Sidelobe Modification for Reflector Antennas by Electronically-Reconfigurable Rim Scattering
aa r X i v : . [ a s t r o - ph . I M ] F e b Sidelobe Modification for Reflector Antennas byElectronically-Reconfigurable Rim Scattering
S.W. Ellingson and R. Sengupta
Abstract —Dynamic modification of the pattern of a reflectorantenna system traditionally requires an array of feeds. This pa-per presents an alternative approach in which the scattering froma fraction of the reflector around the rim is passively modifiedusing, for example, an electronically-reconfigurable reflectarray.This facilitates flexible sidelobe modification, including sidelobecanceling, for systems employing a single feed. Applications forsuch a system include radio astronomy, where deleterious levelsof interference from satellites enter through sidelobes. We showthat an efficient reconfigurable surface occupying about 11% ofthe area around the rim of an axisymmetric circular paraboloidalreflector antenna fed from the prime focus is sufficient tonull interference arriving from any direction outside the mainlobe with at most 0.3% and potentially zero change in mainlobe gain. We further show that the required surface area isindependent of frequency and that the same performance can beobtained using 1-bit phase control of the constituent unit cellsfor a reconfigurable surface occupying an additional 6% of thereflector surface.
I. I
NTRODUCTION R EFLECTOR antenna systems are commonly used whenhigh gain is required to receive weak signals. In theseapplications, the system is vulnerable to interference arrivingthrough sidelobes. This vulnerability can be ameliorated bysidelobe modification; and in particular, by sidelobe canceling.Sidelobe canceling involves placing a pattern null within thesidelobe so as to reject the interference, and ideally withoutsignificantly affecting the main lobe. Traditional sidelobecanceling requires an array; see e.g. [1]. Such an array canbe implemented in a reflector antenna system using an arrayof feeds in lieu of the traditional single feed; see e.g. [2] andreferences therein. However feed arrays entail compromisesincluding greater aperture blockage and dynamic variabilityin gain and the shape of the main lobe.An example is radio astronomy, where telescopes are com-monly implemented as either large reflectors or arrays oflarge reflectors (see e.g., [3]). These systems are vulnerable tointerference from satellite transmissions received through side-lobes. Of particular concern are transmissions at frequenciesnear astrophysical spectral lines near 1.6 GHz and 10.7 GHz.This long-standing problem is expected to become dramati-cally worse in coming years as new generation of satellitesbegin transmitting at these frequencies; see e.g., [4], [5]. Mostradio telescopes now operational or planned do not employfeed arrays, as such arrays are not scientifically necessary in
The authors are with the Dept. of Electrical and Computer Engineering,Virginia Tech, Blacksburg, VA, 24061 USA, e-mail: [email protected] material is based upon work supported in part by the National ScienceFoundation under Grant ECCS 2029948. Fig. 1. On-axis ( top ) and side ( bottom ) views of an electronically-reconfigurable rim scattering system. most applications and introduce onerous performance limita-tions in applications where they are not required. In particular,top-tier radio telescopes such as the Expanded Very LargeArray (EVLA) [6] and Green Bank Telescope (GBT) [7],and new telescopes in development including Next GenerationVLA (ngVLA) [8] and Deep Synoptic Array (DSA) [9] aresingle-feed systems. This paper presents an alternative approach to sidelobe mod-ification which can be implemented in single-feed systems.The concept is illustrated in Figure 1. For simplicity, we limitthe scope of this paper to prime focus-fed circular axisymmet-ric paraboloidal reflectors; however the concept is applicable toother types of systems, including those employing Cassegrainor Gregorian optics, and those using shaped reflectors. In thisconcept, scattering from a fraction of the reflector surfacearound the rim is electronically modified by manipulating thephase of scattering from the unit cells comprising the surface.The unit cells are envisioned to be contiguous elements havingsub-wavelength dimension. For example, the surface could beimplemented as a reflectarray, with unit cells implemented as We note that the GBT has a mechanically-selectable feed system whichincludes both single feed and array feed configurations.
