A review on array mutual coupling analysis



[1] An overview about mutual coupling analysis in antenna arrays is given. The relationships between array impedance matrix and embedded element patterns, including beam coupling factors, are reviewed while considering general-type antennas; approximations resulting from single-mode assumptions are pointed out. For regular arrays, a common Fourier-based formalism is employed, with the array scanning method as a key tool, to explain various phenomena and analysis methods. Relationships between finite and infinite arrays are described at the physical level, as well as from the point of view of numerical analysis, considering mainly the method of moments. Noise coupling is also briefly reviewed.

1. Introduction

[2] Dense antenna arrays have been a subject of intense research for about half a century [Hansen, 1966; Mailloux, 1965; Serracchioli and Levis, 1959; Stein, 1962]. Such arrays have been primarily considered for radar systems but they see increasing applications in the fields of communications [Oestges and Clerckx, 2007], near-field sensing [Gibbins et al., 2010], radiometry [Camps et al., 1998; Weissman and Le Vine, 1998] and radio astronomy [van Ardenne et al., 2009]. Also, it is interesting to notice that more recent fields of research also benefit from concepts and methods developed in the area of antenna arrays. This is the case of frequency-selective surfaces [Munk, 2000], as well as metamaterials (the Yagi antenna is sometimes regarded as the first metamaterial). Many applications consider very dense arrays, i.e., arrays with interelement distances smaller than the wavelength. For instance, to avoid grating lobes in regular antenna arrays, the interelement distance is generally kept smaller than half a wavelength. Irregular arrays [Manica et al., 2009] may mitigate those grating lobes, but when such arrays are limited to very few elements, a proper control of the level of sidelobes may require quite small average spacings [Bucci et al., 2010]. Consequently, in view of the scattering that takes place among the elements of the array, the behavior of an antenna in its array environment may be significantly different from that of the element taken in isolation.

[3] The goal of the present paper is to review concepts and methods developed for the analysis of mutual coupling. In view of the very wide literature on this subject, the task appears as a major challenge and an exhaustive treatment seems impossible. This is why we preferred to select a number of complementary aspects of mutual coupling, which either have known a fast development over the recent decade, or are key to the understanding of the limits of some models. Among those aspects, we shall note mutual coupling in irregular and regular arrays, beam coupling factors, fast finite-array analysis and noise coupling.

[4] The main goal being physical analysis tools, the vast subject of mutual coupling correction will be treated very succinctly. In the following, we will assume that the reader is familiar with fundamental electromagnetic concepts (equivalence principle, reciprocity, unicity, radiation integrals, …) as well as with fundamental array theory (embedded element pattern, array factor, grating lobes, visible region, array impedance matrix, etc.). Regarding the N-port coupling matrix, an impedance (rather than scattering) matrix approach will be adopted, because we believe this formulation is closer to field representations (this is mainly a matter of taste). Apart from a few comments on limit cases, we will avoid the single-mode approximation for current distributions on antennas. Indeed, although some aspects of mutual coupling can benefit from rich insights under that approximation [Wasylkiwskyj and Kahn, 1970; Capolino and Albani, 2009], (among which the asymptotic decay rate of mutual coupling [Galindo and Wu, 1968]), they can also be the source of some misunderstandings. Finally, when it comes to considering full-wave approaches, integral-equation methods will be preferred to differential-equation approaches, because physical interpretations are already possible at the level of periodic Green's functions and because those functions implicitly involve radiation conditions. Whenever possible, developments presented by different authors will be described under a common notation such that the advantages of different analysis methods can be clearly identified. Throughout the paper, a phasor notation will be adopted, with an implicit exp(j ω t) time dependence. Free-space impedance and wave number will be denoted by η and k°, respectively. The materials composing the array will be assumed linear and reciprocal.

[5] This paper is structured as follows. In section 2, generalities are provided about simple interpretations of mutual coupling. Starting from the definition of open-circuit patterns, we explain why descriptions based only on mutual coupling matrices, as often considered in the communications community, are bound to single-mode approximations. In section 3, we show that exact relationships however exist between the array N-port impedance matrix and the beam coupling factors, which are receiving increasing attention among the astronomic and radiometric communities. An original extension to the case of the unmatched array is provided. Section 4 defines basic quantities for infinite regular arrays. This is done primarily for linear arrays, with the help of Fourier methods relying on distributions theory, and the array scanning method is introduced. In section 5, in the mathematical framework recalled in section 4, two important relationships are given regarding the embedded element pattern in an infinite terminated array. Numerical methods for array analysis are reviewed in section 6, with special emphasis on integral-equation approaches and on good approximations based on macro basis functions (MBF). This approach allows one to dramatically reduce the number of effective degrees of freedom per antenna when large arrays of identical and complex elements are considered. The special case of eigenmodes on regular structures is shortly reviewed in section 7, while section 8 reviews a number of increasingly finer methods for the analysis of finite arrays, that exploit quantities provided by infinite-array calculations. Since ultrawideband arrays form an often extreme study case in terms of mutual coupling, for which most approximations fail, we found it useful to devote a specific part (section 9) to such arrays. Finally, over the recent decade, the issue of noise coupling in high-sensitivity arrays has received great attention; the basic analysis tools for such arrays are reviewed in section 10. Conclusions and further prospects are drawn in section 11.

2. Generalities About Mutual Coupling in Finite Arrays

[6] Two of the most important characteristics of an antenna are its input impedance and its radiation pattern, both being frequency-dependent quantities. Mutual coupling simply corresponds to the fact that, in view of the presence of another object in the vicinity of the antenna (a supporting element, a human body, another antenna…), its near-field configuration is different from the one found when the antenna is isolated in unbounded free space. As a result of the change in boundary conditions, new currents (or different equivalent currents) appear on the neighboring objects. The currents on the antenna itself, including its port current, are also modified. As a result, both the antenna radiation pattern and its input impedance change. Another way of looking at this problem is to see the neighboring objects as being part of the antenna (the “augmented” antenna, as named below), which hence must have new radiation characteristics. When the neighboring element is another antenna, even passively terminated, currents may flow through its termination, such that energy is being dissipated in it. This will, for instance, impose constraints on the efficiency of an antenna located in a dense array [Kahn, 1967].

[7] When considering an array with only one element excited, the neighboring elements may also be regarded as an extension of the excited antenna. The impedance of the resulting antenna is sometimes called the “passive” impedance (as opposed to the “active” impedance; see sections 4, 5 and 10) and its radiation pattern is called the “embedded element pattern.” A simple approximation to finite array effects on embedded patterns is presented by Kelley and Stutzman [1993], where a distinction is made between edge and core elements in the array. An N-element array can also be regarded as an N-port device with its own impedance matrix (or, equivalently, admittance or scattering matrix). The elements of the N-port scattering matrix are sometimes called the “coupling coefficients.” However, it is important to bear in mind that, except for special cases, the knowledge of the “coupling coefficients” does not provide a complete picture of the effects of mutual coupling. This means that, in general, the knowledge of the radiation pattern of the isolated element and of the N-port scattering matrix is not sufficient to obtain all the embedded element patterns. This will be made more clear below, when we will treat the limit case of single-mode antennas.

[8] The top row of Figure 1 shows the general configuration of an array, with Thevenin equivalent representations of the generators. The embedded element pattern (EEP) of element n, equation imagene(equation image), is defined here assuming a Thevenin equivalent source with unit voltage (v°,n = 1) and series impedance ZL,n, while the other elements are passively terminated. At distance R in direction equation image, the far electric field then reads:

equation image

In this section and in the next one, the antennas of the array may be considered different from each other. The loads terminating the elements may also be different and they may be modified, for instance when the array is connected to a new transmitter or receiver. Hence, it appears useful to provide a description of patterns in array conditions that would be independent from the array terminations. The (embedded) open-circuit patterns [Kahn, 1977; Camps et al., 1998; Wallace and Jensen, 2004; Warnick and Jensen, 2005] provide such a representation. It is clear that arrays are practically never operated in open-circuit form, however, as will be shown below, this classical concept helps to understand the assumptions underlying some often encountered approximations. Hence, it will be briefly reviewed here, by providing a short alternative proof.

Figure 1.

Array with (top) generators model and (bottom) open-circuit pattern model.

[9] The open-circuit pattern, equation imageno.c.(equation image), of a given element in an array is the radiation pattern obtained when the element of interest is excited with a unit current source, while all other elements are left open. The far field then reads:

equation image

where in is the imposed current flowing through port n.

