Recommendation ITU-R F.1336-4 (/14) Reference radiation patterns of omnidirectional, sectoral and oter antennas for te fixed and mobile services for use in saring studies in te frequency range from 4 MHz to about 7 GHz F Series Fixed service
ii Rec. ITU-R F.1336-4 Foreword Te role of te Radiocommunication Sector is to ensure te rational, equitable, efficient and economical use of te radio-frequency spectrum by all radiocommunication services, including satellite services, and carry out studies witout limit of frequency range on te basis of wic Recommendations are adopted. Te regulatory and policy functions of te Radiocommunication Sector are performed by World and Regional Radiocommunication Conferences and Radiocommunication Assemblies supported by Study Groups. Policy on Intellectual Property Rigt (IPR) ITU-R policy on IPR is described in te Common Patent Policy for ITU-T/ITU-R/ISO/IEC referenced in Annex 1 of Resolution ITU-R 1. Forms to be used for te submission of patent statements and licensing declarations by patent olders are available from ttp://www.itu.int/itu-r/go/patents/en were te Guidelines for Implementation of te Common Patent Policy for ITU-T/ITU-R/ISO/IEC and te ITU-R patent information database can also be found. Series of ITU-R Recommendations (Also available online at ttp://www.itu.int/publ/r-rec/en) Series BO BR BS BT F M P RA RS S SA SF SM SNG TF V Title Satellite delivery Recording for production, arcival and play-out; film for television Broadcasting service (sound) Broadcasting service (television) Fixed service Mobile, radiodetermination, amateur and related satellite services Radiowave propagation Radio astronomy Remote sensing systems Fixed-satellite service Space applications and meteorology Frequency saring and coordination between fixed-satellite and fixed service systems Spectrum management Satellite news gatering Time signals and frequency standards emissions Vocabulary and related subjects Note: Tis ITU-R Recommendation was approved in Englis under te procedure detailed in Resolution ITU-R 1. Electronic Publication Geneva, 14 ITU 14 All rigts reserved. No part of tis publication may be reproduced, by any means watsoever, witout written permission of ITU.
Rec. ITU-R F.1336-4 1 RECOMMENDATION ITU-R F.1336-4 * Reference radiation patterns of omnidirectional, sectoral and oter antennas for te fixed and mobile services for use in saring studies in te frequency range from 4 MHz to about 7 GHz (Question ITU-R 4/5) (1997--7-1-14) Scope Tis Recommendation gives reference models of antennas used in te fixed service and in te mobile service. It gives peak and average patterns of omnidirectional and sectoral antennas in te frequency range 4 MHz to about 7 GHz, as well as of low gain directional antennas in te frequency range 1 GHz to about 3 GHz, to be used in saring studies in te relevant frequency range. Keywords Fixed service, land mobile service, reference radiation pattern, sectoral antenna, omni-directional antenna, peak side-lobe pattern, average side-lobe pattern Te ITU Radiocommunication Assembly, considering a) tat, for coordination studies and for te assessment of mutual interference between point-to-multipoint (P-MP) fixed wireless systems (FWSs) or systems in te land mobile service (LMS), and between stations of suc systems and stations of space radiocommunication services saring te same frequency band, it may be necessary to use reference radiation patterns for FWS or LMS base station antennas; b) tat, depending on te saring scenario, it may be appropriate to consider te peak envelope or average side-lobe patterns in te saring studies; c) tat it may be appropriate to use te antenna radiation pattern representing average side-lobe levels in te following cases: to predict te aggregate interference to a geostationary or non-geostationary satellite from numerous fixed wireless stations or LMS base stations; to predict te aggregate interference to a fixed wireless station or LMS base stations from many geostationary satellites; to predict interference to a fixed wireless station or LMS base stations from one or more non-geostationary-satellites under continuously varying angles; in any oter cases were te use of te radiation pattern representing average side-lobe levels is appropriate; d) tat reference radiation patterns may be required in situations were information concerning te actual radiation pattern is not available; e) tat te use of antennas wit te best available radiation patterns will lead to te most efficient use of te radio-frequency spectrum; * Tis Recommendation sould be brougt to te attention of Radiocommunication Study Groups 4, 6 and 7.
Rec. ITU-R F.1336-4 f) tat at large angular distances from te main beam pattern gain may not fully represent te antenna emissions because of local ground reflections, noting tat Recommendations ITU-R F.699 and ITU-R F.145 give te peak and average reference patterns respectively, for directional antennas to be used in coordination studies and interference assessment in cases not referred to in recommends 1 to 4 below, recommends 1 tat, in te absence of particular information concerning te radiation pattern of te P-MP FWS or LMS base station antenna involved (see Note 1), te reference radiation pattern as stated below sould be used for: 1.1 interference assessment between line-of-sigt (LoS) P-MP FWSs or LMS base stations; 1. coordination studies and interference assessment between P-MP LoS FWSs or LMS base stations and oter stations of services saring te same frequency band; tat, in te frequency range from 4 MHz to about 7 GHz, te following reference radiation patterns sould be used in cases involving stations tat use omnidirectional (in azimut) antennas:.1 in te case of peak side-lobe patterns referred to in considering b), te following equations sould be used for elevation angles tat range from 9 to 9 (see Annex 1): wit: θ G 1 for θ < θ4 θ3 G ( θ) = G 1 + 1 log( k + 1) for θ4 θ < θ3 (1a) 1.5 θ G 1 + 1 log + θ θ k for 3 9 θ3.1 G θ = 17.6 (1b) 3 1 1 θ4 = θ3 1 log( k + 1) (1c) 1. were: G(θ) : G : gain relative to an isotropic antenna (dbi) te maximum gain in te azimut plane (dbi) θ : elevation angle relative to te angle of te maximum gain (degrees) ( 9 θ 9 ) θ 3 : te 3 db beamwidt in te elevation plane (degrees)
