This is the author s ersion of a work that was subitted/accepted for publication in the following source: haloupka, Heinz J, Wang, Xin, & oetzee, Jacob (3) superdirectie 3-eleent array for adaptie beaforing. Microwae and Optical Technology Letters, 36(6), pp. 45-43. This file was downloaded fro: https://eprints.qut.edu.au/48949/ c opyright 3 Wiley Periodicals, Inc. Notice: hanges introduced as a result of publishing processes such as copy-editing and foratting ay not be reflected in this docuent. For a definitie ersion of this work, please refer to the published source: https://doi.org/./op.78
SUPERDIRETIVE 3-ELEMENT RR FOR DPTIVE EMFORMING H.J. haloupka *, X. Wang * and J.. oetzee ** * Departent of Electrical Engineering and Inforation Technology, Uniersity of Wuppertal, Rainer-Gruenter Str., D-49, Wuppertal, Gerany ** Departent of Electrical and oputer Engineering, National Uniersity of Singapore, 4 Engineering Drie 3, Singapore 7576 STRT: sall array coposed of three onopole eleents with ery sall eleent spacing on the order of 6 to / is considered for application in adaptie beaforing. The properties of this 3-port array are goerned by strong utual coupling. It is shown that for signal-to-noise axiization, it is not sufficient to adjust the weights to copensate for the effects of utual coupling. The necessity for a RF-decoupling network (RF-DN) and its siple realization are shown. The array with closely spaced eleents together with the RF-DN represents a superdirectie antenna with a directiity of ore than di. It is shown that the required fractional frequency bandwidth and the aailable unloaded Q of the antenna and RF-DN structure deterine the lower liit for the eleent spacing. Key words: Superdirectie arrays, adaptie beaforing, utual coupling. INTRODUTION The nuber of utually orthogonal radiation patterns (MORPs) associated with an antenna array equals the nuber of its (operational) radiation eleents, M. daptie digital beaforing with M- degrees of freedo is based on a linear cobination of these array-specific MORPs with ariable coplex-alued weights. choice of eleent spacing substantially saller than / results in strong utual coupling, which gies rise to seere frequency bandwidth and efficiency liitations []-[3]. This letter addresses these probles for a specific array coposed of three identical onopole antennas with an array footprint diaeter of / to / 6. The array produces three MORPs associated with three distinct eigenodes: one low-order quasi-oni-directional pattern and two degenerated higher-order patterns. It is shown that a reduction of eleent spacing results in both an
increased radiation quality factor of the higher-order patterns and an increased deiation between the low- and higher-order pattern radiation resistance. Without a decoupling-network, the low-order and higher-order odes cannot be siultaneously atched, resulting in a signal-to-noise degradation, which cannot be copensated for by digital beaforing. y choice of a proper onopole height, the decoupling network reduces to siple reactie cross coupling between the antenna ports. In this way siultaneous atching of all three odes can be achieed, resulting in an optiu signal-to-noise ratio (SNR). In this new configuration, the two higher-order radiation patterns becoe superdirectie. The allowed iniu eleent spacing is goerned by the required fractional frequency bandwidth.. PROPERTIES OF 3-ELEMENT RR The array structure coprising of three onopole antenna eleents is depicted in Fig.. The frequency-dependent adittance atrix characterises the utual coupling between the eleents of the array, and according to 3 3 V V V I I I () relates the port-currents n I to the driing port-oltages n V. Port oltages ay be expressed as a linear cobination of the orthogonal eigenectors of the adittance atrix: V V V U 6 3 3 () with as the ode-oltages. The port currents then becoe, 6 3 3 I I I U (3) with eigenalues j G and j G as the frequency dependent ode-adittances. The three utually orthogonal far-field radiation patterns associated with the eigenodes can be characterised by the three ector functions ), (, noralized such that
3 * l (, ) (, ) d 4l (4) where l denotes the Kronecker delta function. In order to study the effect of a change in the eleent spacing a on the frequency response of the ode-adittances ( f ) and ( f ) and on the corresponding radiation patterns, nuerical siulations were carried out with the coputer codes SuperNE Lite [4] and Zeland IE3D [5]. In these siulations, the etallic ground plane was assued to be infinite in size and etallic losses were neglected. In the presentation of the obtained results, f and c / f denote the reference frequency and the corresponding free space waelength. The onopole length and onopole diaeter were chosen as l / 4 and d / 4. The eleent spacing was aried between a / and a / 6. For the nuerical siulations, a reference frequency of f. 45 GHz was chosen, but the obtained results were noralized to allow conclusions to be drawn for arbitrary operating frequencies. In Figs. and 3, the results for relatiely sall eleent spacing a / are shown as an exaple. Fig. depicts the noralized radiation patterns ( /, ) /,, at D, f f c / 4l. The axiu directiity for the three odes were coputed to be ax D 3.4 5. di, D 7.94 9 di and D 7.76 8.9 di. s a ax ax consequence of utual coupling, ode adittance for the lower-order ode with dipole-like radiation pattern substantially differs fro the ode input adittance of the degenerated higher-order odes and. Fig. 3 shows the frequency dependence of the odal adittances as a function of the noralized frequency f / f. The low-order and higher-order odes resonate at different frequencies, but the ost notable difference is in the radiation quality factor Q rad, where the higher-order odes display a significantly higher alue. The relation between the radiation quality factor of higher-order odes and the eleent spacing a / is shown in Fig. 4, indicating a sharp increase with reduced spacing.
