Dolph-Chebyshev Pattern Synthesis for Uniform Circular Arrays

Transcription

1 1 Dolh-Chebyshev Pattern Synthesis for Unifor Circular Arrays Tin-Ei Wang, Russell Brinkan, and Kenneth R. Baker, Sr. Meber, IEEE Interdiscilinary Telecounications Progra UCB 530, University of Colorado, Boulder, CO {tin-ei.wang, Fidelity Cotech, Inc Kansas Avenue, Suite 2D, Longont, Colorado Abstract The synthesis of Dolh-Chebyshev atterns for the Unifor Circular Array (UCA) has been docuented riarily for isotroic antenna eleents. We have focused on UCA attern synthesis with directional antenna eleents. The radiation roerties of a UCA were analyzed by hase ode and a ethod for attern synthesis was develoed. The synthesis rocess erits generation of colex weights for each eleent to roduce the Chebyshev atterns with desired beawidth or sidelobe level. Synthesis and verification easureents were erfored on the Phocus Array Syste, a coercial 8-eleent circular array designed for use in the 2.45 GHz ISM band for systes. Measureents were erfored on a novel, off-the-shelf outdoor range to coare redicted to actual atterns. Results indicate good agreeent between redicted and easured atterns. Measured atterns show a iniu sidelobe level of around 20 db to 25 db relative to the ain bea. A coarison of sidelobe beawidths in the range fro 22.5 to 135 exhibits good agreeent between theoretical and easured results. Beyond 135, side lobes are ill defined due to the lack of colex valued inforation for the eleent attern. The redicted and easured results show wider half ower beawidths than the theoretical values but are within accetable liits. Index Ters Dolh-Chebyshev, Unifor Circular Array, Array Synthesis, Phase Mode C I. INTRODUCTION oared to the Unifor Linear Array (ULA), the Unifor Circular Array (UCA) has the advantage of attern stability around the lane of the array. Such stability and the ability to odify and control the attern for circular arrays has traditionally been considered for direction finding alications [1]. Cobining direction finding with interference anageent is desirable for any alication that requires the caability to detect, discriinate, and locate RF eitters. The adatability rovides a wide set of ossible alications for the UCA. Recently, UCA have been used in MIMO satial ultilexing [2] and esh networking alications [3]. Key to this utilization is the ability to aniulate the attern of the array using software, ossibly on a acket-by-acket basis. This aer describes the efficient ileentation of UCA attern synthesis. Manuscrit received Aril 18, This work was suorted by a grant fro Fidelity-Cotech, Inc. Longont, Colorado. Phocus Array is a tradeark of Fidelity-Cotech, Inc. Dolh-Chebyshev arrays were first introduced by Dolh in 1946 [4]. The coutation and synthesis of current distributions for Dolh-Chebyshev atterns for linear arrays were further develoed by others [5-8]. Using a transforation ethod to coute current distributions for the UCA fro the ULA was first roosed by Davies [9]. The technique was then further develoed in [1] utilizing the concet of a virtual array that transfors UCA to virtual ULA for the urose of satial soothing in direction finding alications. These works concentrated on the analysis of arrays with isotroic eleents. Secial cases of directive eleents were discussed in [10] and [11]. The virtual array transforation concet was later used for Chebyshev attern synthesis in [12]. The Chebyshev attern has the liitation that only one variable, either beawidth or sidelobe level, can be secified at a tie. A odified Chebyshev attern synthesis ethod that overcoes such liitations, at the exense of non-equal rile sidelobes, was described in [13]. The synthesis rocesses in [12] and [13] are docuented for isotroic eleents only. The develoent in [14] introduced the hase ode ethod attern synthesis, which can deal with both isotroic eleents and directive eleents cases on attern synthesis for UCA. We ileented the develoed hase ode ethod of attern synthesis on the Phocus Array Syste, and tested the synthesized atterns using free-sace easureents. Our synthesis algorith is sensitive to the realities of this hardware as well as considering the actual antenna eleent attern. The resulting atterns are useful without being a burden to rocessing ower within the array structure since a non-iterative aroach is being used. The reainder of this aer reviews hase odes, hase ode synthesis, Dolh-Chebyshev attern synthesis with hase odes, and easureent results. Section II covers hase odes and our hase ode synthesis ethod. Section III rovides the stes used for synthesizing Dolh-Chebyshev atterns on the UCA. Section IV resents our results, coaring redicted to easured atterns. Section V is the conclusion. II. PHASE MODE SYNTHESIS The continuous for of the far-field attern of a UCA, in the lane of the array, can be exressed as E(!) = 1 W (#)EL(!!#)e jkrcos(!!# ) d# 2 (1)! where W( ) and EL( ) are the colex weight and the antenna attern of each antenna eleent resectively [14]. The notation of (1) is that ϕ designates the angle of the array