[10] It is possible to prove that, once those patterns are known, along with the array impedance (or scattering) matrix Z (or S), the embedded element patterns can be obtained for any set of generator impedances. Assuming a shielded generator (for instance beneath an infinite ground plane) and a set of shielded transmission lines (waveguides, coaxial cables …) feeding all the elements, we may imagine a surface that wraps the generators and part of the feeding transmission lines and cuts the latter transversally, at a point where a unique transmission line mode may be defined (see four dashed boxes in Figure 1). From unicity, if all currents (or voltages) are known at the level of the cuts (horizontal arrows in Figure 1), then all the fields outside the surface are uniquely determined. Now, assuming a given set of generators, represented by their Thevenin equivalent circuits, it is easy to determine the voltages necessary to obtain a unit current through cut n and zero currents through the other cuts. As recalled above, for element n, this corresponds to the configuration from which open-circuit patterns are calculated (e.g., see current source on element n = 1 in bottom array of Figure 1). If we denote the port currents by en, i.e., a column vector with N entries, all equal to zero, except for a unit entry at element n, then the corresponding voltage sources of the generators must be:

equation image

where ZL is a diagonal matrix whose diagonal elements contain all the array terminations. Superimposing the embedded element patterns (defined with unit voltage sources) weighted by coefficients v°n, we obtain the (embedded) open-circuit pattern for element n:

equation image

where index p refers to a given polarization and superscript “T” stands for transposed. For a given direction, fn,po.c. is a scalar quantity, while fpe is an N-element column vector (one entry per antenna). Noticing that enT acts as a selection operator (in view of its definition, it selects line n) among lines of matrix (Z + ZL), the generalization to all open-circuit patterns in the direction of interest is immediate:

equation image

It should be noted here that all patterns are defined with a common reference point for the phase. Upon establishing (5) for two different sets of terminations, the corresponding sets of embedded element patterns can be linked to each other. On receive, for an incident plane wave, the voltages across terminals can be obtained from the above by reciprocity applied to the antenna of interest in its array environment (i.e., to the “augmented” antenna mentioned above, in the open-circuited or terminated array). When all elements are open-circuited, the voltage across terminals of antenna n, vno.c., is obtained from the corresponding open-circuit pattern equation imageno.c. (in vector form to include both polarizations) and the incident electric field equation image° as vno.c. = 4πequation image° · equation imageno.c./(j k°η). When all the elements of the array are terminated, the voltage across the load of antenna n is obtained as vne = 4π ZL,nequation image° · equation imagene/(j k°η). Hence, it immediately follows from (5) that:

equation image

which is consistent with open-circuit voltages defined for an N-port circuit fed from the outside [Gupta and Ksienski, 1983]. Equality (6) can also be immediately obtained from the equivalent Thevenin representation of an array on receive, recalled in detail by Maaskant [2010] and derived by de Hoop [1975] using Lorentz's reciprocity theorem. An alternative model for antennas on receive, based on the definition of a new impedance matrix, specific to receiving conditions, has been proposed by Hui [2007]. Although this new model has its own advantages (see below), contrary to Hui [2007], we think that (6) exactly matches the real situation in a receiving antenna array, independently from the type of elements (the controversy may arise from the difference between mutual impedances considered in equation (2) of Hui [2007] and in equation (4) of Gupta and Ksienski [1983]).

[11] An approximation of (5) is often found in the literature [Clerckx et al., 2007], although not always announced as such:

equation image

where fpis. corresponds to the isolated element patterns. It should be stressed that fpis.fpo.c. only holds for single-mode antennas, i.e., for antennas whose current distribution may be considered constant, within a multiplying factor. Indeed, under this strong assumption, open antennas do not support any current and may be considered invisible; the antennas are also named minimum scattering versus impedance parameters [Roscoe and Perrott, 1994]. Unfortunately, only a few antenna types satisfy this condition; among them are, in good approximation, thin dipole antennas, except when placed in very dense arrays (spacings less than about λ/4). When this approximation holds, it can be seen that the embedded element patterns can be obtained from the isolated element pattern, with the help of the array impedance matrix. This means, by reciprocity, that for an incident plane wave coming from a given direction, the received voltages in the uncoupled case can be obtained from voltages received in the real situation. Indeed, the former are proportional to fpis., while the latter are proportional to fpe, and those patterns are linked to each other through (7). Still under the single-mode assumption, from a signal-processing point of view, (Z + ZL) can be regarded as a coupling matrix [Gupta and Ksienski, 1983; Clerckx et al., 2007], whose effects are easily undone as soon as the array impedance matrix is known. Such coupling matrices can then be used to obtain direction-of-arrival estimates that account for the effects of mutual coupling [Filik and Tuncer, 2010].

[12] When antennas deviate from the minimum scattering model (e.g., wideband dipoles), they rapidly loose their minimum-scattering property, such that the decoupling procedure recalled above no longer holds. In this case, the knowledge of the embedded element patterns still allows one to undo the effects of mutual coupling, through an equalization procedure. However, in this case, the direction of the source needs to be determined. If several sources are present in different directions, a more global inversion procedure, that exploits the a priori knowledge of the embedded patterns, needs to be carried out. If embedded element patterns present dips in certain directions, the equalization procedure may be ill-conditioned and, for given levels of noise and intensity of incoming wave, estimated parameters can suffer from a larger error, assuming the best signal processing and inversion techniques. It is thus among the goals of the array design to avoid such dips (except for very special cases, where the array needs to be made insensitive in certain directions).

[13] Finally, for some class of antennas, a finer method for correction of the effects of mutual coupling [Hui, 2004] consists of still considering a single-mode current distribution, practically independently from the incidence angle, without however implicity assuming the same distribution when other loadings are considered. Here, one attempts to remove from the antenna of interest, denoted by index 1, the current induced by the fields radiated by the other antennas, denoted by indices n with n ≠ 1. Denoting by I1n the corresponding contribution to the port current on antenna 1, this operation entails removing from V1 all voltages V1n = I1nZL,1. For a given type of antenna, if the current distribution on the receiving antennas is always the same, then Z1nt = V1n/In only depends on the array configuration. For dipoles and other simple radiating structures, supporting nearly a single mode on receive, the new mutual impedances Zmnt (with mn) are almost independent from the incidence angle of the plane wave, and can be estimated numerically or experimentally [Lui and Hui, 2010]. It has been shown by Hui [2004], Wang et al. [2010], and Wu and Nie [2008], in the case of monopoles and dipoles, that the mutual coupling correction achieved in that way is much better than with the approach based on open-circuit voltages.

3. Beam Coupling Factors

[14] Another exact relationship between the embedded element patterns (EEPs) and the array impedance matrix can be established for lossless arrays. That type of links was first studied by Stein [1962]. Such relations were further studied for the case of just two elements by Ludwig [1976]. This approach, based on energy-conservation conditions, will be generalized below for the case of N elements. Expressions relating the EEPs with the array impedance or scattering matrix are useful for MIMO applications [Stjernman, 2005; Oestges and Clerckx, 2007] and for noise estimations in receiving arrays [Warnick et al., 2010]. An extension will be provided for arrays whose elements have different termination loads. It is interesting to notice that those relations are valid for any type of lossless arrays. This means that those relations also hold for arrays made of different elements, as well as for focal-plane arrays [Warnick and Jensen, 2005; Hayman et al., 2010; Brisken and Craeye, 2004], involving a large reflector, which transforms embedded element patterns into directive secondary patterns. In the latter application, beam coupling factors are used for the calculation of directivity and G/T ratios (gain over noise temperature).

[15] Assuming a common point serving as a reference for the phase, when computing the EEPs, the Beam Coupling Factor (BCF) between elements k and l is defined as:

equation image

where Ω denotes the unit sphere. The BCF can also be defined in terms of open-circuit patterns (OCPs):

equation image

[16] For antenna arrays where the element patterns cannot be approximated by a cosq(θ) variation [Angeletti et al., 2008], one has to resort, in a first instance, to numerical integration in order to obtain the BCFs. The numerical computation of (8) and (9) may require a large number of points to yield accurate results in complex cases.

[17] For lossless antennas, these quantities can be obtained from the array impedance or scattering matrix, thereby avoiding the numerical integration. In this respect, Ludwig [1976] provides a relation between the BCF and the S matrix for the case of two antennas, while Stjernman [2005] extends this expression for an arbitrary number of antennas terminated with identical loads ZL. Appendix A recalls how energy conservation leads to the link between the quantities referred to above. The resulting expression in terms of the array impedance matrix is [de Hoop, 1975]:

equation image

where B is a matrix containing all the BCFs, B(k, l) = bk,l, R is the real part of the array impedance matrix Z, and Y is an admittance matrix defined as (Z + ZL)−1. In case one considers BCFs between the open-circuit patterns we arrive to:

equation image

and is thus purely real. The equivalence between (11) and (10) can be established by noticing that Y links port currents i with source voltages v°. As shown by Ludwig [1976], the BCF between two embedded patterns is also real when only two identical elements with identical loads are in presence. Finally, the expression of B in terms of the array scattering matrix S is:

equation image

[18] Below is an original contribution which provides an extension to the nonmatched case (varying values of ZL,n). It can be handled if one just replaces the scattering matrix S in the previous equations by the generalized scattering matrix S′ [Collin, 2000], defined as:

equation image

where Γ is a N × N diagonal matrix whose diagonal entries are:

equation image

where Z° is an arbitrary real reference impedance and F is a N × N diagonal matrix whose diagonal entries are:

equation image

where Γn is defined in (14), and with the following definition for eequation image:

equation image

Then, we can also write b = Sa, where the elements of vector a are now

equation image

Defining a matrix C with the following entries:

equation image
equation image

[19] Then, the radiated power P = v°TB v° (see equation (A9)) becomes in the nonmatched case:

equation image

[20] Hence, considering arbitrary values of v°, we must have B = C.

[21] In the following example, BCFs are obtained in the two different ways described above: using (8), and using the definition (18) for the elements of the C matrix. Method of moments simulations have been used for that purpose. The problem under study corresponds to a large irregular array consisting of 60 antennas, randomly distributed over a square area. The radiating element chosen for this example is a broadband dipole with integrated balun [Edward and Rees, 1987]; it is shown in Figure 2. The antenna surface has been discretized with the help of 433 RWG [Rao et al., 1982] and 1 RWG-rectangular hybrid elementary basis functions. A delta gap generator is placed between the ground plane and the antenna foot to excite the structure. At the simulation wavelength λ, L = 0.175 λ, w = 0.055 λ and the height H above an infinite ground plane is 0.256 λ. The minimum spacing between the antennas center points is 0.44 λ, the furthest neighbor is at 0.66 λ and the average minimum spacing is 0.47 λ. The load impedance for each antenna in the array is defined as ZL,k = R + jX, where R is a random resistance uniformly distributed between 25 Ω and 75 Ω and X is a random reactance uniformly distributed between −25 Ω and 25 Ω. The normalized BCF bk1, between element 1 in Figure 3 and all other elements in the array, i.e., the first column of matrix B for this problem, is shown in Figure 4. The BCF is normalized such that bn,n, as defined in (8), is equal to the inverse of the directivity of an isolated element. The integral (8) has been carried out by sampling the EEPs every degree in both azimuth and elevation. The error between the integration and the generalized scattering matrix approach is computed in a relative sense, taking the BCFs given by the C matrix as a reference. A very good agreement can be observed when comparing BCFs computed with the generalized S matrix approach and BCFs obtained from numerical integration of EEPs.

