetric variables is required to provide meaningfulinformation for reliable hydrometeor classifications andimproved quantitative precipitation estimations For exam-ple it is desirable that the measurement error for the differ-ential reflectivity 119885DR be on the order of 01 dB In additionit is desirable that the copolar correlation coefficient 120588ℎVerror be less than 001 In previous research [9ndash13] the pola-rimetric biases of weather radars with mechanically scanningantennas have been widely discussed A detailed literaturereview of the bias analysis and calibration methods waspresented in [9] Generally in order to make accurate polari-metric measurements by using a mechanically scanningantenna a narrowbeamwith low sidelobes low coaxial cross-polarization and high polarization isolation are indispens-able

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 340704 12 pageshttpdxdoiorg1011552015340704

2 International Journal of Antennas and Propagation

x

y

z

r

ar

a120579

a120593

120579

120593

rarrE2

rarrE1

rarrrmn

Figure 1 The array configuration and spherical coordinate systemused for radiated electric fields

Although the weather radar polarimetry has matured foryears there are some challenges for the planar polarimetricphased array radar (PPPAR) [14] As shown in Figure 1 thearray is placed on the 119910119911 plane When the beam is away fromthe principle planes the electric field 1 from the horizontal(119867) port and 2 from the vertical (119881) port are not necessarilyorthogonal which will introduce polarimetric biases thatare not negligible The nonorthogonality of the 119867 and 119881

polarizations is called the polarization distortion in thispaper Meanwhile the polarization distortion also includesmismatches in the power levels of 119867 and 119881 beams as afunction of the scan angle The calibration matrix that relieson the measured array patterns is needed to calibrate thepolarimetric bias due to the polarimetric distortion As themeasured antenna pattern always contains measurementerrors the calibration matrix cannot completely calibratethe bias due to the polarization distortion which is notthoroughly analyzed in previous research [1 15 16] andwill bediscussed in this study Moreover in [1 15 16] it implies thatthe beam is thin enough so that the calibration performedat the boresight is sufficient to retrieve the polarimetricvariables However in practice the finite beamwidth alsocontributes to the polarimetric bias which will be evaluatedin this paper Besides the antenna the imperfect 119867119881channels can still bias the polarimetric variables which willbe modeled and analyzed Actually other factors such asthe mismatch between element patterns spatial variationsof cross-polarization patterns mutual coupling edge effectsdiffracted fields and surfacewaves can significantly affect theoverall accuracy of a PPPAR To simplify the analysis thesefactors are ignored

Usually there are two operation modes chosen forweather observations the alternately transmitting and simul-taneously receiving (ATSR) mode and the simultaneouslytransmitting and simultaneously receiving (STSR) modeEach mode has its advantages and disadvantages With aldquoperfectrdquo antenna the STSR mode is vastly superior to theATSR mode in the worst-case polarimetricspectral situa-tions Thus the STSR mode is the preferred mode froma meteorological standpoint However both the theoretical

analysis and measurement experiments have shown that theSTSRmode has higher accuracy requirements than the ATSRmode This paper is mainly focused on the ATSR mode as itis simple for the analysis

The remainder of this paper is organized as followsSection 2 presents the array model Sections 3 and 4 give thedetailed analysis in the ATSR and STSR modes respectivelySummaries and conclusions are made in Section 5

2 Array Model

The coordinate system and array configuration are shownin Figure 1 It is common that in antenna measurements theantenna is placed on the 119909119910 plane In this situation the119867 and119881 vectors correspond to the second definition in [17] In thispaper the array with119872 rows and119873 columns is placed on the119910119911 plane which is different from the typical situation Thereason is that in meteorological applications when the arrayis placed on the 119910119911 plane the expressions of the horizontaland vertical polarization basis are simple which are writtenas

aℎ = a120593

aV = minus a120579(1)

where aℎ aV is the horizontal and so-called ldquoverticalrdquopolarization basis and a119903 a120579 and a120593 are unit vectors in thespherical coordinate system

We consider the array has a 90∘ angular range in azimuthand a 30∘ range in elevation which is applicable for weatherobservations Thus in Figure 1 120593 is from minus45∘ to 45∘ and120579 is from 60∘ to 90∘ For a well-designed array it wouldbe symmetrical with respect to 120593 So in this paper we onlyconsider 120593 from 0∘ to 45∘ and 120579 from 60∘ to 90∘ Accordinglythe beam direction (120579119878 120593119878) = (90

∘ 0∘) is the broadside of the

array

21 Element Pattern The element pattern in a dual-polarizedphased array can be written as

f (120579 120593) = [119891ℎℎ (120579 120593) 119891ℎV (120579 120593)

119891Vℎ (120579 120593) 119891VV (120579 120593)] (2)

where

(i) 119891ℎℎ(120579 120593) is the horizontal electric field componentwhen only the119867 port is excited

(ii) 119891ℎV(120579 120593) is the horizontal electric field componentwhen only the 119881 port is excited

(iii) 119891Vℎ(120579 120593) is the vertical electric field component whenonly the119867 port is excited

(iv) 119891VV(120579 120593) is the vertical electric field component whenonly the 119881 port is excited

For a practical dual-polarized antenna element the cross-polarization components 119891ℎV(120579 120593) and 119891Vℎ(120579 120593) are not 0 inthe beam scan area

International Journal of Antennas and Propagation 3

MCCFEC

RX V

TX HV

RX H

Control unit and SPI

H

V

(a)

MCCFEC

RX V

TX V

RX H

TX H


H

V

(b)

Figure 2 Dual-polarized 119879119877modules for polarimetric phased array weather radars (a) is for the ATSRmode and (b) is for the STSRmodeIn the ATSRmode one channel is shared for transmitting119867 and119881 signals and two independent channels are used for reception In the STSRmode four independent channels two for transmission two for reception and one 119879119877 switch to commute the transmission and receptionsignals are required

Transmission Reception

H

VV

HHH

HV

VH

VV

HH

HV

VH

VV

ahh

ahv

avh

avv

bhh

bhv

bvh

bvv

Figure 3 Channel imbalance and channel coupling

22 Transmission and Reception Patterns Figure 2 from [18]shows dual-polarized 119879119877 modules for polarimetric phasedarray weather radars in the ATSR and STSR modes Asexplained in [4] the 119879119877module connected to each elementmay have cross-coupling between its 119867 and 119881 channels aswell as complex gainphase imbalances These cross-coup-lings and imbalances can be modeled by a matrix multipli-cation of the 119867 and 119881 signals presented to the 119879119877 moduleon both transmission and reception with components asdesignated in Figure 3 In this paper we use the term ldquochannelisolationrdquo to express the cross-coupling between the119867 and119881channelsMeanwhile we use the term ldquochannel imbalancerdquo toexpress complex gainphase imbalances between the119867 and119881channels

For each element we use a 2 times 2 complex matrix A tomodel the channel imbalance and channel isolation for thetransmission while B is for the receptionA and B are writtenas

A = [

119886ℎℎ 119886ℎV

119886Vℎ 119886VV] (3)

B = [119887ℎℎ 119887ℎV

119887Vℎ 119887VV] (4)

where 119886ℎℎ 119886VV 119887ℎℎ and 119887VV describe the channel imbalanceand 119886ℎV 119886Vℎ 119887ℎV and 119887Vℎ give the channel isolation For

simplicity we assume 119886ℎℎ = 119887ℎℎ = 1 Then the channel imbal-ance CIM is defined as

CIM = max100381610038161003816100381610038161003816100381610038161003816

20 log 11003816100381610038161003816119886VV

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

20 log 11003816100381610038161003816119887VV

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

(5)

Note that if |119886VV| gt 1 there will be 20 log(1|119886VV|) lt 0 Thuswe use the expression |20 log(1|119886VV|)| such that CIM remainspositive

Similarly the channel isolation CIS can be defined as

CIS = 20 log(max 1

1003816100381610038161003816119886Vℎ1003816100381610038161003816

1003816100381610038161003816119886VV1003816100381610038161003816

1003816100381610038161003816119886ℎV1003816100381610038161003816

1

1003816100381610038161003816119887Vℎ1003816100381610038161003816

1003816100381610038161003816119887VV1003816100381610038161003816

1003816100381610038161003816119887ℎV1003816100381610038161003816

) (6)

which means CIS is defined as the worst value among 1|119886Vℎ||119886VV||119886ℎV| 1|119887Vℎ| and |119887VV||119887ℎV|

The array transmission pattern T(120579 120593) and receptionpattern R(120579 120593) are expressed as [19]

T (120579 120593) =119872

sum

119898=1

119873

sum

119899=1119883119898119899 (120579119878 120593119878) f119898119899 (120579 120593) sdotA119898119899

R (120579 120593) =119872

sum

119898=1

119873

sum

119899=1119884119898119899 (120579119878 120593119878) f119898119899 (120579 120593) sdotB119898119899

(7)

where (120579119878 120593119878) is the beam direction 119883119898119899(120579119878 120593119878) and119884119898119899(120579119878 120593119878) are weighting coefficients with respect to eachelement A119898119899 and B119898119899 model the imperfect channel effects

The mutual coupling between array elements is compli-cated so that a thorough analysis of the mutual couplingusually includes the full-wave electromagnetic computationand measurement experiments which is beyond the scope ofthis paper Moreover for a large array most of the elementsare far from an edge Therefore except for the phase centerdisplacement all of the central element patterns are nearlythe same So it is reasonable to use the array average elementpattern to replace the single element pattern Hence T(120579 120593)and R(120579 120593) reduce to

T (120579 120593) = fave (120579 120593) F119879 (8)

R (120579 120593) = fave (120579 120593) F119877 (9)


where fave(120579 120593) is called the array average element patternThe subscripts 119879 and 119877 in (8) and (9) represent the transmis-sion and reception respectively F119879 and F119877 are written as

F119879 =119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903)119883119898119899 (120579119878 120593119878)A119898119899

F119877 =119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903) 119884119898119899 (120579119878 120593119878)B119898119899

(10)

3 Array Analysis in ATSR Mode

31 Formulation For a point target with the polarizationscattering matrix (PSM) S in the direction (120579 120593) at the range119903 the received voltage matrix can be written as

V = 119862exp (minus1198952119896119903)

41205871199032R119905 sdot S sdotT sdotEinc

ATSR (11)

where119862 is a gain term Here the superscript ldquo119905rdquo meansmatrixtranspose 119896 = 2120587120582 and 120582 is the wavelength Einc

ATSR is the unitexcitation for119867 and 119881 ports which is written as

EincATSR = [

1 00 1

] (12)

The received voltage matrix for distributed precipitationscan be expressed as an integral Consider

V = intΩ

R119905 (120579 120593) sdot S (120579 120593) sdotT (120579 120593) 119889Ω (13)

where Ω is the solid angle and 119889Ω = sin 120579119889120579 119889120593 In (13) thegain term 119862 and the term related to range 119903 are dropped forthe sake of simplicity To retrieve S(120579 120593) the calibrated volt-age matrix can be expressed as

V = C119877 sdotV sdotC119879 = intΩ

C119877 sdotR119905sdot S sdotT sdotC119879119889Ω (14)

where the calibration matrices C119879 and C119877 are expressed as

C119879 = (T (120579 120593)1003816100381610038161003816120579=120579119878120593=120593119878

)minus1

C119877 = (R119905(120579 120593)

10038161003816100381610038161003816120579=120579119878120593=120593119878)

minus1

(15)

C119879 and C119877 can be obtained through array pattern measure-ments By defining

R = R sdot (C119877)119905= [

ℎℎ ℎV

Vℎ VV

]

T = T sdotC119879 = [ℎℎ ℎV

Vℎ VV

]

(16)

the calibrated voltage matrix is written as

V = intΩ

R119905 sdot S sdot T 119889Ω (17)

Assuming

S (120579 120593) = [119878ℎℎ (120579 120593) 0

0 119878VV (120579 120593)] (18)

the intrinsic differential reflectivity 119885DR is defined as

119885DR = 10 log⟨1003816100381610038161003816119878ℎℎ

1003816100381610038161003816

2⟩

⟨1003816100381610038161003816119878VV

1003816100381610038161003816

2⟩

(19)

where ⟨sdot⟩ means the ensemble average The bias of 119885DR canbe calculated as

119885119887

DR = 10 log119875ℎℎ

119875VVminus119885DR (20)

where 119875119894119895 (119894 119895 = ℎ V) is the received power Meanwhile theintegrated cross-polarization ratio (ICPR) is calculated as

ICPR = 10 log119875Vℎ

119875ℎℎ

(21)

which is the minimal linear depolarization ratio 119871DR thatcan be measured by a weather radar After some trivialmathematical derivations we get

119885119887

DR = 10 log1 +1198821119885

minus12dr

1198822 +119882311988512dr

ICPR = 10 log1198824 +1198825119885

minus1dr +1198826119885

minus12dr

1 +1198821119885minus12dr

(22)

where

120588ℎV exp (119895120601DP) =⟨119878lowast

ℎℎ119878VV⟩

radic⟨1003816100381610038161003816119878ℎℎ

1003816100381610038161003816

2⟩ ⟨1003816100381610038161003816119878VV

1003816100381610038161003816

2⟩

(23)

1198821 =intΩ2Re (VℎVℎ

lowast

ℎℎlowast

ℎℎ120588ℎV exp (119895120601DP)) 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(24)

