Far-field errors due to random noise in cylindrical near-field measurements

19 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 40, NO. 1, JANUARY 1992 Fa -Field E o s Due o Random Noise in Cylind ical Nea -Field Measu emen s Jo di Romeu, S uden Membe , IEEE, Luis Jo e, Membe , IEEE, and Angel Ca dama, Membe , IEEE Abs ac -A ull cha ac e iza ion o he a - ield noise ob- ained om cylind ical nea - o a - ield ans o ma ion, o a whi e Gaussian, space s a iona y, nea - ield noise is de i ed. A possible sou ce o such noise is he ecei e addi i e noise. The noise cha ac e iza ion is done by ob aining he au oco ela ion o he a - ield noise, which is shown o be easily compu ed du ing he ans o ma ion p ocess. E en o his simple case, he a - ield noise has complex beha io dependen o he mea- su emen p obe. Once he s a is ical p ope ies o he a - ield noise a e de e mined, i is possible o compu e uppe and lowe bounds o he adia ion pa e n o a gi en p obabili y. These bounds de ine a s ip wi hin he adia ion pa e n is ound wi h he desi ed p obabili y. This may be used as pa o a comple e nea - ield e o analysis u a pa icula cylind ical nea - ield acili y. I. INTRODUCTION EAR-FIELD an enna pa e n measu emen has become N a widely employed echnique o e he pas yea s. As he a - ield pa e n is ob ained a e p ocessing he nea - ield da a, i is di icul o o esee he e ec o a gi en nea - ield e o on he a - ield pa e n. Thus a , exp essions a e a ailable [l], [2] ha p edic he e ec o andom e o s o he plana case. These exp essions a e no alid o he cylind ical case, due o he di e en na u e o he ans o ma- ion p ocess. The a - ield noise is a s ochas ic p ocess, and o each measu emen ha is pe o med a new ealiza ion o he p ocess is ob ained. The e o e, i is no possible o speci y he a - ield e o in e ms o a de e minis ic alue, bu a he i has o be looked as a andom a iable whose dis ibu ion unc ion has o be de e mined. This app oach o he p oblem allows he compu a ion o bounds o he e o o a gi en p obabili y, ha is a alue ha i is no exceeded o a chosen p obabili y. These bounds o e a be e unde s anding o he e ec s o andom e o s in he a - ield pa e n. Mo eo e , i will be shown ha o a whi e Gaussian, space s a iona y, nea - ield noise, he a ield is a Gaussian, nons a iona y in ele a ion, and colo ed in azimu h p ocess. Thus, a pa ame e such as he a - ield signal- o-noise a io, al hough being a qui e appealing simple pa ame e , hides in i s simplici y a complex phenomenon. The pape is o ganized as ollows. In Sec ion 11 he au oco ela ion o he a - ield noise o a whi e Gaussian Manusc ip ecei ed Ap il 16, 1991; e ised Augus 22, 1991. The au ho s a e wi h he Depa men o Signal heo y and Communica- ions, Telecommunica ion Enginee ing School o Ba celona, Poly echnic Uni e si y o Ca alonia, 08080 Ba celona, Spain. IEEE Log Numbe 9105280. nea - ield noise is de i ed. In Sec ion 111 is shown how o e alua e he au oco ela ion om he nea - ield measu emen , and i s applica ion o p edic he e o bounds. Sec ion IV shows a simula ed as well as expe imen al e i ica ion. II. FAR-FIELD NOISE AUTOCORRELATION The heo y o he cylind ical nea - o a - ield ans o m has been well de eloped elsewhe e [3]-[5]. In his sec ion, we will employ hose exp essions necessa y o de elop he e o o mula ion. A he momen , only he case o