Fig. 2. Geometry for analysis of reflector antenna systems in this paper. patch antennas whose scattering is controlled by manipulatingthe impedances presented to the antenna terminals; see e.g.[10].The concept of pattern modification by altering the reflectorsurface is not entirely new. The use of fixed mechanicalmodification of the surface as a means to optimize sidelobes isdescribed in [11], and as a means to implement dual-frequencybeam optimization is described in [12]. Novel aspects of thepresent work include (1) electronic reconfigurability, enablingdynamic pattern modification; (2) analysis addressing areathat must be given over to reconfigurability; and (3) analysisdemonstrating that the method is robust to errors in controland hardware failure.II. M
ETHOD OF A NALYSIS
The directivity, main lobe shape, and characteristics of thelargest sidelobes of an electrically-large reflector antenna canbe accurately determined using the theory of physical optics(PO; see e.g. [13]). Consider the case shown in Figure 2 of anaxisymmetric paraboloidal reflector with a single feed locatedat its focus. The receive directivity and pattern of this systemis equal to the transmit directivity and pattern, and the transmitpattern is determined using the following procedure:1) Calculate the field incident at the surface ( s i = ˆ s i s i ) ofthe reflector, due to the feed.2) Calculate the PO equivalent surface current distribution J ( s i ) = 2ˆ n ( s i ) × H i ( s i ) (1)where ˆ n is the unit normal to the surface and H i is theincident magnetic field intensity.3) In the far field, the electric field intensity E s scatteredby the reflector is determined by integrating over thereflecting surface: E s = − jωµ e − jkr πr Z θ θ f =0 Z πφ =0 J ( s i ) e − jk ˆ r · s i ds (2)where j = √− , ω is π times frequency, µ is thepermeability of free space, k is wavenumber, ˆ r pointsfrom the origin of the global coordinate system towardthe field point, θ f is the angle measured from thereflector axis of rotation toward the rim (thus, θ f = θ c o - po l a r i z ed pa tt e r n [ d B i ]
18 m17 m
Fig. 3.
Solid:
H-plane co-polarized pattern of the reflector system describedin Section II, D = 18 m, 1.5 GHz. Dashed:
Same, except diameter reducedto m (addressed in Section III). at the rim), φ is the angular coordinate orthogonal toboth θ f and the reflector axis, and ds is the differentialelement of surface area.4) The far-field power pattern P (ˆ r ) is determined in the tra-ditional way by eliminating the radial ( ˆ r ) component of E s , computing the resulting power density, and dividingby the power density averaged over a sphere boundingthe system. This yields units of directivity.To establish a baseline of performance, consider a traditionalreflector antenna system having diameter D = 18 m and focalratio f /D = 0 . , with a feed modeled as follows: H i ( s i ) = I ˆ y × ˆ s i | ˆ y × ˆ s i | e − jks i s i (cos θ f ) q , θ f ≤ π/ (3)and H i = 0 for θ f > π/ . In this model, I represents thesource magnitude and phase and q is used to set the directivityof the feed. Setting q = 1 . yields edge illumination (EI); i.e.,ratio of field intensity in the direction of the rim relative to thefield intensity in the direction of the vertex, to approximately − dB, yielding aperture efficiency of about 81.5%.Figure 3 shows the H-plane pattern of this system at1.5 GHz, limiting view to the first few sidelobes aroundthe main beam where the PO approximation is known to beaccurate. The PO integral was computed numerically usingapproximately square surface elements having side length . λ , where λ is wavelength. Directivity and first sidelobelevel are found to be . dBi and − dB, respectively.Next we consider a system which is identical except thata reconfigurable surface replaces the portion of the reflectorsurface between radial distances 8.5 m and 9 m. The newsystem may be viewed as a traditional reflector having diam-eter D = 17 m, with a reconfigurable surface that followsthe same paraboloidal surface and extends the diameter to D = 18 m. In this case the scattered field is E s = E s + E s (4) where E s is the field scattered by the non-reconfigurablecenter portion of the reflector, and E s is the field scattered bythe reconfigurable surface. The former is given by Equation 2with θ = θ , where θ is the angle to the rim of thenon-reconfigurable part of the reflector. Similarly, the fieldscattered