1198822 =intΩ

10038161003816100381610038161003816VV10038161003816100381610038161003816

2 10038161003816100381610038161003816VV10038161003816100381610038161003816

2

119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(25)

1198823 =intΩ2Re [VVVV

lowast

ℎVlowast

ℎV120588ℎV exp (119895120601DP)] 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(26)

1198824 =intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎV1003816100381610038161003816

2119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(27)

1198825 =intΩ

1003816100381610038161003816119879Vℎ1003816100381610038161003816

2 1003816100381610038161003816119877VV1003816100381610038161003816

2119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(28)

1198826 =intΩ2Re [119879Vℎ119877VV119879

lowast

ℎℎ119877lowast

ℎV120588ℎV exp (119895120601DP)] 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(29)


In (23) 120588ℎV is called the copolar correlation coefficient and120601DP represents the differential phase The symbol lowast meanscomplex conjugate

According to (22) it is clear that119885119887DR and ICPR are relatedto both the array patterns and the intrinsic119885drThe impacts of119885dr and 120588ℎV exp(119895120601DP) on119885

119887

DR have been thoroughly analyzedin [9] and those conclusions can be directly applied for theanalysis of a PPPAR Hence in this paper we assume 119885dr = 1and 120588ℎV exp(119895120601DP) = 1 so that we can focus on the biases dueto the radar system

32 Array with Perfect HV Channels An array with perfect119867119881 channels means that the channel imbalance and isola-tion can be ignored Hence the transmission and receptionpatterns can be written as

T (120579 120593)

= fave (120579 120593)119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903)119883119898119899 (120579119878 120593119878)

R (120579 120593)

= fave (120579 120593)119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903) 119884119898119899 (120579119878 120593119878)

(30)

For the transmission pattern the radiation power isprincipal Thus a uniform illumination is applied For thereception pattern a beamwith low sidelobes is desired Herewe choose the Taylor weighting So R and T are written as

T = T sdotC119879 = 119865uni (120579 120593) fave (120579 120593) sdotC119879

R = R sdot (C119877)119905= 119865tay (120579 120593) fave (120579 120593) sdot (C119877)

119905

(31)

where 119865uni(120579 120593) and 119865tay(120579 120593) are the array factors of theuniform and Taylor weightings According to the definitionsof C119879 and C119877 T and R can be modeled as

T = 119865uni [1 + 120576119879ℎℎ(120579 120593) 120576

119879

ℎV (120579 120593)

120576119879

Vℎ (120579 120593) 1 + 120576119879VV (120579 120593)] (32)

R = 119865tay [1 + 120576119877ℎℎ(120579 120593) 120576

119877

ℎV (120579 120593)

120576119877

Vℎ (120579 120593) 1 + 120576119877VV (120579 120593)] (33)

where 120576119894119895 are the error terms after the calibration 119865uni and119865tay are the normalized array factors The superscripts 119879 and119877 represent the transmission and reception which are usuallydropped for simplicity To simplify the analysis 120576119894119895 is modeledas

120576119894119895 (120579 120593) asymp 120572119894119895 (120579 minus 120579119878) + 120573119894119895 (120593 minus120593119878) + 120575119894119895 (34)

where 120572119894119895 120573119894119895 and 120575119894119895 are complex numbers The unit of 120579and 120593 is radian If C119879 and C119877 have no error there will be120576119894119895 = 0 at (120579119878 120593119878) that is 120575119894119895 = 0 However due to theantenna pattern measurement errors 120575119894119895 is not 0 In addition120572119894119895 and 120573119894119895 indicate the polarization variation near (120579119878 120593119878)

0 002 004 006 008 010

02

04

06

08

1

Simulation

|Zb D

R|

(dB)

Δ

10Δ

Figure 4 The relation between Δ and |119885119887DR|

It should be pointed out that the linear error model (34) ismost appropriate for well-behaved elements making up anarray that is large enough to ensure that it is accurate overthe beamwidth of the overall array

According to Appendix we know that the upper boundof 120575119894119895 has the same level as the relative error upper bound 119864119891of the antenna measurements Therefore in the rest of thispaper we just focus on 120575119894119895

First we analyze a simple case to get some insightstowards 120575119894119895We assume there is only one spherical scatterer inthe beam direction (120579119878 120593119878) indicating119885DR = 0 dB and 119871DR =minusinfin dB So we can ignore the impacts of the finite beamwidthand sidelobes Consequently 119885119887DR can be calculated as

119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119877

ℎV120575119879

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(35)

Furthermore we assume |120575119894119895| = Δ and the phase of 120575119894119895 isuniformly distributed in [0 2120587] Using Taylor expansion andignoring the second and higher order terms we can get theapproximation of the average bias of |119885DR|

1003816100381610038161003816119885119887

DR1003816100381610038161003816 asymp 10Δ (36)

where ∙ means mathematical expectation Since 119885119887DR has asymmetric distribution centered at 0 there is119885119887DR = 0Hencewe use |119885119887DR| other than 119885119887DR Figure 4 shows the relationbetween Δ and |119885119887DR| The red line is calculated from (36)while the blue line is obtained through Monte Carlo simula-tion inwhich 120575119879

119894119895and 120575119877119894119895are generated from a randomnumber

generator and 119885119887DR is calculated from (35) In Figure 4 we seethat the approximation from (36) agrees well with the resultfromMonte Carlo simulation

Using the same procedure the average ICPR is derived in(37) Figure 5 shows the relation between Δ and ICPR

ICPR asymp 20 log (Δ) (37)


Table 1 Array parameters

Array size 64 times 64Elements separation 1205822Sidelobe level of 119865tay minus40 dB

Simulation

Δ

0 002 004 006 008 01

ICPR

(dB)

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

20log(Δ)

Figure 5 The relation between Δ and ICPR

From Figures 4 and 5 we know that the calibration error120575119894119895 has great impacts on119885119887DR and ICPR For a single sphericalscatterer to achieve |119885119887DR| lt 01 dB Δ should be less than001 which is really demanding for antenna pattern mea-surements Moreover we see that the relation between Δ and|119885119887

DR| is linear while the relation between Δ and ICPR islogarithmic

As revealed in [9 12 13] the finite beamwidth has consid-erable impacts on themeasurement of polarimetric variablesIn order to evaluate the bias under different conditions amethod based on Monte Carlo simulation is developed sothat we can evaluate the polarimetric bias with differentparameters The array parameters are shown in Table 1 Thesimulation procedure is shown below

Step 1 Specify the polarization distortion calibration error|120575119894119895|

Step 2 Generate 120572119894119895 120573119894119895 and 120575119894119895 through a random numbergenerator

Step 3 Calculate T from (32) and R from (33)

Step 4 Calculate V from (17)

Step 5 Calculate 119885119887DR from (20) and ICPR from (21)

The simulation parameters are listed in Table 2 119880(119886 119887)means the uniform distribution in [119886 119887] and Arg(119911) repre-sents the phase of 119911 It should be pointed out that a minus40 dBTaylor weighting is not practical for the implementation

Table 2 Simulation parameters

119885DR 0 dB119871DR minusinfin dB|120575119894119895| 001|120572119894119895| 119880(0 2)|120573119894119895| 119880(0 2)

Arg(120572119894119895) 119880(0 2120587)Arg(120573119894119895) 119880(0 2120587)Arg(120575119894119895) 119880(0 2120587)

Table 3 The model parameters of a pair of crossed dipoles in thebeam direction (60∘ 45∘)

120572 120573 120575 1198772

120576hh 00000 minus11092 minus00013 099120576hv 0 0 0 None120576vh 11526 minus04989 00007 099120576vv 05769 00000 minus00013 099

According to [14] for weather observations the two-waysidelobe level of a PPPAR is expected to be under minus54 dBwhich is equal to that of the WSR-88D Thus the simulatedresults with a minus13 dB uniform weighting and minus40 dB Taylorweighting are more comparable to those of radars withmechanical scanning antennas

The ranges of 120572119894119895 and 120573119894119895 in Table 2 are determined basedon the radiation pattern of a pair of crossed dipoles which iswritten as

fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579] (38)

The calibrated pattern fdipole(120579 120593) is then written as

fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579]

sdot [

cos120593119878 0cos 120579119878 sin120593119878 sin 120579119878

]

minus1

equiv [

1 + 120576ℎℎ (120579 120593) 120576ℎV (120579 120593)

120576Vℎ (120579 120593) 1 + 120576VV (120579 120593)]

(39)

Choosing (120579119878 120593119878) = (60∘ 45∘) we calculate 120576119894119895(120579 120593) By using

Matlab Curve Fitting Toolbox we calculate the parameters120572119894119895 120573119894119895 and 120575119894119895 and show them in Table 3 1198772 is called thecoefficient of determination which is a number that indi-cates how well data fit a statistical model As shown inTable 3 1198772 with respect to 120576119894119895 is 099 indicating a very goodapproximation performance of the linear error modelAccording toTable 3we know120572119894119895 120573119894119895 sim 119880(0 2) is validUsingthe same procedure we calculate the parameters 120572119894119895 120573119894119895 and120575119894119895 for a pair of crossed dipoles with the length of 120582 and showthem in Table 4 Actually the practical phased array usuallyhas an element spacing of about 1205822 Thus the ranges of 120572119894119895


Table 4Themodel parameters of a pair of crossed dipoles with thelength of 120582 in the beam direction (60∘ 45∘)

120572 120573 120575 1198772

120576hh minus10078 minus27556 00076 099120576hv 0 0 0 None120576vh minus25394 minus03672 00003 099120576vv 23362 00000 minus00010 099

60

65

70

75

80

85

900094

0096

0098

01

0102

0104

0106

120593 (deg)0 10 20 30 40

120579(d

eg)

Figure 6 |119885119887DR| in dB with |120575119894119895| = 001 |120572119894119895| |120573119894119895| sim 119880(0 2)

and 120573119894119895 should be better than the worst value in Table 4 Inthis paper we assume |120572119894119895| lt 4 and |120573119894119895| lt 4

Figures 6 and 7 show the simulated |119885119887DR| and ICPR inthe beam scan area [60∘ 90∘] times [0∘ 45∘] In Figure 6 mostof |119885119887DR| are between 0095 dB and 0105 dB which agreeswith the approximation of (36) Figure 6 indicates that theimpact of the finite beamwidth on 119885119887DR is not obvious and119885119887

DR is not sensitive to the beam expansion due to the beamscan On the contrary in Figure 7 the impact of the beamexpansion on ICPR is obvious with about a 25 dB differencebetween the broadside and the beam direction (60∘ 45∘)Furthermore the ICPR of minus326 dB at the broadside inFigure 7 is much larger than that of minus40 dB calculated from(37) indicating that the finite beamwidth considerably affectsthe measurement of 119871DR

We then set |120575119894119895| = 002 |120572119894119895| |120573119894119895| isin 119880(0 4) and keepother parameters the same as those in Tables 1 and 2 Figures8 and 9 show the simulation results In Figure 8most of |119885119887DR|are between 019 dB and 021 dB which also agrees with (36)very well In Figure 9 the difference between the broadsideand the beam direction (60∘ 45∘) is about 25 dB and theminimal ICPR at the broadside (90∘ 0∘) is about minus265 dBincreasing by about 75 dB compared with that calculatedfrom the approximation of (37)

60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

120579(d

eg)

minus325

minus32

minus315

minus31

minus305

Figure 7 ICPR in dB with |120575119894119895| = 001 |120572

119894119895| |120573119894119895| sim 119880(0 2)

60

65

70

75

80

85

900185

019

0195

02

0205

021

0215

120593 (deg)0 10 20 30 40

120579(d

eg)


60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

minus265

minus26

minus255

minus25

minus245

minus24

120579(d

eg)

Figure 9 ICPR in dB with |120575119894119895| = 002 |120572119894119895| |120573119894119895| sim 119880(0 4)


33 Array with Imperfect HV Channels With imperfect119867119881 channels R and T can be written as

T (120579 120593) = [1 + 120576119879

ℎℎ120576119879

ℎV

120576119879

Vℎ 1 + 120576119879

VV

] F119879

R (120579 120593) = [1 + 120576119877

ℎℎ120576119877

ℎV

120576119877

Vℎ 1 + 120576119877

VV

] F119877

(40)

A and B are expressed as

A = [

1 120578ℎV exp (119895120574ℎV)120578Vℎ exp (119895120574Vℎ) 120578VV exp (119895120574VV)

]

B = [1 120591ℎV exp (119895120595ℎV)

120591Vℎ exp (119895120595Vℎ) 120591VV exp (119895120595VV)]

(41)

First we analyze the case with a single spherical scattererin the beam direction (120579119878 120593119878) Thus we just need to considerT(120579 120593)|120579=120579119878120593=120593119878 and R(120579 120593)|120579=120579119878120593=120593119878 Since 119883119898119899(120579119878 120593119878) and119884119898119899(120579119878 120593119878) compensate the phase displacement exp(119895119896 119903119898119899 sdota119903) we can get

T (120579119878 120593119878) = [1 + 120575119879

ℎℎ120575119879

ℎV

120575119879

Vℎ 1 + 120575119879

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119883119898119899 (120579119878 120593119878)

10038161003816100381610038161003816A119898119899

R (120579119878 120593119878) = [1 + 120575119877

ℎℎ120575119877

ℎV

120575119877

Vℎ 1 + 120575119877

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119898119899 (120579119878 120593119878)