Figure 2.

Geometry of the broadband dipole considered for the simulations.

Figure 3.

Positions for the broadband dipoles in the random array.

Figure 4.

Normalized beam coupling factors bk1.

4. Mutual Coupling in Infinite Arrays

[22] Fields in finite and infinite regular arrays have been a subject of intense research for quite a few decades. Among early compilations, one should note chapters 2–4 of Hansen [1966], written by Oliner and Malech. In the subsequent sections, we will try to explain several analysis methods that relate finite and infinite arrays in an intuitive way, by relying as much as possible on the field superposition principle. This will be naturally expressed in terms of Fourier-based relationships and will be limited here to one-dimensional arrays with period a along X. The extension to planar arrays is straightforward. The first part of this section will recall the necessary mathematical tools, based on Fourier transforms seen in the framework of distributions theory. At the same time, this reminder will also set the notation used in subsequent sections.

[23] Let us first define the following comb function:

equation image

which may be regarded as the expression of an infinite regular array of scalar point sources with spacing a and phase shift ψ between successive elements. Then, the following two fundamental results hold:

equation image
equation image

While (22) will be used below to obtain the solution for infinite arrays excited at one element only (array scanning method), the more commonly accepted equation (23) will be exploited to decompose infinite-array field solutions in terms of cylindrical waves for linear arrays and of in terms of plane waves for planar arrays (Floquet waves). As shown in Appendix B, given for completeness, the proofs of those identities are straightforward.

[24] Our starting point for the analysis of both finite and infinite arrays will be the current (or equivalent-current) distributions on an infinite array when only one element is excited. This allows the easy determination of coupling between ports and of the embedded element patterns. When a single element is excited, different current distributions are found on successive elements, with magnitudes which, most of the time, decay away from the excited element. This is schematically, i.e., without referring to the details of the individual current distributions, represented in Figure 5 (top) by a continuous line.

Figure 5.

Fields in infinite array. (top) One element excited. (bottom) All elements excited with constant amplitudes and linear phase progression. Solid lines schematically represent port currents. For a given excited element, different current distributions are created on successive antennas.

[25] Let us denote by equation images(x, y, z) the current density along the array axis X for excitation of element 0 only and by equation image(x, y, z, ψ) the currents obtained with constant-magnitude excitation and interelement phase shift ψ. To alleviate the notation, for the case of a linear array along X, the y and z dependences will be omitted. By simple superposition (see Figure 5, bottom), one obtains for the currents in infinite arrays with constant magnitudes and interelement phase shift ψ:

equation image

From there, the array scanning method [Munk and Burrell, 1979] is easily proven by convoluting both sides of (22) by equation images:

equation image

Coordinate x does not necessarily lie in the reference unit cell (0 < x < a) of the array and can be rewritten as x = x° + m a, where x° lies in the unit cell. From there, (25) can also be written as:

equation image

A similar expression allows one to derive the elements of the array scattering matrix from the active reflection coefficient over the reciprocal lattice [Galindo and Wu, 1968].

[26] The infinite-array solution can also be written as:

equation image

where equation image°(x, ψ) = equation image(x, ψ) for x within the unit cell and equation image°(x, ψ) = 0 for x outside the unit cell. In this case, the infinite-array solution is just regarded as the repetition of its value within the unit cell, with the proper phase shift. Using the convolution theorem, the right-hand side can now be written as the inverse Fourier transform (see choice for Fourier transform pair in Appendix B) of the product between two Fourier transforms. Hence, with the help of (23):

equation image
equation image


equation image


equation image
equation image

Expression (29) is a Fourier-series decomposition of the infinite-array current along the direction of space in which the structure is periodic.

[27] Each term of (29) is responsible for radiation of a given cylindrical wave (“Floquet mode”). Below, we shortly provide expressions for the radiated magnetic vector potential, from which electric and magnetic fields can be easily derived. The vector potential is obtained from the convolution of currents over the unit cell with the scalar Green's function of the infinite linear array, which may be written as [Mailloux, 1982]:

equation image

where ρ = equation image with equation image = (x, y, z) the coordinates of observation point and equation image′ = (x′, y′, z′) those of the source points. The wave number kρ,p is such that kρ,p2 + kp2 = k°2. Then, denoting by μ the free-space permeability, the vector potential can be written as:

equation image

Inserting (33) into (34), interchanging summations with integration along x′ and using (32) we obtain:

equation image
equation image

where (vy, vz) is a unit vector in the Y Z plane, in the projected direction of observation and R = equation image. Equation (36) provides a far field approximation in which equation image corresponds to the values of p leading to main or grating lobes (∣kp∣ < k°). Hence, an explicit link between every cylindrical wave and every spectral component is now established. Similarly, a double spectral series can be established for the plane wave decomposition of fields radiated by planar arrays.

5. Active Element Pattern and Array Scanning Method

[28] The active element pattern is the embedded element pattern for an antenna in an infinite array. As mentioned by Hansen [1998], the name “active” may be a little misleading and the expression “scan element pattern” may be more appropriate. Here, the far-field radiation integral can be expressed as:

equation image

where R is the distance between antenna and observer located in direction equation image. Expressing the summation explicitly over successive cells, with equation image′ = equation image°n a âx, we obtain:

equation image

with K(R) = μ exp(−jk°R)/(4πR). Interchanging the orders of summation and integration, we obtain:

equation image

where, with the help of (24), the sum between parentheses is identified as the infinite-array current estimated for an interelement phase shift ψ = k°a ux that matches the direction in which the array is scanned. Therefore, the scan element pattern can be obtained simply from the usual radiation integrals taken over the unit cell only, provided that the considered current be the infinite-array current for the array scanned in the direction of interest. An alternative proof is obtained from the ASM. If in (38)equation images(x′ = x°n a) is replaced by its expression (26), we obtain, after interchanging the orders of summation and integration:

equation image
equation image

in which, as can be delineated from (B2) and (B3), the sum of exponentials can be rewritten as 2πp=−∞δ(ψ′ − kpa) where kp is given by (30). Only for p = 0 does ψp = kpa fall inside the [0, 2π] interval. From there, we obtain:

equation image

which is equivalent to (39).

[29] Another important relationship regarding infinite arrays is closely connected; it concerns the equivalent circuit on receive and the subsequent value of the receiving cross section for an antenna embedded in an infinite array. The result is linked to the active antenna impedance, i.e., the port voltage-to-current ratio for the array scanned in a given direction. A general proof, i.e., avoiding single-mode assumptions, is given by Craeye and Arts [2004], based on the reciprocity approach presented by de Hoop and de Jong [1974]. First, we consider an infinite linear array of antennas terminated by loads ZL and an incident plane wave with amplitude E°b and polarization vector equation imageb coming from direction −equation image. On the other hand, the infinite array is assumed transmitting, scanned in direction equation image. The interelement distance a is such that there are no grating lobes. Based on Lorentz reciprocity integrals evaluated over the unit cell (excluding the generators), a model for the receiving antenna is obtained. The equivalent circuit in Thevenin form has a series impedance equal to the active impedance of the array and the following open-circuit voltage:

equation image

where E°aequation imagea is a normalized field radiated by the array scanned in direction equation image. The normalization is such that the total far field reads:

equation image

where Ia is the port current on transmit. From there, the power dissipated on the load ZL can be computed. Its ratio with the incident power density gives the receiving cross section:

equation image

where Γa is the active reflection coefficient, between impedance ZL on one side and scan-angle dependent active input impedance of the fully active array, on the other side. Dθ(ϕ) is the azimuthal directivity of the radiation pattern of the infinite array, which is contained in an infinitely thin cone (see Figure 6) forming an angle θ with the X axis, such that k°a cos θ = ψ. That pattern corresponds to a cut between the embedded element pattern and the cone (shaded area).

Figure 6.

Radiation cone and pattern of infinite linear array.

[30] For planar arrays, a similar approach leads, still in the absence of grating lobes, to a better known result [Hannan, 1964; Pozar, 1994]:

equation image

where a and b are the interelement distances along x and y, respectively, and Γa is again the active reflection coefficient. This result is valid only for arrays radiating in one half plane only (e.g., array backed by an infinite conducting ground plane). Extensions to grating-lobe conditions can be found in the work of Craeye and Arts [2004].

[31] By reciprocity, expression (46) indicates important constraints imposed by mutual coupling on the embedded element pattern of an antenna located in dense arrays. Indeed, the effective area is now constrained by the area alotted by the element in the array, which may be much smaller than the effective area of the element taken in isolation. This effect becomes very strong when the spacings are smaller than half a wavelength, as derived by Kahn [1967], who obtained a πab/λ2 maximum efficiency under those conditions for a square lattice (here, the mismatch factor at the excited element is included in the efficiency). This more recently led to the observation of efficiency limitations for wideband focal-plane arrays [Ivashina et al., 2009], which tend to be very dense to produce overlapping secondary beams over the whole frequency range.