Rec. ITU-R F.1336-4 3 k: parameter wic accounts for increased side-lobe levels above wat would be expected for an antenna wit improved side-lobe performance (see recommends.3 and.4).. in te case of average side-lobe patterns referred to in considering c), te following equations sould be used for elevation angles tat range from 9 to 9 (see Annex 1 and Annex 4): wit: θ G 1 for θ < θ3 θ3 G ( θ) = G 15 + 1 log( k + 1) for θ3 θ < θ5 (1d) 1.5 θ G 15 + 1 log + θ θ k for 5 9 θ3 1 θ5 = θ3 1.5 log( k + 1) 1. were θ, θ 3, G and k are defined and expressed in recommends.1;.3 in cases involving typical antennas operating in te 4 MHz to 3 GHz range, te parameter k sould be.7;.4 in cases involving antennas wit improved side-lobe performance in te 4 MHz to 3 GHz range, and for all antennas operating in te 3-7 GHz range, te parameter k sould be ;.5 in cases were te antennas in recommends.1 troug. operate wit a downward electrical tilt, all of te equations in tose recommends are valid wit te definitions of te following variables (see 3 in Annex 5): θ e : θ : elevation angle (degrees) by wic te tilted radiation patterns are calculated using equations in recommends.1 and. elevation angle (degrees) measured from te orizontal plane at te site of te antenna ( 9 θ 9 : were 9 is te zenit and 9 is te nadir) β: downward tilt angle, te positive angle (degrees) tat te main beam axis is below te orizontal plane at te site of te antenna. Tese are interrelated as follows: ( θ + β) 9 θ = e for θ + β (1e) 9 + β ( θ + β) 9 θ = e for θ + β < 9 β An electrically tilted radiation gain at θ is calculated by using θ e of equation (1e) instead of θ at te equations in recommends.1 and., respectively; 3 tat, in te frequency range from 4 MHz to about 7 GHz, te following reference radiation patterns sould be used in cases involving stations tat use sectoral antennas; 3.1 in te frequency range from 4 MHz to about 6 GHz (see Annex 7):
4 Rec. ITU-R F.1336-4 3.1.1 in te case of peak side-lobe patterns referred to in considering b), te following equations sould be used for elevation angles tat range from 9 to 9 and for azimut angles tat range from 18 to 18 : were: G r (x ): x = φ /φ 3 G(ϕ,θ) = G + G r (x ) + R G vr (x v ) (dbi) (a1) relative reference antenna gain in te azimut plane at te normalized direction of (x,) (db) ϕ: azimut angle relative to te angle of te maximum gain in te orizontal plane (degrees) ϕ 3 : te 3 db beamwidt in te azimut plane (degrees) (generally equal to te sectoral beamwidt) G vr (x v ): relative reference antenna gain in te elevation plane at te normalized direction of (, x v ) (db) x v = θ /θ 3 R: orizontal gain compression ratio as te azimut angle is sifted from to ϕ, as sown below: Oter variables are as defined in recommends.1; ( x ) Gr (18 / ϕ3 ( ) G (18 / ϕ ) Gr ) R = (a) G 3.1.1.1 te relative minimum gain, G 18, can be calculated as follows: were: k p : r r 3 18 G18 = 1 + 1log (1+ 8k p ) 15log (b1) θ3 parameter wic accomplises te relative minimum gain for peak side-lobe patterns; 3.1.1.1.1 in cases involving typical antennas te parameter k p sould be.7 (see Note ); 3.1.1.1. in cases involving antennas wit improved side-lobe performance te parameter k p sould be.7, wic also applies for IMT base station antennas (see Note ); 3.1.1. te relative reference antenna gain in te azimut plane; G G ( ) 1 r x x r = for x.5 ( k ) ( x ) = 1x λ for.5 < x (b) k were: G r (x ) G 18 k : azimut pattern adjustment factor based on leaked power ( k 1) k λ = 3 (1.5 ) ; k 3.1.1..1 in cases involving typical antennas te parameter k sould be.8 (see Note );
Rec. ITU-R F.1336-4 5 3.1.1.. in cases involving antennas wit improved side-lobe performance te parameter k sould be.7, wic also applies for IMT base station antennas (see Note ); 3.1.1.3 te relative reference antenna gain in te elevation plane: G ( ) 1 vr xv xv = for x v < x k 1.5 v G ( x ) = 1+ 1log( x + k vr v v ) for x k x v < 4 (b3) were: G x ) = λ C log( x ) for 4 x v < 9 /θ 3 vr( v kv v G vr (x v ) = G 18 for x v = 9 /θ 3 k v : elevation pattern adjustment factor based on leaked power ( k v 1) x k = 1.36 kv λ kv = 1 Clog(4) 1log(4 1.5 + k v ) te attenuation incline factor of C is represented as follows: 1.5 18 θ3 1log 1+ 8k p C =.5 log θ3 1.5 ( 4 + k ) 3.1.1.3.1 in cases involving typical antennas te parameter k v sould be.7 (see Note ); 3.1.1.3. in cases involving antennas wit improved side-lobe performance te parameter k v sould be.3, wic also applies for IMT base station antennas (see Note ); 3.1. in te case of average side-lobe patterns referred to in considering c), for use in a statistical interference assessment, te following equations sould be used for elevation angles tat range from 9 to 9 and for azimut angles tat range from 18 to 18 : G(ϕ,θ) = G + G r (x ) + R G vr (x v ) 3.1..1 te relative minimum gain, G 18, calculated as follows: were: k a : v (dbi) 18 G18 = 15 + 1log (1+ 8ka ) 15log (c1) θ3 parameter wic accomplises te relative minimum gain for average side-lobe patterns; 3.1..1.1 in cases involving typical antennas te parameter k a sould be.7 (see Note ); 3.1..1. in cases involving antennas wit improved side-lobe performance te parameter k a sould be also.7, wic also applies for IMT base station antennas (see Note );
