Accuracy Analyses of Synthesized Reference-Wave Holography for Determining Antenna Radiation Characteristics

Transcription

1 Accuracy Analyses f Synthesized Reference-Wave Hlgraphy fr Determining Antenna Radiatin Characteristics Vladimir Schejbal 1, Jan Pidanic 1, Vlastimil Kvarill, and Dusan Cermak 1 1University f Pardubice, Jan Perner Transprt Faculty Studentska 95, 5321 Pardubice, Czech Republic vladimir.schejbal@upce.cz;jan.pidanic@upce.cz;dusan.cermak@upce.cz 2Retia, a.s. Prazska 341, Zelene Predmesti, 53 2 Pardubice, Czech Republic vkvarik@retia.cz Abstract Synthesized-reference-wave hlgraphic techniques ffer relatively simple and cst-effective measurements f antenna radiatin characteristics, and recnstructin f cmplex aperture fields using near-field intensity-pattern measurements. These methds allw utilizatin f the advantages f the methds fr prbe cmpensatin fr near-field amplitude and phase measurements fr planar and cylindrical scanning, including accuracy analyses. Accuracy analyses using mathematical mdels cnsidering randm prcesses with crrelatin intervals are presented. Numerical simulatins, taking int accunt randm as well as deterministic errrs and the prcessing f measurement statistics, are als presented. t is demnstrated that the given analyses crrespnd t ur measurements and/r numerical simulatins. Keywrds: Antenna measurements; antenna radiatin patterns; accuracy analyses; near-field sampling; hlgraphy; nearfield measurements; antenna prbe cmpensatin 1. ntrductin Near-field (NF) measurements prvide a fast and accurate methd fr determining an antenna's gain, pattern, plarizatin, beam pinting, etc. n cntrast t cnventinal far-field methds, near-field antenna measurement methds make use fa measuring prbe in the radiating near-field regin f the antenna under test (AUT). The far-field radiatin pattern fthe AUT must then be indirectly cmputed frm the measurements made in a near-field regin. Several methds fr near-field antenna measurements have been described, such as [1-1 7]. Synthesized-reference-wave hlgraphy was prpsed in the PhD thesis [9], where several aspects were thrughly analyzed. (These methds allw emplying the advantages' f methds fr prbe cmpensatin fr amplitude and phase near-field measurements fr planar and cylindrical scanning, including accuracy analyses [1-7]). Mst f the results have been published [1-16], such as a descriptin fhlgraphic near-field measurements using prbe cmpensatin fr planar and cylindrical scanning; a cmparisn f hlgraphic near-field and far-field measurements with and withut prbe cmpensatin; accuracy analyses using mathematical mdels cnsidering randm prcesses with crrelatin intervals; numerical simulatins cnsidering randm errrs as well as deterministic errrs; and the prcessing f measurement statistics. Paper [17] summarized the cited results (as smef the papers and the PhD thesis were nly in Czech), and gave sme new results (especially numerical simulatins). This paper analyzes the accuracy, using mathematical mdels cnsidering randm prcesses with crrelatin intervals, the prcessing fmeasurement statistics, and numerical simulatins cnsidering randm as well as deterministic errrs. The analyses given crrespnd t ur measurements and/r numerical simulatins. 2. Accuracy Analyses Near-field measurements using synthesized-reference-wave hlgraphy with prbe cmpensatin fr planar and cylindrical scanning allw emplying the advantages f methds fr prbe cmpensatin fr the amplitude and phase f near-field measurements fr planar -and cylindncal scanning, including accuracy analyses [1-7], which shuld be slightly mdified. Hwever~ accuracy analyses using mathematical mdels cnsidering randm prcesses with crrelatin intervals, measurements using statistical prcessing, and numerical simulatins cnsidering randm as well as deterministic errrs have been dne, and they are described belw. EEE Antennas and Prpagatin Magazine, Vl. 5, N.6, December 28 SSN /$25 28 EEE 89 Authrized licensed use limited t: University f Pardubice. Dwnladed n December 5, 29 at 2:13 frm EEE Xplre. Restrictins apply.