4 Fro an inspection of the nuerical results, it was found that for a / 6, the radiation patterns can be approxiated by the following analytical expressions deried under the assuption of a sinusoidal line current distribution along the onopole axes while eploying a Taylor series approxiation:, and cos cos sin (, ).8 exp( j ) e (5a) (, ).83exp( j ) cos cos sin e (5b) (, ).83exp( j ) cos cos cos e (5c) in eqs. (5) are phase angles that depend on the frequency. 3. EFFETS OF MUTUL OUPLING y eans of the eigenode representation, the actual array with three utually coupled eleents can forally be replaced with a set of three equialent antennas with radiation patterns corresponding to the three MORPs. In the receie ode, each of the three equialent antennas can be odelled by eans of a current source (see left part of Fig. 5) with source adittance equal to the corresponding ode adittance G j and source current i, which in case of a spectru of hoogeneous plane waes incident on the antenna becoes i G (, ) inc (, ) d,, Z E (6), For digital beaforing, three receier channels with input adittance in are connected to the three antenna ports. onsequently the load in the equialent circuits for the three odes in Fig. 5 has a alue of in, too. Due to the orthogonality of the oltage ectors defined in eq. (), the total power deliered to the receier channels is the su of the power receied by the three odes. In case of a power isatch (i.e. * in ), the power deliered by eigenode is reduced by a factor of copared to the total power aailable fro that ode, where
5 GGin. (7) This is equialent to a reduction in gain of the receier channels with respect to this ode. Due to in utual coupling, the ode adittance substantially differs fro. If (theoretically) lossless two-port atching networks as shown in Fig. 6(a) are inserted between the antenna ports and the three receier channels, the ode adittances, and are transfored to new adittances ~, ~ and ~ ia ~ jp q, {,, } (8) jr with real alued paraeters p, q and r. Howeer, since the sae transforation applies for each of the three ode adittances, siultaneous atching of all odes cannot be achieed. If for exaple the adittance for ode is power-atched to the input adittance of the receier channels (i.e. ~ * ), the effectie gain of the receier channels with respect to the quasi-oni-directional pattern in of ode is axiized, but the effectie gain for the two higher-order patterns is significantly reduced. For the purpose of this discussion, the frequency-conerted and digitised output oltages of the receier channels are assued to be related to the input port oltages ia V out, n Vin, n. Fro a linear cobination of these output oltages, an unliited nuber of siultaneous beas can be fored by eans of a weighted su of these output oltages with the port-weights W W, W, ) [6]. The beaforer output oltage thus becoes (with + denoting transjungation) t ( W3 V bea W V in. (9) With ode-weights t w U W where U is defined in eq. (), this ay be transfored into a linear cobination of the input ode oltages in, : V bea w in. ()
6 Introducing eq. (6) and calculating in fro the circuit shown in Fig. 5, one finds the beaforer oltage excited by an incident spectru of hoogeneous plane waes E (, ) as V in * w,, inc (, ) Einc (, ) d. () bea 4Z G The effectie weights for a linear cobination of the eigenpatterns are thus gien by If the coplex alued coefficients ~ * w w. () are known, the weights can be adjusted in accordance with eq. () to account for utual coupling. This represents a easure of copensation for utual coupling ia digital signal processing, as discussed in detail in a ariety of papers [7-]. Howeer, this easure does not copensate for the SNR degradation due to utual coupling. The oltages produced by the waes receied by the antenna are superiposed onto noise oltages due to noise sources within the receier channels. The noise properties of each receier channel can be represented by a noise oltage and a noise current source (partially correlated to the noise oltage) at the input port of the receier channel, as shown in Fig. 5. The effectie receier noise teperature is a function of the source adittance. The iniu receier noise teperature T eff T eff, in is achieed when the source adittance equals an optiu alue opt G opt j opt. In the structure under consideration, three parallel receier chains are connected to the utually coupled antenna ports. Due to the utual orthogonality of the three different odes, the noise odel for each ode can be shown to be identical to the noise odel for a single receier channel. The effectie noise teperature for ode therefore becoes [] T eff, opt Teff, in T Req (3) G with T eff, in, opt and R eq representing the iniu noise teperature, optiu source adittance and equialent noise resistance of the three receier channels, and T denoting roo teperature. Fro eq. (3), it is clear that a deiation of the ode adittance fro the optiu source
