Massachusetts Institute of Technology Haystack Observatory WESTFORD, MASSACHUSETTS DATE 07/15/2009

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1 BBD Memo #033 Maachuett Intitute of Technolog Hatack Obervator WESTFORD, MASSACHUSETTS 0886 DATE 07/5/2009 To: Broadband Development Group From: C. J. Beaudoin Subject: Holographic Proceing and Conideration for VLBI200 Antenna Diagnotic Holographic Proceing Conider the antenna aperture hown in figure, radiating (or receiving b reciprocit) at frequenc f, and having a ingularl () polarized electric field ditribution E, ; the unknown aperture ditribution. Given ample of the far-field given b ( apt apt ) receive pattern of ( ) E apt, apt, the aperture ditribution function can be recontructed b the following formulation: E j( apt F + apt F ) ( apt, apt ) E ( F, F ) e θ, φ df df f F in( θ ) co( φ ) c (b) f F in( θ ) in( φ ) (c) c Where θ, φ are a depicted in figure and define the location of the far-field ource relative to the aperture. The integration of equation a i performed over the range of θ and φ for which there i knowledge of the far-field radiation pattern E. Equation a relate the aperture ditribution function to the two-dimenional Fourier tranform of the far-field pattern; normalization contant have been uppreed. Equation b and c are formatting equation which indicate where the far-field ample mut be located in Fourier pace in order to uccefull recontruct the aperture ditribution uing the Fourier-baed method. A benefit of incorporating thi method i that one can emplo propertie of the multi-dimenional Fourier tranform when analzing imager generated from uch holographic proceing. Data collected for holographic proceing are done o b canning the antenna under tet (AUT) through a ource at ome azimuth and elevation angle in the antenna coordinate tem which i fied relative to the phical ground plane. The angle θ and φ, on the other hand, are defined in the aperture coordinate tem and change a a (a)

2 Source Pointing Vector ( ) E, apt apt θ φ Figure : Geometr depicting the antenna aperture in the aperture coordinate Stem. function of antenna pointing. A uch, the ource pointing vector defined b θ and φ mut be related to the antenna azimuth and elevation pointing angle in order to give them meaning for the application at hand. Figure 2 dipla thi relationhip which i defined b the following tranformation: F F F z f c in co in( φ ) co( φ ) 0 ( ψ ) co( φ ) in( ψ ) in( φ ) co( ψ ) ( ψ ) co( φ ) co( ψ ) in( φ ) in( ψ ) P ant (2) Here, P ant i the antenna pointing unit vector in the antenna coordinate tem and ψ and φ are the elevation and azimuth pointing angle, repectivel, to the ource being ued to meaure the far-field pattern. Baed on figure 2 and equation 2, one will oberve that at ψ φ 0, the ant ai coincide with the z apt ai (ource pointing vector) of the aperture tem. In figure 3, the ource pointing vector (green vector in figure 2) coincide with z apt. The ignificance of the tranformation decribed b equation 2 i that it decribe how a a uniform azimuth and elevation ampling grid in the antenna tem i deformed into a nonuniform ample grid in the Fourier pace. A an eample, aume that a holographic data collection i performed whereb a ampling grid i contructed uing 5 tep panning 40 in azimuth and elevation and the ource i located at 45 elevation and 0 azimuth. Figure 3 dipla the F,F patial frequenc mapping of the ample given the pecified antenna rater can. 2

3 ψ φ φ Figure 2: Relationhip of antenna (gre) and aperture (red) coordinate tem through the ource azimuth (φ p ) and elevation (ψ p ) angle when the antenna i pointed on-ource. ΔF ΔF δf δf Figure 3: Eample mapping of nonuniforml ditributed Fourier pace ample obtained from uniform ampling in antenna azimuth and elevation. 3

4 In order to recontruct a hologram given a ample mapping uch a that hown in Figure 3, two method ma be incorporated. The firt method, which i technicall traightforward et computationall intenive, eplicitl evaluate equation. Thi amount to calculating a dicrete Fourier tranform which require that a 2D comple weighting and umming of E be performed for each piel evaluated in the hologram. A more eicient and widel ued approach i to reample the nonuniform Fourier pace grid to one that i uniform and then emplo a 2D FFT routine. In the interet of development time, the firt method ha been implemented in MATLAB code. If proceor peed become an iue uing thi method, a inc interpolant can be developed to peed up the calculation b implementing the econd method. Reolution of a holographic image recontruction i an important parameter to conider when diagnoing the performance of an antenna. Since the Fourier patial frequenc counterpart to apt and apt are F and F, repectivel, the apt, apt holographic image reolution, δ and δ, repectivel, are inverel proportional to the patial frequenc pan of the data in each dimenion. Formall epreed: δ ΔF where ΔF and ΔF are the pan of the data in the F and F dimenion, repectivel. In regard to a patial frequenc mapping, ΔF and ΔF are taken a hown in figure 3. Converel to image reolution, the unambiguou ize of the hologram that can be recontructed i dependant on the pacing of the ample in the Fourier pace which i alo depicted in figure 3. The larget pacing of the ample in the F and F dimenion i ued to determine the unambiguou image ize in the apt and apt dimenion: Here Δ and Δ are the unambiguou ize of the image in the apt and apt dimenion and δf and δf are the maimum F and F ample pacing, repectivel. Latl, the image phae of a holographic recontruction can be ued to infer information about height relative to the aperture plane. Thi i done uing the following relation: c z p Φ p f (5) where Φ p i the phae of an given piel in the image and z p i the inferred height of that particular piel out of the aperture plane. Thi technique i onl applicable to thoe piel which have appreciable amplitude in the hologram. δ ΔF Δ δf Δ δf (3a) (3b) (4a) (4b) 4