10038161003816100381610038161003816B119898119899

(42)

As sum119872119898=1sum

119873

119899=1 |119883119898119899(120579119878 120593119878)| = 1 and sum119872

119898=1sum119873

119899=1 |119898119899(120579119878120593119878)| = 1 the double summations on the right sides of (42)are close to the mathematical expectations of A and B If120574ℎV 120574Vℎ 120595ℎV 120595Vℎ sim 119880(0 2120587) we have 119886ℎV = 0 119886Vℎ = 0 119887ℎV = 0and 119887Vℎ = 0 In this situation 119885119887DR can be calculated as

119885119887

DR

= 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1

119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119879

ℎV120575119877

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(43)

If 120575119879119894119895= 120575119877

119894119895= 0 (43) reduces to

119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

(44)

Assuming |119886VV| = |119887VV| lt 1 based on the definition of CIM in(5) (44) can be expressed as

119885119887

DR = 2CIM (45)

0 001 002 003 004 005 006 007 008 009 0102

03

04

05

06

07

08

09

1

11

|Zb D

R|

(dB)

Δ

Figure 10 |119885119887DR| of a single spherical scatterer with imperfect119867119881channels

Figure 10 shows the simulated |119885119887DR| of a single sphericalscatterer based on (43) in which 120578VV = 120591VV = 099 120574VV 120595VV sim

119880(minus10∘ 10∘) 120578ℎV = 120578Vℎ = 120591ℎV = 120591Vℎ = 0 |120575119894119895| = Δ andArg(120575119894119895)sim 119880(0 2120587) In Figure 10 whenΔ lt 001 |119885119887DR| almostremains constant around 026 dB and when Δ gt 002 |119885119887DR|increases linearly with a slope less than 10 When Δ is smallthe channel imbalance has the main contribution to 119885119887DROtherwise the polarization distortion calibration error is thedominant bias source

In order to evaluate the bias under different conditionswe use the Monte Carlo simulation method again which isshown below

Step 1 Specify |120575119894119895|

Step 2 Generate 120572119894119895 120573119894119895 and 120575119894119895 from a random numbergenerators and then calculate 120576119894119895

Step 3 Generate A119898119899 and B119898119899 for each element

Step 4 Calculate R and T

Step 5 Calculate 119885119887DR and ICPR

Figures 11 and 12 show simulated |119885119887DR| and ICPR in thebeam direction (90∘ 0∘) in which |120572119894119895| = |120573119894119895| = 2 120578ℎV = 120578Vℎ =120591ℎV = 120591Vℎ = 0 120578VV 120591VV sim 119873(099 001

2) and 120574VV 120595VV sim 119880(minus10

∘

10∘) In Figure 11 the results with the finite beamwidthmatchthe results of a single spherical scatterer well HoweverFigure 12 indicates the finite beamwidth considerably affectsICPR when Δ is small

We then set 120578ℎV 120578Vℎ 120591ℎV 120591Vℎ sim 119873(001 0012) 120574ℎV 120574Vℎ 120595ℎV

120595Vℎ sim 119880(minus10∘ 10∘) and keep other parameters the same asthose in Figures 11 and 12 to evaluate the impact of the channelisolation The simulated |119885

119887

DR| and ICPR considering theimperfect channel isolation are given in Figures 13 and 14with the blue squares while the simulated |119885119887DR| and ICPRfrom Figures 11 and 12 are still shown with the red triangles


0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11

Single spherical targetFinite beamwidth

minus001

|Zb D

R|

(dB)

Δ

Figure 11 |119885119887DR| of a single spherical scatterer and finite beamwidth


0 001 002 003 004 005 006 007 008 009 01minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

ICPR

(dB)

Δ

Figure 12 ICPR of a single spherical scatterer and finite beamwidth

In Figure 13 the impact of the imperfect channel isolationon 119885119887DR is not obvious However in Figure 14 the impact onICPR is obvious when Δ is small

4 Array Analysis in STSR Mode

In the STSR mode the received voltages for distributed pre-cipitations are expressed as

[

119881ℎ (119905)

119881V (119905)]

= intΩ

R119905 (120579 120593) sdot S (120579 120593) sdotT (120579 120593) sdot [119904ℎ (119905)

119904V (119905)] 119889Ω

(46)

0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11

Without channel couplingWith channel coupling

minus001

|Zb D

R|

(dB)

Δ

Figure 13 The impact of the channel coupling on |119885119887DR|

0 001 002 003 004 005 006 007 008 009 01 011


minus001minus36

minus33

minus30

minus27

minus24

minus21

minus18

Δ

ICPR

(dB)

Figure 14 The impact of the channel coupling on ICPR

where 119904ℎ(119905) and 119904V(119905) represent two waveforms from the 119867and 119881 channels Assuming 119878ℎV(120579 120593) = 119878Vℎ(120579 120593) = 0 (46) canbe written as

119881ℎ (119905) = (119877ℎℎ119878ℎℎ119879ℎℎ +119877Vℎ119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎℎ119878ℎℎ119879ℎV +119877Vℎ119878VV119879VV) 119904V (119905)

119881V (119905) = (119877ℎV119878ℎℎ119879ℎℎ +119877VV119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎV119878ℎℎ119879ℎV +119877VV119878VV119879VV) 119904V (119905)

(47)

As shown in (47) 119881ℎ(119905) is contaminated by both the first-and second-order terms of the cross-polar patterns In theATSR mode the received voltages are just contaminated bythe second-order terms of the cross-polar patterns Thus


the accuracy requirement in the STSRmode should be higherthan that in the ATSR mode As discussed in Section 3 inthe ATSR mode the relative error of the antenna patternmeasurement should be under 1 to achieve 119885119887DR lt 01 dBHence the accuracy requirement in the STSR mode is moredemanding

The orthogonal waveforms are usually employed toimprove the polarimetric performance [20ndash23] The receivedvoltages after passing the matched filters of the 119867 and 119881channels can be written as

[

119881ℎℎ (119905) 119881ℎV (119905)

119881Vℎ (119905) 119881VV (119905)] = [

119881ℎ (119905) otimes 119904ℎ (119905) 119881ℎ (119905) otimes 119904V (119905)

119881V (119905) otimes 119904ℎ (119905) 119881V (119905) otimes 119904V (119905)]

= intΩ

R119905 sdot S sdotT sdot [119904ℎ (119905) otimes 119904ℎ (119905) 119904ℎ (119905) otimes 119904V (119905)

119904V (119905) otimes 119904ℎ (119905) 119904V (119905) otimes 119904V (119905)] 119889Ω

(48)

where 119904ℎ(119905) = 119892ℎ119904lowast

ℎ(1199050 minus 119905) and 119904V(119905) = 119892V119904

lowast

V (1199050 minus 119905) are thematched filters of the119867 and119881 channels respectivelyotimesmeanssignal convolution If 119904ℎ(119905) and 119904V(119905) are completely ortho-gonal we can get

[


119904V (119905) otimes 119904ℎ (119905) 119904V (119905) otimes 119904V (119905)] prop [

1 00 1

] (49)

In this situation (48) is equivalent to (13) derived in theATSRmode Thus the same calibration procedure and analysis inSection 3 can be applied

In practice 119904ℎ(119905) and 119904V(119905) cannot be completely orthogo-nal Then we define

P (119905) = [119904ℎ (119905) otimes 119904ℎ (119905) 119904ℎ (119905) otimes 119904V (119905)

119904V (119905) otimes 119904ℎ (119905) 119904V (119905) otimes 119904V (119905)] (50)

So (48) can be written as

V (119905) = intΩ

R119905 (120579 120593) sdot S (120579 120593) sdotT (120579 120593) sdotP (119905) 119889Ω (51)

Accordingly the calibrated voltagematrix V(119905) can be writtenas

V (119905) = C119877 sdotV (119905) sdotPminus1(119905) sdotC119879 (52)

where C119879 and C119877 are defined in (15)Once the waveforms 119904ℎ(119905) and 119904V(119905) are known P(119905) can

be calculated from (50)Then the calibration procedure in theSTSR mode is still the same as that in the ATSR mode

5 Conclusions

In this paper we analyze the accuracy requirements of aPPPAR in the ATSR and STSR modes Among many factorswe focus on the polarization distortion due to the intrinsiclimitation of a dual-polarized antenna element the antennapattern measurement error the entire array patterns and theimperfect 119867119881 channels Other factors such as the mutualcoupling between the array elements are also important forthe accurate weather measurement However these factorsare ignored to simplify the analysis in this study

The polarization distortion calibration error 120575119894119895 that hasthe same level as the relative error upper bound 119864119891 of theantenna pattern measurement is critical for the biases of 119885DRand 119871DR 120575119894119895 should be under 001 to achieve 119885DR lt 01 dBindicating that 119864119891 should be under 1 The imperfect119867 and119881 channels have considerable contributions to the biases of119885DR and 119871DRThe channel isolation CIS should be over 40 dBso that the impact of the channel isolation is negligiableAccording to (45) the channel imbalance CIM should beunder 005 dB to ensure 119885119887DR lt 01 dB

The finite beamwidth considerably affects the measure-ment of 119871DR However the measurement of 119885DR is notsensitive to the finite beamwidth Moreover the impact ofthe beam scan that results in the beam expansion is justobvious on the measurement of 119871DR Therefore for a largearray with a narrow beam that is commonly used for weatherradars the measurement performance of 119885DR can be directlyestimated through the measurement at the boresight whichcan significantly simplify the analysis for the measurement of119885DR

In the STSR mode if orthogonal waveforms are appliedthe analysis is the same as that in the ATSRmode Otherwisethe measurement performance may be worse than that inthe ATSR mode In the future research those ignored factorscould be taken into account to have a better understandingabout the accuracy requirements of a PPPAR

Appendix

Polarization Distortion Calibration Error

Themeasured element pattern f119898 can be expressed as

f119898 = f + e = [119891ℎℎ 119891ℎV

119891Vℎ 119891VV]+[

119890ℎℎ 119890ℎV

119890Vℎ 119890VV] (A1)

where e represents the absolute measurement error and f isthe real element pattern It should be pointed out that themeasured element pattern f119898 does not include the errorsassociated with the imperfect 119867 and 119881 channels whichare characterized by the matrices A and B in (3) and (4)respectively The matricesA and B are integrated in the arraytransmission and reception patterns to account for the biascontributions of the imperfect119867 and 119881 channels Thereforewe can just focus on the measurement error of f119898

The calibrated element pattern f can be expressed as

f = f sdot (f119898)minus1 (A2)

By using the Matrix Inversion Lemma we can get

(f119898)minus1= (f + e)minus1 = fminus1 minus fminus1 (fminus1 + eminus1)

minus1

fminus1 (A3)

Substituting (A3) in (A2) we can get

f = Iminus (fminus1 + eminus1)minus1fminus1 (A4)


where I is the identitymatrixThus the polarization distortioncalibration error can be written as

f minus I = [120575ℎℎ 120575ℎV

120575Vℎ 120575VV] = minus (fminus1 + eminus1)

minus1fminus1 (A5)

Equation (A5) shows that the polarization distortion cali-bration error 120575119894119895 is related to both the absolute measurementerror e and the element pattern f itself

For awell-designed antenna element the copolar patternsare larger than the cross-polar patterns In addition since|119891119894119895| ≫ |119890119894119895| the matrix elements of fminus1 are far less than thematrix elements of eminus1 Thus (A5) can be approximated as

f minus I = minus (fminus1 + eminus1)minus1fminus1 asymp minus (eminus1)

minus1fminus1

asymp minus e sdot fminus1(A6)

It should be pointed out that (A6) is only valid under thecondition that the matrix elements of fminus1 are far less than thematrix elements of eminus1 Fortunately this condition is usuallysatisfied for a well-designed antenna element in the beamscan area

Since the relative error is more essential to represent themeasurement accuracy than the absolute error we define therelative error upper bound 119864119891 of e as

119864119891 = max119894119895=ℎV

10038161003816100381610038161003816119890119894119895

1003816100381610038161003816100381610038161003816100381610038161003816119891119894119895

10038161003816100381610038161003816

(A7)

According to (A6) the polarization distortion calibrationerror can be estimated as

f minus I asymp minus1119891ℎℎ119891VV minus 119891ℎV119891Vℎ

sdot[[[

[

119890ℎℎ

119891ℎℎ

119891ℎℎ119891VV minus119890ℎV

119891ℎV119891ℎV119891Vℎ 119891ℎℎ119891ℎV (

119890ℎV

119891ℎVminus119890ℎℎ

119891ℎℎ

)

119891Vℎ119891VV (119890Vℎ

119891Vℎminus119890VV

119891VV)

119890VV

119891VV119891ℎℎ119891VV minus

119890Vℎ

119891Vℎ119891ℎV119891Vℎ

]]]

]

(A8)

Consequently we can get

1003816100381610038161003816120575ℎℎ1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575ℎV1003816100381610038161003816 le

21003816100381610038161003816119891ℎℎ119891ℎV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575Vℎ1003816100381610038161003816 le

21003816100381610038161003816119891Vℎ119891VV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575VV1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

(A9)

For a well-designed antenna element (A9) reveals that119864119891 and |120575119894119895| are at the same level Thus we just focus onthe polarization distortion calibration error 120575119894119895 in this paperWhen we need to focus on a specific array the analysis hereincan be applied

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grant nos 61201330 61490690and 61490694 The authors sincerely express their gratitudeto the anonymous reviewers