a com- plex whi e Gaussian, space s a iona y, wi h ze o mean, and a a iance U* nea - ield noise is conside ed. This model as- sumes ha all andom e o s a e o Gaussian na u e, an assump ion ha i is ue o ce ain sou ces o e o like ecei e loo noise, bu inco ec o o he sou ces as quan i- za ion e o s. Ne e heless, he Gaussian model is accu a e enough o mos measu emen si ua ions whe e lack o dy- namic ange is he undamen al limi a ion. The cylind ical nea - o a - ield ans o ma ion is based on ob aining he cylind ical modal coe icien s o he ields adi- a ed by he an enna unde es (AUT). Once he coe icien s a e known, he asymp o ic exp ession o he a ields as a unc ion o he modal coe icien s is employed o compu e he AUT a ield. Two measu emen s co esponding o wo di e en measu emen p obes a e equi ed o ob ain he modal expansion o he AUT ields. F om a heo e ical poin o iew e y ew cons ain s a e placed on he choice o he measu emen p obes, bu in p ac ice he p obes a e chosen so each one esponds basically o one o hogonal pola iza ion. In he ollowing analysis he s a is ical cha ac e is ics o he a - ield noise due o he ans o ma ion o he nea - ield noise a e ound. The analysis is pe o med by conside ing nea - ield da a con aining only noise, and applying he cylin- d ical nea - o a - ield ans o m o his da a. Le nn (zi, 4,) be he measu ed nea ield, whi e Gauss- ian noise o e he cylind ical measu emen su ace. The measu emen g id is de ined by he sample poin s zi and +/. The e ical and azimu hal sampling spacing is Az and A4, and hey a e assumed o be cons an . The numbe o mea- su ed ings is N,, and he numbe o measu ed poin s pe ings is N+,. The i s s ep in he ans o ma ion p ocess is he Fou ie ans o m o he nea ield. Le k, and n be he a iables in he ans o med domain. The disc e e Fou ie ans o m (DFT) o he nea - ield noise is 0018-926X/92$03.00 0 1992 IEEE 80 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 40, NO. 1, JANUARY 1992 The au oco ela ion o he DFT o he noise is [6] This au oco ela ion unc ion co esponds o a whi e Gauss- ian noise, because he spec al powe densi y, ha is de ined as he Fou ie ans o m o he au oco ela ion, is la . The cylind ical modal coe icien s co esponding o he expansion o he nea - ield noise in cylind ical wa es a e ela ed o he nea - ield Fou ie ans o m as k,, n) bL2)( - k,) - i;,( k,, n) b;)( - k,) U!)( - k,) bL2)( - k,) - U',"'( - k,) b:')( - k,) k bn(kz) = - ~ 16? 2k: i: (k,, n)a$)(-k,) - ii (k,, n)bL2'(-k CZ~)( - k,) bL2'( - k,) - @( - k,) b!)( - k,) whe e uz)(kz) and b?)(k,) a e ela ed o he modal coe i- cien s o he measu emen p obe, and can be compu ed as shown in [3] o [4], and and ii a e he DFT o he noise in he ho izon al and e ical measu emen . 7 is he medium impedance, k = w m, and k, = dm. In his case he modal coe icien s a e s ochas ic p ocesses, and assuming ha ii and ii a e s a is ically independen , he au oco ela ion o he coe icien s is Raj P7 4) (4) Now an( k,) and bn( k,) a e Gaussian andom a iables wi h ze o mean, bu a e no s a iona y as hei au oco ela ion is a unc ion o n and k,. Each a - ield componen is ob ained as PI N. E,(k,, 4) = -2ksin81 jn+'bn(kz)ejnd n N. E+( k, ,4) = j2 k sin 0 1 jn+lan( k,) e'"+ (6) n wi h k, = k COS 8 k, = ksin8. (7) The au oco ela ion o he 8 a - ield noise componen is Rn0(7,E) =E{+,+ 774+ E)n;(k,A)} = (2ksi118)~E E{bn(kz + T)b;(k,)} mn . jn + lj- (m+ l)ejn(++€)e- jm0 as R,jn - n, 7) is ze o i n # m I al ),( - k,) 1 + I @( - k,) I c I a )( -k,)bL2)( -k,) - @( -k,)bL')( -k,) I - ejn'6(7). (9) The au oco ela ion o he 4 componen is de i ed in an analogous way. The a - ield noise is whi e Gaussian nons a- iona y in kz, because he au oco ela ion is a unc ion o kz, and s a iona y and colo ed in 4 because he au oco ela- ion is nonze o o [ # 0. The a - ield noise a iance o each pola iza ion is o, @(kz 4) = R,,(O, 0) 81 ROMEU e al.: FAR-FIELD ERRORS DUE TO RANDOM NOISE F om he abo e exp essions, he e ec o an addi i e Gauss- ian whi e noise in he nea ield is an addi i e Gaussian noise wi h a iance p opo ional o he nea - ield noise a iance and unc ion o kz. Mo eo e , he a iance depends on he p obe cylind ical coe icien s ( adia ion pa e n), so he e - ec s o he noise will be dependen on he p obe and in gene al di e en o each pola iza ion. III. FAR-FIELD ERROR BOUNDS The noise-con amina ed adia ion pa e n will be o he o m -- 4) = E@,, 4) + Ti(‘,, 4) ( 12) whe e s(kz, 4) is a Gaussian noise wi h a iance u;(k,) gi en by (10) and (11). The a - ield magni ude is o a gi en pola iza ion a - ield noise a iance. This can be easily accomplished du ing he ans o ma ion p ocess. I should be no ed ha he p obe co ec ion coe icien s needed o e alua e (10) and (1 1) a e also compu ed o p oduce he nea - o a - ield ans o m. Ne e heless, one unknown emains le o be ound as he a ield noise a iance is p opo ional o he nea - ield noise a iance u2. The nea - ield a iance can be es ima ed om he nea - ield measu emen by aking in o accoun he bandlimi ed p ope - ies o he AUT ields in he spec al domain n - k,. The AUT ield ep esen a ion is limi ed o he isible domain de ined by ( ;)2 + ( i2 < 1 whe e (I is he adius o he minimum cylinde ha encloses he AUT. Conside ing (2), he nea - ield noise a iance can be es ima ed as whe e X and Y a e he eal and imagina y pa s o E( k,, 4) + n(k,, 4). X and Y a e Gaussian andom a iables wi h mean alue % (E) and (E), and a iance U: and uk, espec i ely, and (14) =I 2 im 2 “ . I E I is a andom a iable wi h a Rice p obabili y densi y unc ion (pd ) gi en by [7] whe e Io is he modi ied Bessel unc ion o o de ze o. When I E I ’/U; >> 1, he Rice pd can be app oxima ed by a Gaussian pd ‘wi h mean alue I E 1, on he o he hand i I E I ’/U: << 1 we ha e a Rayleigh pd wi h mean alue U, J(? /2). No ice ha in he la e case he mean alue o he ob ained a - ield pa e n is independen o he ac ual alue o he adia ion pa e n I E 1. Knowing he pd o he module o he a ield, an uppe and lowe bound M and m o 1 E I can be calcula ed by whe e En is he DFT o AUT nea - ield plus he whi e Gaussian nea - ield noise nn,(z, +), and he summa ion is ex ended o e he e anescen egion o he spec al domain. The a iance is compu ed as he ensemble a e age o he noise powe . This es ima ion p ocedu e assumes ha he noise in he spec al domain is s a iona y, which in his case is ue. I should be no ed ha he pd o he noise pe u bed a ield (15) is a unc ion o he a - ield noise a iance U; and he unpe u bed a - ield magni ude 1 E I. In o de o compu e he bounds m and M gi en by (16) and (17) i is necessa y o know he unpe u bed alue o he a - ield pa e n. Ob iously his is no a ailable om a noise-con- amina ed nea - ield measu emen , bu usually a ce ain a p io i knowledge o he AUT adia ion pa e n is a ailable. I mus be no ed ha he a - ield signal o noise a io de ined as can be easily compu ed a e he ans o ma ion p ocess. . (~l?