2 The thery f near-field antenna measurements applies rigrusly t linear antennas. n the thery f prbe-crrected near-field measurements, multiple reflectins between the prbe and test antennas are cnsidered negligible, but experimental errrs als limit the accuracy f near-field techniques. The experimental measurements intrduce prbe-psitining errrs, instrumentatin errrs, and - fr the planar and cylindrical (r truncated spherical) scanning gemetries - finite scanning errrs. Errrs are als created by rm reflectins, uncertainties in the far field f the prbe, and uncertainties in the measurement f the insertin lss between the test antenna and the prbe when abslute gain is required. The NST 18-term errr mdel can be applied [4, 7]. f the sample spacing and cmputer accuracy are adequate, aliasing and cmputatinal errrs shuld be negligible cmpared t the experimental errrs. The imprtance f the varius near-field measurement errrs depends n the antenna under test, the frequency f peratin, the measurement facility, and the prbe. Hwever, results shw that the finite scanning area, the z psitin f the prbe fr planar scanning, receiver nnlinearities in measuring the near-field amplitude, and, smetimes, multiple reflectins are usually the mst imprtant surces ferrr. n general, planar scanning is limited t determining the fields within the frward slid angular regin subtended by the edges f the test antenna and the finite scanning area. Cylindrical scanning mits nly the bicnical angular regin frmed by the uter edges f the test antenna and the cylindrical scanning area f finite height. t is pssible t use a priri infrmatin n the gemetry f the antenna under test (AUT) and t evaluate the farfield pattern frm the estimated near-field data. The methd f [3] takes advantage f the pssibility - present in mst f the scanning setups - fmving the prbe nt nly n the scanning surface, but als alng the axis perpendicular t the scanning surface. These effects are the same, even fr synthesized-reference-wave hlgraphy[17]. The z-psltn inaccuracies, i.e., the deviatin, g, frm planarityfthe prbe transprt ver the scanning area, can prduce relatively large errrs in the sidelbe levels fthe far field. Variatins in the z psitin f the prbe prduce crrespnding variatins in the near-field phase. Thus, large errrs in the sidelbes ccur in the far-field directins crrespnding t the predminant spatial frequencies fthe variatin in z psitin acrss the scanning area. n the main-beam directin, this is much less critical: the reductin in gain is given by g2/2, frm the familiar Ruze relatin [1]. The errrs in the sidelbes caused by inaccurate z psitining can be reduced by measuring the deviatin f the prbe frm the scanning plane and crrecting the near-field phase. t shuld als be mentined that receiver phase errrs generally have a much smaller effect n the far field than d phase errrs caused by inaccurate z psitining. This is because typical receiver phase errrs are negligible at the maximum near-field amplitude, and increase mntnically with decreasing amplitude. Hwever, receiver nnlinearities in the measurement f near-field amplitude can cause significant errrs in the main beam and sidelbes fthe far fields. The cntributin t the utput f the prbe frm the multiple reflectins can be estimated by changing the separatin distance between the prbe and test antenna, and recrding the amplitude ~ariatins that ccur in the received signal with a perid fabut A12. fmultiple reflectins prve significant, they may be reduced by the sensible use f absrbing material, by decreasing the size f the prbe, by increasing the prbe separatin distance, by averaging the far fields cmputed frm the near-field data taken n scanning planes that are separated by a small fractin f a wavelength (abut A4 r less), r by using specially designed prbes that filter the main beam and emphasize the sidelbes. Hwever, the upper-bund errr frmulas shuld be applied with care. They are dependent upn underlying (usually, explicitly stated) assumptins that are satisfied by mst antennas and nearfield measurement cnditins, but which may nt be fulfilled in certain circumstances. Fr example, it is well knwn that phase errrs intrduced int the main near-field beam f directive antennas cause a reductin in the cmputed n-axis gain. Hwever, this gain reductin strictly applies t near-field beams f unifrm phase. When the underlying assumptins can be relaxed, lwer values fr upper bunds can usually be btained. Fr example, an estimate f the specific z-psitin errrs fr a particular measurement facility allws ne t mre accurately estimate their effect upn far-field sidelbes. Accuracy analyses using mathematical mdels cnsidering randm prcesses with crrelatin intervals have been perfrmed [9, 11] using the results f[8, 19]. Therefre, nly a briefsummary is given. The measurement scanning [17] is dne n a finite surface, S, and the mst imprtant part f the pwer radiates thrugh that surface. The surface S can be transfrmed nt a square regin, [-1,1] x [-1,1], and nrmalized crdinates xl' x2 culd be used fr bth planar and cylindrical scanning. f the theretically measured values are v(xl' X2) and the values affected by errrs are vf ( Xl' X2 ), then the fllwing relatinship can be cnsidered: where the values f B (xl' X2) are the amplitude errrs, and the values f q (xl, X2) are the phase errrs. As requirements n antennaparameter measurements are usually quite strict, nly the case f small errrs (i.e., B(xj,x2)1,lq(x,x2)1«) can be cnsidered. The randm errrs, 1B ( Xl ' x2)1 and q (Xl, x2 ), are assumed t have nrmal distributins, zer means, cnstant variances given by u~ and u~ (i.e., u~, u~ d nt depend n Xl' x2), and crrelatin cefficients rb cnsidered: where cl and c2 are crrelatin intervals, Xl =Xl -1]1 and X 2 =X2-172 are differences, and Xl and x2 are nrmalized crdinates. Similarly, the crrelatin cefficient r q (Xl' X 2) can be used fr phase errrs. and r q. The fllwing crrelatin cefficients are The radiatin patterns are cmpletely described by angular wave spectra A(K, K 2 ), which are given by the Furier transfrm f the measured field, F [v (Xl' X2)] ' and similarly fr vf (Xl' X2). t shuld be nted that the surface S is transfrmed, as well as K, K 2, and A(KJ,K 2 ). t is well knwn (the scaling therem) that (2) (3) 9 EEE Antennas and Prpagatin MagaZine, Vl. 5, N.6, December 28 Authrized licensed use limited t: University f Pardubice. Dwnladed n December 5, 29 at 2:13 frm EEE Xplre. Restrictins apply.