6. Numerical Methods

[32] In this section, we provide a brief overview of the numerical methods that can be used to simulate finite and infinite antenna arrays, with a special emphasis on integral-equation methods. Basically, finite (regular or not) antenna arrays may be analyzed with any of the numerical methods used in high-frequency electromagnetics, i.e., for the full-wave analysis of complex objects from about 0.1 to 100 wavelengths. Those methods are classified according to two criteria: along one axis, differential-equation versus integral-equation; along the other axis, time domain versus frequency domain. Among differential-equation approaches, one finds the FDTD method [Veysoglu et al., 1993; Holter and Steyskal, 1999; Yu and Mittra, 2003; Taflove and Hagness, 2000] and the FEM (finite elements method) [Jin, 1993; Bossavit, 1998; Davidson, 2003]. In those cases, the whole computational volume needs to be discretized, and absorbing boundary conditions are necessary to account for radiation. The main advantages of those methods is that they can handle complex media, like nonuniform dielectrics or the combination of many different materials and, above all, that the obtained systems of equations are very sparse, and can hence be solved with very fast specialized methods. One should also note the FIT (finite integration technique) [Weiland, 1977], on which the CST Microwave Studio® software is mainly based. This method is another form of differential-equation method and is sometimes compared with the FDTD method, although it has some fundamental differences, since it is based on the verification of several integral relations defined over elementary cells of the mesh [Schuhmann and Weiland, 2001]. The finite difference method is mainly defined in time domain, although frequency domain forms are also possible. The FEM is generally more applied in frequency domain, but time domain implementations are also possible.

[33] The other important class of methods is the integral-equations (IE) method [Harrington, 1967], generally implemented in the form of the method of moments in frequency domain, although time domain approaches are also subject of intense research [Weile et al., 2004]. Here, with the help of the surface-equivalence approach, unknowns are only defined on the interfaces between piecewise homogeneous media. When zero-thickness metal sheets (either closed or open) are considered in a homogeneous medium, the equivalent currents looked for correspond to the sum of physical electric currents on both sides of the sheets. When penetrable (magnetic or dielectric) materials are considered, the equivalent currents are obtained from both tangential electric and magnetic fields [Harrington, 1961]. When the supporting materials have canonic shapes (e.g., planar, cylindrical, spherical stratified media), their presence can be accounted for through the use of appropriate Green's functions [Tai, 1971]. This probably remains the most successful way of analyzing microstrip antennas [Pozar, 1982; Skrivervik and Mosig, 1993; Vandenbosch and Demuynck, 1998], for which the boundary conditions on the metallic patches are enforced with the help of the electric field integral equation, while having unknowns defined only on the metalized surfaces. Another important advantage of integral-equation approaches is that the Green's function implicitly entails radiation. Unfortunately, integral-equation methods generally lead to quite dense systems of equations. An exception to this drawback appears when a complex object is entirely contained inside another one or inside a fictitious surface [van de Water et al., 2005]; this however does not immediately alleviate the challenge of solution for large open objects. The dense systems of equations a priori lead, for large problems (more than a few thousands of unknowns) to a complexity that grows with the third power of the number of surface unknowns. Nevertheless, those methods could be dramatically accelerated, with the combined help of Krylov-subspace based iterative solution techniques, like GMRES [Saad and Schultz, 1986; Poirier et al., 1998], and of specialized techniques for fast matrix-vector multiplication (see below). Proper normalization [Yla-Oijala and Taskinen, 2007] and use of good preconditioners [Bagci et al., 2009] may also strongly reduce the number of iterations.

[34] In view of the above, it is difficult to say which method is most efficient in general. Also, for some types of problems, combinations of such methods may prove very efficient, like the combination of finite element and integral-equation approaches [Eibert et al., 1999]. For instance, in the works of Vouvakis et al. [2004] and Lee et al. [2005], finite element approaches are used to treat the complex feeding details of the elements, while integral-equation approaches are used to analyze mutual coupling between elements without resorting to absorbing boundary conditions. Besides, domain decomposition [Stupfel, 1996; Vouvakis et al., 2006; Zhao et al., 2007] appears as a powerful tool for the analysis of complex structures, in which the meshes of contiguous regions do not necessarily need to be overlapping. Finite regular arrays of complex antennas with impressive dimensions are solved by Peng and Lee [2010], with a combination of finite element and boundary-integral methods. An elegant solution combining FEM for interior problems and spherical mode expansions for the array mutual coupling is also presented by Rubio et al. [2005].

[35] In the following, we will provide more details about the method of moments (MOM). In a nutshell, the unknown surface currents distributions (either electric or, equivalent, magnetic) are discretized into a number of elementary basis functions, defined over domains that are in general overlapping each other. Taking the example of an electric current distribution, equation image(equation image′), we can write:

equation image

where the equation imagei(equation image′) are known basis functions, while the coefficients xi need to be determined. This is carried out by imposing that, on the interfaces between homogeneous media, the boundary conditions expressed in terms of tangential fields are satisfied. This is generally imposed in weak form, i.e., the integral equation is weighted with a set of testing functions. The Galerkin testing procedure consists of having the sets of basis and testing functions correspond to each other. It is interesting to notice that, besides leading to generally better posed systems of equations, this methodology also exactly ensures reciprocity in antenna problems (this, on its own, is of course not a guarantee for accuracy). Probably the most popular basis functions are the Rao-Wilton-Glisson [Rao et al., 1982] ones, although a very vast class of basis functions have been defined. For detailed descriptions of possible basis functions and for the establishment of the system of equations, using the EFIE and MFIE (electric and magnetic field integral equations) and combinations thereof, we refer the reader to a number of classical papers and books [Rao and Wilton, 1990; Harrington, 1993; Peterson et al., 1998; Wilton, 2002; Gibson, 2008; Kolundzija and Djordjevic, 2002]. We will just keep in mind that the unknowns xi are obtained from the linear system of equations Z x = v, where Z is called the MOM impedance matrix when the EFIE is used. Analytical solutions for entries of the MOM impedance matrix (or admittance matrix when slots are treated), obtained with spatial or spectral domain approaches, are found for several canonical elements in sections 7.3 and 7.4 of Hansen [1998]. Vector v is the excitation vector, whose entries correspond to the projection of incident fields on the different testing functions. The incident fields may be a plane wave, fields radiated by another antenna or fields impressed by a generator, the simplest representation of which is the delta-gap generator (voltage discontinuity between two infinitely close conductors).

[36] A very large number of examples of array problems treated with the MOM is available in the literature, such that it does not sound realistic to give a fair overview of them. A priori, for those methods to be accurate, a sufficiently fine discretization should be considered on each element of the array, with a special attention given to the representation of the feed region. Involving part of the feeding transmission line in the antenna model is a good way of improving accuracy. This rapidly leads to a prohibitive number of unknowns, such that simplifications and accelerations need to be found. In the case of regular arrays, important time savings can be obtained by exploiting the periodicity of the structure. First, the number of different impedance matrix blocks that need to be computed is limited, since the interactions between basis and testing functions only depend on the vectorial distance between elements. Contrary to the case of irregular arrays, the number of different blocks to be computed grows only linearly with the number of cells in the regular array. Second, in iterative schemes, block-diagonal preconditioners [Ubeda et al., 2006], with blocks corresponding to self-cell interactions, form a relatively easy-to-implement and efficient option. Also, the block-Toeplitz structure of the MOM impedance matrix allows fast FFT-based matrix-vector multiplications [Fasenfest et al., 2004; Janpugdee et al., 2006]. An alternative option is the use of fast multipole approaches [Chew et al., 2001]. Finally, interactions between distant blocks can be obtained at lower computational cost, by exploiting the rank deficiency of the corresponding block, through the use of the Adaptive Cross Approximation (ACA) [Bebendorf and Rjasanow, 2003] or incomplete QR [Ozdemir and Lee, 2004] methods. The comments made above regarding preconditioning and matrix compression also hold for nonregular arrays.

[37] An important problem in the analysis of finite arrays with iterative techniques is that, in order to provide a complete description, i.e., the full array impedance matrix and all embedded element patterns, it is necessary to run a new series of iterations to obtain a solution for excitation at each individual port. This may rapidly become prohibitive. This is why noniterative methods, based on a reduction of the effective number of unknowns, have been developed over the last decade [Ooms and De Zutter, 1998; Suter and Mosig, 2000; Prakash and Mittra, 2003; Lu et al., 2005; Matekovits et al., 2007]. Those methods consist of “aggregating” basis functions into relatively small sets defined over every unit cell of the array. They build on the limited spatial bandwidth of fields radiated or scattered by finite-size objects [Bucci and Franceschetti, 1987]. The aggregated basis functions are sometimes named macro basis functions (MBFs). In some of their variants, MBFs are also named CBFs: characteristic basis functions. The new basis is generally made of solutions for smaller problems, limited to a unit cell, or to a few such cells. The corresponding excitations are of four different types: (1) multiple-scattering solutions [Yeo et al., 2003], with a primary MBF resulting from port excitation of a single element, and secondary MBFs obtained from the fields radiated by the primary MBF (higher-order solutions have also been investigated [Hay and O'Sullivan, 2008; Craeye et al., 2009a]), (2) excitation by a wide spectrum of plane waves [Maaskant et al., 2008], (3) by elementary sources placed all around the antenna of interest and disposed over an equivalence surface [Crocco et al., 2007; Matekovits et al., 2007] and (4) for regular structures, solutions obtained with the array scanning method [Craeye and Sarkis, 2008]. More details about the latter approach are provided in section 8. MBF-type basis functions, specific to certain types of canonic radiators and transitions, have also been extracted from spectral representations in the works of Neto et al. [2003] and Bruni et al. [2005]. So far, to set the notations, we just summarize the formulation of the reduced system of equations. On every block i, the solution xi is approximated by:

equation image

where each column of Q corresponds to an MBF, i.e., to a solution, in terms of elementary basis functions, of one of the smaller problems referred to above and vector yi corresponds to the new list of unknowns. The main expectation is that the number of columns of Q, i.e., the number of MBFs, be much smaller than the number of unknowns on the unit cell, which is often the case by 1–2 orders of magnitude. In the same way as macro basis functions are defined, macro testing functions are determined. Again, a Galerkin approach is generally considered. This allows one to reduce each block of the MOM impedance matrix according to:

equation image

and the segments of the excitation vector corresponding to different antennas are reduced in the following way:

equation image

It should be noted that the reduction procedure in (49) is generally computationally very expensive and that faster methods, based on multipole decompositions [Coifman et al., 1993; Craeye, 2006] and on the ACA [Bebendorf and Rjasanow, 2003; Zhao et al., 2005; Maaskant et al., 2008] have been developed to dramatically reduce the computation time of the reduced system of equations, without computing explicitly all the blocks of the MOM impedance matrix. For the same purpose, an interpolatory approach is developed by González-Ovejero and Craeye [2011] for nonregular arrays. While linear independence between MBFs may be ensured through the use of the SVD [Matekovits et al., 2001], the completeness of the MBF basis remains a subject of research. The method based on the equivalence surface in principle answers this question but it is difficult to apply when unit cells are electrically interconnected and it may be relatively expensive in terms of number of unknowns. This discussion will be continued in section 8, where it is shown that the ASM approach is particularly well suited to the introduction of physical bases for the current distributions in regular arrays. Finally, some insight is provided by Craeye [2009] regarding the completeness of bases formed in a multiple-scattering fashion.

