The University of Calgary - Focal Plane Field Calculator...

18 June 2009 Dominion Radio Astrophysical Observatory (DRAO)

The University of Calgary - Focal Plane Field Calculator (UC-FPFC) and Its Applications

Thushara Gunaratne & Dr. Len BrutonMDSP Group

Department of Electrical and Computer EngineeringShulich School of EngineeringUniversity of Calgary, Canada.

18 June 2009 DRAO 2

OutlineIntroduction & Motivation

Focal plane arrays (FPAs) for SKAWhy an accurate model for the focal field is required?

Calculation of focal region electric field in UC-FPFCPhysical optics approximationSampling density of the reflector surfaceComparison with GRASP®

Applications of UC-FPFCSpectral analysis of the synthesized fieldBroadband conjugate field matching (BB-CFM)

Conclusions and future work

18 June 2009 DRAO 3

Focal Plane Arrays (FPAs) for the Square Kilometer Array (SKA)

The proposed receiver technology for the SKA for the bandwidth 0.3 – 3 GHz.

Combines the Large collection area of a paraboloidal reflectorMulti-beam synthesis capability of a phased array The complete 180-element PHAD

array assembly – DRAO.http://www.casca.ca/ecass/issues/2007-ae/features/ska/ska.htm

18 June 2009 DRAO 4

Motivations for Focal Field Modeling

Accurate modeling of focal field is essential

Predict the best achievable selectivity with FPAsPredict suitable receiver elements for FPAsPredict suitable arrangements of elements of FPAsPredict the maximum power recovery by FPAsWays of achieving optimum SNR and radio interference mitigation through FPA signal processing

TICRA – GRASP9® is the industry standard for focal field calculations involving parabolic reflectors

But!!….it is too Expensive!!…..

18 June 2009 DRAO 5

University of Calgary – Focal Plane Field Calculator (UC-FPFC)

UC-FPFC is a custom MATLAB® based software tool that evaluates the electromagnetic fields in the focal region of a prime focus paraboloidal reflector in response to an impinging monochromatic plane-wave.

Uses the “physical-optics” approximation in determining the reflector currentsEnables straight-forward secondary analysis of fields

Agree well with the calculated fields with GRASP9®

18 June 2009 DRAO 6

Orientation of the Objects & Notations Used

18 June 2009 DRAO 7

Calculating the Electric Field in the Focal Region

( )ˆ( ) = 2 jkn e ′− ⋅′ × incr RincJ r H

Physical Optics Approximation

( ) ( )ˆ( ) = 2 jkn e ′− ⋅′ × × incr Rinc incJ r E R

( )ˆ ˆ( ) = ( ) - ( ( ) )4

jk

refl

k ej dsρη

π ρ

−

′ ′ ′− ⋅∫∫E r J r J r R R

x

y

z

Rinc

r

′r

ˆρR

n̂

18 June 2009 DRAO 8

Sampling the Reflector Surface

There is no direct application of sampling theory for the sampling of induced current-density on the reflector surface.

Hence, and can be changed to have a tread off between the accuracy and the computational complexity.

m∆ n∆ Sample grid on the

reflector surface

Solve by 2D numerical integration

( ),

, , , ,,

ˆ ˆ( )= ( ) - ( ( ) )4

m njk

m n m n m n m n m nm n m n

k ej Jρη

π ρ

−

Σ′ ′− ⋅ ∆ ∆∑∑E r J r J r R R

18 June 2009 DRAO 9

Some Results Achieved with UC-FPFC

Parabolic Reflector : = 6m; = 10m:Plane Wave : Pol = [0,1,0]; = 1.5 GHz; 1.5 ; 0 .

F Df θ φ= ° = °

FPA0.75×0.75 m2

Sampled into a grid of 51×51

18 June 2009 DRAO 10

Comparing UC-FPFC and GRASP®

Plane Wave : Pol = [0,1,0]; = 1 GHz; 1.5 ; 0 .f θ φ= ° = °

Comparison for Co-polar components Maximum relative error of Ey = 0.011826



Comparison for cross-polar components Maximum relative error of Ex = 0.064326




Comparison for longitudinal componentsMaximum relative error of Ez = 0.022712




2.27457.60711.23653.0

2.29016.48551.20252.5

2.28416.56141.18482.0

2.27126.43261.18261.5

2.25886.31531.18351.0

2.24616.96861.19000.5

2.23356.24731.06550.0

Longitudinal Component

Cross-Polar Component

Co-Polar Component

Percentage Maximum Relative Error (%)Angleθ in degrees

(φ = 0°)

Note: Sample grid size on the reflector surface is (32×32).