~)d~k~ =p (16) IV. VERIFICATION Nume ical and expe imen al e i ica ion o he abo e o - mula ion has been done. To e i y (10) and (ll), he a - ield noise a iance has been es ima ed by pe o ming a high numbe o nea - o a - ield ans o ms, wi h only noise as inpu . The a - ield noise a iance is es ima ed as m P(JE’I >m) = 1 - L ’( I E I)d = ‘ (17) whe e p and q a e he p obabili ies o exceed he bound and no o exceed m, espec i ely. The p obabili y ha he a ield is be ween M and m is (18) To compu e he a - ield bounds, i is equi ed o e alua e he whe e nR is he i h ans o med a - ield noise, and N is he numbe o ans o ma ions. Figs. 1 and 2 show he esul a e lo00 ans o ma ions, compa ed o he e alua ion o - 02 5 5 XI0 5 4 *IO 5 3 *IO IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 40, NO. 1, JANUARY 1992 ~ ~ - heo y esiimaied - - . - . -. . OL - 6 *IO5 heo y es ima ed - -. -. -. . a:[ 5 o2 - 9 *I0 20 40 60 80 100 120 140 160 he a (deg ees) es ima ed - -. -. -. . heo y OX 5 O2 - 6 *IO 2 *I0 80 100 120 140 160 20 40 60 he a (deg ees) (b) Fig. 1. Ra io be ween he a - ield noise a iance and he nea - ield noise a iance o an ideal magne ic p obe. The measu emen pa ame e s a e case 1 o Table I. (a) The a pola iza ion, magne ic p obe, = 3 GHz. (b) Phi pola iza ion, magne ic p obe, = 3 GHz. exp essions 10 and 11. Two di e en cases and speci ics a e shown in Table I. The p obe employed in case 2 is a o a ionally symme ic p obe ha is o a ed 90" along i s axis. Due o he symme y o he p obe he a - ield a iance is he same o bo h pola iza ions. I is impo an o no ice ha he a - ield a iance ac ually depends on he measu e- men p obe and he ele a ion angle. Fig. 3 shows he nume ically simula ed a ield o a low sidelobe a ay and he p edic ed uppe and lowe bounds de ining an 80% p obabili y s ip when he nea - ield noise is 40 dB. Fig. 4 shows he a ield when he nea - ield S/N is ac ually 40 dB. Fig. 5 shows he adia ion pa e n o a 50 dB S/N a io and he compu ed bounds. F om Figs. 4 and 5 i is concluded ha in his case a 40 dB S/N a io p ac ically smea s he sidelobes, while o 50 dB an e o o app oxi- ma ely +2 dB is expec ed in he i s sidelobes. The expe imen al e i ica ion has been done by pe u bing a nea - ield measu emen wi h a nea - ield S/N g ea e han 50 dB. The cylind ical nea - ield measu emen ange is de- sc ibed in [8]. The a - ield pa e n ob ained om he ini ial measu emen has been conside ed an e o - ee pa e n, and used as unpe u bed a ield o compu e he e o bounds when he nea - ield S/N is smalle han 50 dB. The nea - ield da a has been pe u bed by adding noise o achie e he I 20 40 60 80 100 120 140 160 ele a ion (deg ees) (a) heo y es ima ed - -. -. -. . OL 5 U* - 9 *io 5 8 *IO 5 7 XI0 5 6 *IO 5 5 *I0 20 40 60 80 100 120 140 160 he a (deg ees) (b) Fig. 2. Ra io be ween he a - ield noise a iance and he nea - ield noise a iance o eal p obe. The measu emen pa ame e s a e case 2 o Table I. (a) The a pola iza ion, eal p obe, = 10 GHz. (b) Phi pola iza ion, eal p obe, = 10 GHz. TABLE I MEASUREMENT PARAMETERS EMPLOYED IN THE EVALUATION OF FIGS. 1 AND 2 Pn Case 1 3 64 64 0.5 