[38] For sufficiently large arrays [Holter and Steyskal, 2002], useful insight into the behavior of the elements may be achieved with the help of infinite-array approaches. Again, while finite element (e.g., with the HFSS® commercial software) and FDTD [Holter and Steyskal, 1999] approaches have been developed for this configuration, we will mainly focus on integral-equation approaches. Here, the main difference with respect to finite-array approaches lies in the use of periodic Green's functions, the convergence of which is also subject of intensive research. Only the main points will be covered here; the reader is referred to the review chapter [Craeye et al., 2009b] for more details and advanced techniques. The infinite-array solution can be obtained with the MOM by simply replacing the single-source scalar Green's function Gs(R) = exp(−jk°R)/(4πR), where R is the distance between source point and observation point, by an infinite series. This series involves the same type of terms, each term being related to a “copy” of the source in other cells of the array, and multiplied by a phase factor that takes into account the linear phase progression along the array. This explicit space-domain summation converges very poorly. It can be replaced by a spectral-domain series, each term of which corresponds to a cylindrical wave for the case of the linear array and to a plane wave in the case of a planar array. The former has already been provided above (33) and the latter can be expressed as:

equation image

where a and b are the periods along X and Y, respectively, while ψx and ψy are the corresponding interelement phase shifts, kp = ψx/a + p2π/a, kq = ψy/b + q2π/b and kpq2 = k°2kp2kq2. From the expression of kpq, it can be delineated that only a few values will be real (kpq2 > 0), leading to propagating plane waves, while all the others are evanescent. For very large values of p or q, the exponential factor versus z can be approximated by exp(−2πequation imagezz′∣), which means that the convergence versus p and q will be slower for small values of ∣zz′∣. This is generally referred to as the “on-plane” convergence problem [Jorgenson and Mittra, 1990]. Similarly, an “online” convergence difficulty appears for the linear-array Green's function (33). In this case, other techniques need to be developed. Among them, we note the use of formulas for series accelerations, like the Shanks formula and the Levin-T method [Singh and Singh, 1993]. Several methods are based on a combination of space-domain and spectral-domain approaches, like the Ewald method [Ewald, 1921; Capolino et al., 2005a; Stevanovic and Mosig, 2007; Valerio et al., 2007]. Two-dimensional arrays of point sources can also be viewed as a subset of 3-D arrays, which led Silveirinha and Fernandes [2005] to find a method with exponential convergence. For the planar-array case, another method [Craeye et al., 2003; Craeye and Capolino, 2006; Guérin et al., 2009] consists of regarding the planar array as an array of linear arrays, for which formula (33) is used and where the space-domain summation is accelerated with the help of the Levin-T formula applied to each spectral term separately. For this method, as well as for the Ewald method, a rapid exponential convergence is achieved for all positions of the observation point with respect to the unit cell. A detailed review regarding periodic Green's functions can be found in the work of Craeye et al. [2009b]. Special attention needs to be given to arrays in which elements are interconnected. For metallic interconnections, it is sufficient to create basis functions on the edge of the unit cell that perfectly overlap those in the next cell (or, equivalently, on the opposite side of the same cell). For penetrable objects, when there are no interconnections between cells, the Green's function used inside the surface that bounds the object should be the Green's function of a single source in a homogeneous medium. When there are interconnections, two strategies can be adopted. The first one consists of explicitly imposing the continuity of fields (within the proper phase shifts) at intersections between cell boundaries and dielectric material [Usner et al., 2007]. This in general also requires a careful treatment of the basis and testing functions located at the junction [Yla-Oijala et al., 2005] between connected regions. This is not necessary when the medium inside the penetrable object is itself considered infinite [Dardenne and Craeye, 2008]. For instance, for arrays involving infinite dielectric slabs parallel to each other (e.g., in the case of planar single-polarized arrays of tapered-slot antennas), inside the dielectric slab, the structure may be considered infinite in one direction. This can be treated by using the Green's function of a linear array inside the dielectric material (33), while the Green's function used outside is the doubly periodic Green's function in free space (51).

7. Eigenmodes, Surface Waves, and Leaky Waves

[39] Eigenmodes correspond to field distributions than can, in principle, exist while there is no excitation. From a MOM point of view, such modes correspond to solutions to the homogeneous system of equations Zx = 0, where Z is the MOM impedance matrix for the periodic medium. In other words, this will occur when at least one of the eigenvalues is zero. The corresponding eigenvector then represents the current distribution in terms of elementary basis functions. Strictly speaking, this situation only occurs for lossless and infinite structures, for which energy does not get dissipated nor radiated. For slightly lossy infinite arrays, eigenmodes may exist for complex frequencies, the inverse imaginary part of the frequency giving an idea of the lifetime of eigen fields once the excitation is interrupted. Under those more realistic conditions, where some eigenvalue may be very small, but not exactly equal to zero, eigenmodes will be excited as soon as there is a nonzero projection between the incident fields on a field distribution associated to the eigen fields. This becomes explicit by writing the solution of the Zx = b system of equations in terms of eigenvectors of Z.

equation image

where the λi are the eigenvalues of Z and the vi are the corresponding eigenvectors arranged according to the columns of a matrix V, while b is the excitation vector. The wi are the columns of the associated matrix W = (VH)−1. From there, it becomes clear that the excitation of eigenfields represented by vi, corresponding to a very small eigenvalue (a “quasi-eigenmode”) will be very effective if the excitation has a high projection on the associated fields, represented by wi. Since the MOM impedance matrix is symmetric but not hermitian, viwi. However, our numerical experiments showed that, for the smallest eigenvalues, vi and wi often correspond to very similar current distributions. This point requires further investigation.

[40] The analysis of eigenmodes can be carried out through the determination of the dispersion surfaces of the periodic structure, defined as follows. For a doubly periodic array, we may define kx = ψx/a and ky = ψy/b, where ψx and ψy are the interelement phase shifts along X and Y, respectively. The dispersion surfaces correspond to surfaces in the three-dimensional (ω, kx, ky) space for which eigenmodes are obtained. In the discretized MOM formulation, this means that ∣Z(ω, kx, ky)∣ = 0. Such a field solution is also called a Block mode, while the expression “first Brillouin zone” is borrowed from the field of solid-state physics to denote the domain defined by 0 < kx < 2π/a and 0 < ky < 2π/b. For some frequencies, there may be no solution, whatever is equation image = (kx, ky); the periodic structure then has a bandgap. Mutual coupling between two antennas can be reduced by introducing between them a periodic medium with a bandgap in the frequency band of interest [Gonzalo et al., 1999].

[41] When there are eigenmodes, their phase velocity is given by equation imageph = equation imageω/∥k2, while the group velocity is ∇kω. For lossless structures, the eigenmodes, or “trapped waves” must have a phase velocity slower than speed of light, such that they do not radiate into visible space. For an array scanned in a given direction, slow waves may be excited although the array is scanned into visible space: although the phase velocity of trapped waves is slow, the excitation fields may have a phase progression that matches that of the eigenmode fields. This may be regarded as an aliasing phenomenon (see Figure 7 for illustration in the case of a linear array): the condition to be met is that, at the level of the excitation points of successive antennas, the phase of the surface wave is consistent with that of the excitation of the antennas. In the planar array case, the condition reads:

equation image
equation image

where superscript “e” denotes the eigenmodes, p and q are integers. The left-hand sides correspond to the interelement phase shifts of the eigenmodes, while the right-hand sides correspond, within multiple increments of 2π, to the phase shifts for radiation in direction equation image = (ux, uy, uz). When eigenmodes are isotropic, (kxe)2 + (kye)2 = β2 with β > k. From there, and with the help of (53) and (54), it is readily obtained that, in the (kx, ky) plane, eigenmode conditions are satisfied on circles centered on 2π(p/a, q/b) and of radius β. Whenever portions of those circles enter the visible region (kx2 + ky2 < k°2), their position in the plane corresponds to a scan angle for which the eigenmode can be excited. The reader is referred to Pozar and Schaubert [1984] for a graphical representation in the wave number space. In that reference, (kx, ky) is denoted by (u, v)k°. Surface waves corresponding to waves trapped on a grounded dielectric slab are isotropic; however, the presence of printed antennas on the dielectric slab will distort the iso-frequency contours of the dispersion surfaces. Llombart et al. [2005] show the dispersion diagrams of dielectric slabs (grounded or not) loaded by strip gratings and associated bandgaps. Reference Enoch et al. [2003] is recommended for further reading on dispersion surfaces and their interpretation. The excitation of eigenmodes can also occur when a single element is excited. Indeed, as shown by the ASM (26), fields obtained under those conditions can be viewed as a spectral superposition of periodic-excitation fields for all possible scan angles. In fully excited infinite arrays, we may say that the eigen waves excited by different elements add up in phase when the conditions (53) and (54) are satisfied, while they cancel each other when these conditions are not met. It should be noted that, in finite arrays, this cancelation may be incomplete, which leads to the onset of eigenmodes even when the scan angle does not match the conditions above. When the scan angle meets the eigenmode condition, the infinite-array active impedance (with electric-type “dipoles” in the very broad sense of the word) tends toward zero, which leads to a major matching problem for that particular scan angle. The array is then said to have a “blind spot” in that particular direction.

