Optimized Aperiodic Concentric Ring Arrays

Transcription

1 24th Aual Review of Progress i Applied Computatioal Electromagetics March 30 - April 4, iagara Falls, Caada 2008 ACES Optimized Aperiodic Cocetric Rig Arrays Rady L Haupt The Pesylvaia State Uiversity Applied Research Laboratory P O Box 30 State College, PA haupt@ieeeorg Abstract: This paper uses a geetic algorithm to optimize the elemet spacigs i a cocetric rig array Each rig has equally spaced elemets The umber of elemets i a rig ad/or the spacig betwee rigs is optimized i order to miimize the maximum sidelobe level Keywords: cocetric rig array, thied array, aperiodic array, geetic algorithm 1 Itroductio A rig array has elemets lyig o a circle Two or more rig arrays with differet radii ad a commo ceter form a cocetric rig array A cocetric rig array with rig havig elemets ad a radius of r is show i Figure 1 The distace betwee elemets o rig is give by d Figure 1 A cocetric rig array The usual approach to obtaiig low sidelobes i the array factor of a cocetric rig array is amplitude weightig the elemets [1], [2] Aother approach is to use aperiodic spacig A circular Taylor amplitude taper projected oto a lie i the azimuth plae relates to the elemet desity i the cocetric rig array [3] The radii of the rigs ad the elemet desities i the rigs cotribute to give the desired low sidelobe taper High sidelobes appear i the azimuth patter of a Taylor weighted rig array whe the spacig betwee elemet i a rig exceeds 04 [4] The steepest descet method ca fid the weights ad rig spacig that results i desirable sidelobes ad mai lobe characteristics [5] A ivestigatio was also doe to fid the rig spacig of small cocetric rig arrays that results i either the miimum 275

2 24th Aual Review of Progress i Applied Computatioal Electromagetics March 30 - April 4, iagara Falls, Caada 2008 ACES maximum sidelobe level or the maximum directivity [6] Rig spacig determies the samplig of a low sidelobe Taylor taper for cocetric rig arrays Adjustig the rig spacig ca sigificatly reduce the maximum sidelobe level of a cocetric rig array whe the mai beam is steered from boresight [7] Milliga provides a excellet overview of geeratig a space taper for cocetric rig arrays He demostrates calculatig the spacig to obtai approximate low sidelobe Taylor tapers [8] This paper presets results from optimizig the rig spacig ad/or the umber of elemets i a rig i order to produce a low sidelobe array factor as a fuctio of 2 Uiform Cocetric Rig Arrays The array factor for the cocetric rig array with a sigle elemet at the ceter (Figure 1) is give by where r k 2 / wavelegth w elemet weights for rig r radius of rig x, y locatio of elemet x y m r si m u sicos v sisi 2 m1 / m umber of elemets i rig umber of rigs r cos r AF 1 w e 1 m1 jkr cosmusi mv (1) I this formula, all the elemets i the same rig have the same weight A uiform array has equal elemet spacig ad weightig For a uiform cocetric rig array, the rig spacig, r, is a costat times the rig umber ad the spacig betwee elemets withi a rig, d, is approximately costat for all rigs Figure 2 is a diagram of a 279 elemet cocetric rig array There are ie rigs with havig the characteristics listed i Table 1 A ie rig cocetric rig array has the array factor show i Figure 3 The array has a directivity of 2935 db ad a peak relative sidelobe level of -174 db Table 1 Rig radius ad umber of elemets per rig for a 9-rig uiform cocetric rig array r ()