6 Rec. ITU-R F.1336-4 3.1.. te relative reference antenna gain in te azimut plane: G G ( ) 1 r x x r = for x.5 ( k ) ( x ) = 1x λ for.5 < x (c) k were: G r (x ) G 18 k λ = 3 (1.5 ). k 3.1...1 in cases involving typical antennas te parameter k sould be.8 (see Note ); 3.1... in cases involving antennas wit improved side-lobe performance te parameter k sould be.7, wic also applies for IMT base station antennas (see Note ); 3.1..3 te relative reference antenna gain in te elevation plane: G ( ) 1 vr xv xv = for x v < x k 1.5) v G ( x ) = 15 + 1 log( x + k vr v v ) for x k x v < 4 (c3) G x ) = λ 3 C log( x ) for 4 x v < 9 /θ 3 vr( v kv v were: G vr (x v ) = G 18 for x v = 9 /θ 3 x = 1.33. 33 k k v λ kv = 1 Clog(4) 1log(4 1.5 + k v ); te attenuation incline factor of C is represented as follows: 1.5 18 θ3 1log 1+ 8ka C =.5 log θ3 1.5 ( 4 + k ) 3.1..3.1 in cases involving typical antennas te parameter k v sould be.7 (see Note ): 3.1..3. in cases involving antennas wit improved side-lobe performance te parameter k v sould be.3, wic also applies for IMT base station antennas (see Note ); 3. in te frequency range from 6 GHz to about 7 GHz (see Annex 6): 3..1 in te case of peak side-lobe patterns referred to in considering b), te following equations sould be used for elevation angles tat range from 9 to 9 and for azimut angles tat range from 18 to 18 : v. G( ϕ, θ) = G ( x) (d1) ref tanθ α = arctan 9 α 9 (d) sinϕ
Rec. ITU-R F.1336-4 7 1 ψ α = for ψ 9 (d3) cosα sinα + ϕ3 θ3 ψ α = cosθ ϕ3m 1 ψ = arccos cos sinθ + θ3 ( ϕcosθ) for 9 < ψ 18, ψ 18 (d4) x = ψ / (d5) ψ α were: ϕ 3m : te equivalent 3 db beamwidt in te azimut plane for an adjustment of orizontal gains (degrees); ϕ3 m = ϕ 3 for ϕ ϕt (d6) ϕ 3m = ϕ ϕ cos 18 ϕ ϕ3 ϕ t : t t 9 1 ϕ ϕ sin 18 ϕ + θ3 te boundary azimut angle (degrees) t t 9 for ϕt < ϕ 18 (d7) ϕ t = ϕ 3 Oter variables and parameters are as defined in recommends.1 and 3.1.1; G ref G ref ( x) = G 1x for x < 1 (e) ( x) = G 1 15log( x) for 1 x 3.. in te case of average side-lobe patterns referred to in considering c), for use in a statistical interference assessment, te following equations sould be used for elevation angles tat range from 9 to 9 and for azimut angles tat range from 18 to 18 (see Note 3): G ref G ref ( x) = G 1x for x < 1.15 (f) ( x) = G 15 15 log( x) for 1.15 x In tis case, as for ϕ t in equations (d6) and (d7), ϕ t = 1.15ϕ 3. 3.3 in cases involving sectoral antennas wit a 3 db beamwidt in te azimut plane less tan about 1, te relationsip between te maximum gain and te 3 db beamwidt in bot te azimut plane and te elevation plane, on a provisional basis, is (see Annex and Notes 4 and 5): θ 3 31 = ϕ 1 3.1 G were all parameters are as defined under recommends 3.1; (3a)
8 Rec. ITU-R F.1336-4 3.4 in cases were te antennas in recommends 3.1 troug 3. operate wit a downward mecanical tilt, all of te equations in tose recommends are valid wit te definitions and redefinitions of te following variables (see in Annex 5): θ: elevation angle (degrees) measured from te plane defined by te axis of maximum gain of te antenna and te axis about wic te pattern is tilted (θ 3 is also measured from tis plane) φ: azimut (degrees) measured from te azimut of maximum gain in te plane defined by te axis of maximum gain of te antenna and te axis about wic te pattern is tilted θ : elevation angle (degrees) measured from te orizontal plane at te site of te antenna ( 9 θ 9 ) φ : azimut angle (degrees) in te orizontal plane at te site of te antenna measured from te azimut of maximum gain ( 18 ϕ 18 ) β: downward tilt angle, te positive angle (degrees) tat te main beam axis is below te orizontal plane at te site of te antenna. Tese are interrelated as follows: θ = arcsin(sin θ cosβ + cosθ cosϕ sinβ), 9 θ 9 (3b) ( sin θ sin β + cos θ cos ϕ cosβ) ϕ = arccos cos θ, ϕ 18 (see Note 1 in Annex 5) (3c) 3.5 in cases were te antennas in recommends 3.1 troug 3. operate wit a downward electrical tilt, an electrically tilted radiation gain at θ is also calculated by using θ e of equation (1e) in recommends.5 instead of θ at te equations in recommends 3.1 and 3., respectively; 4 tat, in te frequency range from 1 GHz to about 3 GHz, te following reference radiation patterns sould be used in cases involving stations tat use low-gain antennas wit circular symmetry about te 3 db beamwidt and wit a main lobe antenna gain less tan about dbi: 4.1 te following equations sould be used in te case of peak side-lobe patterns referred to in considering b) (see Note 6): were: θ G 1 for θ < 1.8 ϕ3 ϕ3 G ( θ) = G ϕ θ < ϕ 14 for 1.8 3 1 (4) θ G 14 3 log for ϕ1 θ < ϕ ϕ1 8 for ϕ θ 18 G(θ) : G : gain relative to an isotropic antenna (dbi) te main lobe antenna gain (dbi) θ : off-axis angle (degrees) ( θ 18 ) ϕ 3 : te 3 db beamwidt of te low-gain antenna (degrees).1 G = 7 1 (degrees)
Rec. ITU-R F.1336-4 9 ϕ 1 ϕ = 1.9 ϕ 3 (degrees) = ϕ 1 1 (G 6)/3 (degrees); 4. in te case of average side-lobe patterns referred to in considering c), te antenna pattern given in Recommendation ITU-R F.145 sould be used; 5 tat te following Notes sould be regarded as part of tis Recommendation: NOTE 1 It is essential tat every effort be made to utilize te actual antenna pattern in coordination studies and interference assessment. NOTE Te values of parameter k, k v, k a and k p in recommends 3.1 were based on statistical data wic were derived from many measured sectoral antenna patterns in te 7 MHz to around 6 GHz frequency range. NOTE 3 Measured results of a specially designed sectoral antenna for use around GHz indicate te possibility of compliance wit a more restrictive reference side-lobe radiation pattern. Furter studies are required to develop suc an optimized pattern. NOTE 4 In a case involving an antenna wose 3 db beamwidt in te elevation plane is already known, it is recommended to use te known θ 3 as an input parameter. NOTE 5 As discussed in Annex, an exponential factor as been replaced by unity. As a result, te teoretical error introduced by tis approximation will be less tan 6% for 3 db beamwidts in te elevation plane less tan 45. NOTE 6 Te reference radiation pattern given in recommends 4.1 primarily applies in situations were te main lobe antenna gain is less tan or equal to dbi and te use of Recommendation ITU-R F.699 produces inadequate results. Furter study is required to establis te full range of frequencies and gain over wic te equations are valid.