7 adittance increases the effectie noise teperature. With these results, the gain-to-noise-teperature ratio can be forulated explicitly. With a certain set of weights dependent antenna gain function g (, ) and receier noise teperature becoes w ~, the ratio between the direction g(, ) Teff T eff,in,, w~,, w~ T R (, ) w~ eq,, G opt. (4) If all ode adittances are atched to the optiu source adittance opt, the second ter in the noinator anishes, resulting in the highest SNR. This optiu case can only be achieed if all ode adittances are identical, which necessitates the use of a decoupling network. Without a decoupling network, the ode adittances cannot be siultaneously atched to the optiu source adittance, and soe odes will be badly noise-atched. If these odes are needed for foring the desired radiation characteristic, the SNR will be reduced substantially. In contrast to the correction of the weights as shown aboe, this effect cannot be copensated for by eans of digital signal processing. 4. DEOUPLING OF NTENN PORTS In order to oercoe the probles described aboe, an RF decoupling-network (RF-DN) has to be introduced between the antenna ports and the receier channels. For the considered configuration of three onopoles, a noel and ery siple solution for this decoupling network is proposed. s seen fro Fig. 3, there are two alues F and F of the noralized frequency F f / f 4lf / c where the real parts of and are equal to each other. For a gien operational center frequency f ctr, the noralized frequency and thus the length l of the onopole can be adjusted to yield and therefore G f ) G ( f ) G ( f ) 4 f ctrl F F (5) c or F ( ctr 3 ctr ctr. With ( f ctr ) j ( f ctr ), the decoupling network reduces to reactie cross coupling between the adjacent antenna ports. Each coupling eleent has an adittance of j at the center frequency (see Fig. 6), thus yielding a 3-port adittance atrix
8 j j j DN j j j (6) j j j for the RF-DN. The adittance atrix of the array and the attached decoupling network, ~, then becoes ~ DN ( f ctr ). (7) ~ Each antenna port thus has an input adittance of j at the center frequency, irrespectie of the excitation of the other ports. It therefore becoes possible to siultaneously atch each port (and hence each ode) to the receier input ipedance, in or the optiu source ipedance, opt. This ay be accoplished in a straightforward anner using standard ipedance atching techniques. Receier channels are often internally noise-atched and possess a real alued input adittance in that coincides with the characteristic adittance of cables (e.g. /5 ). In such cases * opt in, so that optiu noise atching and power atching can be achieed siultaneously. For the exaple with a / and d / 4, the noralized frequency F. 9998, so that the length l of the onopole needs to be decreased by.% relatie to / 4 in order to yield ctr j. 5 at the center frequency. If, for exaple, the center frequency is chosen as f ctr.45 GHz, decoupling can be accoplished by connecting a capacitor of. 98 pf between adjacent antenna ports. With this siple decoupling network, ports to 3 becoe decoupled at f ctr. This results in a decoupled port ipedance of ~ / 4.3 j. 7, which ay easily be transfored into the desired load ipedance, e.g. Z in 5. The frequency bandwidth of decoupling and atching for this case is indicated in Fig. 7, which shows the frequency response of ~ S
9 (fraction of power reflected) and ~ S (fraction of power coupled to the loads at the other two ports). Note that the loss due to reflection and cross coupling is theoretically zero at the center frequency. If the frequency bandwidth is defined ia the requireent that 89 % of the power incident to the array is to be radiated (equialent with the VSWR< bandwidth for single antennas), a fractional bandwidth of.6% is obsered. If a wae is fed into port of the new structure in Fig. 6(b) while the other two ports are terinated in atched loads, a linear cobination of odes and is excited, resulting in a radiation pattern (, ) (, ) / 3, (, ) (7) with a axiu directiity of D.4. di. Fig. 8 depicts the noralised aziuth ax pattern for /. If port or 3 is excited, the radiation pattern is rotated about the z-axis by degrees. These radiation patterns are characterized by the fact that een in direction where the axiu directiity occurs, a partially destructie interference between the field contributions of the onopole currents occurs, which is a feature of a superdirectie array []. The onopole eleents together with the RF-DN for a resonator that proides the resonant current enhanceent needed for superdirectie radiation properties. 