1 0.51 magne ic Case2 10 64 64 0.5 1 0.15 eal desi ed S/N. In Table I1 is shown he nea - ield S/N and he es ima ed alue ob ained as desc ibed in Sec ion 111. The e o in he es ima ion is in gene al lowe han 1 dB and inc eases o la ge and small alues o he S/N. Fo la ge alues o he S/N a io he noise es ima ion is mo e subjec o e o , because he assump ion ha he e is a null con ibu ion o he ield in he e anescen egion i is no comple ely ue in he nea - ield zone, and as he noise a iance becomes smalle he e o is g ea e . On he o he hand, o low S/N he alue o he maximum ield becomes mo e co up ed by he noise, esul ing in a w ong es ima ion o he S/N. Fig. 6 shows he e o bounds and he e o in he adia- ion pa e n o a 40 dB S/N. The esul s show how a good p edic ion o he e o due o andom noise can be accom- plished. 83 ROMEU e al.: FAR-FIELD ERRORS DUE TO RANDOM NOISE CO-POLAR COMPONENT dynamic ange 40 dB CO-POLAR COMPONENT dynamic ange 50 dB module IdBl module (dB) 0 -20 -20 - -40 -40 - 45 75 60 90 W5 120 135 / -60 -60 45 60 75 90 105 120 135 ele a ion (deg ees) ele a ion (deg ees) (a) (a) CROSS-POLAR COMPONENT CROSS-POLAR COMPONENT dynamic ange 50 dB dynamic ange 40 dB module IdB) -20 I ele a ion (deg ees) (b) Fig. 3. In solid lines co-pola (a) and c oss-pola (b) adia ion pa e n o a low sidelobe an enna and in do ed lines 90% bounds o a 40 dB nea - ield S/N a io. The bounds de ine an 80% p obabili y s ip. CO-POLAR COMPONENT dynamic ange 40 dB module (dBl ele a ion (deg ees) (a) dynamic ange 40 dB CROSS-POLAR COMPONENT module (dB) -20, I -80 ..,.,..,..,..,..I 45 60 75 90 105 120 135 ele a ion (deg ees) (b) Fig. 4. Compu ed adia ion pa e n when he nea - ield S/N a io is 40 dB. Fig. 5. 45 60 75 90 105 120 135 ele a ion (deg ees) (b) Compu ed adia ion pa e n when he nea - ield S/N a io is 50 dB and he compu ed bounds de ining an 80% p obabili y s ip. CO-POLAR COMPONENT nea ield SIN = 40 dB elm, IOWW - . “DDW -. e o (dBl I I 6 go 92 94 96 98 100 102 WI 106 108 110 azimu h (deg ees) Fig. 6. E o o a eal measu emen o a 40 dB nea - ield SIN a io and compu ed bounds o a 90% p obabili y, de ining an 80% p obabili y s ip. TABLE I1 NEAR-FIELD SIN RATIO AND &TIMATION FROM THE NEAR-FIELD MEASUREMENT Nea -Field SIN Es ima ed SIN E o (dB) (dB) (dB) 15 20 25 30 35 40 45 16.78 1.78 20.57 0.57 25.26 0.26 30.01 0.01 34.98 - 0.02 39.71 - 0.29 44.02 - 0.98 84 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. VOL. 40, NO. 1, JANUARY 1992 V. CONCLUSION In his pape an exp ession o he a - ield noise au oco - ela ion o a nea - ield Gaussian andom noise is de i ed. The exp ession conside s he e ec o he measu emen p obe and i is shown ha he p obe plays an impo an ole in he de e mina ion o he a - ield noise a iance. I is impo an o no ice ha he a - ield noise a iance is p opo ional o he numbe o sample poin s, and p opo ional o he squa e Jo di Romeu (S’88) was bo n in Ba celona, Spain, in 1962. He ecei ed he Ingenie o and Doc o Ingenie o deg ees in elecommunica ion enginee - ing, bo h om he Poly echnic Uni e si y o Ca - alonia (UPC), Ba celona, Spain, in 1986 and 1991, espec i ely. In 1985 he joined he An enna-Mic owa e-Rada g oup o he Signal Theo y and Communica ions Depa men he e. Cu en ly he is Associa e P o- esso a UPC, whe e he is engaged in esea ch in an enna nea - ield measu emen s, an