2 2 attern in the far-field, centered at the center of the array. The sybol φ designates the angle around the array and also centered at the center of the uniforly arranged set of eleents. The sybol φ is thus used for designating the location of the eleents and for describing the current distribution around the array. Due to the circular nature of the array, the weight function and the far-field antenna attern are eriodic functions with eriod 2π. Thus one can exand W( ) and E( ) into a Fourier series as follows: E(!) =! A e j! (2) where: A = 1! E()e! j d 2! (3)!! A in (2) and (3) is known as the far-field hase odes of the attern alitude function. The weight function reresents the relative alitude and hase of the current excitation around the array. W (!) =! C e j! (4) where: C = 1! W ()e! j d 2! (5)!! The Fourier coefficients of the weight function, C, are the hase odes of the excitation function. Thus, each in (2) through (5) reresents a different hase ode. Concetually, the hase ode ethod of array synthesis sily equates the near-field (excitation) hase odes to the far-field attern hase odes. For isotroic eleents, EL( )=1. So, by inserting the integral for of the Bessel function exression into the far-field integral, the hase ode for of the far-field attern becoes: 1 E iso (!) = 2 C e j! # e j(#!! ) e krcos(#!! ) d#! (6) = C j e j! J (kr) # In (6) J is the Bessel function of the first kind, defined as: 1! J (kr) = e j(!# ) e jkrcos(!# ) d 2! j (7)!! k is the wavenuber defined as k=ω/c=2π/λ and r is the radius of the UCA. In the isotroic eleent case, considering (2) and (6), we can now relate the far-field hase ode to the excitation hase ode as: C iso = A j J ( kr) ( )!1 (8) A. Directional Eleent Patterns In order to accoodate directional eleents into the above develoent, the eleent attern ust be transfored into a set of Fourier series coefficients, D, as shown in (9) and (10). EL(! ) =! D e j! (9) where: D = 1! EL( )e! j d 2! (10)!! The eleent attern, in suation for, can be substituted into (1). After silification, the relation aong weight function, eleent attern and far field attern can then be found by equating the far field attern, (1), with the Fourier exansion coefficients in (2) and (4) as: 1 E dir (!) = ## C D e j(!!# ) e j# e jkrcos(!!# ) d# 2! 1 = ## C e j! D 2 ##! = C e j! D j (!) J! (kr) e j(!)(!!# ) e jkrcos(!!# ) d# (11) Relating the excitation hase odes, C, in (11) that now include the eleent attern Fourier coefficients, D, to the far field hase odes, A, we arrive at the following: C dir = A # % ( ) D j (!) J (!) kr & ( '!1 (12) The Fourier series exansions of (2) and (4) are infinite series and will need to be truncated to the axiu ode nuber that can be sustained by the array. B. Mode Nuber The analysis of [14] stated that in order to avoid aliasing within the hase ode sectru, and to excite only useful odes, the nuber of odes has the criteria such that 2h +1 < N (13) h < kr (14) where 2h+1 is total nuber of odes. in (2) through (12) is then ranged fro h to h. Here, h reresents the highest ositive ode nuber and N is the nuber of eleents. Thus, adding the criteria for deterining the correct ode nuber in (8) and (12) is iortant for successful ileentation. For exale, the Phocus Array Syste has a diaeter of one wavelength. Alying the criteria in (13) and (14) we find: N = 8 kr =!! 3.14 h = 3 C. An Alternate Mode Nuber Definition (15) In the virtual array transforation techniques, the authors used different criteria to decide ode nuber, which deterines the size of the virtual array [1], [12], [13]. In the transforation fro actual array to virtual array, the ode nuber ust be chosen to ake the aroxiation in (16) valid. a (!) = N j +ln J +ln ( kr)e j( +ln)!! N j J ( kr)e j! (16) l where a reresents the steering vector of the unifor circular array. The ideal case is that if N! kr then l in (16) is negligible, i.e., l=0. To ake the aroxiation in (16) ore reliable, the authors also take the second significant ters into account, i.e., l = -1, 1. In this way the value of Bessel function for l = -1, 1 should be fairly sall. The criterion for h is then:

3 3 ( ) #% Max h h! N 1 2, J kr ' hn % <! ( (17) &% J h ( kr) )% where ε is a redeterined sall nuber. If the eleent nuber is an even nuber, then the size of virtual array should be saller than actual array size. Using the criteria fro [1], [12], [13] and letting ε =0.05 for the Phocus Array Syste, we see fro Fig. 1 that the reasonable nuber of odes is again 2h+1 = 7. Fig. 1 The ε values corresond to the ossible ode nubers, h. III. PATTERN SYNTHESIS The goal of our synthesis rocess is to roduce a set of colex weights for the circular array, considering directional antenna eleents, to roduce a far field Chebyshev attern. The initial ste is to synthesize the far field Chebyshev attern with the desired beawidth or SideLobe Level (SLL). Fro this redicted attern we derive the set of Fourier coefficients A as defined in (2). This set of A is used to define a set of C using (8) or (12), for directive and isotroic eleents resectively. Fro the set of C we can define the continuous eleent weight function W(φ) as defined in (4). The weight values are W(φ=φ n ), where each φ n is the angular value of an array eleent. In suary, attern synthesis is accolished by equating the Fourier coonents of the near and far field hase odes. We have coe to refer to this constraint as the Fourier ethod; although it ight also be referred to as hase ode analysis. A. Far Field Pattern Synthesis Based on SLL Our derivation of the desired far field Chebyshev attern is taken fro the develoent found in [12]. The far field attern, and the associated far field hase odes, A, are found using (18). The nuber of odes is deterined by the ethods described in Section φl φ 1 φl φ E( φ) = T 2 h, z0cos = cosh 2hcosh z0cos 2 2 (18) In (18), φ l reresents the direction of look angle of the desired attern, φ is the aziuth angle ranged fro 0 to 2π and z 0 is defined by the desired SLL as defined in (19). For exale, if the desired sidelobe level is 40 db, then SLL=100. In other words, secifying the SLL can be the first ste of synthesis of the far field Chebyshev attern. T(2h, z 0 ) = cosh(2hcosh!1 (z 0 )) = SLL z 0 = cosh 1 % (19) 2h cosh!1 (SLL) # & ' Alternatively, the SideLobe BeaWidth (SLBW) can be secified as described in the next section. Note that the Chebyshev olynoial will not erit one to secify both the SLL and the Half Power BeaWidth (HPBW) siultaneously. Please note that the far field attern defined by (18) fors the theoretical Chebyshev attern used in later analysis of the results. B. Far Field Pattern Synthesis Based on Beawidth Instead of secifying the SLL, one can alternatively choose to secify beawidth. In this definition, beawidth is related to the sidelobe beawidth as described below. Consider the HPBW and SLBW as define here: z HPBW = 4cos!1 1 % # z 0 & ' (20) Z 1 is deterined by the half ower oint which is defined in (21). T(2h, z 1 ) = cosh(2hcosh!1 (z 1 )) = SLL 2 z 1 = cosh 1 2h cosh!1 ( SLL # 2 ) % & ' (21) The SLBW is the beawidth of oints at both sides of the ain lobe where the value is equal to the sidelobe level. Fro the Chebyshev olynoial definition this can be seen to be: 1 % SLBW = 4cos!1 # & ' (22) z 0 Fro the equations for both HPBW and SLBW, it is clear to see that beawidths deend on SLL only. The beawidths of the Chebyshev atterns are invariant with the look angle. Thus, the Chebyshev attern can be generated by secifying either the desired beawidth, or the sidelobe level, but not both indeendently. C. Directive Antenna Eleents For directive eleents, the Fourier exansion coefficients of the eleent attern, D, are needed in (12) for deriving C. We have used, and recoend the use of, easured antenna eleent atterns. The easured attern will also include any utual couling effects between eleents. The total nubers of antenna eleent Fourier coefficients to be used, 2+1, is deterined by the eleent attern itself. In general, the 2+1 ters should be the ost significant ters of the Fourier series exansion of the eleent attern. Fig. 2a shows the eleent attern alitude of the Phocus Array Syste and the Fourier series exansion of that attern is shown in Fig. 2b. A total of 17 Fourier coefficients, D, are used in this case.