References

[1] G Zhang R J Doviak D S Zrnic J Crain D Staiman andY Al-Rashid ldquoPhased array radar polarimetry for weathersensing a theoretical formulation for bias correctionsrdquo IEEETransactions on Geoscience and Remote Sensing vol 47 no 11pp 3679ndash3689 2009

[2] R H Medina E J Knapp J L Salazar and D J McLaughlinldquoTr module for casa phase-tilt radar antenna arrayrdquo in Pro-ceedings of the 7th European Microwave Integrated CircuitsConference (EuMIC rsquo12) pp 913ndash916 IEEE October 2012

[3] C Fulton andW Chappell ldquoA dual-polarized patch antenna forweather radar applicationsrdquo in Proceedings of the IEEE Inter-national Conference onMicrowaves Communications Antennasand Electronic Systems (COMCAS rsquo11) pp 1ndash5 November 2011

[4] C Fulton and W J Chappell ldquoCalibration of a digital phasedarray for polarimetric radarrdquo in Proceedings of the IEEE MTT-SInternational Microwave Symposium Digest (MTT rsquo10) pp 161ndash164 IEEE May 2010

[5] G Zhang R J Doviak D S Zrnic R Palmer L Lei and YAl-Rashid ldquoPolarimetric phased-array radar for weather mea-surement a planar or cylindrical configurationrdquo Journal ofAtmospheric and Oceanic Technology vol 28 no 1 pp 63ndash732011

[6] C Fulton G Zhang W Bocangel L Lei R Kelley and MMcCord ldquoCylindrical Polarimetric phased array radar a multi-function demonstrator and its calibrationrdquo in Proceedings ofthe IEEE International Conference on Microwaves Communica-tions Antennas and Electronic Systems (COMCAS rsquo13) pp 1ndash5October 2013

[7] M Sanchez-Barbetty RW Jackson and S Frasier ldquoInterleavedsparse arrays for polarization control of electronically steeredphased arrays for meteorological applicationsrdquo IEEE Transac-tions on Geoscience and Remote Sensing vol 50 no 4 pp 1283ndash1290 2012

[8] JDongQ Liu andXWang ldquoNewpolarization basis for polari-metric phased array weather radar theory and polarimetricvariables measurementrdquo International Journal of Antennas andPropagation vol 2012 Article ID 193913 15 pages 2012

[9] D Zrnic R Doviak G Zhang and A Ryzhkov ldquoBias in dif-ferential reflectivity due to cross coupling through the radiationpatterns of polarimetric weather radarsrdquo Journal of Atmosphericand Oceanic Technology vol 27 no 10 pp 1624ndash1637 2010

[10] V Chandrasekar and R J Keeler ldquoAntenna pattern analysisandmeasurements formultiparameter radarsrdquo Journal of Atmo-spheric amp Oceanic Technology vol 10 no 5 pp 674ndash683 1993

[11] D N Moisseev C M H Unal H W J Russchenberg and LP Ligthart ldquoImproved polarimetric calibration for atmosphericradarsrdquo Journal of Atmospheric and Oceanic Technology vol 19no 12 pp 1968ndash1977 2002


[12] Y Wang and V Chandrasekar ldquoPolarization isolation require-ments for linear dual-polarization weather radar in simulta-neous transmission mode of operationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 44 no 8 pp 2019ndash20282006

[13] M Galletti and D S Zrnic ldquoBias in copolar correlation coeffi-cient caused by antenna radiation patternsrdquo IEEE Transactionson Geoscience and Remote Sensing vol 49 no 6 pp 2274ndash22802011

[14] C Fulton J Herd S Karimkashi G Zhang and D ZrnicldquoDual-polarization challenges in weather radar requirementsfor multifunction phased array radarrdquo in Proceedings of the 5thIEEE International Symposium on Phased Array Systems andTechnology (ARRAY rsquo13) pp 494ndash501 October 2013

[15] D S Zrnic G Zhang and R J Doviak ldquoBias correction anddoppler measurement for polarimetric phased-array radarrdquoIEEE Transactions on Geoscience and Remote Sensing vol 49no 2 pp 843ndash853 2011

[16] L Lei G Zhang and R J Doviak ldquoBias correction for pola-rimetric phased-array radar with idealized aperture and patchantenna elementsrdquo IEEETransactions onGeoscience andRemoteSensing vol 51 no 1 pp 473ndash486 2013

[17] A Ludwig ldquoThe definition of cross polarizationrdquo IEEE Trans-actions on Antennas and Propagation vol 21 no 1 pp 116ndash1191973

[18] J L Salazar R HMedina and E Loew ldquoTRmodules for activephased array radarsrdquo in Proceedings of the IEEE InternationalRadar Conference (RadarCon rsquo15) pp 1125ndash1133 Arlington VaUSA May 2015

[19] C Fulton andWChappell ldquoCalibration of panelized polarimet-ric phased array radar antennas a case studyrdquo in Proceedings ofthe 4th IEEE International Symposium on Phased Array Systemsand Technology (ARRAY rsquo10) pp 860ndash867 October 2010

[20] D Giuli M Fossi and L Facheris ldquoRadar target scatteringmatrix measurement through orthogonal signalsrdquo IEE Proceed-ings Part F Radar and Signal Processing vol 140 no 4 pp 233ndash242 1993

[21] O A Krasnov and L P Ligthart ldquoRadar polarimetry usingsounding signals with dual orthogonality-PARSAX approachrdquoin Proceedings of the 7th European Radar Conference (EuRADrsquo10) pp 121ndash124 September 2010

[22] G Galati G Pavan and S Scopelliti ldquoMPAR waveform designfor the weather functionrdquo in Proceedings of the European RadarConference (EuRAD rsquo10) pp 152ndash155 IEEE Paris FranceOctober 2010

[23] Z Wang F Tigrek O Krasnov F van der Zwan P vanGenderen andA Yarovoy ldquoInterleavedOFDMradar signals forsimultaneous polarimetric measurementsrdquo IEEE Transactionson Aerospace and Electronic Systems vol 48 no 3 pp 2085ndash2099 2012

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



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Civil EngineeringAdvances in

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



x

y

z

r

ar

a120579

a120593

120579

120593

rarrE2

rarrE1

rarrrmn

Figure 1 The array configuration and spherical coordinate systemused for radiated electric fields

Although the weather radar polarimetry has matured foryears there are some challenges for the planar polarimetricphased array radar (PPPAR) [14] As shown in Figure 1 thearray is placed on the 119910119911 plane When the beam is away fromthe principle planes the electric field 1 from the horizontal(119867) port and 2 from the vertical (119881) port are not necessarilyorthogonal which will introduce polarimetric biases thatare not negligible The nonorthogonality of the 119867 and 119881

polarizations is called the polarization distortion in thispaper Meanwhile the polarization distortion also includesmismatches in the power levels of 119867 and 119881 beams as afunction of the scan angle The calibration matrix that relieson the measured array patterns is needed to calibrate thepolarimetric bias due to the polarimetric distortion As themeasured antenna pattern always contains measurementerrors the calibration matrix cannot completely calibratethe bias due to the polarization distortion which is notthoroughly analyzed in previous research [1 15 16] andwill bediscussed in this study Moreover in [1 15 16] it implies thatthe beam is thin enough so that the calibration performedat the boresight is sufficient to retrieve the polarimetricvariables However in practice the finite beamwidth alsocontributes to the polarimetric bias which will be evaluatedin this paper Besides the antenna the imperfect 119867119881channels can still bias the polarimetric variables which willbe modeled and analyzed Actually other factors such asthe mismatch between element patterns spatial variationsof cross-polarization patterns mutual coupling edge effectsdiffracted fields and surfacewaves can significantly affect theoverall accuracy of a PPPAR To simplify the analysis thesefactors are ignored

Usually there are two operation modes chosen forweather observations the alternately transmitting and simul-taneously receiving (ATSR) mode and the simultaneouslytransmitting and simultaneously receiving (STSR) modeEach mode has its advantages and disadvantages With aldquoperfectrdquo antenna the STSR mode is vastly superior to theATSR mode in the worst-case polarimetricspectral situa-tions Thus the STSR mode is the preferred mode froma meteorological standpoint However both the theoretical

analysis and measurement experiments have shown that theSTSRmode has higher accuracy requirements than the ATSRmode This paper is mainly focused on the ATSR mode as itis simple for the analysis

The remainder of this paper is organized as followsSection 2 presents the array model Sections 3 and 4 give thedetailed analysis in the ATSR and STSR modes respectivelySummaries and conclusions are made in Section 5

2 Array Model

The coordinate system and array configuration are shownin Figure 1 It is common that in antenna measurements theantenna is placed on the 119909119910 plane In this situation the119867 and119881 vectors correspond to the second definition in [17] In thispaper the array with119872 rows and119873 columns is placed on the119910119911 plane which is different from the typical situation Thereason is that in meteorological applications when the arrayis placed on the 119910119911 plane the expressions of the horizontaland vertical polarization basis are simple which are writtenas

aℎ = a120593

aV = minus a120579(1)

where aℎ aV is the horizontal and so-called ldquoverticalrdquopolarization basis and a119903 a120579 and a120593 are unit vectors in thespherical coordinate system

We consider the array has a 90∘ angular range in azimuthand a 30∘ range in elevation which is applicable for weatherobservations Thus in Figure 1 120593 is from minus45∘ to 45∘ and120579 is from 60∘ to 90∘ For a well-designed array it wouldbe symmetrical with respect to 120593 So in this paper we onlyconsider 120593 from 0∘ to 45∘ and 120579 from 60∘ to 90∘ Accordinglythe beam direction (120579119878 120593119878) = (90

∘ 0∘) is the broadside of the

array

21 Element Pattern The element pattern in a dual-polarizedphased array can be written as

f (120579 120593) = [119891ℎℎ (120579 120593) 119891ℎV (120579 120593)

119891Vℎ (120579 120593) 119891VV (120579 120593)] (2)

where

(i) 119891ℎℎ(120579 120593) is the horizontal electric field componentwhen only the119867 port is excited

(ii) 119891ℎV(120579 120593) is the horizontal electric field componentwhen only the 119881 port is excited

(iii) 119891Vℎ(120579 120593) is the vertical electric field component whenonly the119867 port is excited

(iv) 119891VV(120579 120593) is the vertical electric field component whenonly the 119881 port is excited

For a practical dual-polarized antenna element the cross-polarization components 119891ℎV(120579 120593) and 119891Vℎ(120579 120593) are not 0 inthe beam scan area


MCCFEC

RX V

TX HV

RX H


H

V

(a)

MCCFEC

RX V

TX V

RX H

TX H


H

V

(b)



H

VV

HHH

HV

VH

VV

HH

HV

VH

VV

ahh

ahv

avh

avv

bhh

bhv

bvh

bvv




A = [

119886ℎℎ 119886ℎV

119886Vℎ 119886VV] (3)

B = [119887ℎℎ 119887ℎV

119887Vℎ 119887VV] (4)



CIM = max100381610038161003816100381610038161003816100381610038161003816

20 log 11003816100381610038161003816119886VV

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

20 log 11003816100381610038161003816119887VV

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

(5)



CIS = 20 log(max 1

1003816100381610038161003816119886Vℎ1003816100381610038161003816

1003816100381610038161003816119886VV1003816100381610038161003816

1003816100381610038161003816119886ℎV1003816100381610038161003816

1

1003816100381610038161003816119887Vℎ1003816100381610038161003816

1003816100381610038161003816119887VV1003816100381610038161003816

1003816100381610038161003816119887ℎV1003816100381610038161003816

) (6)



T (120579 120593) =119872

sum

119898=1

119873

sum

119899=1119883119898119899 (120579119878 120593119878) f119898119899 (120579 120593) sdotA119898119899

R (120579 120593) =119872

sum

119898=1

119873

sum

119899=1119884119898119899 (120579119878 120593119878) f119898119899 (120579 120593) sdotB119898119899

(7)



T (120579 120593) = fave (120579 120593) F119879 (8)

R (120579 120593) = fave (120579 120593) F119877 (9)



F119879 =119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903)119883119898119899 (120579119878 120593119878)A119898119899

F119877 =119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903) 119884119898119899 (120579119878 120593119878)B119898119899

(10)



V = 119862exp (minus1198952119896119903)


ATSR (11)



EincATSR = [

1 00 1

] (12)


V = intΩ

R119905 (120579 120593) sdot S (120579 120593) sdotT (120579 120593) 119889Ω (13)





C119879 = (T (120579 120593)1003816100381610038161003816120579=120579119878120593=120593119878

)minus1

C119877 = (R119905(120579 120593)

10038161003816100381610038161003816120579=120579119878120593=120593119878)

minus1

(15)


R = R sdot (C119877)119905= [

ℎℎ ℎV

Vℎ VV

]


Vℎ VV

]

(16)


V = intΩ

R119905 sdot S sdot T 119889Ω (17)

Assuming

S (120579 120593) = [119878ℎℎ (120579 120593) 0

0 119878VV (120579 120593)] (18)


119885DR = 10 log⟨1003816100381610038161003816119878ℎℎ

1003816100381610038161003816

2⟩

⟨1003816100381610038161003816119878VV

1003816100381610038161003816

2⟩

(19)


119885119887

DR = 10 log119875ℎℎ

119875VVminus119885DR (20)