3 Cnsidering the assumptins f synthesized-reference-wave hlgraphy [17], the fllwing equatin can be derived fr a suitable interval kxl < k l : =F[E(X'X2t e-jaxl +lcl 2 e- jax1 +C E(XX2)+ CE (x,x2)e- 2jaxl ] (4) where C exp (- jaxl) is the reference signal, and E(x) is the near field. The fllwing expressin is valid (the bars dente the mean values) where the higher-pwer functins f (]'q have been neglected. This means that the mean value fthe angular spectra f vf (Xl' X2) are equal t the angular spectra f the riginal field v(xl' X2), except fr a multiplicative cnstant. The variances fthe angular spectra are [8, 19] where the cefficients a Band a q individual B (Xl, X2) and q (Xl' X2) errrs. (5) (6) determine the effects f the f the individual errrs are cnsidered t be independent errrs and ct,c2 «1, then [8, 19] ab(kl,k2)"~ ==--2 f JJJV(Xl,X2)V (17,1]2)(]'~rB(X b X 2 ) (27l") e-j(ktxt+k2,x2)dxldx2d1]ld1]2 The variances fthe angular spectra are given by Equatin (6). The individual terms, a B(j~ and aq(j~, are given by Equatins (7) and (8). This allws the determinatin f the variances f the angular spectra fr individual errrs. The randm-errr assumptins used generally cannt be fulfilled, but they are fulfilled fr varius imprtant cases. This culd simplify the analyses f randm-errr effects (quite general assumptins wuld nt allw any analysis). t was shwn [19] that nearly the same results can be btained if rb (r, pssibly, r q ) is given nt by Equatin (2), but it is nly required that the crrelatin cefficients decrease mntnically with increasing X 1 1 and/r X21. This means that even ifthe abve assumptins are nt exactly fulfilled, Equatins (5) t (8) are apprximately valid. Other parameter changes were analyzed [19], such as the extremes f the angular-spectrum mean values, and the directin changes fr angular-spectrum maximum and beam-width changes. Linear and surface apertures were als cmpared. t is clear that the requirements n the measurement accuracy fr a linear antenna aperture are higher than the requirements fr tw-dimensinal apertures. This fllws frm the derivatin fequa~ins (5) and (6) fr the ne-dimensinal case. This culd be explained by cnsidering the fact that statistical analyses are perfrmed fr tw crdinates fr the tw-dimensinal case, whereas fr linear antennas they are perfrmed nly fr ne crdinate. All given equatins crrespnd t the case where radiatin patterns are determined using ne realizatin f near-field measurements. f the mean values are calculated using n realizatins, then the angular-spectrum variances are reduced n times, i.e., tne standard deviatin by the.j;" times. The cmputatinal accuracies are very high and they shuld nt be cnsidered if the accuracies fthe measurement are cnsidered. The abve analyses culd be applied fr generatr-frequency instabilities, which change the phase differences between the direct. and reference signals, as well as fr generatr-amplitude instabilities and scalar receiver (r pssibly pwer-meter) errrs. Similarly, they can be used fr prbe-psitin inaccuracies. Peridic errrs can be created during scanning. The phase shifters d nt change phases in an arbitrary range, but nly fr the interval f [, 27l"]. That means nly the values <> - 2m7l" are set (where m is an integer), instead f <1>. Systematic phase-shifter errrs create peridic errrs, the effects f which have been analyzed by several authrs, such as in [2]. f the errrs are d s =f3 sin <1>, where <> =-ax is the required phase value, then the fllwing equatin is valid fr the interval kxl <k 1 : Similarly, a q (KbK2)(]'~ 2 [(K 2 c 2 +K 2 c 2 )/4] 2 == C t : 2 ;q e- t t 2 2 J J A(K 1,K 2 )1 dk 1 dk 2 -- (8) since the ther terms are equal t zer fr the interval kxl < k 1. The span f the spatial perid N M 1 determines if the last term will appear in the interval kx< k 1 (see [17] fr details). The selectin f Ax r dz scanning fr the hlgraphic methd is a difficult prblem, as this determines the angular space [17], and therefre 'the measurement accuracy. EEE Antennas and Prpagatin Magazine, Vl. 5, N.6, December Authrized licensed use limited t: University f Pardubice. Dwnladed n December 5, 29 at 2:13 frm EEE Xplre. Restrictins apply.