5 Proceor Performance with Simulated Data A a demontration of the proceor performance, a imulated holographic data et wa created b modeling the far-field pattern of a 5 meter circular aperture poeing a 2m circular ditortion: E J 2.5 (.5k in( θ )) k in( θ ) where θ and φ are a defined in figure and J i the firt-order Beel function of the firt kind. In the model decribed b equation 6, the ditortion i oet from the center of the aperture b meter in apt and apt, poee one half the field amplitude of the ret of the aperture, and i diplaced 0 mm out of the aperture plane. The data were calculated over elevation and azimuth pan of 0 and 3.25, repectivel, with the ource located at 45 elevation angle and 0 azimuth angle. For thi particular collection, δ δ 0.4 m. Figure 4 dipla the magnitude and height profile of the aperture ditribution a recontructed b holographic proceing of the imulated far-field data. In oberving figure 4, one will oberve that the circular hape and relative intenitie of the aperture and the ditorted area are preerved in the recontruction. Furthermore, the proceor faithfull reproduce the height profile of the radiating aperture relative to the apt, apt plane. Conideration in Meauring Far-Field Data ( k in( θ )) in( θ ) 2 2 J jk in co k ( 0.5e ) ( θ )( co( φ ) + in ( φ )) j(0.00) k ( θ ) Up to thi point, the aumption wa made that ample of the tet antenna farfield pattern were available. Alo, the development of the holographic proceor in the previou ection wa done o for a ingle frequenc ignal. That being aid, the phae and magnitude of the far-field ample are obtained b cro-correlating the ignal received b the reference antenna (fied pointing on the ource) and that b the AUT (canned through the ource). In accordance with the development of the previou ection, the farfield meaurement i then the magnitude and phae of the cro-power pectrum (CPS) at a ingle frequenc. Though conitent with the proceor development, uing onl one frequenc from the CPS i unatifactor becaue the majorit of the data i dicarded and SNR i lot. However, if the reciprocal bandwidth of the ignal ued to generate the cro-correlation function (CCF) i uicientl large with repect to time dela variation acro the aperture (due to antenna imperfection), there will be little phae variation acro the CPS and the magnitude/phae of the CCF at it peak(magnitude) i repreentative of the comple far-field ample at each point in the rater can. In practice, the bandwidth of the CCF will probabl be 32 MHz, for which the time-dela reolution i 3 n. The time dela error acro the antenna aperture would have to be etremel gro in order to looe coherence acro uch a narrow bandwidth. Thi being the cae, it i permiible to ue the peak of the CCF to provide the comple far-field ample ued to recontruct the hologram. e (6) 5

6 (a) (b) Figure 4: Holographic recontruction of data imulating 5m circular aperture with m circular ditortion. (a) dipla the magnitude of the electric field in the aperture and (b) dipla the height profile of the aperture. 6

7 SNR Conideration It i alo worthwhile to evaluate the amount of recording time needed for each can to achieve a pecified maimum rm phae uncertaint (at minimum AUT area when pointed o-ource) and hence minimal SNR in the far-field data. Baed on rough power meaurement of a randoml elected atellite ource uing one of AEER VSRT, the noie power in the ource wa oberved to be 0 db tronger than that of the LNB. Thi particular VSRT poeed an LNB with noie temperature of 00K and an 8 aperture with 50% eicienc. Baed on the oberved power level of the ource relative to the LNB noie and the aperture area, in 32 MHz bandwidth the power denit available in Hatack parking lot i approimatel -82 dbm/m 2. Baed on thi power denit, auming a reference antenna poeing m aperture with 50% eicienc uing a front-end having temperature 00K, the SNR in the reference meaurement will be 3.5 db. Auming MV3 a the tet antenna, that it i operating at 40% eicienc, and that it i pointed +/- 5 deg o-ource, the eective area of the antenna when pointed full o-ource, auming an Air pattern, i 0. m 2. Given that the MV3 antenna temperature i currentl 00K and the o-ource eective area, the minimum SNR in the AUT obervation i epected to be.5 db. Given the ingle dih SNR in each antenna and auming 2-bit ampling, the correlation amplitude, ρ o, i calculated with the following: ρo nr + nr + nr nr (4) r t r t where nr r and nr t are the ingle dih ignal-to-noie ratio of the reference and tet antenna, repectivel. The factor /.3 i the lo a a reult of 2-bit ampling. Given the previoul calculated ingle dih SNR quantitie, ρ o Having the correlation amplitude, the total recording time needed to achieve a pecified phae uncertaint, σ φ, in the far-field data i given b: T r f ( ρ ) 2 σ φ o (5) where f i the data ample rate. With 32 MHz channel, the data ample rate i 64 MS/ and taking σ φ deg, the recording time needed to achieve the required SNR i 73 u. Thi record time correpond to a total ample count of 4672 which i uicientl mall enough to allow the reference/tet antenna correlation to be performed on the ame computer running the holographic proceor. 7