119875ℎℎ

(21)


119885119887

DR = 10 log1 +1198821119885

minus12dr

1198822 +119882311988512dr

ICPR = 10 log1198824 +1198825119885

minus1dr +1198826119885

minus12dr

1 +1198821119885minus12dr

(22)

where

120588ℎV exp (119895120601DP) =⟨119878lowast

ℎℎ119878VV⟩

radic⟨1003816100381610038161003816119878ℎℎ

1003816100381610038161003816

2⟩ ⟨1003816100381610038161003816119878VV

1003816100381610038161003816

2⟩

(23)


lowast

ℎℎlowast

ℎℎ120588ℎV exp (119895120601DP)) 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(24)

1198822 =intΩ

10038161003816100381610038161003816VV10038161003816100381610038161003816

2 10038161003816100381610038161003816VV10038161003816100381610038161003816

2

119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(25)


lowast

ℎVlowast

ℎV120588ℎV exp (119895120601DP)] 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(26)

1198824 =intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎV1003816100381610038161003816

2119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(27)

1198825 =intΩ

1003816100381610038161003816119879Vℎ1003816100381610038161003816

2 1003816100381610038161003816119877VV1003816100381610038161003816

2119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(28)

1198826 =intΩ2Re [119879Vℎ119877VV119879

lowast

ℎℎ119877lowast

ℎV120588ℎV exp (119895120601DP)] 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(29)




119887



T (120579 120593)

= fave (120579 120593)119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903)119883119898119899 (120579119878 120593119878)

R (120579 120593)

= fave (120579 120593)119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903) 119884119898119899 (120579119878 120593119878)

(30)




119905

(31)


T = 119865uni [1 + 120576119879ℎℎ(120579 120593) 120576

119879

ℎV (120579 120593)

120576119879

Vℎ (120579 120593) 1 + 120576119879VV (120579 120593)] (32)

R = 119865tay [1 + 120576119877ℎℎ(120579 120593) 120576

119877

ℎV (120579 120593)

120576119877

Vℎ (120579 120593) 1 + 120576119877VV (120579 120593)] (33)




0 002 004 006 008 010

02

04

06

08

1

Simulation

|Zb D

R|

(dB)

Δ

10Δ





119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119877

ℎV120575119879

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(35)


1003816100381610038161003816119885119887

DR1003816100381610038161003816 asymp 10Δ (36)









Simulation

Δ

0 002 004 006 008 01

ICPR

(dB)

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

20log(Δ)













Arg(120572119894119895) 119880(0 2120587)Arg(120573119894119895) 119880(0 2120587)Arg(120575119894119895) 119880(0 2120587)


120572 120573 120575 1198772




fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579] (38)


fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579]

sdot [

cos120593119878 0cos 120579119878 sin120593119878 sin 120579119878

]

minus1

equiv [

1 + 120576ℎℎ (120579 120593) 120576ℎV (120579 120593)

120576Vℎ (120579 120593) 1 + 120576VV (120579 120593)]

(39)





120572 120573 120575 1198772


60

65

70

75

80

85

900094

0096

0098

01

0102

0104

0106

120593 (deg)0 10 20 30 40

120579(d

eg)






60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

120579(d

eg)

minus325

minus32

minus315

minus31

minus305


119894119895| |120573119894119895| sim 119880(0 2)

60

65

70

75

80

85

900185

019

0195

02

0205

021

0215

120593 (deg)0 10 20 30 40

120579(d

eg)


60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

minus265

minus26

minus255

minus25

minus245

minus24

120579(d

eg)




T (120579 120593) = [1 + 120576119879

ℎℎ120576119879

ℎV

120576119879

Vℎ 1 + 120576119879

VV

] F119879

R (120579 120593) = [1 + 120576119877

ℎℎ120576119877

ℎV

120576119877

Vℎ 1 + 120576119877

VV

] F119877

(40)


A = [


]

B = [1 120591ℎV exp (119895120595ℎV)

120591Vℎ exp (119895120595Vℎ) 120591VV exp (119895120595VV)]

(41)


T (120579119878 120593119878) = [1 + 120575119879

ℎℎ120575119879

ℎV

120575119879

Vℎ 1 + 120575119879

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119883119898119899 (120579119878 120593119878)

10038161003816100381610038161003816A119898119899

R (120579119878 120593119878) = [1 + 120575119877

ℎℎ120575119877

ℎV

120575119877

Vℎ 1 + 120575119877

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119898119899 (120579119878 120593119878)

10038161003816100381610038161003816B119898119899

(42)

As sum119872119898=1sum

119873

119899=1 |119883119898119899(120579119878 120593119878)| = 1 and sum119872

119898=1sum119873


119885119887

DR

= 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1

119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119879

ℎV120575119877

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(43)

If 120575119879119894119895= 120575119877

119894119895= 0 (43) reduces to

119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

(44)


119885119887

DR = 2CIM (45)

0 001 002 003 004 005 006 007 008 009 0102

03

04

05

06

07

08

09

1

11

|Zb D

R|

(dB)

Δ





Step 1 Specify |120575119894119895|







∘




119887



0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ



0 001 002 003 004 005 006 007 008 009 01minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

ICPR

(dB)

Δ





[

119881ℎ (119905)

119881V (119905)]

= intΩ


119904V (119905)] 119889Ω

(46)

0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ


0 001 002 003 004 005 006 007 008 009 01 011


minus001minus36

minus33

minus30

minus27

minus24

minus21

minus18

Δ

ICPR

(dB)



119881ℎ (119905) = (119877ℎℎ119878ℎℎ119879ℎℎ +119877Vℎ119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎℎ119878ℎℎ119879ℎV +119877Vℎ119878VV119879VV) 119904V (119905)

119881V (119905) = (119877ℎV119878ℎℎ119879ℎℎ +119877VV119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎV119878ℎℎ119879ℎV +119877VV119878VV119879VV) 119904V (119905)

(47)





[

119881ℎℎ (119905) 119881ℎV (119905)

119881Vℎ (119905) 119881VV (119905)] = [



= intΩ



(48)

where 119904ℎ(119905) = 119892ℎ119904lowast

ℎ(1199050 minus 119905) and 119904V(119905) = 119892V119904

lowast


[



1 00 1

] (49)






V (119905) = intΩ






5 Conclusions





Appendix



f119898 = f + e = [119891ℎℎ 119891ℎV

119891Vℎ 119891VV]+[

119890ℎℎ 119890ℎV

119890Vℎ 119890VV] (A1)






minus1

fminus1 (A3)





f minus I = [120575ℎℎ 120575ℎV


minus1fminus1 (A5)




minus1fminus1




119864119891 = max119894119895=ℎV

10038161003816100381610038161003816119890119894119895

1003816100381610038161003816100381610038161003816100381610038161003816119891119894119895

10038161003816100381610038161003816

(A7)



sdot[[[

[

119890ℎℎ

119891ℎℎ

119891ℎℎ119891VV minus119890ℎV

119891ℎV119891ℎV119891Vℎ 119891ℎℎ119891ℎV (

119890ℎV


119891ℎℎ

)

119891Vℎ119891VV (119890Vℎ


119891VV)

119890VV

119891VV119891ℎℎ119891VV minus

119890Vℎ

119891Vℎ119891ℎV119891Vℎ

]]]

]

(A8)


1003816100381610038161003816120575ℎℎ1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575ℎV1003816100381610038161003816 le

21003816100381610038161003816119891ℎℎ119891ℎV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575Vℎ1003816100381610038161003816 le

21003816100381610038161003816119891Vℎ119891VV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575VV1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

(A9)




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










MCCFEC

RX V

TX HV

RX H


H

V

(a)

MCCFEC

RX V

TX V

RX H

TX H


H

V

(b)



H

VV

HHH

HV

VH

VV

HH

HV

VH

VV

ahh

ahv

avh

avv

bhh

bhv

bvh

bvv




A = [

119886ℎℎ 119886ℎV

119886Vℎ 119886VV] (3)

B = [119887ℎℎ 119887ℎV

119887Vℎ 119887VV] (4)



CIM = max100381610038161003816100381610038161003816100381610038161003816

20 log 11003816100381610038161003816119886VV

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

20 log 11003816100381610038161003816119887VV

1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

(5)



CIS = 20 log(max 1

1003816100381610038161003816119886Vℎ1003816100381610038161003816

1003816100381610038161003816119886VV1003816100381610038161003816

1003816100381610038161003816119886ℎV1003816100381610038161003816

1

1003816100381610038161003816119887Vℎ1003816100381610038161003816

1003816100381610038161003816119887VV1003816100381610038161003816

1003816100381610038161003816119887ℎV1003816100381610038161003816

) (6)



T (120579 120593) =119872

sum

119898=1

119873

sum

119899=1119883119898119899 (120579119878 120593119878) f119898119899 (120579 120593) sdotA119898119899

R (120579 120593) =119872

sum

119898=1

119873

sum

119899=1119884119898119899 (120579119878 120593119878) f119898119899 (120579 120593) sdotB119898119899

(7)



T (120579 120593) = fave (120579 120593) F119879 (8)

R (120579 120593) = fave (120579 120593) F119877 (9)



F119879 =119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903)119883119898119899 (120579119878 120593119878)A119898119899

F119877 =119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903) 119884119898119899 (120579119878 120593119878)B119898119899

(10)



V = 119862exp (minus1198952119896119903)


ATSR (11)



EincATSR = [

1 00 1

] (12)


V = intΩ

R119905 (120579 120593) sdot S (120579 120593) sdotT (120579 120593) 119889Ω (13)





C119879 = (T (120579 120593)1003816100381610038161003816120579=120579119878120593=120593119878

)minus1

C119877 = (R119905(120579 120593)

10038161003816100381610038161003816120579=120579119878120593=120593119878)

minus1

(15)


R = R sdot (C119877)119905= [

ℎℎ ℎV

Vℎ VV

]


Vℎ VV

]

(16)


V = intΩ

R119905 sdot S sdot T 119889Ω (17)

Assuming

S (120579 120593) = [119878ℎℎ (120579 120593) 0

0 119878VV (120579 120593)] (18)


119885DR = 10 log⟨1003816100381610038161003816119878ℎℎ

1003816100381610038161003816

2⟩

⟨1003816100381610038161003816119878VV

1003816100381610038161003816

2⟩

(19)


119885119887

DR = 10 log119875ℎℎ

119875VVminus119885DR (20)



119875ℎℎ

(21)


119885119887

DR = 10 log1 +1198821119885

minus12dr

1198822 +119882311988512dr

ICPR = 10 log1198824 +1198825119885

minus1dr +1198826119885

minus12dr

1 +1198821119885minus12dr

(22)

where

120588ℎV exp (119895120601DP) =⟨119878lowast

ℎℎ119878VV⟩

radic⟨1003816100381610038161003816119878ℎℎ

1003816100381610038161003816

2⟩ ⟨1003816100381610038161003816119878VV

1003816100381610038161003816

2⟩

(23)


lowast

ℎℎlowast

ℎℎ120588ℎV exp (119895120601DP)) 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(24)

1198822 =intΩ

10038161003816100381610038161003816VV10038161003816100381610038161003816

2 10038161003816100381610038161003816VV10038161003816100381610038161003816

2

119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(25)


lowast

ℎVlowast

ℎV120588ℎV exp (119895120601DP)] 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(26)

1198824 =intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎV1003816100381610038161003816

2119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(27)

1198825 =intΩ

1003816100381610038161003816119879Vℎ1003816100381610038161003816

2 1003816100381610038161003816119877VV1003816100381610038161003816

2119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(28)

1198826 =intΩ2Re [119879Vℎ119877VV119879

lowast

ℎℎ119877lowast

ℎV120588ℎV exp (119895120601DP)] 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(29)




119887



T (120579 120593)

= fave (120579 120593)119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903)119883119898119899 (120579119878 120593119878)

R (120579 120593)

= fave (120579 120593)119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903) 119884119898119899 (120579119878 120593119878)

(30)




119905

(31)


T = 119865uni [1 + 120576119879ℎℎ(120579 120593) 120576

119879

ℎV (120579 120593)

120576119879

Vℎ (120579 120593) 1 + 120576119879VV (120579 120593)] (32)

R = 119865tay [1 + 120576119877ℎℎ(120579 120593) 120576

119877

ℎV (120579 120593)

120576119877

Vℎ (120579 120593) 1 + 120576119877VV (120579 120593)] (33)




0 002 004 006 008 010

02

04

06

08

1

Simulation

|Zb D

R|

(dB)

Δ

10Δ





119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119877

ℎV120575119879

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(35)


1003816100381610038161003816119885119887

DR1003816100381610038161003816 asymp 10Δ (36)









Simulation

Δ

0 002 004 006 008 01

ICPR

(dB)

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

20log(Δ)













Arg(120572119894119895) 119880(0 2120587)Arg(120573119894119895) 119880(0 2120587)Arg(120575119894119895) 119880(0 2120587)


120572 120573 120575 1198772




fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579] (38)


fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579]

sdot [

cos120593119878 0cos 120579119878 sin120593119878 sin 120579119878

]

minus1

equiv [

1 + 120576ℎℎ (120579 120593) 120576ℎV (120579 120593)

120576Vℎ (120579 120593) 1 + 120576VV (120579 120593)]

(39)





120572 120573 120575 1198772


60

65

70

75

80

85

900094

0096

0098

01

0102

0104

0106

120593 (deg)0 10 20 30 40

120579(d

eg)