4 3. Measurements 2 lg F[v(x)exp(-jax)] We have dne statistical prcessing fmeasurements [9, 15]. Near-field measurements perfrmed by line scanning take less time than thse perfrmed by planar r cylindrical scanning. Therefre, line scanning can be perfrmed many times, and statistical methds can be used t supprt bth the analyses f the accuracy f the hlgraphic methd fr measuring near-field micrwave antenna parameters, and the results fnumerical simulatins...1 We chse a slt antenna (a waveguide with five slts in the brad wall, terminated by a matched lad) fr the measurement. Measurements were carried ut with the reference plarizatin at a wavelength f A = 32 mm. The measuring arrangement was described in detail in [9, 13]. Every measurement was perfrmed by line scanning with 64 samples. After taking a fast Furier transfrm (FFT), 64 values fthe angular spectrum culd be fund. A -2 maximum f ne-third f this number, i.e., 21 values, culd be used. These crrespnded t the values f the angular spectrum (Furiertransfnn), F[C*E(x)]. Meas.l Meas.. 5 vvvvv Meas. 9 Ten measurements were carried ut with a diple prbe and a spacing f Ax = 5mm, with x values ranging frm -16 rm t 155 mm, i.e., N =64 samples. The measured values (Meas. 1, Meas. 5, and Meas. 9) are shwn in Figure 1. The ther measurements were mitted fr the sake f clarity. The angular-spectrum values f these measurements are shwn in Figure 2 (t be mre accurate, these are the amplitudes in db, nrmalized t the first angular-spectrum value). Fr clarity, nly the angular-spectrum values crrespnding t the measurements in Figure 1 are shwn. 1 Meas.l Meas. 5 vvvvv Meas Figure 2. The values f the angular spectrum, 2gF[v(x)e- jtn ]. measured with a diple prbe using Ax=5mm. When testing the statistical hypthesis that the variance, u 2, f a randm variable, x, is u 2 =u6, the critical value f the F distributin and a suitable level fsignificance are cnsidered. The sample mean, v n - l L n x =- "- xi, n 1- (1) and sample variance, 2 1 Ln ( _)2 S =-- "- Xi - x, n-l 1- (11) fr randm samples Xl' x2'..., X n can be calculated. -SO -1 -s xl...) Figure 1. A measurement with a diple prbe, Ax = 5mm, alix =2n/3, alng a straight line The sample mean values, Equatin (1), f the angular spectrum using 1 measurements are pltted in Figure 3. The mean values were nrmalized t the maximum mean f the angular spectrum. The mean values fthe angular spectrum, measured with a diple prbe and Ax = 5 mm spacing, are shwn as a slid line. Each f the values was pltted as a functin f the angle,.9. Fr cmparisn, the radiatin pattern measured in the far field is shwn by a dashed line. The radiatin pattern differs frm the angular spectrum (see [1 ]), yet it is evident that in the vicinity f the main lbe, the far-field measurements and the near-field measurements 92 EEE Antennas and Prpagatin MagaZine, Vl. 5, N.6, December 28 Authrized licensed use limited t: University f Pardubice. Dwnladed n December 5, 29 at 2:13 frm EEE Xplre. Restrictins apply.