1 Rec. ITU-R F.1336-4 Annex 1 Reference radiation pattern for omnidirectional antennas as used in P-MP fixed wireless systems 1 Introduction An omnidirectional antenna is frequently used for transmitting and receiving signals at central stations of P-MP fixed wireless systems. Studies involving saring between tese types of fixed-wireless systems and space service systems in te GHz bands ave used te reference radiation pattern described ere. Analysis Te reference radiation pattern is based on te following assumptions concerning te omnidirectional antenna: tat te antenna is an n-element linear array radiating in te broadside mode; te elements of te array are assumed to be dipoles; te array elements are spaced 3λ/4. Te 3 db beamwidt θ 3 of te array in te elevation plane is related to te directivity D by (see Annex 3 for te definition of D): [ 191..818 + 1/ θ 17.4] dbi D = 1 log 3 (5a) Equation (5a) may be solved for θ 3 wen te directivity is known: 1 θ 3 = (5b) α.818 1. 1 D + 17.4 α = (5c) 191. Te relationsip between te 3 db beamwidt in te elevation plane and te directivity was derived on te assumption tat te radiation pattern in te elevation plane was adequately approximated by: m f ( θ) = cos ( θ) were m is an arbitrary parameter used to relate te 3 db beamwidt and te radiation pattern in te elevation plane. Using tis approximation, te directivity was obtained by integrating te pattern over te elevation and azimut planes. Te intensity of te far-field of a linear array is given by: ET ( θ) = Ee( θ) AF( θ) (6) were: E T (θ) : total E-field at an angle of θ normal to te axis of te array
Rec. ITU-R F.1336-4 11 E e (θ) : E-field at an angle of θ normal to te axis of te array caused by a single array element AF(θ) : array factor at an angle θ normal to te axis of te array. Te normalized E-field of a dipole element is: Te array factor is: E ( θ) = cos ( θ) (7) e were: N : d : number of elements in te array ψ 1 = spacing of te radiators ψ sin N 1 AF N = (8) N ψ sin d π sin θ λ λ : wavelengt. Te following procedure as been used to estimate te number of elements N in te array. It is assumed tat te maximum gain of te array is identical to te directivity of te array. Given te maximum gain of te omnidirectional antenna in te elevation plane, compute te 3 db beamwidt, θ 3, using equations (5b) and (5c); Ignore te small reduction in off-axis gain caused by te dipole element, and note tat te ψ array factor, AF N, evaluates to.77 ( 3 db) wen N ; = 1.396 and N is ten determined as te integer value of: were x means te maximum integer value not exceeding x. Te normalized off-axis discrimination ΔD is given by: 1.3916 N = (9) d θ3 π sin λ [ AF cos ( ) ] ΔD = log θ db (1) N Equation (1) as been evaluated as a function of te off-axis angle (i.e., te elevation angle) for several values of maximum gain. For values in te range of 8 dbi to 13 dbi, it as been found tat te envelope of te radiation pattern in te elevation plane may be adequately approximated by te following equations:
1 Rec. ITU-R F.1336-4 [ G ( θ), G ( )] G ( θ) = max 1 θ (11a) ( θ ) θ = G θ3 G 1 1 dbi (11b) 1.5 θ G + ( θ) = G 1 + 1 log max, 1 k dbi (11c) θ3 k is a parameter wic accounts for increased side-lobe levels above wat would be expected for an antenna wit improved side-lobe performance. Figures 1 to 4 compare te reference radiation envelopes wit te teoretical antenna patterns generated from equation (11), for gains from 8 dbi to 13 dbi, using a factor of k =. Figures 5 to 8 compare te reference radiation envelopes wit actual measured antenna patterns using a factor of k =. In Figs 7 and 8, it can be seen tat te side lobes are about 15 db or more below te level of te main lobe, allowing for a small percentage of side-lobe peaks wic migt exceed tis value. However practical factors suc as te use of electrical downtilt, pattern degradations at band-edges and production variations would furter increase te side lobes to about 1 db below te main lobe in actual field installations. Te k factor, mentioned above, in equation (11), is intended to caracterize tis variation in side-lobe levels. Figures 9 and 1 provide a comparison of a 1 dbi and a 13 dbi gain antenna, at.4 GHz, wit te reference radiation pattern envelope, using k =.5. A factor of k =.5 represents side-lobe levels about 15 db below te main-lobe peak. However, to account for increases in side-lobe levels wic may be found in field installations, for typical antennas a factor of k =.7 sould be used, representing side-lobe levels about 13.5 db below te level of te main lobe. Finally, Figs 11 and 1 illustrate te effect on elevation patterns of using various values of k. FIGURE 1 Normalized radiation pattern of a linear array of dipole elements compared wit te approximate envelope of te radiation pattern G = 1 dbi, k = 5 Discrimination (db) 1 15 5 3 1 3 4 5 6 7 8 9 Elevation angle (degrees) F.1336-1
Rec. ITU-R F.1336-4 13 FIGURE Normalized radiation pattern of a linear array of dipole elements compared wit te approximate envelope of te radiation pattern G = 11 dbi, k = 5 Discrimination (db) 1 15 5 3 1 3 4 5 6 7 8 9 Elevation angle (degrees) F.1336- FIGURE 3 Normalized radiation pattern of a linear array of dipole elements compared wit te approximate envelope of te radiation pattern G = 1 dbi, k = 5 Discrimination (db) 1 15 5 3 1 3 4 5 6 7 8 9 Elevation angle (degrees) F.1336-3
14 Rec. ITU-R F.1336-4 FIGURE 4 Normalized radiation pattern of a linear array of dipole elements compared wit te approximate envelope of te radiation pattern G = 13 dbi, k = 5 Discrimination (db) 1 15 5 3 1 3 4 5 6 7 8 9 Elevation angle (degrees) F.1336-4 FIGURE 5 Comparison of measured pattern and reference radiation pattern envelope for an omnidirectional antenna wit 11 dbi gain and operating in te band 98-944 MHz, k = 75 6 45 3 15 Relative power (db) 4 8 1 16 4 8 3 36 15 3 45 6 75 9 9 15 15 1 1 135 135 15 165 18 165 15 F.1336-5
Rec. ITU-R F.1336-4 15 FIGURE 6 Comparison of measured pattern and te reference radiation pattern envelope for an omnidirectional antenna wit 8 dbi gain and operating in te band 1 85-1 99 MHz, k = 75 6 45 3 15 Relative power (db) 4 8 1 16 4 8 3 36 15 3 45 6 75 9 9 15 15 1 1 135 135 15 165 18 165 15 F.1336-6 FIGURE 7 Comparison of measured pattern and te reference radiation pattern envelope wit k = for an omnidirectional antenna wit 1 dbi gain and operating in te 1.4 GHz band 5 1 Relative gain (db) 15 5 3 35 4 7 3 33 3 6 9 Angle (degrees) k = Antenna A Antenna B F.1336-7
16 Rec. ITU-R F.1336-4 FIGURE 8 Comparison of measured pattern and te reference radiation pattern envelope wit k = for an omnidirectional antenna wit 13 dbi gain and operating in te 1.4 GHz band 5 1 Relative gain (db) 15 5 3 35 4 7 3 33 3 6 9 Angle (degrees) k = Antenna A F.1336-8 FIGURE 9 Comparison of measured pattern and te reference radiation pattern envelope wit k =.5 for an omnidirectional antenna wit 1 dbi gain and operating in te.4 GHz band 5 1 Relative gain (db) 15 5 3 35 4 7 3 33 3 6 9 Angle (degrees) k =.5 Antenna A Antenna B F.1336-9
Rec. ITU-R F.1336-4 17 FIGURE 1 Comparison of measured pattern and te reference radiation pattern envelope wit k =.5 for an omnidirectional antenna wit 13 dbi gain and operating in te.4 GHz band 5 1 Relative gain (db) 15 5 3 35 4 7 3 33 3 6 9 Angle (degrees) k =.5 Antenna A F.1336-1 Relative gain (db) 5 1 15 5 3 FIGURE 11 Reference radiation pattern envelopes for various values of k for an omnidirectional antenna wit 1 dbi gain k = 4.. 1..5.. 35 4 7 3 33 3 6 9 Angle (degrees) F.1336-11
18 Rec. ITU-R F.1336-4 Relative gain (db) 5 1 15 5 3 FIGURE 1 Reference radiation pattern envelopes for various values of k for an omnidirectional antenna wit 13 dbi gain k = 4.. 1..5.. 35 4 7 3 33 3 6 9 Angle (degrees) F.1336-1 3 Summary, conclusions and furter analyses A reference radiation pattern as been presented for omnidirectional antennas exibiting a gain between 8 dbi and 13 dbi. Te reference radiation pattern as been derived on te basis of teoretical considerations of te radiation pattern of a collinear array of dipoles. Te proposed pattern as been sown to adequately represent te teoretical patterns and measured patterns over te range from 8 dbi to 13 dbi. Furter work is required to determine te range of gain over wic te reference radiation pattern is appropriate especially wit regard to antennas operating in frequency bands above 3 GHz. Annex Relationsip between gain and beamwidt for omnidirectional and sectoral antennas 1 Introduction Te purpose of tis Annex is to derive te relationsip between te gain of omnidirectional and sectoral antennas and teir beamwidt in te azimutal and elevation planes. Section is an analysis of te directivity of omnidirectional and sectoral antennas assuming two different radiation intensity functions in te azimutal plane. For bot cases, te radiation intensity in te elevation plane was assumed to be an exponential function. Section 3 provides a comparison between te gain-beamwidt results obtained using te metods of Section and results contained in te previous versions of tis Recommendation for omnidirectional antennas. Section 4 summarizes te results, proposes a provisional equation for gain-beamwidt for omnidirectional and sectoral antennas, and suggests areas for furter study.