60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

120579(d

eg)

minus325

minus32

minus315

minus31

minus305


119894119895| |120573119894119895| sim 119880(0 2)

60

65

70

75

80

85

900185

019

0195

02

0205

021

0215

120593 (deg)0 10 20 30 40

120579(d

eg)


60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

minus265

minus26

minus255

minus25

minus245

minus24

120579(d

eg)




T (120579 120593) = [1 + 120576119879

ℎℎ120576119879

ℎV

120576119879

Vℎ 1 + 120576119879

VV

] F119879

R (120579 120593) = [1 + 120576119877

ℎℎ120576119877

ℎV

120576119877

Vℎ 1 + 120576119877

VV

] F119877

(40)


A = [


]

B = [1 120591ℎV exp (119895120595ℎV)

120591Vℎ exp (119895120595Vℎ) 120591VV exp (119895120595VV)]

(41)


T (120579119878 120593119878) = [1 + 120575119879

ℎℎ120575119879

ℎV

120575119879

Vℎ 1 + 120575119879

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119883119898119899 (120579119878 120593119878)

10038161003816100381610038161003816A119898119899

R (120579119878 120593119878) = [1 + 120575119877

ℎℎ120575119877

ℎV

120575119877

Vℎ 1 + 120575119877

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119898119899 (120579119878 120593119878)

10038161003816100381610038161003816B119898119899

(42)

As sum119872119898=1sum

119873

119899=1 |119883119898119899(120579119878 120593119878)| = 1 and sum119872

119898=1sum119873


119885119887

DR

= 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1

119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119879

ℎV120575119877

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(43)

If 120575119879119894119895= 120575119877

119894119895= 0 (43) reduces to

119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

(44)


119885119887

DR = 2CIM (45)

0 001 002 003 004 005 006 007 008 009 0102

03

04

05

06

07

08

09

1

11

|Zb D

R|

(dB)

Δ





Step 1 Specify |120575119894119895|







∘




119887



0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ



0 001 002 003 004 005 006 007 008 009 01minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

ICPR

(dB)

Δ





[

119881ℎ (119905)

119881V (119905)]

= intΩ


119904V (119905)] 119889Ω

(46)

0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ


0 001 002 003 004 005 006 007 008 009 01 011


minus001minus36

minus33

minus30

minus27

minus24

minus21

minus18

Δ

ICPR

(dB)



119881ℎ (119905) = (119877ℎℎ119878ℎℎ119879ℎℎ +119877Vℎ119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎℎ119878ℎℎ119879ℎV +119877Vℎ119878VV119879VV) 119904V (119905)

119881V (119905) = (119877ℎV119878ℎℎ119879ℎℎ +119877VV119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎV119878ℎℎ119879ℎV +119877VV119878VV119879VV) 119904V (119905)

(47)





[

119881ℎℎ (119905) 119881ℎV (119905)

119881Vℎ (119905) 119881VV (119905)] = [



= intΩ



(48)

where 119904ℎ(119905) = 119892ℎ119904lowast

ℎ(1199050 minus 119905) and 119904V(119905) = 119892V119904

lowast


[



1 00 1

] (49)






V (119905) = intΩ






5 Conclusions





Appendix



f119898 = f + e = [119891ℎℎ 119891ℎV

119891Vℎ 119891VV]+[

119890ℎℎ 119890ℎV

119890Vℎ 119890VV] (A1)






minus1

fminus1 (A3)





f minus I = [120575ℎℎ 120575ℎV


minus1fminus1 (A5)




minus1fminus1




119864119891 = max119894119895=ℎV

10038161003816100381610038161003816119890119894119895

1003816100381610038161003816100381610038161003816100381610038161003816119891119894119895

10038161003816100381610038161003816

(A7)



sdot[[[

[

119890ℎℎ

119891ℎℎ

119891ℎℎ119891VV minus119890ℎV

119891ℎV119891ℎV119891Vℎ 119891ℎℎ119891ℎV (

119890ℎV


119891ℎℎ

)

119891Vℎ119891VV (119890Vℎ


119891VV)

119890VV

119891VV119891ℎℎ119891VV minus

119890Vℎ

119891Vℎ119891ℎV119891Vℎ

]]]

]

(A8)


1003816100381610038161003816120575ℎℎ1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575ℎV1003816100381610038161003816 le

21003816100381610038161003816119891ℎℎ119891ℎV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575Vℎ1003816100381610038161003816 le

21003816100381610038161003816119891Vℎ119891VV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575VV1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

(A9)




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











F119879 =119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903)119883119898119899 (120579119878 120593119878)A119898119899

F119877 =119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903) 119884119898119899 (120579119878 120593119878)B119898119899

(10)



V = 119862exp (minus1198952119896119903)


ATSR (11)



EincATSR = [

1 00 1

] (12)


V = intΩ

R119905 (120579 120593) sdot S (120579 120593) sdotT (120579 120593) 119889Ω (13)





C119879 = (T (120579 120593)1003816100381610038161003816120579=120579119878120593=120593119878

)minus1

C119877 = (R119905(120579 120593)

10038161003816100381610038161003816120579=120579119878120593=120593119878)

minus1

(15)


R = R sdot (C119877)119905= [

ℎℎ ℎV

Vℎ VV

]


Vℎ VV

]

(16)


V = intΩ

R119905 sdot S sdot T 119889Ω (17)

Assuming

S (120579 120593) = [119878ℎℎ (120579 120593) 0

0 119878VV (120579 120593)] (18)


119885DR = 10 log⟨1003816100381610038161003816119878ℎℎ

1003816100381610038161003816

2⟩

⟨1003816100381610038161003816119878VV

1003816100381610038161003816

2⟩

(19)


119885119887

DR = 10 log119875ℎℎ

119875VVminus119885DR (20)



119875ℎℎ

(21)


119885119887

DR = 10 log1 +1198821119885

minus12dr

1198822 +119882311988512dr

ICPR = 10 log1198824 +1198825119885

minus1dr +1198826119885

minus12dr

1 +1198821119885minus12dr

(22)

where

120588ℎV exp (119895120601DP) =⟨119878lowast

ℎℎ119878VV⟩

radic⟨1003816100381610038161003816119878ℎℎ

1003816100381610038161003816

2⟩ ⟨1003816100381610038161003816119878VV

1003816100381610038161003816

2⟩

(23)


lowast

ℎℎlowast

ℎℎ120588ℎV exp (119895120601DP)) 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(24)

1198822 =intΩ

10038161003816100381610038161003816VV10038161003816100381610038161003816

2 10038161003816100381610038161003816VV10038161003816100381610038161003816

2

119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(25)


lowast

ℎVlowast

ℎV120588ℎV exp (119895120601DP)] 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(26)

1198824 =intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎV1003816100381610038161003816

2119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(27)

1198825 =intΩ

1003816100381610038161003816119879Vℎ1003816100381610038161003816

2 1003816100381610038161003816119877VV1003816100381610038161003816

2119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(28)

1198826 =intΩ2Re [119879Vℎ119877VV119879

lowast

ℎℎ119877lowast

ℎV120588ℎV exp (119895120601DP)] 119889Ω

intΩ

1003816100381610038161003816119879ℎℎ1003816100381610038161003816

2 1003816100381610038161003816119877ℎℎ1003816100381610038161003816

2119889Ω

(29)




119887



T (120579 120593)

= fave (120579 120593)119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903)119883119898119899 (120579119878 120593119878)

R (120579 120593)

= fave (120579 120593)119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903) 119884119898119899 (120579119878 120593119878)

(30)




119905

(31)


T = 119865uni [1 + 120576119879ℎℎ(120579 120593) 120576

119879

ℎV (120579 120593)

120576119879

Vℎ (120579 120593) 1 + 120576119879VV (120579 120593)] (32)

R = 119865tay [1 + 120576119877ℎℎ(120579 120593) 120576

119877

ℎV (120579 120593)

120576119877

Vℎ (120579 120593) 1 + 120576119877VV (120579 120593)] (33)




0 002 004 006 008 010

02

04

06

08

1

Simulation

|Zb D

R|

(dB)

Δ

10Δ





119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119877

ℎV120575119879

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(35)


1003816100381610038161003816119885119887

DR1003816100381610038161003816 asymp 10Δ (36)









Simulation

Δ

0 002 004 006 008 01

ICPR

(dB)

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

20log(Δ)













Arg(120572119894119895) 119880(0 2120587)Arg(120573119894119895) 119880(0 2120587)Arg(120575119894119895) 119880(0 2120587)


120572 120573 120575 1198772




fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579] (38)


fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579]

sdot [

cos120593119878 0cos 120579119878 sin120593119878 sin 120579119878

]

minus1

equiv [

1 + 120576ℎℎ (120579 120593) 120576ℎV (120579 120593)

120576Vℎ (120579 120593) 1 + 120576VV (120579 120593)]

(39)





120572 120573 120575 1198772


60

65

70

75

80

85

900094

0096

0098

01

0102

0104

0106

120593 (deg)0 10 20 30 40

120579(d

eg)






60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

120579(d

eg)

minus325

minus32

minus315

minus31

minus305


119894119895| |120573119894119895| sim 119880(0 2)

60

65

70

75

80

85

900185

019

0195

02

0205

021

0215

120593 (deg)0 10 20 30 40

120579(d

eg)


60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

minus265

minus26

minus255

minus25

minus245

minus24

120579(d

eg)




T (120579 120593) = [1 + 120576119879

ℎℎ120576119879

ℎV

120576119879

Vℎ 1 + 120576119879

VV

] F119879

R (120579 120593) = [1 + 120576119877

ℎℎ120576119877

ℎV

120576119877

Vℎ 1 + 120576119877

VV

] F119877

(40)


A = [


]

B = [1 120591ℎV exp (119895120595ℎV)

120591Vℎ exp (119895120595Vℎ) 120591VV exp (119895120595VV)]

(41)


T (120579119878 120593119878) = [1 + 120575119879

ℎℎ120575119879

ℎV

120575119879

Vℎ 1 + 120575119879

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119883119898119899 (120579119878 120593119878)

10038161003816100381610038161003816A119898119899

R (120579119878 120593119878) = [1 + 120575119877

ℎℎ120575119877

ℎV

120575119877

Vℎ 1 + 120575119877

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119898119899 (120579119878 120593119878)

10038161003816100381610038161003816B119898119899

(42)

As sum119872119898=1sum

119873

119899=1 |119883119898119899(120579119878 120593119878)| = 1 and sum119872

119898=1sum119873


119885119887

DR

= 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1

119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119879

ℎV120575119877

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(43)

If 120575119879119894119895= 120575119877

119894119895= 0 (43) reduces to

119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

(44)


119885119887

DR = 2CIM (45)

0 001 002 003 004 005 006 007 008 009 0102

03

04

05

06

07

08

09

1

11

|Zb D

R|

(dB)

Δ





Step 1 Specify |120575119894119895|







∘




119887



0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ



0 001 002 003 004 005 006 007 008 009 01minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

ICPR

(dB)

Δ





[

119881ℎ (119905)

119881V (119905)]

= intΩ


119904V (119905)] 119889Ω

(46)

0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ


0 001 002 003 004 005 006 007 008 009 01 011


minus001minus36

minus33

minus30

minus27

minus24

minus21

minus18

Δ

ICPR

(dB)



119881ℎ (119905) = (119877ℎℎ119878ℎℎ119879ℎℎ +119877Vℎ119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎℎ119878ℎℎ119879ℎV +119877Vℎ119878VV119879VV) 119904V (119905)

119881V (119905) = (119877ℎV119878ℎℎ119879ℎℎ +119877VV119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎV119878ℎℎ119879ℎV +119877VV119878VV119879VV) 119904V (119905)

(47)





[

119881ℎℎ (119905) 119881ℎV (119905)

119881Vℎ (119905) 119881VV (119905)] = [



= intΩ



(48)

where 119904ℎ(119905) = 119892ℎ119904lowast

ℎ(1199050 minus 119905) and 119904V(119905) = 119892V119904

lowast


[



1 00 1

] (49)






V (119905) = intΩ






5 Conclusions





Appendix



f119898 = f + e = [119891ℎℎ 119891ℎV

119891Vℎ 119891VV]+[

119890ℎℎ 119890ℎV

119890Vℎ 119890VV] (A1)






minus1

fminus1 (A3)





f minus I = [120575ℎℎ 120575ℎV


minus1fminus1 (A5)




minus1fminus1




119864119891 = max119894119895=ℎV

10038161003816100381610038161003816119890119894119895

1003816100381610038161003816100381610038161003816100381610038161003816119891119894119895

10038161003816100381610038161003816

(A7)



sdot[[[

[

119890ℎℎ

119891ℎℎ

119891ℎℎ119891VV minus119890ℎV

119891ℎV119891ℎV119891Vℎ 119891ℎℎ119891ℎV (

119890ℎV


119891ℎℎ

)

119891Vℎ119891VV (119890Vℎ


119891VV)

119890VV

119891VV119891ℎℎ119891VV minus

119890Vℎ

119891Vℎ119891ℎV119891Vℎ

]]]

]

(A8)


1003816100381610038161003816120575ℎℎ1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575ℎV1003816100381610038161003816 le

21003816100381610038161003816119891ℎℎ119891ℎV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575Vℎ1003816100381610038161003816 le

21003816100381610038161003816119891Vℎ119891VV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575VV1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