4 4 targeted at reventing aliasing when saling the continuous colex weight function by ensuring that nuber of odes roerly constrained. Siultaneously, odes are not defined for an array that will not roagate or will not contribute to the far field attern. A good discussion of the theory behind hase ode analysis for UCA can be found in [14]. IV. MEASURED RESULTS Fig. 2 Tyical Eleent Pattern; Eleent Pattern in Fourier Doain D. Colex Weight Definition The set of C can be defined by inserting the A and D into (12). The continuous angle weight function is then found using (4). Fig. 3 shows an exale weight function W(φ) for the Phocus Array Syste as calculated by (4) and then saled at the eight Phocus Array eleent locations. Fig. 3 Exale of derived weights saled at eleent location The ode nuber definition as defined in Section 2 is A. Measureent Syste Tyically, antenna easureents are erfored in an anechoic chaber with a calibrated network analyzer. In this way, unwanted reflections and interference are iniized during the easureent rocess. In our case, an outdoor range in an office ark environent was constructed using coercial WiFi radio cards. While the syste is inexensive and easy to create, the easureent results contain known errors as described below. The syste records agnitude easureents, using RSSI readings recorded fro acket values returned by WiFi cards, and a turntable to sin the antenna under test. Our easureents were ade using a Fidelity Cotech Phocus Array Syste, using b channel 1. The array under test was set to a known angular osition and then each bea attern was easured by actively eliciting a resonse fro the WiFi access oint. Each attern easureent oint was reeated five ties before oving on to the next angular osition. The five easureents for each bea osition were then cobined to create the easured results analyzed below. RSSI levels in db were first converted to a linear ower unit, averaged, and then converted back to db. Due to the easureent technique, soe distinguishing features can be seen in the raw easureent results. For exale, in Fig. 7, soe outlier dis aeared in the results. This tye of outlier is likely due to the radio card reorting a zero RSSI value. This error condition is likely caused by a acket collision with a acket fro a nearby WiFi network. Since a zero value is several standard deviations fro the ean of the readings, each zero has a large effect on the results. Also, these zero readings often occur in grous at a articular angle in the bea attern, coounding the ean error. This grouing ight be due to the relatively quick (over a few seconds) bea attern easureent interval eritting soe systeatic fors of external interference to have a reeated effect. The discontinuous jus aearing in Fig. 5 are likely due to the undocuented behavior of the radio card s autoatic gain control (AGC) circuit. These jus are due to the AGC adatation, which is not directly controllable in the easureent syste used. B. Pattern Analysis Figures 5 through 12 show easured ower attern results along with a coarison to theoretical Chebyshev atterns. Theoretical and easured results are resented for look angles of 0 and 22.5 for various SLBW values ranging fro 22.5 to 180. The theoretical Chebyshev attern is defined by (18) and does not include any consideration of a liited nuber of hase odes or antenna eleents. It is sily the far field attern as defined by the Chebyshev olynoial. The