by the reconfigurable surface can be calculated asfollows: E s = − jωµ e − jkr πr Z θ θ f = θ Z πφ =0 J ( s i ) e − jk ˆ r · s i ds (5)where J is the equivalent surface current for scattering fromthe reconfigurable surface.The function J depends on the technology used to modifythe scattering and the state of the reconfigurable surface. Asan initial assessment of feasibility, it is useful to be able tocalculate estimates of performance that are independent ofthe technology used to implement the surface. To accomplishthis, the reconfigurable surface is modeled as a contiguoussurface of approximately square flat plates having side length . λ (area ∆ s = 0 . λ ). These plates represent the unitcells. Scattering is calculated using Equation 5, except nowquantizing the integrand to the dimensions of the plates. Thus: E s = − jωµ e − jkr πr X n J ( s in ) e − jk ˆ r · s in ∆ s (6)where s in is the center of cell n , and n indexes the cells.Further, we assume J ( s in ) = c n J ( s in ) (7)that is, J is the PO surface current at the center of theplate, times a complex constant c n to account for reconfig-urability (e.g., phase shifting), imperfect efficiency, and anyother effects associated with whatever technology is used toimplement the surface. This is similar to the physics-basedmethods used in [14] and [15] to model reconfigurable scat-tering surfaces in other applications. The motivation for . λ quantization of the surface is simply that technologies thatmight be used to implement reconfigurability would normallyconsist of unit cells having approximately this periodicity inorder to satisfy the Nyquist condition for full sampling of theavailable aperture.In a practical system, there should be negligible modifi-cation of the main lobe when the reconfigurable surface isuncontrolled; e.g., when c n = 1 for all cells, either by choiceor due to system failure. In this case the only differencebetween the original ( D = D = 18 m) system and themodified ( D = 18 m, D = 17 m) system is that the outer0.5 m of the reflector surface consists of 2756 contiguous half-wavelength-square flat plates (representing the uncontrolledcells) as opposed to a continuously-varying surface. The resultin this case is not significantly different from the result shownin Figure 3, and can be detected only as a ∼ . dB differencein the peak level of the second sidelobe.III. F EASIBILITY FOR S IDELOBE M ODIFICATION
This section addresses the following question: How muchof the surface of the reflector must be given over to recon-figurability in order to obtain effective sidelobe modification? Consider the patterns P (ˆ r ) and P (ˆ r ) associated with E s and E s , respectively. For full ability to modify sidelobes over theentire angular span outside the main lobe, it must be possibleto achieve a value of P which is at least as large as P overthis span. Let P m (ˆ r ) be the largest value of P (ˆ r ) that canbe achieved by manipulating the reconfigurable surface. Then,canceling of interference located at the peak of the largestsidelobe requires the P ,m (ˆ r ) to be at least as large as P (ˆ r ) in the direction of this peak. If this condition is satisfied, thenthe area allocated to the reconfigurable surface is sufficient inthe sense that any technology and control algorithm providingsufficiently fine control of the scattered field can be used toimplement effective sidelobe canceling.Thus, P m (ˆ r ) is of primary interest. Note that P m is nota pattern corresponding to a single set of c n ’s (i.e., P ), butrather the value of P for whatever choice of c n ’s maximize P in each direction. Since P (ˆ r ) varies monotonically withthe magnitude of the field scattered by the reconfigurablesurface, P m (ˆ r ) is obtained when all contributions to theassociated field integral add in phase in direction ˆ r . Thiscan be determined from the scattered field calculated fromEquations 6 and 7 with c n = exp n − j arg h J ( s in ) e − jk ˆ r · s in io (8)In other words, the field scattered from each cell is assumed tobe phase shifted such that all terms of the sum add in phase.This assumes that the phase of each reconfigurable elementis continuously variable; the effect of phase quantizationwill be addressed in Section IV. This also assumes that thereconfigurable surface is 100% efficient; i.e., all incident poweris scattered. Thus we obtain an upper limit on performancewhich depends only on the