5 btained by the diple prbe using a step size f ax = 5mm are in gd agreement. T shw the influence f an incrrect chice f the spacing Ax, ten measurements were perfrmed with the diple prbe, a spacing Ax=7.5mm, and a number f samples N=64, Le., Ax ranged frm -24 mm t mm. n this case, the individual cmpnents fthe angular spectrum affected each ther (aliasing). The sample mean values f these measured values are shwn as circles in Figure 3. Ten measurements were carried ut t shw the influence f the prbe. An pen-ended 5 x mm waveguide was used as a prbe, with a step size f Ax = 5 mm, and N = 64 samples, i.e., Ax ranged frm -16mm t 155 mm. The angular spectrum f the prbe will change the resulting spectrum, as fllwed frm [1]. Therefre, a prbe crrectin is necessary fr cmputing radiatin patterns. The sample mean values f these measurements are pltted as crsses in Figure 3. The sample variances f the angular spectrum were calculated frm Equatin (11) using 1 measurements every time. The 2 /.,,/ 1. 5, ~v{x)e-jllf M1{x,-JOrlf,,,, "". variances (6) with Cl = 1/32 variances (6) with Cl =1/8,,,,,,, 5 --, " 2 lg F[v(x)exp(-jax)) Figure 4. The variances f the measurements with a diple, Ax=5mm.,"', l',1+ ' -2.. r ' ', f.. " " tl ~, :~ -3 1 f ': ~., a, ~"., 1 '. " f,.. 1,,, 11 ' f' ' 11 + V f >[] OpeD-elided wavepide Ax = S mdl " 5 ~[v(x)t-)drr ~pll{l ~llr variances (6) with Cl = 1/32 variances (6) with Cl = Far-field radiati. pattel1l F[l(:z;)cap(- jax)] (db) Diple Ax =S mm Diple 4x=7.5 mm Figure 3. The sample means f the angular spectrum, 21gF[v(x)e-jaxJ. EEE Antennas and Prpagatin Magazine, Vl. 5, N.6, December 28 8[ ] t Figure 5. The variances f the measurements with a diple, ax=7.5mm Authrized licensed use limited t: University f Pardubice. Dwnladed n December 5, 29 at 2:13 frm EEE Xplre. Restrictins apply.

6 angular spectra, Fr cmparisn with thery, it is necessary t estimate the variances f individual errr surces. The measurements (see belw) indicated that the lngitudinal z-psitin errrs with u~ ~ 1-3 were the mst imprtant. The variances can be calculated using Equatin (6). Assuming (see [8]) that ci = 2/N = 1/32, the values shwn in Figures 4 and 5 by dts were btained. Obviusly, the assumptin ci = 2/N is nt crrect (in fact, the crrelatin interval is larger). fwe assume that the crrelatin interval is detennined by the prperties f the scanner (scanning equipment), we arrive at mre realistic values. f the crrelatin interval was 2 mm fr nn-nrmalized crdinates, i.e., C = 1/8 fr Ax = 5mm spacing and ci = 1/12 fr a step size f 7.5 nun, the values btained are as shwn in Figures 4 and 5 by the shrter dashes. The values Ct = 1/8 and ci = 1/12 are als nt quite satisfactry. Hwever, the agreement between the measured and calculated (Equatin (6)) results using cnsiderable simplificatin became wrse with increasing ci : see [9, 14]. Therefre, the values f Equatin (6) d nt agree well with the variances calculated fr an increasing Ct using Equatin (11). Fr the limiting case Cl ~, the variances are given by 1L1F[v(x)e- jaxr'f the measurements perfrmed with the diple prbe and with step sizes f 5 m.m and 7.5 mm are shwn in Figures 4 and 5 as slid lines. Fr cmparisn, the sample variances f the angular-spectrum amplitudes, {1L1F[v(x)e-jaxJlr' are pltted in Figure 4 and 5 with dashed lines. The variances were nrmalized t the maximum mean fthe angular spectrum. -SO -SO -1 Diple Ax = 7.5 lu -1 -SO Diple 4x =5 lu,5 wl ( r arg (w) C xl...) xl...] SO Figure 6. The sample means f the measured fields (calculated using angular spectra). s 1 1 SO Opea-eaded wavepicle Ax=5 18m (12) Hwever, this estimate is t lw in the main-lbe regin and t high in the sidelbe regin. f the prperties f the scanning equipment are nt knwn, we can use a cmbinatin f Equatin (6) fr ci ~ 2/N (in the sidelbe regin) with Equatin (12) having effect particularly in the main-lbe regin. We can thus cnclude that the use f Equatin (6), derived with cnsiderable simplificatins, results in an apprximate estimate fr the variance, prvided that the crrelatin interval, ct, is detennined by the scanning-equipment characteristics. As mentined abve, nly apprximately ne-third f the angular-spectrum values can be used (mre accurately, thse fr n = 1,2,...,11 and n = 55,56,...,64). When replacing the remaining values by zers (fr n = 12,13,...,54) and perfrming the inverse Furier transfrm, the apprximate field amplitude and phase distributins alng a line can be btained; detailed analyses can be fund in [17]. 1-6 The sample mean values f the field distributin, calculated with Equatin (1) using 1 values each time, are shwn in Figure 6. The slid line (r the circles) represent the values detennined by measurements with a diple and Ax = 5 nun (r Ax=7.5mm), and the crsses shw the results f measurements with an penended waveguide. ndividual cases differed, due t an imprper chice f the sample interval, Ax (measurement with the diple x[...) SO SO 1 SO Diple Ax =5 Dm Opea-eaded W3vepide Ax = 5 DUll Diple 4x = Figure 7. The sample variances f the field-amplitude errrs, uj, nrmalized t the maximum mean fthe field amplitude. 94 EEE Antennas and Prpagatin MagaZine, Vl. 5, N.6, December 28 Authrized licensed use limited t: University f Pardubice. Dwnladed n December 5, 29 at 2:13 frm EEE Xplre. Restrictins apply.