Figure 7.

Slow-wave propagation and coherent excitation of that wave by an array scanned into visible space.

[42] Blind spots have been observed in waveguide-fed slots covered by a dielectric layer; the role of surface waves and of leaky waves in such cases is described in more detail by Knittel et al. [1968]. Surface waves cannot be excited by incident plane waves in infinite arrays; an argument based on reciprocity is given for this by Munk [2003, section 4.4], where it first observed that the infinite array affected by the surface wave does not radiate. On the contrary, finite arrays supporting surface waves can radiate, as explained by Janning and Munk [2002], which in turn explains why such waves can be excited by a plane wave incident on a finite array.

[43] In the work of Çivi and Pathak [2006], surface waves are analyzed numerically and asymptotically, as well as through a Fourier analysis of currents over a dipole array, for both free-space arrays and arrays on grounded dielectric slabs.

[44] While eigenmodes often represent a problem for antenna arrays, they may also be exploited to create materials with new properties, like a negative refraction index [Enoch et al., 2003]. In that case, energy is carried by the eigenmodes in a direction (group velocity direction) that has a negative projection on the phase velocity vector (backward waves). However, an important challenge for negative refraction is the strong spatial dispersion (dependence of effective parameters on kx and ky) of structured materials. Another interesting application of eigenmode analysis corresponds to leaky-wave radiation from periodic structures. In this case, the array scanning method is again very useful to the analysis of the periodic structure excited by just one source [Capolino et al., 2005b, 2007]. Following Oliner [1993], leaky wave antennas can be defined as waveguiding structures that can leak power along their length. A first category of periodic leaky-wave antennas corresponds to periodic modulations in a guiding structure supporting a slow wave. What will make the structure leaky is that some space harmonics (or “Floquet modes,” see equations (29), (30) and (36)) created by the periodic modulation may correspond to fast waves, which will hence radiate. In general, the structure is designed such that only the p = −1 harmonic radiates. More precisely, if the wave number of the slow guided wave is denoted by ψg/a = βg > k°, then, the wave number of the p = −1 space harmonic is βl = βg + p2π/a = βg − 2π/a, which may lie between −k° and k° and hence correspond to a fast wave, with backward radiation for βl < 0 and forward radiation else. In the latter case, the forward radiation angle with respect to broadside may be limited by the appearance of a second (p = −2), usually undesired, fast harmonic. Obviously, mutual coupling between successive cells of a periodic leaky-wave antenna plays a major role in the radiation mechanism.

[45] As the space harmonic radiates, its amplitude decreases exponentially along the leaky transmission line with an attenuation constant equal to α Neper/m. The attenuation constant may have a behavior difficult to control when β is very small (antenna scanned near broadside), which leads to matching difficulties. This is referred to as the open stopband effect [Jackson et al., 2005]. The value of ∣β∣ determines the scan angle (far from broadside for ∣β∣ large), while α determines the width of the radiation pattern (broad for α large). For a localized source that launches leaky waves in two opposite directions, α < ∣β∣ will lead to the appearance of two distinct patterns pointing oppositely with respect to broadside. When αβ, broadside radiation occurs, since the width of the two patterns is about as large as their tilt angle, such that they overlap to form only one main lobe. The leaky wave along the structure may be regarded as exhibiting a complex interelement phase shift ψx = (β)/b. This needs to be taken into account when evaluating the corresponding periodic Green's functions [Paulotto et al., 2010]. The excitation of such structures with isolated or nonperiodic sources may be also studied with the array scanning method, for which the integration contour may be extended to the complex ψx plane [Capolino et al., 2005b]. Finally, more recently, leaky-wave antennas have been designed, based on the alternation of left- and right-handed sections of transmission lines [Casares-Miranda et al., 2006]. The latter offer full scanning capability in their fundamental harmonic.

8. Finite-Array Analysis: From Infinite to Finite

[46] In a first instance, fields in a finite array may be regarded as the superposition of fields due to excitation at every single element and “scattered” by the edges of the array. A first approach for finite-array analysis [Roederer, 1971] is equivalent to superimposing results from the ASM considering the excitation coefficient sn of each element. Denoting by f any current or field component, with the help of (26), we obtain:

equation image

with the following weighting function:

equation image

In the particular case where the array is scanned with constant magnitudes in a given direction, corresponding to an interelement phase shift equal to ψs, the weighting function can be written as:

equation image

where SN(x) equation image sin(N x)/sin(x). SN is a periodic function of period 2π and the width of its main lobe is inversely proportional to N. Hence, for smaller arrays, (small value of N), the weighting function W is broad, such that solutions for the array scanned in directions other than ψs play a more significant role. This method, however, does not take into account scattering by the edges of the array, since the complementary parts of the infinite array, although passive, are still present. The method presented by Ishimaru et al. [1985], when applied to periodic arrays, may be regarded as a special case of (55) for current-source excitations (“forced” excitation) and for single-mode currents on the antennas. Under those conditions, in the linear-array case, the voltage at port n can be written as:

equation image

where V is the infinite-array solution at port n and W(ψ) is the windowing function (56) where the sn now correspond to the coefficients of the current-source excitations. A one-to-one link can readily be established between (58) and equation (15) of Ishimaru et al. [1985]. Starting from Ishimaru et al. [1985], one should simplify to the linear-array case, replace D by W and avoid division by currents, in order to keep results in terms of voltages instead of active impedances.

[47] A second windowing approach is based on a perturbation methodology. Two equivalent formulations, one in space domain [Skrivervik and Mosig, 1997], one in spectral domain [Skrivervik and Mosig, 1993], have been presented. The idea consists of considering that, when solving for fields and currents on a particular element of the array, the current distributions on the other elements are the same, except for an amplitude factor which follows the excitation law. It is important to notice that this assumption will lead to different results for different antennas within the array. When exploited in space domain, the implementation of this method is straightforward. Here, we assume that all blocks of the MOM impedance matrix are available. In view of the regularity of the array, the total number of different blocks grows only linearly with the number of elements in the array. Then, the above assumption simply consists of linearly combining different blocks before solving the system of equations, whose dimension is then the same as the one obtained for an isolated element. The spectral-domain formulation of this approach is also interesting. It can be also explained with the help of the Fourier relationships recalled above. In a method of moments perspective, this method is equivalent to considering finite-array Green's functions, i.e., replacing the infinite-array Green's functions by the Green's function of a finite array of dipoles with amplitudes sn corresponding to the excitation law. Denoting by gs(x) the Green's function for an infinitesimal dipole located on x = 0, the finite-array Green's function can be written as:

equation image

With the help of the ASM (26), we can write:

equation image

which leads to:

equation image

where the weighting function W(ψ) has been defined in (56). In other words, the infinite-array Green's function is convoluted with the Fourier transform of the excitation window. Despite the similarity with (55), it is very important to underscore that, in this case, the windowing approach is applied to the Green's function and not to the MOM solution. This means that the interpretation given above in terms of waves propagating in an infinite array is no longer valid.

[48] Nevertheless, windowing methods provide a good accuracy in many circumstances [Roscoe and Perrott, 1994], essentially when the effects of truncation can be viewed a priori as “moderate.” This is for instance the case for mutual coupling between colinear arrays of dipoles or mutual coupling between microstrip antennas printed on thin substrates [Skrivervik and Mosig, 1993]. This motivated the research on fast calculation of “finite-array Green's functions,” which correspond to fields radiated by finite rectangular arrays made of infinitesimal dipoles with constant amplitudes and linear phase progression [Capolino et al., 2000a, 2000b; Mariottini et al., 2006]. Another model for the effects of array truncation is based on the Gibbs phenomenon, well known in Fourier analysis [Hansen and Gammon, 1996]. Modulations appearing in the active impedance of finite arrays have been analyzed by the same author in the work of Hansen [1999] and have been explained by Craeye and Arts [2003] as due to the fact that the currents diffracted by the array ends are sampled by the successive elements of the array at a rate very close to the Nyquist sampling rate.