5 5 redicted attern is that attern defined by our UCA architecture, with a finite set of hase odes and our directive antenna eleents. As such, the redicted attern is effectively a realization of (2) for a finite nuber of odes. Coarison of any two sets of easureent results for look angles of 0 and 22.5 reveals that the atterns for look angle of 22.5 show better erforance. This is due to the relative geoetric location of the antenna eleents. At a 22.5 look angle (and by extension, any look angle that is odulo 45 fro 22.5 around the array), the look angle is coincident with an antenna eleent and the adjacent antenna eleents are 45 offset fro the look angle. In contrast, for a look angle of 0 (and by extension, any look angle that is odulo 45 fro 0 around the array), the look angle is between two antenna eleents and the adjacent antenna eleents are 22.5 offset fro either side of the look angle. In the far field, the closer eleent sacing restricts the attern contribution fro these eleents. Thus, the interference and couling effects that for the far field ode attern for a 0 look angle are seen to be larger than for a look angle of Both sets of results show a iniu sidelobe level of around 20 db to 25 db relative to the ain bea. This is due to the utual couling between eleents cobined with the non-isotroic roerties of the array eleents. Moreover, lacking the colex value inforation of the eleent attern is likely a cause for the synthesized weights to be inaccurate. Overall, there is good agreeent between the redicted and easured attern. AGC ju Fig. 5 Chebyshev, Predicted and Measured Patterns Coarison Sidelobe Beawidth 22.5 Look Angle 0 Look Angle 22.5 Fig. 6 Chebyshev, Predicted and Measured Patterns Coarison Sidelobe Beawidth 45 Look Angle 0 Look Angle 22.5

6 6 Wi-Fi droout or interference Fig. 8 Chebyshev, Predicted and Measured Patterns Coarison Sidelobe Beawidth 90 Look Angle 0 Look Angle 22.5 Wi-Fi droout or interference Fig. 7 Chebyshev, Predicted and Measured Patterns Coarison Sidelobe Beawidth 67.5 Look Angle 0 Look Angle 22.5 Fig. 9 Chebyshev, Predicted and Measured Patterns Coarison Sidelobe Beawidth Look Angle 0 Look Angle 22.5

7 7 Fig. 11 Chebyshev, Predicted and Measured Patterns Coarison Sidelobe Beawidth Look Angle 0 Look Angle 22.5 Fig. 10 Chebyshev, Predicted and Measured Patterns Coarison Sidelobe Beawidth 135 Look Angle 0 Look Angle 22.5 Fig. 12 Chebyshev, Predicted and Measured Patterns Coarison Sidelobe Beawidth 180 Look Angle 0 Look Angle 22.5

8 8 Fig. 13 shows a SLL coarison between the theoretical (ideal) Chebyshev SLL, average redicted SLL and the average easured SLL for a look angle of Measureent data used in this analysis had the WiFi droouts reoved. The SLL of redicted and easured results are derived fro the average of the eak sidelobe values. Due to the fact that there are no distinct sidelobe eak values at SLBW of 135, and 180 in the easured results, the average SLL values of these SLBWs are derived fro the average of all data oints between the ain bea sidelobes. The lack of defined SLBW for 135, and 180 can be observed in Fig The SLL coarison shows that SLBWs ranged fro 22.5 to 135 exhibit good agreeent aong theoretical and easured results. The SLBWs of redicted and easured results can be coared by using the average SLL and theoretical SLL values, as shown in Fig. 14. Fig. 14 reresents the coarison result using average SLLs while Fig. 14 is the coarison using theoretical SLL. The data oints of large SLBW are not shown in Fig. 14 since easured and calculated redictions cannot achieve SLLs as low as the theoretical values for large SLBW. Both lots show that the SLBWs of both redicted and easured data agree well with theoretical SLBWs. As described in Section III, each SLBW corresonds to a certain SLL value, and each SLL can also corresond to certain HPBW. By utilizing such roerties, HPBW coarisons can also be erfored using the sae easured and redicted data. The relation of SLLs, SLBWs, and HPBW is shown in Fig. 15. The coarison of HPBW for redicted and easured results is shown in Fig. 15. The redicted and easured results show wider HPBW than the theoretical values in a reasonable range. Fig. 14 Predicted and Measured SLBW Coarison of Look Angle 22.5 Average SLLs Theoretical SLLs Fig. 13 Chebyshev, Predicted and Measured SLL Coarison of Look Angle 22.5