(A9)




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119887



T (120579 120593)

= fave (120579 120593)119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903)119883119898119899 (120579119878 120593119878)

R (120579 120593)

= fave (120579 120593)119872

sum

119898=1

119873

sum

119899=1exp (119895119896 119903119898119899 sdot a119903) 119884119898119899 (120579119878 120593119878)

(30)




119905

(31)


T = 119865uni [1 + 120576119879ℎℎ(120579 120593) 120576

119879

ℎV (120579 120593)

120576119879

Vℎ (120579 120593) 1 + 120576119879VV (120579 120593)] (32)

R = 119865tay [1 + 120576119877ℎℎ(120579 120593) 120576

119877

ℎV (120579 120593)

120576119877

Vℎ (120579 120593) 1 + 120576119877VV (120579 120593)] (33)




0 002 004 006 008 010

02

04

06

08

1

Simulation

|Zb D

R|

(dB)

Δ

10Δ





119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119877

ℎV120575119879

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(35)


1003816100381610038161003816119885119887

DR1003816100381610038161003816 asymp 10Δ (36)









Simulation

Δ

0 002 004 006 008 01

ICPR

(dB)

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

20log(Δ)













Arg(120572119894119895) 119880(0 2120587)Arg(120573119894119895) 119880(0 2120587)Arg(120575119894119895) 119880(0 2120587)


120572 120573 120575 1198772




fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579] (38)


fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579]

sdot [

cos120593119878 0cos 120579119878 sin120593119878 sin 120579119878

]

minus1

equiv [

1 + 120576ℎℎ (120579 120593) 120576ℎV (120579 120593)

120576Vℎ (120579 120593) 1 + 120576VV (120579 120593)]

(39)





120572 120573 120575 1198772


60

65

70

75

80

85

900094

0096

0098

01

0102

0104

0106

120593 (deg)0 10 20 30 40

120579(d

eg)






60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

120579(d

eg)

minus325

minus32

minus315

minus31

minus305


119894119895| |120573119894119895| sim 119880(0 2)

60

65

70

75

80

85

900185

019

0195

02

0205

021

0215

120593 (deg)0 10 20 30 40

120579(d

eg)


60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

minus265

minus26

minus255

minus25

minus245

minus24

120579(d

eg)




T (120579 120593) = [1 + 120576119879

ℎℎ120576119879

ℎV

120576119879

Vℎ 1 + 120576119879

VV

] F119879

R (120579 120593) = [1 + 120576119877

ℎℎ120576119877

ℎV

120576119877

Vℎ 1 + 120576119877

VV

] F119877

(40)


A = [


]

B = [1 120591ℎV exp (119895120595ℎV)

120591Vℎ exp (119895120595Vℎ) 120591VV exp (119895120595VV)]

(41)


T (120579119878 120593119878) = [1 + 120575119879

ℎℎ120575119879

ℎV

120575119879

Vℎ 1 + 120575119879

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119883119898119899 (120579119878 120593119878)

10038161003816100381610038161003816A119898119899

R (120579119878 120593119878) = [1 + 120575119877

ℎℎ120575119877

ℎV

120575119877

Vℎ 1 + 120575119877

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119898119899 (120579119878 120593119878)

10038161003816100381610038161003816B119898119899

(42)

As sum119872119898=1sum

119873

119899=1 |119883119898119899(120579119878 120593119878)| = 1 and sum119872

119898=1sum119873


119885119887

DR

= 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1

119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119879

ℎV120575119877

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(43)

If 120575119879119894119895= 120575119877

119894119895= 0 (43) reduces to

119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

(44)


119885119887

DR = 2CIM (45)

0 001 002 003 004 005 006 007 008 009 0102

03

04

05

06

07

08

09

1

11

|Zb D

R|

(dB)

Δ





Step 1 Specify |120575119894119895|







∘




119887



0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ



0 001 002 003 004 005 006 007 008 009 01minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

ICPR

(dB)

Δ





[

119881ℎ (119905)

119881V (119905)]

= intΩ


119904V (119905)] 119889Ω

(46)

0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ


0 001 002 003 004 005 006 007 008 009 01 011


minus001minus36

minus33

minus30

minus27

minus24

minus21

minus18

Δ

ICPR

(dB)



119881ℎ (119905) = (119877ℎℎ119878ℎℎ119879ℎℎ +119877Vℎ119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎℎ119878ℎℎ119879ℎV +119877Vℎ119878VV119879VV) 119904V (119905)

119881V (119905) = (119877ℎV119878ℎℎ119879ℎℎ +119877VV119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎV119878ℎℎ119879ℎV +119877VV119878VV119879VV) 119904V (119905)

(47)





[

119881ℎℎ (119905) 119881ℎV (119905)

119881Vℎ (119905) 119881VV (119905)] = [



= intΩ



(48)

where 119904ℎ(119905) = 119892ℎ119904lowast

ℎ(1199050 minus 119905) and 119904V(119905) = 119892V119904

lowast


[



1 00 1

] (49)






V (119905) = intΩ






5 Conclusions





Appendix



f119898 = f + e = [119891ℎℎ 119891ℎV

119891Vℎ 119891VV]+[

119890ℎℎ 119890ℎV

119890Vℎ 119890VV] (A1)






minus1

fminus1 (A3)





f minus I = [120575ℎℎ 120575ℎV


minus1fminus1 (A5)




minus1fminus1




119864119891 = max119894119895=ℎV

10038161003816100381610038161003816119890119894119895

1003816100381610038161003816100381610038161003816100381610038161003816119891119894119895

10038161003816100381610038161003816

(A7)



sdot[[[

[

119890ℎℎ

119891ℎℎ

119891ℎℎ119891VV minus119890ℎV

119891ℎV119891ℎV119891Vℎ 119891ℎℎ119891ℎV (

119890ℎV


119891ℎℎ

)

119891Vℎ119891VV (119890Vℎ


119891VV)

119890VV

119891VV119891ℎℎ119891VV minus

119890Vℎ

119891Vℎ119891ℎV119891Vℎ

]]]

]

(A8)


1003816100381610038161003816120575ℎℎ1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575ℎV1003816100381610038161003816 le

21003816100381610038161003816119891ℎℎ119891ℎV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575Vℎ1003816100381610038161003816 le

21003816100381610038161003816119891Vℎ119891VV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575VV1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

(A9)




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Simulation

Δ

0 002 004 006 008 01

ICPR

(dB)

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

20log(Δ)













Arg(120572119894119895) 119880(0 2120587)Arg(120573119894119895) 119880(0 2120587)Arg(120575119894119895) 119880(0 2120587)


120572 120573 120575 1198772




fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579] (38)


fdipole (120579 120593) = [cos120593 0

cos 120579 sin120593 sin 120579]

sdot [

cos120593119878 0cos 120579119878 sin120593119878 sin 120579119878

]

minus1

equiv [

1 + 120576ℎℎ (120579 120593) 120576ℎV (120579 120593)

120576Vℎ (120579 120593) 1 + 120576VV (120579 120593)]

(39)





120572 120573 120575 1198772


60

65

70

75

80

85

900094

0096

0098

01

0102

0104

0106

120593 (deg)0 10 20 30 40

120579(d

eg)






60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

120579(d

eg)

minus325

minus32

minus315

minus31

minus305


119894119895| |120573119894119895| sim 119880(0 2)

60

65

70

75

80

85

900185

019

0195

02

0205

021

0215

120593 (deg)0 10 20 30 40

120579(d

eg)


60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

minus265

minus26

minus255

minus25

minus245

minus24

120579(d

eg)




T (120579 120593) = [1 + 120576119879

ℎℎ120576119879

ℎV

120576119879

Vℎ 1 + 120576119879

VV

] F119879

R (120579 120593) = [1 + 120576119877

ℎℎ120576119877

ℎV

120576119877

Vℎ 1 + 120576119877

VV

] F119877

(40)


A = [


]

B = [1 120591ℎV exp (119895120595ℎV)

120591Vℎ exp (119895120595Vℎ) 120591VV exp (119895120595VV)]

(41)


T (120579119878 120593119878) = [1 + 120575119879

ℎℎ120575119879

ℎV

120575119879

Vℎ 1 + 120575119879

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119883119898119899 (120579119878 120593119878)

10038161003816100381610038161003816A119898119899

R (120579119878 120593119878) = [1 + 120575119877

ℎℎ120575119877

ℎV

120575119877

Vℎ 1 + 120575119877

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119898119899 (120579119878 120593119878)

10038161003816100381610038161003816B119898119899

(42)

As sum119872119898=1sum

119873

119899=1 |119883119898119899(120579119878 120593119878)| = 1 and sum119872

119898=1sum119873


119885119887

DR

= 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1

119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119879

ℎV120575119877

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(43)

If 120575119879119894119895= 120575119877

119894119895= 0 (43) reduces to

119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

(44)


119885119887

DR = 2CIM (45)

0 001 002 003 004 005 006 007 008 009 0102

03

04

05

06

07

08

09

1

11

|Zb D

R|

(dB)

Δ





Step 1 Specify |120575119894119895|







∘




119887



0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ



0 001 002 003 004 005 006 007 008 009 01minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

ICPR

(dB)

Δ





[

119881ℎ (119905)

119881V (119905)]

= intΩ


119904V (119905)] 119889Ω

(46)

0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ


0 001 002 003 004 005 006 007 008 009 01 011


minus001minus36

minus33

minus30

minus27

minus24

minus21

minus18

Δ

ICPR

(dB)



119881ℎ (119905) = (119877ℎℎ119878ℎℎ119879ℎℎ +119877Vℎ119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎℎ119878ℎℎ119879ℎV +119877Vℎ119878VV119879VV) 119904V (119905)

119881V (119905) = (119877ℎV119878ℎℎ119879ℎℎ +119877VV119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎV119878ℎℎ119879ℎV +119877VV119878VV119879VV) 119904V (119905)

(47)





[

119881ℎℎ (119905) 119881ℎV (119905)

119881Vℎ (119905) 119881VV (119905)] = [



= intΩ



(48)

where 119904ℎ(119905) = 119892ℎ119904lowast

ℎ(1199050 minus 119905) and 119904V(119905) = 119892V119904

lowast


[



1 00 1

] (49)






V (119905) = intΩ






5 Conclusions





Appendix



f119898 = f + e = [119891ℎℎ 119891ℎV

119891Vℎ 119891VV]+[

119890ℎℎ 119890ℎV

119890Vℎ 119890VV] (A1)






minus1

fminus1 (A3)





f minus I = [120575ℎℎ 120575ℎV


minus1fminus1 (A5)




minus1fminus1




119864119891 = max119894119895=ℎV

10038161003816100381610038161003816119890119894119895

1003816100381610038161003816100381610038161003816100381610038161003816119891119894119895

10038161003816100381610038161003816

(A7)



sdot[[[

[

119890ℎℎ

119891ℎℎ

119891ℎℎ119891VV minus119890ℎV

119891ℎV119891ℎV119891Vℎ 119891ℎℎ119891ℎV (

119890ℎV


119891ℎℎ

)

119891Vℎ119891VV (119890Vℎ


119891VV)

119890VV

119891VV119891ℎℎ119891VV minus

119890Vℎ

119891Vℎ119891ℎV119891Vℎ

]]]

]

(A8)


1003816100381610038161003816120575ℎℎ1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575ℎV1003816100381610038161003816 le

21003816100381610038161003816119891ℎℎ119891ℎV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575Vℎ1003816100381610038161003816 le

21003816100381610038161003816119891Vℎ119891VV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575VV1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

(A9)




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120572 120573 120575 1198772


60

65

70

75

80

85

900094

0096

0098

01

0102

0104

0106

120593 (deg)0 10 20 30 40

120579(d

eg)






60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

120579(d

eg)

minus325

minus32

minus315

minus31

minus305


119894119895| |120573119894119895| sim 119880(0 2)

60

65

70

75

80

85

900185

019

0195

02

0205

021

0215

120593 (deg)0 10 20 30 40

120579(d

eg)


60

65

70

75

80

85

90

120593 (deg)0 10 20 30 40

minus265

minus26

minus255

minus25

minus245

minus24

120579(d

eg)




T (120579 120593) = [1 + 120576119879

ℎℎ120576119879

ℎV

120576119879

Vℎ 1 + 120576119879

VV

] F119879

R (120579 120593) = [1 + 120576119877

ℎℎ120576119877

ℎV

120576119877

Vℎ 1 + 120576119877

VV

] F119877

(40)


A = [


]

B = [1 120591ℎV exp (119895120595ℎV)

120591Vℎ exp (119895120595Vℎ) 120591VV exp (119895120595VV)]

(41)


T (120579119878 120593119878) = [1 + 120575119879

ℎℎ120575119879

ℎV

120575119879

Vℎ 1 + 120575119879

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119883119898119899 (120579119878 120593119878)

10038161003816100381610038161003816A119898119899

R (120579119878 120593119878) = [1 + 120575119877

ℎℎ120575119877

ℎV

120575119877

Vℎ 1 + 120575119877

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119898119899 (120579119878 120593119878)

10038161003816100381610038161003816B119898119899

(42)

As sum119872119898=1sum

119873

119899=1 |119883119898119899(120579119878 120593119878)| = 1 and sum119872

119898=1sum119873


119885119887

DR

= 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1

119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119879

ℎV120575119877

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(43)