[49] The methods referred to above are essentially limited to the analysis of arrays with periodic excitation, or with smooth tapering [Mariottini et al., 2005]. A priori, this does not provide all the embedded element patterns and the array impedance matrix. Regarding patterns, however, solutions can be obtained by reciprocity from periodic-excitation problems, by computing the voltage across the antenna loads of all antennas for plane wave incidence from directions covering the whole upper hemisphere. This approach has for instance been adopted by Craeye and Dardenne [2006] to compute the embedded element patterns in ultrawideband arrays, based on the solution of finite-by-infinite arrays [Scharstein, 1990; Ellgardt and Norgren, 2010]. The latter can be viewed as the solutions from infinite arrays, from which one withdraws the currents induced by the complementary parts of the infinite array [Denison and Scharstein, 1995], that occupy two semi-infinite planar arrays (i.e., arrays with one edge only) and supporting the infinite-array currents. This is why the asymptotic analysis of semi-infinite arrays has been analyzed more specifically [Capolino et al., 2000a; Craeye and Capolino, 2006; Capolino and Albani, 2009]. The latter reference extends the works of, among others, Wasylkiwskyj [1973] and Carin and Felsen [1993] and treats semi-infinite arrays of strips excited by plane waves. A highly accurate solution is obtained by writing the integral-equation with a z-transform formalism and by solving it with the help of a Wiener-Hopf factorization. For arrays of relatively simple elements, like slots, asymptotic solutions for semi-infinite arrays enabled the representation of currents resulting from edge effects with the help of basis functions that cover the whole array domain [Neto et al., 2000a, 2000b]. Corrections for finite rectangular arrays can then be obtained by accounting for truncation in the principal directions of the array. By doing so, the effects of the four complementary sectoral arrays (see Figure 8) are accounted for twice. This is why asymptotic Green's functions for those arrays also have been developed [Capolino et al., 2000b]. In this way, embedded element patterns and active input impedances can be approximated for finite rectangular arrays. A similar approach, based on ray-type field behavior, and applied to the case of arrays of dipoles, is presented by Çivi et al. [2000].

Figure 8.

(left) Finite-by-infinite array of tapered-slot antennas. (right) Finite array (white), complementary semi-infinite arrays (light gray) and complementary sectoral arrays (dark gray). From Craeye and Dardenne [2006], © 2006 IEEE.

[50] However, as explained in section 2, a full-wave analysis of finite arrays should also yield the array impedance matrix. Among others, this allows a precise matching analysis (including noise matching, see section 10) for arrays whose beamforming weights are far from periodic. Here again, the experience accumulated in the development of infinite-array Green's functions and propagation in infinite arrays can be exploited efficiently. The main idea here is to be able to recuperate most of the side products of infinite-array analysis, carried out in the first stage of array design, to efficiently analyze finite arrays. Again, the ASM will be a key component here. Indeed, what is looked for is the current distribution over the array for excitation at any element of the array. Considering port currents only, columns of the array admittance matrix are obtained, while fields radiated by currents induced over the whole array by single-port excitation yield the embedded element patterns. When only one element is excited, the main difference between infinite and finite arrays is that, in the latter, currents may be “reflected” or “scattered” from the array edges. In the work of Craeye et al. [2006], a simple model has been used for the forward propagation and reflections by the array ends, in the case of broadband dipoles. These phenomena are sketched by the solid lines in Figure 9, where current and field distributions on successive elements are not explicitly represented and should, in general, be considered different. It is expected that, when the “current wave” propagates back toward the inside of the array (and may bounce on other edges of the array), the new current distributions created in the array are similar to those obtained in the forward wave case. The latter correspond to the distributions provided by the ASM. In some configurations, exceptions may appear for elements located right on the edges of the array. This is for instance the case for arrays of electrically connected elements: the outer elements may support current distributions that cannot be found in infinite arrays. Among those distributions, for arrays of tapered-slot antennas, one should note large vertically oriented currents that flow along the edges of the outer elements [Ellgardt and Norgren, 2010].

Figure 9.

Fields in finite and infinite arrays excited at one element. Solid line with outgoing arrows: infinite array. Solid line with ingoing arrows: fields scattered by edges. Dashed lines: aliased sources resulting from discretized ASM calculation.

[51] For arrays of complex elements, it is difficult to track the multiple reflections on the edges of the array and to find the corresponding reflection coefficients (as far as such quantities can be defined, which would assume the clear definition of modes). However, the above reasoning allows us to think that the current distributions obtained on successive elements with the ASM will form a good basis for current distributions in finite arrays. In practice, the ASM integral will be discretized, which leads to an aliasing phenomenon (implicit repetition of the source; see dashed lines in Figure 9). Nevertheless, the ASM discretized with very few samples forms a very good basis for currents in infinite arrays. This basis can also be augmented with current distributions obtained with a minimal array, e.g., a 2 × 2 array, to account for current distributions that cannot be found in infinite arrays, as mentioned above. Those current distributions, obtained from infinite and finite arrays, form an excellent set of macro basis functions (see end of section 6). This method is known under the name ASM-MBF technique [Craeye and Sarkis, 2008]. Validations carried out with the ASM-MBF for arrays of tapered-slot antennas showed that very high accuracy can be achieved with the application of the ASM with a very low order, i.e., with the ASM integral computed from typically 2 to 4 points in both directions in the Brillouin zone. Hence, the ASM-MBF will allow for the reduction of the effective number of unknowns per antenna by typically 1–2 orders of magnitude when complex elements are considered. This means that relatively large arrays (up to about 16 × 16) can now be treated in direct form, i.e., by direct solution of a reduced system of equations, whose matrix of coefficients can be inverted (or LU-decomposed) once and for all, such that solutions for all possible excitations are readily obtained, including for instance single-port excitation. An example consisting of a 6 × 6 array of TEM horns [McGrath and Baum, 1999], like the one shown in Figure 10, is provided below. The element is meshed with 633 basis functions per antenna (630 RWG [Rao et al., 1982], 1 rooftop and 2 hybrids). Hence, the problem involves 22788 unknowns. At the simulation wavelength λ, A = 0.5 λ, L = 0.25 λ and the angle α is equal to 35°. The generators have a 200 Ω series impedance. Figure 11 presents the normalized radiation pattern obtained with the currents computed using Gaussian elimination (in the very large original MOM system of equations), as well as the errors, defined as e = ∣equation imageexact(θ, ϕ) − equation imageapprox(θ, ϕ)∣ and obtained with the ASM-MBF method, when considering two different dimensions for the space of ASM-MBFs. On the logarithmic scale, the solutions, normalized with respect to maximum fields, obtained with the ASM-MBF cannot be distinguished from the exact solution; the errors are about 60 dB below the exact solution, while the error produced with the infinite-array approximation (dash-dot) is very high.

Figure 10.

Three-dimensional view of the 6 × 6 array of TEM horns electrically connected in the Y direction. The geometry of the TEM horn element is shown on the right.

Figure 11.

Normalized E plane exact radiation pattern obtained for the array in Figure 10, and error, with respect to the exact solution, for the infinite-array solution and for the ASM-MBF solution (using two different grids).

[52] It is interesting to notice that the ASM-MBF method remains very effective when the finite array supports eigenmodes. This is true even when the regular sampling in the reciprocal domain, as needed by the ASM, does not involve a point very close to the eigenmode condition, in terms of interelement phase shifts. As discussed by Craeye [2010], this may be due to the fact that the eigensolution, in good approximation, remains among the eigenvectors of the infinite-array MOM matrix, even for the array scanned relatively far from the eigenmode condition. In a sense, the ASM-MBF benefits from the ability to catch both local effects (through the definition of MBFs) and global effects, like eigenmodes, which are relatively well localized in spectral domain. An open-source code (http://www.mathworks.com/matlabcentral/fileexchange/26823-arraymutualcouplingasmmbf) for the ASM-MBF applied to the simple case of linear arrays of dipoles has been described by Craeye et al. [2010].

[53] From a more physical point of view, it is interesting to notice that solutions based on the ASM may help us to understand why finite arrays scanned in a given direction may support eigenmodes that are not expected from infinite-array simulations. Indeed, since finite-array solutions may be built, via the ASM, on infinite-array solutions for a very broad spectrum of scan angles, including angles outside the visible space, such eigenmodes may in principle always appear. For finite arrays, even when the excitation does not correspond exactly to the eigenmode condition, the cancelation of eigenmodes excited by individual elements will be incomplete, as opposed to the infinite-array case.

[54] Finally, an approach related to the MBF method is the eigencurrent method [Bekers et al., 2009a, 2009b], in which a set of current distributions is defined over different elements, based on an eigenvalue decomposition of the impedance operator, or, in discretized form, of the blocks of the MOM impedance matrix standing for mutual interaction. This method may be regarded as an array extension of the theory of characteristic modes [Harrington and Mautz, 1971]. Current bases are decomposed into modes that do or do not significantly couple with neighboring elements. In the work of Bekers et al. [2009b], for the case of dipoles, a performance comparison is provided with the method of CBFs (characteristic basis functions [Yeo et al., 2003], similar to MBFs) applied with primary and secondary CBFs.

9. Ultrawideband Arrays

[55] Ultrawideband (UWB) arrays are receiving increasing attention for traditional radar applications, but also for new applications like near-field imaging [Gibbins et al., 2010] and radio astronomy, in particular in the framework of the Square Kilometer Array radio telescope project [van Ardenne et al., 2009]. Besides their very promising applications, UWB arrays are particularly interesting from a mutual coupling perspective. Indeed, at the lowest-frequency end, the spacing in such arrays is much smaller than the wavelength, hence the effects of mutual coupling are overwhelming. Rather than trying to reduce the effects of mutual coupling, it makes sense to exploit them in order to improve the performance of the array. This will be analyzed, in a first instance, in the infinite-array case. The link recalled in section 5 between active input impedance and embedded element pattern allows one to concentrate mainly on good matching from the active impedance point of view. Polarization purity may also form an additional figure of merit. As we will see, tapered-slot antennas are very attractive, since they rely on two concepts favorable to ultrawideband operation.