7 5 :, Numerical Simulatins We perfrmed numerical simulatins, cnsidering randm errrs as well as detenninistic errrs f typical prblems ccurring in near-field measurements [9, 12, 14, 16]. Cmputer-generated randm sequences are needed fr numerical simulatins. Being interested in Gaussian randm sequences, the crrelatin sequence f Equatin (2) was used. The theretical analysis derived abve used cnsiderable simplificatins. Antennas with cnstant amplitude and phase distributins are usually assumed, but this type f distributin des nt cme in questin in near-field measurement. A typical field distributin is ne where the phase varies quickly at the edge and the amplitude is small. Therefre, the input sequence, E(i) (the near-field distributin) was cnsidered t be x(mm] +... S SO Diple 4x = 5 mm Opea-eaded W3-epide 4x = 5 mm Diple t1x = 7.5 mm Figure 8. The sample variances f the field phase errrs, O'~. where i = 1,2,...,64; m = i fr i = 1,2,...,11; m = fr i = 12,13".,,53 and m = -i fr i = 54,55,...,64; a and f3 are cnstants; and q (i) are Gaussian randm sequences with the crrelatin sequence given by Equatin (2). The measurements n a line (ne-dimensinal case) in the near field fr bth the amplitude and phase measuring methd when the Furier transfrm, F[E(i)J, was determined by the FFT, and by the hlgraphic methd using Equatin (4) were simulated fr a = 3 and fr values f f3 f r 2;r. The near-field data were filtered (a quarter fthe transfrmed values was nly inverse transfrmed). Every prblem was simulated with 3 realizatins f E(i). and Ax = 7.5mm), r the influence fthe prbe (measurement with an pen-ended waveguide). The sample variances fthe field-amplitude errrs, a~, were nrmalized t the maximum mean f the field amplitude. The measurement results are shwn in Figure 6 using a slid line (r circles) fr a diple prbe and Ax = 5mm (r 7.5 mm), and using crsses fr an pen-ended waveguide. The sample variances f field phase errrs are pltted in Figure 7. The slid line, circles, and crsses represent the measurements with a diple using Ax = 5 mm, Ax = 7.5 mm, and with an pen-ended waveguide, respectively. Variances determined by filtering and inverse Furier transfrm present nly a rugh idea abut the distributins and magnitudes f the measured field variances (see [9, 14]). On the ther hand, these variances cannt be immediately fund by hlgraphic measurement, althugh this is pssible in the case famplitude and phase measurements. As fllws frm Figures 7 and 8, the amplitude-errr variances were f the rder f 1-4, and the phase-errr (psitinerrr) variances were f the rder f 1-3 Transverse x-psitin errrs caused an increase in phase errrs fr xl > 1, because the phase was nt cnstant fr xl > 1 but varied very quickly: see Figure 6. Because the field amplitudes were t lw in this regin, nly a negligible influence can be assigned t the phase errrs. EEE Antennas and Prpagatin Magazine, Vl. 5, N.6, December /3=..., /3=21! /, ".., /,...,/,1,1,1 ', ', ',,, ~ / ~ " " i Figure 9. The mean values, F[E(i)], fr cl = 1/32 and '= Authrized licensed use limited t: University f Pardubice. Dwnladed n December 5, 29 at 2:13 frm EEE Xplre. Restrictins apply.