Different Sampling Grids on the Reflector Surface

01.1527 06.6644 00.3549

0.08291.1501 0.081526.6654 0.04230.3631

0.13451.1444 0.05877 6.6265 0.07300.3570

0.5694 1.3189 0.35594 6.5669 0.2834 0.3970

1.3307 1.8347 1.60690 6.9921 0.6917 0.7606

3.6163 3.6330 1.81320 6.6468 1.8132 1.7829

7.3415 7.3309 11.8090 12.342 4.2797 4.2402

ToTo GRASP

To To GRASP

ToTo GRASP

Longitudinal ComponentCross-Polar ComponentCo-Polar Component

Percentage Maximum Relative Error for the Normal Incident (θ = 0°; φ = 0°) (%)Sample

Grid Size

for theReflector

(1023 1023)×

(255 255)×

(11 11)×

(1023 1023)× (1023 1023)×

(1023 1023)×

(21 21)×

(35 35)×

(63 63)×

(127 127)×


Applications of UC-FPFC


Verification of the Conjectured Model for the Focal Field

1max 2

8(F/D)tan16(F/D) 1

θ − ⎛ ⎞= ⎜ ⎟−⎝ ⎠


Conjectured Region of Support (ROS) of the Spectrum of the Focal Field

max max[ , ]θ θ θ∈ − [0 ,360 ]φ ∈ ° °1

1/ 2 maxtan (sin( ))α θ−=

As predicted, the iso-surface of the spectrum of the focal-field synthesized for the band 1 – 1.5 GHz using UC-FPFC resembles a section of a cone.

UC-FPFC ( , , ) 0.1x y tFP ω ω ω =


Conjectured ROS of the Spectrum of the Focal Field (Contd.)

F/D=0.6

F/D=0.8


Broadband - Conjugate Field Matching (BB-CFM)

An extended CFM method is proposed where the spectral components of the desired signal is added in-phase compared to the spectral components of the interfering signals and AGWN.

According to the Cauchy-Schwartz inequality the best possible SNR is achieved in recovering from

is when the 3D coefficient matrix has the frequency response

1( )Cw − n( )x n ( )y n

3 31 2 1 2* 3D *1 1DDFT( , , ) ( , , ) ( ) ( )j jj j j j

C CY e e e W e e e y wω ωω ω ω ω− −= ←⎯⎯→ = −n n

Given the sampled focal field

1 2 31

( ) ( ) ; ( , , )P

C pp

x w AWGN n n n−=

= + =∑n n n


Determination of Best Possible Selectivity with BB-CFM

BB-CFM achieves optimum SNR with AWGN, but how would it perform with interference?

Given4

T1

( ) ( );C pp

x w −=

= ∑n n

1 2

3 4

( ) ( 0 , 0 ), ( ) ( 1 , 0 ),( ) ( 2 , 0 ), ( ) ( 3 , 0 ).

C C

C C

w ww w

θ φ θ φθ φ θ φ

− −

− −

⇒ = ° = ° ⇒ = ° = °

⇒ = ° = ° ⇒ = ° = °

n nn n

In order to enhance with respect to other signals, four 3D coefficient matrices were evaluated such that

* *1 1 2 2

* *3 3 4 4

( ) ( ), ( ) ( ),

( ) ( ), ( ) ( ).C C

C C

y w y w

y w y w− −

− −

= − = −

= − = −

n n n n

n n n n

( ); 1,2,3,4c pw p− =n


Results of BB-CFM

T

Test sequence( ) of size

(26 26 2048)Filter coefficients

( ) of size

(26 26 41)p

x

y

× ×

× ×

n

n

( )

correspond to far-field sincfunctions

C pw − n


Conclusions and Future Work

ConclusionsUC-FPFC results agree very well with GRASP®The primary conjecture on the ROS is verifiedBest possible selectivity for FPA under BB-CFM is determined

Future WorkCombine element response patternsGP-GPU implementation for possible speed upPerformance comparison with low-complexity frequency transformed 3D FIR filters


Thank You.

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