3 24th Aual Review of Progress i Applied Computatioal Electromagetics March 30 - April 4, iagara Falls, Caada 2008 ACES Figure 2 Cocetric rig array with ie rigs spaced /2apart ad havig d /2 Figure 3 Array factor due to the array with ie cocetric uiform rigs 3 ouiformly Spaced Cocetric Rig Arrays The first approach keeps d /2 while varyig r I effect, the elemet desity i the radial directio relates to a low sidelobe amplitude taper r r 1 /2 (2) where 0 ad the miimum spacig betwee rigs is /2 A hybrid geetic algorithm that combies a cotiuous geetic algorithm with a elder Mead algorithm is used to fid the r that result i the miimum maximum sidelobe level i the array factor Table 2 has the optimized rig spacig ad the umber of elemets i each rig This ew array has a slightly larger radius tha the ie rig uiform cocetric array but has oly 201 elemets or 72% of the elemets of the uiform array Its directivity is 2738 db with a maximum relative sidelobe level of db Figure 4 is a diagram of the array Figure 5 shows the correspodig array factor The optimized array has a higher directivity ad lower sidelobe level tha the thied array All the sidelobes have early the same height Table 2 Rig radius ad umber of elemets per rig for a ie rig uiform cocetric rig array with optimized r r ()

4 24th Aual Review of Progress i Applied Computatioal Electromagetics March 30 - April 4, iagara Falls, Caada 2008 ACES Figure 4 Optimized r for a array with six cocetric rigs Figure 5 Array factor associated with optimized r for array with six cocetric rigs A secod approach to miimizig the maximum sidelobe level lets r /2ad optimizes the A geetic algorithm is used to fid the optimum listed i Table 3 The first four rigs have the same umber of elemets as the uiform array i Table 1 The resultig array has 183 elemets (656% of the elemets of the uiform array) arraged as show i Figure 6 Its directivity is 285 db with a maximum relative sidelobe level of db The array factor show i Figure 7 does ot have the costat sidelobe level of Figure 5 Figure 6 Optimized i ie cocetric rigs with equal rig spacig 278

5 24th Aual Review of Progress i Applied Computatioal Electromagetics March 30 - April 4, iagara Falls, Caada 2008 ACES Figure 7 Array factor associated with optimized i ie equally spaced cocetric rigs Table 3 Rig radius ad umber of elemets per rig for a ie rig uiform cocetric rig array with optimized r () A fial approach simultaeously optimizes r ad Table 4 has the resultig optimum r ad This array has 142 elemets or 509% of the 9 rig uiform cocetric array Its directivity is 2889 db with a maximum relative sidelobe level of db The optimized array appears i Figure 8 with the correspodig array factor i Figure 9 The directivity ad the sidelobe level exceed the results obtaied through the three other approaches Table 4 Rig radius ad umber of elemets per rig for a ie rig uiform cocetric rig array with optimized r ad r () Figure 8 Optimized r ad i each rig for array with six cocetric rigs 279

6 24th Aual Review of Progress i Applied Computatioal Electromagetics March 30 - April 4, iagara Falls, Caada 2008 ACES Figure 9 Array factor for array with six cocetric rigs ad optimized r ad 4 Coclusios A geetic algorithm ca be used to miimized the maximum sidelobe level of a cocetric rig array by optimizig the rig spacig ad/or umber of elemets i each rig Refereces [1] C Stears ad A Stewart, "A ivestigatio of cocetric rig ateas with low sidelobes," IEEE AP-S Tras, vol 13, o 6, pp , ov 1965 [2] L Yuhog, KC Ho ad C Kwa, "3-D array patter sythesis with frequecy Ivariat property for cocetric rig," IEEE Sigal Processig Tras, vol 54, o 2, pp , Feb 2006 [3] R Das, "Cocetric rig array," IEEE AP-S Tras, vol 14, o 3, pp , May 1966 [4] Goto ad DK Cheg, "O the sythesis of cocetric-rig," IEEE Proc, vol 58, o 5, pp , May 1970 [5] L Biller ad G Friedma, "Optimizatio of radiatio patters for a array of cocetric rig sources," IEEE Tras Audio ad Electroacoustics, vol 21, o 1, pp 57-61, Feb 1973 [6] DA Hueber, "Desig ad optimizatio of small cocetric rig arrays," IEEE AP-S Symposium, pp , 1978 [7] G Holtrup, A Margulaud, ad J Citers, "Sythesis of electroically steerable atea arrays with elemet o cocetric rigs with reduced sidelobes," IEEE AP-S Symposium, pp , 2001 [8] TA Milliga, "Space-tapered circular (rig) array," IEEE AP Mag, vol 46, o 3, pp 70-73, Ju