5. ONLUSION The potential probles associated with strong utual coupling between closely spaced array eleents hae been highlighted. It was shown that without a decoupling network it would be ipossible to atch all of the characteristic odes siultaneously. lthough the effects of ipedance isatch ay be copensated for in digital beaforing, the inability to copensate for the SNR degradation reains a proble that can only be soled by decoupling the array ports. For the array of three closely spaced onopoles, a noel yet siple decoupling network has been proposed. The decoupled array represents a superdirectie antenna. The drawback of superdirectiity is the frequency bandwidth liitation, which is related to the radiation quality factor Q rad of the higher-
order odes. Since Q rad increases with decreasing eleent spacing, the allowed iniu eleent spacing is goerned by the required fractional frequency bandwidth. Furtherore, dissipatie losses in the antenna and atching network structure (up to now neglected) need to be taken into account. The unloaded quality factor Q of atching and antenna structure ust sufficiently exceed the radiation quality factor of the high-order radiation pattern to ensure that the increased gain due to superdirectiity is not negated by a reduction in efficiency [-3]. REFERENES:. R.. Hansen, Phased array antennas, Wiley, New ork, 998.. R.. Hansen, Fundaental liitations in antennas, Proc. IEEE, Vol. 69, (98), 7-8. 3. H.J. haloupka, HTS antennas, in H. Weinstock and N. Nisenoff (eds.), Microwae superconductiity, Kluwer, Dordrecht,. 4. SuperNE Lite, Version., Poynting Software Pty. Ltd. 5. IE3D, Zeland Software Inc. 6. J. Lita and T.K.. Lo, Digital beaforing in wireless counications, rtech House, oston, 996. 7. H. Steyskal and J.S. Herd, Mutual coupling copensation in sall array antennas, IEEE Trans ntennas Propagat., Vol. 38, (99), 97-975. 8. R.S. de and T.K. Sarkar, opensation for the effects of utual coupling on the direct data doain adaptie algoriths, IEEE. Trans. ntennas Propagation, Vol. 48, (), 86-94. 9. P. Darwood, P. N. Fletcher, and G.S. Hilton, Mutual coupling copensation in sall planar array antennas, IEE Proc Microwaes, ntennas Propagation, Vol. 45, (998), -6.. I.J. Gupta and.. Ksienski, Effect of utual coupling on the perforance of adaptie arrays, IEEE Trans ntennas Propagat., 3 (983), 785-79... Friedlander and. J. Weiss, Direction finding in the presence of utual coupling, IEEE Trans ntennas Propagat., Vol. 39, (99), 73-84.. H.. Haus, Representation of noise in linear two-ports, Proc. IRE, (96), 69-74.
y a 3 x z d l + V + V + V 3 I I I 3 Fig.: onsidered 3-eleent array with onopole eleents (height l, diaeter d) and eleent spacing a.
. 9 6 Mode Mode Mode.5 5 3. 8.5 33. 4 3 7 Fig. : Noralized radiation patterns of the three eigenodes, with axiu directiities D 3. 4, D 7. 94 and D 7. 76. ( /, ) /,, D,
3.5.6.64.8 F F G, -.8 -.64 G G -.6.6.7.8.9.. F = f/f = 4f l/c Fig. 3: oputer siulation results for the frequency response of real and iaginary part of the input adittance for the low-order and of the higher-order odes. F and F indicate alues of the noralized frequency where the conductance G G.
4 8 Qrad 6 4.5..5..5 a / Fig.4: Dependence of the radiation quality factor of the high-order odes on the eleent spacing.
5 noise, + + Receier channel i i noise, in, in Fig. 5: Equialent circuit for the three odes for signal-to-noise considerations: Left: urrent source representing ode of the antenna array with {,, }. Right: Receier channel with equialent noise oltage noise, and noise current i noise,. V bea V in V out W (a) MN MN ' ' Rx Rx D D array 3 MN -port atching networks 3' Rx receier channels D nalog-to-digital onerters V in (b) j j MN MN ' ' j 3 array with decoupling network MN -port atching networks 3' Fig. 6: rray and beaforing syste (a) without and (b) with a decoupling network.
6.5.4 ~ S ~ S ~ S ij.3....94.96.98...4.6 F f / f 4 f l / c Fig. 7: oputer siulation results for the effectieness of decoupling and atching. Shown is the fraction of power reflected and power coupled to other ports as a function of frequency when feeding one port of an array with a /., d /. 5 and onopole length adjusted to l /.4995. Decoupling according to Fig. 6(b).
7 9. 6.5 5 3. 8.5 33. 4 7 3 Fig. 8: Noralized radiation pattern ( /, ) / D at center frequency f ctr obtained by feeding port of the 3-eleent array with attached decoupling network as shown in Fig. 6(b). Maxiu directiity is D. 4.