Rec. ITU-R F.1336-4 19 Analysis Te far-field pattern of te sectoral antenna in te elevation plane is assumed to conform to an exponential function, wereas te far-field pattern in te azimut plane is assumed to conform to eiter a rectangular function or an exponential function. Wit tese assumptions, te directivity, D, of te sectoral antenna may be derived from te following formulation in (sperical coordinates): were: U M : U : ϕ : θ : F(ϕ) : U D = (1) M U π π/ 1 = F( ϕ) F( θ) cos( θ) dθ d 4π π π/ U ϕ (13) maximum radiation intensity radiation intensity of an isotropic source angle in te azimutal plane angle in te elevation plane radiation intensity in te azimutal plane F(θ) : radiation intensity in te elevation plane. Te directivity of omnidirectional and sector antennas is evaluated in te following sub-sections assuming te radiation intensity in te azimutal plane is eiter a rectangular function or an exponential function..1 Rectangular sectoral radiation intensity Rectangular sectoral radiation intensity function, F(ϕ), is assumed to be: were: ϕ F ( ϕ) = U s ϕ (14) ϕ s : beamwidt of te sector, U(x) = 1 for x U(x) = for x < (15) For eiter rectangular or exponential sectoral radiation intensity functions, it is assumed tat te radiation intensity in te elevation plane is given by: were: θ 3 : θ F( θ ) = e a (16). 773 a = ln(.5) = (17) θ3 θ 3 db beamwidt of te antenna in te elevation plane (degrees). 3
Rec. ITU-R F.1336-4 Substituting equations (14) and (16) into equation (13) results in: π π ϕ = 1 / ϕ ϕ θ U s d e a cos( θ) d 4π π π/ U θ (18) Tis double integral may be solved as te product of two independent integrals. Te integral over ϕ is evaluated in a straigtforward way. However, evaluating te integral over θ is somewat more difficult. Te integral over θ could be evaluated numerically wit te results eiter tabulated or a polynomial fitted to te data. However, it is noted tat if te limits of integration are canged to ±, te integral over θ is given in closed-form by: π/ e a θ cos( θ) dθ e a θ cos( θ) dθ = π e 1/4a (19) π/ 1 a Tis is a rater simple and flexible formulation tat, depending on its accuracy, could be quite useful in evaluating te directivity of sector antennas as well as omnidirectional antennas. Te accuracy wit wic te infinite integral approximates te finite integral as been evaluated. Te finite integral, i.e., te integral on te left-and side of equation (19), as been evaluated for several values of 3 db beamwidt using te 4 point Gaussian Quadrature metod and compared wit te value obtained using te formula corresponding to te infinite integral on te rigt-and side of equation (19). (Actually, because of its symmetry, te finite integral as been numerically evaluated over te range to π/ and te result doubled.) Te results for a range of example values of te 3 db beamwidt in te elevation plane are sown in Table 1. Te Table sows tat for a 3 db beamwidt of 45, te difference between te values produced by te finite integral and te infinite integral approximation is less tan.3%. At 5 and below, te error is essentially zero. Equation (18) is now readily evaluated: θ ϕ θ 3 3 π 11.9 = s e U () 4π.773 TABLE 1 Relative accuracy of te infinite integral in equation (19) in te evaluation of te average radiation intensity 3 db beamwidt in te elevation plane (degrees) Finite integral Infinite integral Relative error (%) 45 1.116449558 1.116116449.98 5.6774788.6774788..54974413.54974413. 15.416896869.416896869. 1.8137168.8137168. 5.14734555.14734558.