area given over to reconfigurability.To demonstrate, let us again consider the D = 18 m, D = 17 m example, but now computing P and P m sep-arately, with P m determined from scattered fields calculatedvia Equations 6–8. First, consider the dashed (17 m) line inFigure 3, which is P in this case. It is the sidelobes of thispattern that the reconfigurable surface is attempting to modify.The performance of new system is shown in Figure 4. Here weobserve that the continuously-variable phase shifting schemerepresented by Equation 8 yields P m = P at the peak of thefirst sidelobe. Thus, 10.8% of the surface area of the original D = 18 m reflector must be given over to reconfigurabilityto meet the P m ≥ P criterion in this case.This demonstration reveals an attractive feature of theproposed method: The inability to significantly change themain lobe. From Figures 3 and 4 we see that P m is about − dB (0.3%) relative to P at the center of the mainlobe. This upper-bounds the degradation and variability thatthe reconfigurable surface is able to impart on the mainlobe, regardless of the state (i.e., c n ’s) of the reconfigurablesurface. Since the reconfigurable surface is passive, this is trueregardless of any failures of individual cells; i.e., the scatteringfrom malfunctioning cells cannot be significantly greater thanthat from functioning cells. Therefore, control error and systemfailure cannot significantly alter the directivity and shape ofthe main lobe. Further, the very large number of degrees of c o - po l a r i z ed pa tt e r n [ d B i ] P P , continuous phaseP , 2-bit phaseP , 1-bit phase Fig. 4. Pattern P for the D = 17 m center region and maximum pattern P m for the 0.5 m-wide reconfigurable region around the rim. freedom in this system allow many constraints to be enforcedon P . Thus, the system could null P in along the reflectoraxis in order to prevent change in main beam gain, whilesimultaneously imposing many other constraints on P in orderto accomplish the desired sidelobe modifications.IV. P HASE Q UANTIZATION
Practical reconfigurable surfaces may require quantizationof phase control. That is, it may be necessary to limit thephase of c n to discrete values. Figure 4 shows the effect ofphase quantization. For 1-bit phase quantization, c n is either +1 or − , whichever results in the phase of the associatedterm being closest to . For 2-bit phase quantization, c n iseither +1 , + j , − , or − j . As expected, phase quantizationgenerally reduces P m .To offset the degradation associated with phase quantization,it is necessary to increase the area of the reflector given overto reconfigurability. In the example system, it is found that D = 16 . m (about 17% of the surface made reconfigurable)is needed for 1-bit phase quantization to yield P m = P atthe peak of the first sidelobe of P . Since the number of recon-figurable unit cells is large (2756, as noted in Section II), thequantization of unit cell phase is not expected to significantlylimit the accuracy to which the phase of the field scatteredfrom the reconfigurable surface can be set.V. E DGE I LLUMINATION
For the system considered in previous sections, EI = − dB, which is known to optimize aperture efficiency forprime focus-fed axisymmetric paraboloidal reflector systemswith feeds of the type described by Equation 3 [13]. In someapplications, it is desirable to further reduce EI, which reducessidelobe levels at the expense of aperture efficiency. A systemwith reduced EI will exhibit generally lower P m , since lesspower will be incident on the rim. Thus, it is of interest toconsider the efficacy of reconfigurable rim surface scatteringfor systems with lower EI. c o - po l a r i z ed pa tt e r n [ d B i ] P P , continuous phase Fig. 5. Pattern P and maximum pattern P m for the same system consideredin Figure 4, except with feed q = 1 . . Figure 5 shows the result for a system which is identical inall respects to the system considered in Section III, except nowthe feed parameter q has been increased to 1.85. This decreasesEI to − dB. Comparing to Figure 4, we observe that P m is reduced, as expected. However P is also reduced, and infact the margin P m /P for the peak of the first sidelobe isincreased. Therefore the proposed method does not require anyparticular level of edge illumination for effective operation.VI. F REQUENCY C ONSIDERATIONS
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