8 1-6 t1f[v<xyj<ur ~=O ~=21t-- ~= ~=21t ' &.-.i Figure 1. The sample variances f the angular spectra, ~F[v(x)e-jaxrand (~F[v(x)e-jaxJlr, fr Ct =1/32 u=o.ol. Varius prblems fr several values f c' f3, and standard deviatin u and were simulated. See Figure 9 fr the mean values, F [ E (i)j. The ther cases were nearly the same and they are therefre nt shwn. The sample variances f the angular spectrum, ~F[v(x) e- jaxr'and thse f the angular-spectrum amplitudes, ering randm errrs as well as deterministic errrs, and the prcessing fmeasurement statistics. The imprtance f the varius near-field measurement errrs depends n the antenna under test, the frequency f peratin, the measurement facility, and the prbe. Hwever, the results shw that the finite scan area, the z psitin fthe prbe fr planar scanning, receiver nnlinearities in measuring the near-field amplitude, and, smetimes, multiple reflectins, are usually the mst imprtant. t is necessary t estimate the variances f individual errr surces fr cmparisn with thery. Our measurements (shwn in Figures 1 t 8) indicated that the lngitudinal z-psitin errrs with u~ ~ 1-3 were the mst imprtant fr ur measurement facility. The variances can be calculated using Equatin (6). We have demnstrated that the assumptin c =2/N (given in the references) is nt crrect. The apprximate estimates f the variance present reasnable results (as shwn by analyses f the measurements perfrmed by line scanning) if the crrelatin interval, c' is determined by the prperties fthe scanning equipment. This paper has summarized the analyses f accuracy using mathematical mdels cnsidering randm prcesses with crrelatin intervals, prcessing f measurement statistics, and numerical simulatins. t was demnstrated that the given analyses crrespnded t ur measurements and/r numerical simulatins. t was shwn by filtering and inverse Furier transfrming that the amplitude and phase errr estimates crrespnded t ur measured values. We can cnclude that the described methd is applicable bth t the field measurement, perfrmed by line scanning, and t the errr analyses fthis methd. (~F[v(x) e- jax Jr'are pltted in Figure 1. Fr the sake f brevity, nly sme fthe calculated simulatins are discussed here. Numerical simulatin can be used t slve prblems fmeasuring micrwave antennas in the near field withut the influence f the cnsiderable limitatins needed fr an analytical slutin. t was shwn [9, 11] that theretical estimates can be used fr a gd descriptin fthe variances fr small intervals f c. With increasing C' the differences between the thery and numerical simulatins becme mre significant (particularly fr c > 1/8 ). 5. Cnclusins Synthesized-reference-wave hlgraphic techniques ffer relatively simple and cst-effective measurements f antenna radiatin characteristics, and recnstructin f cmplex aperture fields using near-field intensity-pattern measurement. We have prpsed synthesized-reference-wave hlgraphy. These methds allw emplying the advantages f methds fr prbe cmpensatin fr near-field amplitude and phase measurements fr planar and cylindrical scanning, including accuracy analyses. We have published mst f the results, such as a descriptin f hlgraphic near-field measurements using prbe cmpensatin fr planar and cylindrical scanning, a cmparisn f hlgraphic near-field and far-field measurements with and withut prbe cmpensatin, and accuracy analyses using mathematical mdels cnsidering randm prcesses with crrelatin intervals, numerical simulatins cnsid- 6. Acknwledgements The paper was supprted by the Czech Natinal nstitutinal Research "Thery ftransprt System" N. MSM References 1. A. D. Yaghjian, "An Overview f Near-Field Antenna Measurements," EEE Transactins n Antennas and Prpagatin, AP 34, 1, January 1986, pp D. Slater, "Near-Field Test Facility Design," Antenna Measurement Techniques Assciatin Cnference, O. M. Bucci and M. D. Miglire, "A New Methd fr Aviding the Truncatin Errr in Near-Field Antenna Measurements," EEE Transactins n Antennas and Prpagatin, AP-54, 1, Octber 26,pp A. C. Newell, "Errr Analysis Techniques fr Planar Near-Field Measurements," EEE Transactins n Antennas and Prpagatin, AP-36, 6, June 1988, pp A. D. Yaghjian, "Upper-Bund Errrs in Far-Field Antenna Parameters Determined Frm Planar Near-Field Measurements. Part 1: Analysis," Natinal Bureau f Standards Technical Nte 667, Bulder, Octber EEE Antennas and Prpagatin Magazine, Vl. 5, N.6, December 28 Authrized licensed use limited t: University f Pardubice. Dwnladed n December 5, 29 at 2:13 frm EEE Xplre. Restrictins apply.