9 9 Fig. 15 HPBW Coarison of Look Angle 22.5 Relation of SLL, HPBW and SLBW Predicted and Measured HPBW Coarison. V. CONCLUSION This aer describes the efficient ileentation of Dolh-Chebyshev attern synthesis for the UCA. The attern synthesis is based on equating near field excitation hase odes to far field attern hase odes. The develoent includes synthesis ethods for both isotroic and directive eleents. The synthesis ethod was successfully ileented on a coercial eight eleent circular array oerating in the 2.4 GHz ISM band for WiFi alications. Measureents of array atterns fro this array were erfored on an outdoor range using off the shelf equient. Theoretical and easured results show good agreeent. The coarison of theoretical, redicted and easured atterns were done in various oints of view and they all show good agreeent with ersective of SLL, SLBW and HPBW. Synthesized atterns show a floor to the iniu sidelobe level of around 20 db to 25 db relative to the axiu. Sidelobe beawidths in the range fro 22.5 to 135 exhibit good agreeent between theoretical and easured results. Beyond 135 the beawidths are ill defined due to insufficient eleent attern inforation resulting in a deviation fro redicted erforance. The redicted and easured results show wider half ower beawidths than the theoretical values but are within accetable liits. Future work in this area will include adation of these techniques to other UCA array structures followed by the exeriental use of UCA in icrocell MIMO access oint networking. ACKNOWLEDGMENT The authors would like to thank Joseh Carey of Fidelity-Cotech, Inc. for his suort and helful discussions regarding the theory and analysis of UCA. REFERENCES [1] M. Wax and J. Sheinvald, Direction finding of coherent signals via satial soothing for unifor circular arrays, IEEE Trans. Antennas Pro., 42(5), , [2] Wyne, S., Molisch, A.F., Alers, P., Eriksson, G., Karedal, J., Tufvesson, F., Outdoor-to-Indoor Office MIMO Measureents and Analysis at 5.2 GHz, Vehicular Technology, IEEE Transactions on, vol.57, no.3, , May [3] Stine, J.A., Exloiting sart antennas in wireless esh networks using contention access, Wireless Counications, IEEE, vol. 13, no. 2, , Aril [4] C. L. Dolh, A current distribution for broadside arrays which otiizes the relationshi between beawidth and sidelobe level, Proc. IRE, 34(6), , [5] R. J. Stegen, Excitation coefficients and bea-widths of Tschebyscheff Arrays, Proc. IRE, 40(11), , [6] D. Barbiere, A ethod for calculating the current distribution of Tschebyscheff arrays, Proc. IRE, vol. 40, ; January, [7] T. N. Davidson, A Note on the Calculation of Dolh-Chebyshev Shading for Linear Array, ASPL , Det. Electrical and Electronic Eng., the University of Western Australia, Aug, [8] S. Jazi, A New Forulation for the Design of Chebyshev Arrays, IEEE Trans. Antennas and Pro., 42(3), , [9] D. E. N. Davies, A transforation between the hasing techniques required for linear and circular aerial arrays, Proc. IEE, 112(11), , [10] R. Eiges and H. D. Griffiths, Mode-sace satial sectral estiation for circular arrays, IEE Proc. Radar, Sonar Navig., 141(12), , [11] K. M. Reddy and V. U. Reddy, Analysis of satial soothing with unifor circular arrays, IEEE Trans. Signal Proc., 47(6), , [12] B. K. Lau and Y. H. Leung, A Dolh-Chebyshev aroach to the synthesis of array atterns for unifor circular arrays, Proc. ISCAS 2000, vol. I, , May [13] G. Thadeu Freitas de Abreu and R. Kohno, A odified Dolh-Chebyshev aroach for the synthesis of low sidelobe beaatterns with adjustable beawidth, IEEE Trans. Antennas and Pro., vol.51, no.10, , Oct [14] Lars Josefsson and Patrik Persson, Conforal Array Antenna Theory and Design, IEEE Press Series on Electroagnetic Wave Theory, Cha.2, 10, 2006.