If 120575119879119894119895= 120575119877

119894119895= 0 (43) reduces to

119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

(44)


119885119887

DR = 2CIM (45)

0 001 002 003 004 005 006 007 008 009 0102

03

04

05

06

07

08

09

1

11

|Zb D

R|

(dB)

Δ





Step 1 Specify |120575119894119895|







∘




119887



0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ



0 001 002 003 004 005 006 007 008 009 01minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

ICPR

(dB)

Δ





[

119881ℎ (119905)

119881V (119905)]

= intΩ


119904V (119905)] 119889Ω

(46)

0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ


0 001 002 003 004 005 006 007 008 009 01 011


minus001minus36

minus33

minus30

minus27

minus24

minus21

minus18

Δ

ICPR

(dB)



119881ℎ (119905) = (119877ℎℎ119878ℎℎ119879ℎℎ +119877Vℎ119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎℎ119878ℎℎ119879ℎV +119877Vℎ119878VV119879VV) 119904V (119905)

119881V (119905) = (119877ℎV119878ℎℎ119879ℎℎ +119877VV119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎV119878ℎℎ119879ℎV +119877VV119878VV119879VV) 119904V (119905)

(47)





[

119881ℎℎ (119905) 119881ℎV (119905)

119881Vℎ (119905) 119881VV (119905)] = [



= intΩ



(48)

where 119904ℎ(119905) = 119892ℎ119904lowast

ℎ(1199050 minus 119905) and 119904V(119905) = 119892V119904

lowast


[



1 00 1

] (49)






V (119905) = intΩ






5 Conclusions





Appendix



f119898 = f + e = [119891ℎℎ 119891ℎV

119891Vℎ 119891VV]+[

119890ℎℎ 119890ℎV

119890Vℎ 119890VV] (A1)






minus1

fminus1 (A3)





f minus I = [120575ℎℎ 120575ℎV


minus1fminus1 (A5)




minus1fminus1




119864119891 = max119894119895=ℎV

10038161003816100381610038161003816119890119894119895

1003816100381610038161003816100381610038161003816100381610038161003816119891119894119895

10038161003816100381610038161003816

(A7)



sdot[[[

[

119890ℎℎ

119891ℎℎ

119891ℎℎ119891VV minus119890ℎV

119891ℎV119891ℎV119891Vℎ 119891ℎℎ119891ℎV (

119890ℎV


119891ℎℎ

)

119891Vℎ119891VV (119890Vℎ


119891VV)

119890VV

119891VV119891ℎℎ119891VV minus

119890Vℎ

119891Vℎ119891ℎV119891Vℎ

]]]

]

(A8)


1003816100381610038161003816120575ℎℎ1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575ℎV1003816100381610038161003816 le

21003816100381610038161003816119891ℎℎ119891ℎV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575Vℎ1003816100381610038161003816 le

21003816100381610038161003816119891Vℎ119891VV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575VV1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

(A9)




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











T (120579 120593) = [1 + 120576119879

ℎℎ120576119879

ℎV

120576119879

Vℎ 1 + 120576119879

VV

] F119879

R (120579 120593) = [1 + 120576119877

ℎℎ120576119877

ℎV

120576119877

Vℎ 1 + 120576119877

VV

] F119877

(40)


A = [


]

B = [1 120591ℎV exp (119895120595ℎV)

120591Vℎ exp (119895120595Vℎ) 120591VV exp (119895120595VV)]

(41)


T (120579119878 120593119878) = [1 + 120575119879

ℎℎ120575119879

ℎV

120575119879

Vℎ 1 + 120575119879

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119883119898119899 (120579119878 120593119878)

10038161003816100381610038161003816A119898119899

R (120579119878 120593119878) = [1 + 120575119877

ℎℎ120575119877

ℎV

120575119877

Vℎ 1 + 120575119877

VV

]

sdot

119872

sum

119898=1

119873

sum

119899=1

10038161003816100381610038161003816119898119899 (120579119878 120593119878)

10038161003816100381610038161003816B119898119899

(42)

As sum119872119898=1sum

119873

119899=1 |119883119898119899(120579119878 120593119878)| = 1 and sum119872

119898=1sum119873


119885119887

DR

= 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1

119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

1 + 120575119879

ℎℎ+ 120575119877

ℎℎ+ 120575119879

ℎℎ120575119877

ℎℎ+ 120575119879

Vℎ120575119877

Vℎ

1 + 120575119879VV + 120575119877VV + 120575119879VV120575119877VV + 120575119879

ℎV120575119877

ℎV

1003816100381610038161003816100381610038161003816100381610038161003816

(43)

If 120575119879119894119895= 120575119877

119894119895= 0 (43) reduces to

119885119887

DR = 20 log1003816100381610038161003816100381610038161003816100381610038161003816

1119886VV119887VV

1003816100381610038161003816100381610038161003816100381610038161003816

(44)


119885119887

DR = 2CIM (45)

0 001 002 003 004 005 006 007 008 009 0102

03

04

05

06

07

08

09

1

11

|Zb D

R|

(dB)

Δ





Step 1 Specify |120575119894119895|







∘




119887



0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ



0 001 002 003 004 005 006 007 008 009 01minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

ICPR

(dB)

Δ





[

119881ℎ (119905)

119881V (119905)]

= intΩ


119904V (119905)] 119889Ω

(46)

0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ


0 001 002 003 004 005 006 007 008 009 01 011


minus001minus36

minus33

minus30

minus27

minus24

minus21

minus18

Δ

ICPR

(dB)



119881ℎ (119905) = (119877ℎℎ119878ℎℎ119879ℎℎ +119877Vℎ119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎℎ119878ℎℎ119879ℎV +119877Vℎ119878VV119879VV) 119904V (119905)

119881V (119905) = (119877ℎV119878ℎℎ119879ℎℎ +119877VV119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎV119878ℎℎ119879ℎV +119877VV119878VV119879VV) 119904V (119905)

(47)





[

119881ℎℎ (119905) 119881ℎV (119905)

119881Vℎ (119905) 119881VV (119905)] = [



= intΩ



(48)

where 119904ℎ(119905) = 119892ℎ119904lowast

ℎ(1199050 minus 119905) and 119904V(119905) = 119892V119904

lowast


[



1 00 1

] (49)






V (119905) = intΩ






5 Conclusions





Appendix



f119898 = f + e = [119891ℎℎ 119891ℎV

119891Vℎ 119891VV]+[

119890ℎℎ 119890ℎV

119890Vℎ 119890VV] (A1)






minus1

fminus1 (A3)





f minus I = [120575ℎℎ 120575ℎV


minus1fminus1 (A5)




minus1fminus1




119864119891 = max119894119895=ℎV

10038161003816100381610038161003816119890119894119895

1003816100381610038161003816100381610038161003816100381610038161003816119891119894119895

10038161003816100381610038161003816

(A7)



sdot[[[

[

119890ℎℎ

119891ℎℎ

119891ℎℎ119891VV minus119890ℎV

119891ℎV119891ℎV119891Vℎ 119891ℎℎ119891ℎV (

119890ℎV


119891ℎℎ

)

119891Vℎ119891VV (119890Vℎ


119891VV)

119890VV

119891VV119891ℎℎ119891VV minus

119890Vℎ

119891Vℎ119891ℎV119891Vℎ

]]]

]

(A8)


1003816100381610038161003816120575ℎℎ1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575ℎV1003816100381610038161003816 le

21003816100381610038161003816119891ℎℎ119891ℎV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575Vℎ1003816100381610038161003816 le

21003816100381610038161003816119891Vℎ119891VV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575VV1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

(A9)




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ



0 001 002 003 004 005 006 007 008 009 01minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

ICPR

(dB)

Δ





[

119881ℎ (119905)

119881V (119905)]

= intΩ


119904V (119905)] 119889Ω

(46)

0 001 002 003 004 005 006 007 008 009 01 01102

03

04

05

06

07

08

09

1

11


minus001

|Zb D

R|

(dB)

Δ


0 001 002 003 004 005 006 007 008 009 01 011


minus001minus36

minus33

minus30

minus27

minus24

minus21

minus18

Δ

ICPR

(dB)



119881ℎ (119905) = (119877ℎℎ119878ℎℎ119879ℎℎ +119877Vℎ119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎℎ119878ℎℎ119879ℎV +119877Vℎ119878VV119879VV) 119904V (119905)

119881V (119905) = (119877ℎV119878ℎℎ119879ℎℎ +119877VV119878VV119879Vℎ) 119904ℎ (119905)

+ (119877ℎV119878ℎℎ119879ℎV +119877VV119878VV119879VV) 119904V (119905)

(47)





[

119881ℎℎ (119905) 119881ℎV (119905)

119881Vℎ (119905) 119881VV (119905)] = [



= intΩ



(48)

where 119904ℎ(119905) = 119892ℎ119904lowast

ℎ(1199050 minus 119905) and 119904V(119905) = 119892V119904

lowast


[



1 00 1

] (49)






V (119905) = intΩ






5 Conclusions





Appendix



f119898 = f + e = [119891ℎℎ 119891ℎV

119891Vℎ 119891VV]+[

119890ℎℎ 119890ℎV

119890Vℎ 119890VV] (A1)






minus1

fminus1 (A3)





f minus I = [120575ℎℎ 120575ℎV


minus1fminus1 (A5)




minus1fminus1




119864119891 = max119894119895=ℎV

10038161003816100381610038161003816119890119894119895

1003816100381610038161003816100381610038161003816100381610038161003816119891119894119895

10038161003816100381610038161003816

(A7)



sdot[[[

[

119890ℎℎ

119891ℎℎ

119891ℎℎ119891VV minus119890ℎV

119891ℎV119891ℎV119891Vℎ 119891ℎℎ119891ℎV (

119890ℎV


119891ℎℎ

)

119891Vℎ119891VV (119890Vℎ


119891VV)

119890VV

119891VV119891ℎℎ119891VV minus

119890Vℎ

119891Vℎ119891ℎV119891Vℎ

]]]

]

(A8)


1003816100381610038161003816120575ℎℎ1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575ℎV1003816100381610038161003816 le

21003816100381610038161003816119891ℎℎ119891ℎV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575Vℎ1003816100381610038161003816 le

21003816100381610038161003816119891Vℎ119891VV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575VV1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

(A9)




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












[

119881ℎℎ (119905) 119881ℎV (119905)

119881Vℎ (119905) 119881VV (119905)] = [



= intΩ



(48)

where 119904ℎ(119905) = 119892ℎ119904lowast

ℎ(1199050 minus 119905) and 119904V(119905) = 119892V119904

lowast


[



1 00 1

] (49)






V (119905) = intΩ






5 Conclusions





Appendix



f119898 = f + e = [119891ℎℎ 119891ℎV

119891Vℎ 119891VV]+[

119890ℎℎ 119890ℎV

119890Vℎ 119890VV] (A1)






minus1

fminus1 (A3)





f minus I = [120575ℎℎ 120575ℎV


minus1fminus1 (A5)




minus1fminus1




119864119891 = max119894119895=ℎV

10038161003816100381610038161003816119890119894119895

1003816100381610038161003816100381610038161003816100381610038161003816119891119894119895

10038161003816100381610038161003816

(A7)



sdot[[[

[

119890ℎℎ

119891ℎℎ

119891ℎℎ119891VV minus119890ℎV

119891ℎV119891ℎV119891Vℎ 119891ℎℎ119891ℎV (

119890ℎV


119891ℎℎ

)

119891Vℎ119891VV (119890Vℎ


119891VV)

119890VV

119891VV119891ℎℎ119891VV minus

119890Vℎ

119891Vℎ119891ℎV119891Vℎ

]]]

]

(A8)


1003816100381610038161003816120575ℎℎ1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575ℎV1003816100381610038161003816 le

21003816100381610038161003816119891ℎℎ119891ℎV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575Vℎ1003816100381610038161003816 le

21003816100381610038161003816119891Vℎ119891VV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575VV1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

(A9)




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











f minus I = [120575ℎℎ 120575ℎV


minus1fminus1 (A5)




minus1fminus1




119864119891 = max119894119895=ℎV

10038161003816100381610038161003816119890119894119895

1003816100381610038161003816100381610038161003816100381610038161003816119891119894119895

10038161003816100381610038161003816

(A7)



sdot[[[

[

119890ℎℎ

119891ℎℎ

119891ℎℎ119891VV minus119890ℎV

119891ℎV119891ℎV119891Vℎ 119891ℎℎ119891ℎV (

119890ℎV


119891ℎℎ

)

119891Vℎ119891VV (119890Vℎ


119891VV)

119890VV

119891VV119891ℎℎ119891VV minus

119890Vℎ

119891Vℎ119891ℎV119891Vℎ

]]]

]

(A8)


1003816100381610038161003816120575ℎℎ1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575ℎV1003816100381610038161003816 le

21003816100381610038161003816119891ℎℎ119891ℎV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575Vℎ1003816100381610038161003816 le

21003816100381610038161003816119891Vℎ119891VV

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

1003816100381610038161003816120575VV1003816100381610038161003816 le

1003816100381610038161003816119891ℎℎ119891VV1003816100381610038161003816 +1003816100381610038161003816119891ℎV119891Vℎ

10038161003816100381610038161003816100381610038161003816119891ℎℎ119891VV minus 119891ℎV119891Vℎ

1003816100381610038161003816

119864119891

(A9)




Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article dual-polarized planar phased array

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