9 6. A. C. Newell and C. F. Stubenrauch, "Effect f Randm Errrs in Planar Near-Field Measurement," EEE Transactins n Antennas and Prpagatin, AP-36, 6, June 1988, pp ntrducing the Feature Article Authrs 7. A. C. Newell and D. Lee, "Applicatin f the NST 18 Term Errr Mdel t Cylindrical Near-Field Antenna Measurements," Antenna Measurement Techniques Assciatin Cnference, Philadelphia, Octber V. Kaplun and A. P. Kurchkin, "Analysis f Randm Errrs in Hlgraphic Technique Determinatin f Directinal Antenna Characteristics," Raditekhnika i Elektrnika, XX, 1, Octber 1975,pp V. Schejbal, Micrwave Antenna Near-Field Measurements, PhD Thesis, SAV Bratislava (Slvakia), 1979 (in Czech). 1. V. Schejbal, "Near-Field Antenna Measurement Using Hlgraphic Methd," Slabprudy bzr, 39, 9, September 1978, pp (in Czech). 11. V. Schejbal, "Accuracy f Near-Field Antenna Measurement Using Hlgraphic Methd," Slabprudy bzr, 41, 4, April 198, pp (in Czech). Vladimir Schejbal graduated frm the Czech Technical University' Prague, in 197. He received the PhD degree frm the Slvak Academy f Science, Bratislava, in 198. He was with the Radi Research nstitute Opcinek, the Czech Republic (Antenna Department) frm 1969 t Frm 1983 t 1986, he was n leave with the Higher nstitute f Electrnics (Micrwave Department) Beni Walid, Libya, as a lecturer. He has been with the University fpardubice, the Czech Republic, since 1994, nw as a full prfessr and head f the department. He is interested in micrwave antennas and prpagatin. He has published ver 1 papers. He is a Senir Member fthe EEE. 12. V. Schejbal and J. Hnig, "Hlgraphic Methd fnear-field Antenna Measurements," 1th Eurpean Micrwave Cnference, Warszawa, Pland" 198, pp V. Schejbal and V. Kvarik, "Hlgraphy Methds fr Antenna Near-Field Measurement," Tesla Electrnics, 13, 2, 198, pp V. Schejbal, "Numerical Simulatin f Near-Field Antenna Measurement," Tesla Electrnics, 13, 3, 198, pp V. Schejbal and V. Kvarik, "Accuracy fnear-field Antenna Measurement Using Hlgraphy," Tesla Electrnics, 14, 2, 1981, pp V. Schejbal, "Numerical Simulatins fprblems Occurring in Near-Field Antenna Measurements," 7th Cllquium n Micrwave Cmmunicatin, Budapest, Hungary, 1982, pp Jan Pidanic was brn in He received the MS degree frm the Jan Perner Transprt Faculty, University fpardubice, the Czech Republic, in 25. He has been with the same university since 27, where he is nw wrking twards his PhD. He is interested in the prpagatin f electrmagnetic waves, and in signal and data prcessing. 17. V. Schejbal, V. Kvarik, and D. Cermak, "Synthesized-Reference-Wave Hlgraphy fr Determining Antenna Radiatin Characteristics," EEE Antennas and Prpagatin Magazine, 5, 5, Octber 28, pp D. Smith, M. Leach, M. Elsdn, and S. J. Fti, "ndirect Hlgraphic Techniques fr Determining Antenna Radiatin Characteristics and maging Aperture Fields," EEE Antennas and Prpagatin Magazine, 49, 1, February 27, pp J. S. Shifrin, Prblems fstatistical Antenna Thery, Part, Mscw, S. Radi, 197 (in Russian). 2. K. Bahadri and Y. Rahmat-Samii, "Characterizatin feffects f Peridic and Aperidic Surface Distrtins n Membrane Reflectr Antennas," EEE Transactins n Antennas and Prpagatin, AP-53, 9, September 25, pp Vlastimil Kvarik cmpleted his studies at the Pardubice High Schl f Electrical Engineering, the Czech Republic, in 197. He was with the Radi Research nstitute Opcinek, the Czech Republic (Antenna Department) frm 197 t He has been with Retia a.s., Pardubice, since He is interested in radar antennas and prpagatin. He has published ver 1 papers. EEE Antennas and Prpagatin Magazine, Vl. 5, N.6, December Authrized licensed use limited t: University f Pardubice. Dwnladed n December 5, 29 at 2:13 frm EEE Xplre. Restrictins apply.

10 Dusan Cermak was brn in He received his MS degree frm the Czech Technical University in Prague in 23, and his PhD frm the University f Pardubice, the Czech Republic, in 27. He has been with the University f Pardubice since 23 as an Assistant Prfessr. He is interested in simulatins, measurements, radar antennas, and prpagatin f electrmagnetic waves. He has published ver " EEE Antennas and Prpagatin Magazine, Vl. 5, N.6, December 28 Authrized licensed use limited t: University f Pardubice. Dwnladed n December 5, 29 at 2:13 frm EEE Xplre. Restrictins apply.