Rec. ITU-R F.1336-4 1 From equations (14) and (16), U M = 1. Substituting tese values and equation () into equation (1) yields te directivity of a sector antenna given te beamwidt in te elevation and azimutal planes: 3 θ 3 11.9 11.85 D = e (1) ϕ θ s were te angles are given in radians. Wen te angles are expressed in degrees, equation (1) becomes: 3 θ 3 364 Note tat for an omnidirectional antenna, equation () reduces to: 3875 D = e () ϕ θ s 3 θ 3 364 17.64 D = e (3a) θ If it is assumed tat te radiation efficiency is 1% and tat te antenna losses are negligible, ten.1 te gain, 1 G, and te directivity, D, of te omnidirectional antenna are identical. Additionally, for omnidirectional antennas wit a 3 db beamwidt less tan about 45, te relationsip between te gain and te 3 db beamwidt in te elevation plane may be simplified by setting te exponential factor equal to unity. Te resulting error is less tan 6%..1G 17. 64 1 θ 3 (3b). Exponential sectoral radiation intensity Te second case considered for te sectoral radiation intensity is tat of an exponential function. Specifically: were: ( ϕ) = e b ϕ F (4) ln (.5) b = ϕs (5) and ϕ s is te 3 db beamwidt of te sector. Substituting equations (16) and (4) into equation (13), canging te limits of integration so tat te finite integrals become infinite integrals, integrating and ten substituting te result into equation (1) yields te following approximation: 3 θ 3 11.9 11.9 D = e (6) ϕ θ s
Rec. ITU-R F.1336-4 were te angles are as defined previously and are expressed in radians. Converting te angles to degrees transforms equation (6) into: s 3 θ 3 364 36 4 D = e (7) ϕ θ Comparing equations () and (7), it is seen tat te difference between te directivity computed using eiter of te equations is less tan.3 db. Te results given by equation (7) sould be compared to a number of measured patterns to determine te inerent effect of te radiation efficiency of te antenna and oter losses on te coefficient. At tis time, only two sets of measurements are available for sectoral antennas designed to operate in te 5.5 GHz to 9.5 GHz band. Measured patterns in te azimutal and elevation planes are given, respectively, in Figs 13 and 14 for one set of antennas and Figs 15 and 16, respectively, for te second set. From Figs 13 and 14, te 3 db beamwidt in te azimutal plane is 9 and te 3 db beamwidt in te elevation plane is.5. From equation (7), te directivity is.1 db. Tis is to be compared wit a measured gain of.5-1.4 dbi for te antenna over te range 5.5-9.5 GHz. Assuming te gain G of te antenna in te band around 8 GHz is.7 db less tan its directivity, and te exponential factor is replaced by unity wic introduces an increasing error wit increasing beamwidt. Te error reaces 6% at 45. A larger beamwidt leads to a larger error. Based on tese considerations, te semi-empirical relationsip between te gain and te beamwidt of a sectoral antenna is given by:.1g 31 1 ϕ θ s 3 (8a) Similarly, from Figs 15 and 16, te semi-empirical relationsip between te gain and te beamwidt of tat sectoral antenna is:.1g 34 1 ϕ θ s 3 (8b)
Rec. ITU-R F.1336-4 3 FIGURE 13 Measured pattern in te azimutal plane of a 9 sector antenna. Pattern measured over te band 7.5 GHz to 9.5 GHz. Te band drawn cross marks on te left side of te Figure correspond to values obtained from equation (4) (wen expressed in db) for an assumed 3 db beamwidt of 9 in te azimutal plane Relative power (db) 4 6 8 1 1 14 16 18 4 6 8 3 3 34 36 p9wa. env, P9WA.ENV nbdg 75a. pca, 7.5 GHz nbdg 8a. pca, 8. GHz nbdg 85a. pca, 8.5 GHz nbdg 9a. pca, 9. GHz nbdg 95a. pca, 9.5 GHz 38 4 18 16 14 1 1 8 6 4 4 6 8 1 1 14 16 18 Azimut angle (degrees) F.1336-13
4 Rec. ITU-R F.1336-4 FIGURE 14 Measured pattern in te azimutal plane of a 9 sector antenna. Pattern measured over te band 7.5 GHz to 9.5 GHz 4 6 Relative power (db) 8 1 1 14 16 18 p19e. env, P19E.ENV 1fnl75. pat, 7.5 GHz 1fnl8. pat, 8. GHz 1fnl85. pat, 8.5 GHz 1fnl9. pat, 9. GHz 1fnl95. pat, 9.5 GHz 4 6 8 3 8 81 8 83 84 85 86 87 88 89 9 91 9 93 94 95 96 97 98 99 Elevation angle (degrees) 1 F.1336-14 1 FIGURE 15 Azimut pattern of typical 9 sectoral antenna (V-polarization) 15 dbi alf-value angle: 9 (orn type antenna at 6 GHz) Relative power (db) 1 3 4 5 6 18 16 14 1 1 8 6 4 4 6 8 1 1 14 16 18 Azimut angle (degrees) F.1336-15
Rec. ITU-R F.1336-4 5 1 FIGURE 16 Elevation pattern of typical 9 sectoral antenna (V-polarization) 15 dbi alf-value angle: 1 (orn type antenna at 6 GHz) Relative power (db) 1 3 4 5 6 9 8 7 6 5 4 3 1 1 3 4 5 6 7 8 9 Elevation angle (degrees) F.1336-16 3 Comparison wit previous results for omnidirectional antennas Te purpose of tis section is to compare te results obtained for an omnidirectional antenna given by equation (3) wit previous results reported in and summarized in Annex 1 of tis Recommendation. Te radiation intensity in te elevation plane used in for an omnidirectional antenna was of te form: N F ( θ) = cos θ (9) Substituting equation (9) into equation (13), and assuming tat F(ϕ) = 1, yields: π π = 1 / cos N ( θ) cos( θ) dθ d 4π π π/ U ϕ (3) Tis double integral evaluates to: (N )!! U = (31) (N + 1)!! were (N)!! is te double factorial defined as ( 4 6...(N)), and (N + 1)!! is also a double factorial defined as (1 3 5...(N + 1)). Tus, te directivity becomes: (N + 1)!! D = (3) (N )!! Te 3 db beamwidt in te elevation plane is given by: 1 3 = cos 5 N (. 1/ ) θ (33)