enna diagnos- o he sampling space. So in a e& measu emen si ua ion, whe e he dynamic ange equi emen s may be c i ical, i is ics and mic owa e imaging. impo an o choose hese pa ame e s co ec ly. This exp es- sion can be easily e alua ed du ing he nea - o a - ield ans o ma ion. Once he a - ield noise is cha ac e ized, i is possible o p edic he e ec o he noise on he adia ion pa e n. This equi es a ce ain knowledge o he unpe u bed adia ion pa e n. As he noise is o andom na u e, he a - ield e o is gi en in e ms o an e o wi h a chosen p obabili y. REFERENCES A. C. Newel and C. F. S uben auch, “E ec o andom e o s in plana nea - ield measu emen ,” IEEE T ans. An ennas P opaga ., ol. 36, pp. 769-773, June 1988. J. B. Ho man and K. R. G imm, “Fa - ield unce ain y due o andom nea - ield measu emen e o ,” IEEE T ans. An ennas P opaga ., ol. 36, pp. 774-780, June 1988. W. M. Leach and D. T. Pa is, “P obe compensa ed nea - ield measu emen s on a cylinde ,” IEEE T ans. An ennas P opaga ., G. V. Bo gio i, “In eg al equa ion o mula ion o p obe co ec ed a - ield econs uc ion om measu emen s on a cylinde ,” IEEE T ans. An ennas P opaga ., ol. AP-26, pp. 572-578, July 1978. A. D. Yaghjian, “Nea - ield an enna measu emen s on a cylinde su ace: A sou ce sca e ing ma ix o mula ion,” Na . Bu . S and., NBS Tech. No e 696, July 1977. A. Papoulis, P obabili y, Random Va iables, and S ochas ic P o- cesses. New Yo k: McG aw-Hill, 1965. D. 0. No h, “An analysis o he ac o s which de e mine signal/noise disc imina ion in pulsed-ca ie sys ems,’’ P oc. IEEE, ol. 5 1, pp. J. Romeu e al., “A cylind ical nea - ield es acili y,” Mic owa e Eng. Eu ope, pp. 25-31, Sep ./Oc . 1990. ol. AP-21, pp. 435-446, July 1973. 1016-1027, July 1963. Luis Jo e (S’79-M’83) was bo n in Ma a 6, Spain, in 1956. He ecei ed he Ingenie o and Doc o Ingenie o deg ees in elecommunica ion en- ginee ing, bo h om he Poly echnic Uni e si y o Ca alonia (UPC), Ba celona, Spain, in 1978 and 1982, espec i ely. In 1978 he was Resea ch Assis an in he Elec- ophysics g oup a UPC, whe e he wo ked on he analysis and he nea - ield measu emen o an en- nas. In 1981 he joined he Ecole Sug ieu e d’Elec- ici i in Pa is, whe e he was in ol ed in mi- c owa e imaging echniques o biomedical applica ions. Du ing he pe iod 1986-1987 he was a Visi ing Fulb igh Schola a he Geo gia Ins i u e o Technology, A lan a, wo king on an enna measu emen and elec omagne ic imaging. He is cu en ly P o esso and Di ec o o he Telecommunica ion Enginee ing School a UPC whe e he is engaged in esea ch on an ennas and elec omagne ic sca e ing and imaging, bo h nume ical and expe imen al aspec s. Angel Ca dama (S’67-M’73) was bo n in San i- ago, Spain, on May 13, 1944. He ecei ed Inge- nie o de Telecomunicaci6n deg ee om he Poly- echnic Uni e si y o Mad id, Mad id, Spain, in 1968, and he Sc.M. and Ph.D. deg ees in elec i- cal enginee ing om B own Uni e si y, P o i- dence, RI, in 1970 and 1973, espec i ely. In 1972 he joined he acul y o he E.T.S.I. de elecomunicaci6n a he Poly echnic Uni e si y o Ca alonia, Ba celona, Spain, whe e he holds he posi ion o P o esso . His esea ch in e es s ange om p opaga ion in op ical ibe s, high equency ape u e and a ay an ennas, and nea - ield an enna scanning sys ems o he design o mi- c owa e imaging sys ems and ada an ennas.