6 Rec. ITU-R F.1336-4 A comparison between te directivity computed using te assumptions and metods embodied in equation (3) and tose used in te derivation of equations (3) and (33) is given in Table. It is sown tat results obtained using equation (3a) compare favourably wit te results using equations (3) and (33). In all cases equation (3a) sligtly underestimates te directivity obtained using equations (3) and (33). Te relative error (%) of te estimates, wen expressed in db, is greatest for a 3 db beamwidt in te elevation plane of 65, amounting to.7%. Te error (db) for tis case, expressed in db, is.6 db. For 3 db beamwidt angles less tan 65, te relative error (%) and te error (db), are monotonically decreasing functions as te 3 db beamwidt decreases. For a 16 3 db beamwidt, te relative error (%) is about.1% and te error (db) is less tan about.85 db. An evaluation similar to tat sown in Table for values of N up to 1 (corresponds to a 3 db beamwidt of 1.35 and a directivity of 19. db) confirms tat te results of te two approaces converge. TABLE Comparison of te directivity of omnidirectional antennas computed using equation (3a) wit te directivity computed using equations (3) and (33) N θ 3 (degrees) (equation (33)) Directivity (db) (equation (3)) Directivity (db) (equation (3a)) Relative error (%) Error (db) 9. 1.769 1.7437.98.17 4 65.53.73.6677.8.63 6 54.7 3.3995 3.3419 1.69.576 8 47.161 3.911 3.861 1.8.5 1 4.1747 4.349 4.814 1.1.435 1 38.5746 4.676 4.6343.8.383 14 35.764 4.97 4.9381.69.341 16 33.4873 5.355 5.47.59.37 18 31.5975 5.473 5.443.51.8 9.9953 5.68 5.6565.45.56 8.6145 5.875 5.8516.4.37 4 7.483 6.55 6.35.36. 6 6.348 6.164 6.1959.33.5 8 5.397 6.3688 6.3496.3.19 3 4.5384 6.511 6.4931.8.181 3 3.7649 6.6449 6.678.6.171 34 3.63 6.778 6.7545.4.16 36.4148 6.8897 6.8743..154 38 1.86 7.6 6.9879.1.147 4 1.714 7.198 7.958..14 4.7616 7.1 7.1986.19.134 44.868 7.396 7.967.18.19
Rec. ITU-R F.1336-4 7 N θ 3 (degrees) (equation (33)) Directivity (db) (equation (3)) TABLE (end) Directivity (db) (equation (3a)) Relative error (%) Error (db) 46 19.8431 7.43 7.396.17.14 48 19.474 7.495 7.486.16.119 5 19.367 7.5785 7.5671.15.115 5 18.6687 7.6613 7.65.14.111 54 18.31 7.741 7.73.14.17 56 17.994 7.8178 7.875.13.14 58 17.688 7.891 7.88.13.1 6 17.3847 7.9638 7.9541.1.97 6 17.131 8.333 8.39.1.94 64 16.8347 8.17 8.915.11.9 66 16.5786 8.166 8.1571.11.89 68 16.3338 8.94 8.7.11.87 7 16.996 8.91 8.85.1.85 7 15.8751 8.359 8.346.1.83 74 15.6598 8.49 8.411.1.81 4 Summary and conclusions Equations ave been developed tat permit easy calculation of te directivity and te relationsip between te beamwidt and gain of omnidirectional and sectoral antennas as used in P-MP radiorelay systems. It is proposed to use te following equations to determine te directivity of sectoral antennas: were: k = 3875 k = 364 3 θ 3 364 k D = e (34) ϕ θ s for for ϕ ϕ s s > 1 1 and ϕ s = 3 db beamwidt of te sectoral antenna in te azimutal plane (degrees) for an assumed exponential radiation intensity in azimut and θ 3 is te 3 db beamwidt of te sectoral antenna in te elevation plane (degrees). For omnidirectional antennas, it is proposed to use te following simplified equation to determine te 3 db beamwidt in te elevation plane given te gain in dbi (see equation (3b)): (35) θ 3 17.6 1.1 G It is proposed to use, on a provisional basis, te following semi-empirical equation relating te gain of a sectoral antenna (dbi) to te 3 db beamwidts in te elevation plane and te azimutal plane,
8 Rec. ITU-R F.1336-4 were te sector is on te order of 1 or less and te 3 db beamwidt in te elevation plane is less tan about 45 (see equation (8a)): θ 3 31 ϕ 1 s.1 G Furter study is required to determine ow to andle te transition region implicit in equation (35), and to determine te accuracy of tese approximations as tey apply to measured patterns of sectoral and omnidirectional antennas designed for use in P-MP radio-relay systems for bands in te range from 1 GHz to about 7 GHz. Annex 3 Procedure for determining te gain of a sectoral antenna at an arbitrary off-axis angle specified by an azimut angle and an elevation angle referenced to te boresigt of te antenna 1 Analysis Te basic geometry for determining te gain of a sectoral antenna at an arbitrary off-axis angle is sown in Fig. 17. It is assumed tat te antenna is located at te centre of te sperical coordinate system; te direction of maximum radiation is along te x-axis; te x-y plane is te local orizontal plane; te elevation plane contains te z-axis; and, u is a unit vector wose direction is used to determine te gain of te sectoral antenna. In analysing sectoral antennas in particular, it is important to observe te range of validity of te azimut and elevation angles: 18 ϕ + 18 9 θ + 9 Also observe tat te range of validity of te angle α is 9 α + 9
Rec. ITU-R F.1336-4 9 FIGURE 17 Determining te off-boresigt angle given te azimut and elevation angle of interest z u c d θ y ϕ α b a x adc = ψ F.1336-17 Te two fundamental assumptions regarding tis procedure are tat: te 3 db gain contour of te far-field pattern wen plotted in two-dimensions as a function of te azimut and elevation angles will be an ellipse as sown in Fig. 18; and te gain of te sectoral antenna at an arbitrary off-axis angle is a function of te 3 db beamwidt and te beamwidt of te antenna wen measured in te plane containing te x-axis and te unit vector u (see Fig. 17). Given te 3 db beamwidt (degrees) of te sectoral antenna in te azimut and elevation planes, ϕ 3 and θ 3, te numerical value of te boresigt gain is given, on a provisional basis, by (see recommends 3.3 and equation (8a))..1G 31 1 Te first step in evaluating te gain of te sectoral antenna at an arbitrary off-axis angle, ϕ and θ, is to determine te value of α. Referring to Fig. 17 and recognizing tat abc is a rigt-sperical triangle, α is given by: tanθ α = arctan, sinϕ 9 α + 9 (37a) and te off-axis angle in te plane adc is given by: ψ = arccos cosϕcosθ, ψ 18 (37b) ( ) ϕ θ s 3 (36)
3 Rec. ITU-R F.1336-4 FIGURE 18 Determination of te 3 db beamwidt of an elliptical beam at an arbitrary inclination angle α θ θ 3 / ψ α / α ϕ ϕ 3 / F.1336-18 Given tat te beam is elliptical, te 3 db beamwidt of te sectoral antenna in te plane adc in Fig. 17 is determined from: 1 ψ α = (38) cosα sinα + ϕ3 θ3 Based on tis calculation metod, te alternative approac (see Annex 6) provides te reference radiation pattern in te frequency range from 6 GHz to about 7 GHz (see recommends 3.). Conclusion A procedure as been given to evaluate te gain of a sectoral antenna at an arbitrary off-axis angle as referenced to te direction of te maximum gain of te antenna. Te importance of observing te range of validity of te azimut and elevation angles in modelling te radiation pattern of a sectoral antenna as been empasized. Furter study is required to demonstrate te range of gain and beamwidts in te azimut and elevation planes over wic te reference gain representation used ere (equations (d1)-(f), (3a) and (36)) is valid for sectoral antennas.
Rec. ITU-R F.1336-4 31 Annex 4 Matematical model of generic average radiation patterns of omnidirectional for P-MP FWSs for use in statistical interference assessment 1 Introduction Te main text of tis Recommendation (in recommends.) gives reference radiation patterns, representing average side-lobe levels for omnidirectional (in azimut) antennas, wic can be applied in te case of multiple interference entries or time-varying interference entries. On te oter and, for use in spatial statistical analysis of te interference, e.g., from a few GSO satellite systems into a large number of interfered-wit FWS, a matematical model is required for generic radiation patterns as given in te later sections in tis Annex. It sould be noted tat tese matematical models based on te sinusoidal functions, wen applied in multiple entry interference calculations, may lead to biased results unless te interference sources are distributed over a large range of azimut/elevation angles. Terefore, use of tese patterns is recommended only in te case stated above. Matematical model for omnidirectional antennas In case of spatial analysis of te interference from one or a few GSO satellite systems into a large number of FS stations, te following average side-lobe patterns sould be used for elevation angles tat range from 9 to 9 (see Annex 1): wit: θ G 1 for θ < θ4 θ3 G ( θ) = G 1 + 1 log( k + 1) + F( θ) for θ4 θ < θ3 (39a) 1.5 θ G 1 + 1 log + + θ θ θ k F( ) for 3 9 θ3 3πθ F ( θ) = 1 log.9sin +. 1 (39b) 4θ3 were θ, θ 3, θ 4, G and k are defined and expressed in recommends.1 in te main text. NOTE 1 In cases involving typical antennas operating in te 1-3 GHz range, te parameter k sould be.7. NOTE In cases involving antennas wit improved side-lobe performance in te 1-3 GHz range, and for all antennas operating in te 3-7 GHz range, te parameter k sould be.
3 Rec. ITU-R F.1336-4 Annex 5 Procedure for determining te radiation pattern of an antenna at an arbitrary off-axis angle wen te boresigt of te antenna is mecanically or electrically tilted downward 1 Introduction Tis Annex presents metods to account for te radiation pattern of a sectoral antenna wen tilted downwardly by eiter mecanical or electrical means. Te analysis of te mecanical means is presented in and te electrical means in 3. Analysis of mecanical tilt Te basic geometry for determining te gain of a sectoral antenna at an arbitrary off-axis angle is sown in Fig. 19. It is assumed tat te antenna is located at te centre of te sperical coordinate system; te direction of maximum radiation is along te x-axis. If te antenna is tilted downward, it becomes necessary to distinguis between te antenna-based coordinates (θ, ϕ) and te coordinates referenced to te orizontal plane (θ, ϕ ). Te relationsip between tese coordinate systems is best determined by considering te rectangular coordinate systems attaced to tem. If te antenna is down-tilted to a specified tilt angle by rotating te coordinate system about te y-axis, te x-y plane contains te main beam axis of te sectoral antenna, and tis plane intersects te local orizontal plane along te y-axis. Te tilt angle β is defined as te positive angle (degrees) tat te main beam axis is below te orizontal plane at te site of te antenna. FIGURE 19 Rigt-anded coordinate systems used to account for te radiation pattern of a tilted sectoral antenna z z β u b θ θ y y ϕ c x β x a ϕ F.1336-19
Rec. ITU-R F.1336-4 33 In a rectangular coordinate system located at te antenna, wit its x-axis in te vertical plane containing te maximum gain of te antenna, te coordinates of te unit vector are given as follows: z x y = sinθ = cosθ = cosθ cosϕ sinϕ Note tat tis is a non-standard sperical coordinate system in tat te elevation is measured in te range from 9 to +9 degrees. Tis is te same convention tat was used in recommends in te main text and in te previous annexes. Consider te rectangular coordinate system of Fig. 19, wic contains te main beam axis of te antenna and is rotated downward about te y-axis by an angle of β degrees. Te unit vector in tis system as te coordinates x, y, and z given by: z = z x = z y = y cosβ + x sinβ + x sinβ cosβ In te corresponding sperical coordinate system referenced to te plane defined by te main beam axis and te y-axis, te sperical angles are related to te coordinates x, y and z by sinθ = z and tanϕ = y/x. Te determination of te value of ϕ, wic lies between 18 and +18 degrees, is given by te arctan(y/x) wit possible corrections depending on te algebraic sign of x and y. Alternatively, making use of te fact tat te sum of te squares of x, y and z is unity, it can be sown tat cosϕ = x/cosθ over a restricted range of values of ϕ. Substituting equations (4) into (41) and ten substituting te resultant values of z and x for te relationsips z = sinθ and x = cosθcosϕ, te following expressions for te values of te sperical coordinates are obtained (see Note 1): θ = arcsin( z) = arcsin(sin θ cosβ + cosθ cosϕ sinβ), x ( sinθ sinβ + cosθ cosϕ ϕ = arccos( ) = arccos cosθ cosθ cosβ), 9 θ 9 ϕ 18 NOTE 1 Te range of te function arccos is from to 18. However, tis does not limit te applicability of te metodology because te antenna patterns used exibit mirror symmetry wit respect to te x-z plane and te x-y plane. Te equations in recommends 3.4 come from equation (4). (4) (41) (4) 3 Application of te radiation pattern equations in recommends.5 and 3.5 to electrical tilt antennas In te case of te electrical tilt, te radiation pattern equations sould be teoretically a function of te tilt angle β, wic depends on te pase sift amount of te flux radiated from te vertically placed antenna elements. However, taking into account tat β is actually a small value in general (e.g., witin 15 ), te following assumption could be applied for simplification. Since te tilted radiation gains at te zenit and te nadir ave to remain te same values respectively regardless of te tilt angle β (see Fig. ), te actual radiation pattern, compared to te pattern before tilting, sligtly expands or contracts above te maximum gain axis or below tat axis, respectively, as sown in te solid line pattern in Fig..
34 Rec. ITU-R F.1336-4 Tis radiation pattern s gains (illustrated by te solid line) could be approximated by tose of anoter pattern (illustrated by te broken line in Fig. ) using a parameter conversion. Tis broken line pattern is derived from an ideal uniform elevation angle sift of β for te original pattern calculated from te equations in recommends.1,., 3.1 and 3. in te respective cases. Tus, te electrically tilted radiation patterns are derived using te parameter conversion in te equations in recommends (in.1,., 3.1 and 3.) as follows: Te elevation angle θ from te maximum gain axis can be described as: were, θ : θ = θ + β (43) elevation angle (degrees) measured from te orizontal plane at te site of te antenna for te tilted radiation pattern ( 9 θ 9 ) β: electrical tilt angle as defined in of tis Annex or recommends.5 and 3.4. In order to apply te reference radiation pattern equations in recommends.1,., 3.1 and 3. to te electrically tilt antennas, based on te above assumption, a compression/extension ratio R CE is introduced. Te compression/extension ratio R CE can be defined as: 9 R CE = (44) 9 ± β Elevation angle θ e, by wic te tilted radiation gain at θ are calculated using equations in recommends.1,., 3.1, and 3., can be expressed as follows: ( θ + β) 9 θ 9 θ = θ = = e R CE for θ + β 9 + β 9 + β ( θ + β) 9 θ 9 θ = θ = = e R CE for θ + β < (45) 9 β 9 β Te electrically tilted radiation patterns are calculated by using θ e of equations of (45) instead of θ in te equations in recommends 3.1 and 3. for sectoral antennas and also in recommends.1 and. for omnidirectional antennas.