[56] The first concept is based on complementarity. It is well known that the product between the impedances of two complementary antennas [Rumsey, 1966] is η2/4 where η is the free-space impedance. This means that if a metallic antenna is self-complementary, i.e., the surface occupied by metal and that occupied by air in-between have the same shapes, then the antenna has a frequency-independent input impedance equal to η/2. A well known example for such antennas is the self-complementary spiral antenna. However, in practice, self-complementary antennas do present a low-frequency cutoff, related to the finite dimensions of the structure. That low-frequency cutoff corresponds to a frequency at which the antenna diameter is comparable to the wavelength. This poses a challenge from the array point of view, since, over most of the frequency band of interest, the large size of the antenna makes it difficult to form arrays without grating lobes. This is where self-complementary arrays [McGrath and Baum, 1999] form a very elegant solution. Such arrays may be formed by a tiling of square metallic surfaces, with a feed point between all pairs of contacting corners (see Figure 12, left). In such an arrangement, it is difficult to say where is the antenna. The interesting point is that the array is now self-complementary, such that, in the infinite-array limit and for the array scanned at broadside, the active input impedance is frequency-independent, with a value of η/2. The low-frequency cutoff is now given by the array dimension rather than by the dimension of the antenna. The latter are very strongly connected (and hence coupled) with each other and any arbitrarily taken unit cell of the periodic structure (e.g., dashed box in Figure 12, left) may be regarded as the “antenna.” However, an intrinsic difficulty with this type of arrays is their bilateral radiation, which is not appropriate for most applications. This is why such arrays are subject to intense research efforts, in order to maintain the very wide bandwidth, despite the presence of a ground plane. Without any dielectric material, the magnitude of the active reflection coefficient at broadside is equal to one at any frequency where the array-to-ground plane distance is a multiple of half a wavelength. Optimized configurations led to a bandwidth of the order of 2:1 for this type of antennas [Hay and O'Sullivan, 2008]. At the cost of one of the polarizations, the planar self-complementary structure can also be folded to form an array of TEM horn antennas McGrath and Baum [1999], like the one shown in Figure 10, which will be less influenced by the ground plane.

Figure 12.

(left) Self-complementary array and unit cell. (middle) Linear array of tapered-slot antennas. (right) Array of long slots above ground plane. After Neto and Lee [2006].

[57] TEM waves form another important concept for ultrawideband radiation, for instance launched by current modes traveling along tapered slots [Stockbroeckx and Vander Vorst, 2000]. TEM wave generation is obtained with the so-called “Vivaldi” antenna (see Figure 12, middle), made of a tapered-slot on one side of the feed point and of a cavity on the other side (see Cooley et al. [1991] for arrays of such antennas). To first order, the cavity avoids short-circuiting of the feed and brings an inductive element in parallel with the slot impedance. Many variations of such antennas have been developed, including relatively complex feeding structures, for instance with the help of properly terminated striplines crossing the slot [Shin and Schaubert, 1999]. Such antennas have been very successfully included in arrays by electrically connecting successive elements in single and dual-polarized arrangements [Holter et al., 2000]. In general, those designs were based on infinite-array simulations, since, in view of the strong mutual coupling, optimal designs for isolated Vivaldi antennas have very little in common with optimal designs for dense arrays. It is interesting to notice that, with regard to the size of the element, quite low cutoff frequencies could be obtained in array configurations. This may be regarded as one of the benefits of mutual coupling in the design of low-profile UWB arrays. Of particular importance in such arrays is the analysis and subsequent avoidance of possible parallel-plate waveguide modes [Schaubert, 1996].

[58] A very interesting extension of dual-polarized arrays of Vivaldi antennas corresponds to the sometimes named “bullet array” presented by Holter [2007]. Here, one considers four connected fins in the dual-polarized array and replaces them by one solid object of revolution. A feed point is placed at every contact point between the circular cross sections of the “bullet,” in a way similar to the position of the feed points in the self-complementary arrays referred to above. As in the case of arrays of Vivaldi antennas, the main difference with respect to the case of self-complementary arrays is that the elements have a long extension perpendicular to the array plane, such that wideband traveling waves can be launched. In a sense, this extension allows one to create a preferential direction of propagation and hence enables bandwidths much larger than with self-complementary arrays backed by a ground plane.

[59] The arrays referred to above present a quite high profile, which is not acceptable in every UWB application. Essentially, achieving larger bandwidths requires longer elements, i.e., protruding higher from the ground plane. The link between bandwidth and element height has not yet been precisely established from a theoretical point of view; however, this observation means that, as in the case of isolated elements, larger bandwidths require larger element volumes. More moderate bandwidths, but still in the UWB range, can be achieved with interconnected arrays placed at a certain height above the ground plane. Some of these arrays may still be regarded as variations of the self-complementary array concept, or more generally as the concept of interconnected elements, which received renewed attention after the work carried by Munk and colleagues, who studied capacitively coupled dipoles [Munk, 2003], and the work on slot arrays presented by Lee et al. [2003] and Neto and Lee [2005]. Hansen [2004] studied more systematically the effects of loads between the elements of the array and found out that negative inductance (i.e., non-Foster, and hence, active) terminations may provide matching over very wide bandwidths. In the work of Neto and Lee [2006], a detailed study, based on an analytical Green's functions approach, is presented for arrays made of connected dipole (or strip) arrays, as well as for arrays of slots (see Figure 12, right). The field from a given periodically excited slot is decomposed into cylindrical waves and the fields collectively excited by an infinite array of parallel slots is further decomposed into plane waves, while accounting for mutual coupling between slots and for the very strong impact of a ground plane beneath the slotted conductor. A similar approach can be adopted, by duality, to arrays of connected strips. The latter are found to provide wider scan angles; a theoretical design is presented for a 40% bandwidth array with the lowest possible cross-polarization by Neto et al. [2009]. Regarding this type of connected arrays placed above ground planes, the issue of even-mode currents flowing along the transmission lines feeding the antennas is a topic of ongoing intense research. A possible interpretation is to view these currents as resulting from current loops created between successive elements [Hay and O'Sullivan, 2008]. Possible solutions are proposed by de Lera Acedo et al. [2010] and Cavallo et al. [2010], where the latter addresses cross-polarized fields as a by-product of the even-mode presence in the feeding networks.

10. Noise Coupling

[60] The design of high-sensitivity arrays, like those devoted to remote sensing and radio astronomy, requires a careful treatment of possible sources of noise. Those mainly come from the environment (like noise radiated by the warm ground, which needs to be carefully avoided for radio astronomy applications), from the front-end amplifiers and from the losses that may occur on the antennas themselves. While several seminal papers provide signal-to-noise (or, equivalently, gain-over-noise temperature) ratios for multiple-antenna systems [Demir and Toker, 1999; Kraft, 2000], several more recent studies analyze more specifically the effects of mutual coupling on the noise performance of arrays. In this section, we analyze the effects of mutual coupling on the noise budget of such arrays. More specifically, we focus on noise from front-end amplifiers and from lossy antennas.

[61] Good designs for front-end amplifiers may mean rejection of significant noise power toward the antenna. However, in dense array configurations, this noise may be received, through mutual coupling, by other antennas and hence proceed toward the beamformer. In other words, when considering noise generated by a given amplifier, a component will directly travel through the corresponding channel, while other components couple via the other elements. The noise components from the same amplifier that enter the array's different channels will be partially correlated, such that the final noise budget will also depend on the beamformer's weights. It has been proven by Weem and Popovic [2001] and Craeye et al. [2004] that, from the receiver's point of view (i.e., at the output of the beamformer), the noise appears as if it had been reflected by the antenna, with a reflection coefficient that corresponds to the element's active reflection coefficient. In a nutshell, considering the noise contribution from an amplifier connected to antenna i, the total noise power is the one seen while considering the reflection coefficient Sii, as well as those obtained by multiplication by the S-parameter coupling coefficient Sij, each of them multiplied by the corresponding beamforming weight wi. This series of weighted coupling coefficients simply corresponds to the active reflection coefficient of antenna i. In other words, with respect to the noise contribution of a given amplifier, everything happens “as if” the antenna impedance seen by the amplifier were the active input impedance. Hence, the total noise level at the output of the beamformer depends on the direction in which the array is scanned, or more generally, it depends on the beamforming weights. It is important to underscore that this is just an equivalent representation from the point of view of noise budget at the output of the beamformer, i.e., the amplifier does not effectively see the active impedance at its input. Another possible representation for the noise budget is obtained from a field approach (as opposed to the power-wave approach). Several equivalent representations exist for the noise contribution from low-noise amplifiers; these representations have been compiled and compared to each other by Engberg and Larsen [1995]. Here, we will consider the four-parameter description composed of voltage sources and current sources, represented here by vectors vn and in placed at the input of ideal noise-free amplifiers (in this section, subscript n refers to noise). The noise current flowing through the input impedances of the amplifiers (see Figure 13) may be written as:

equation image

where Ya,L = (Z + ZL)−1 is an admittance matrix defined from the array impedance matrix and from the diagonal matrix containing all the amplifiers input impedances. From there, the contribution from every amplifier to the noise current at the output of the beamformer can be evaluated, while taking into account all the effects of mutual coupling. For identical loads ZL, the contribution of amplifier i to noise power is obtained as:

equation image

where T is the reference temperature, K is Boltzmann's constant, B is the bandwidth and Rn, Gn and Yγ are the noise resistance, conductance and correlation coefficients of the amplifier [Engberg and Larsen, 1995]. a and b are here defined as a = wTYa,Li and b = wTYa,LZi where exponent i refers to column i of the matrix and w is a vector containing the beamforming weights. The link with the active impedance approach of Weem and Popovic [2001] is provided by Craeye et al. [2004]. The impact of noise coupling on direction of arrival estimation has been studied by Kisliansky et al. [2007].

Figure 13.

Equivalent current and voltage representation of noise coupled through an array.

[62] The noise temperature contributed by antenna losses is easily obtained as T(1 − ∣Γ∣2) (1 − ζ) where T is the physical antenna temperature, Γ is the reflection coefficient and ζ is the antenna efficiency. For array problems, regarding the noise contribution at a given port, contributions from the whole array need to be included. In this case, one may consider the other elements of the array as being part of the antenna of interest (see section 2) to obtain the efficiency. It is important to notice that, here, the noise from “losses” in the terminations of the other elements of the array do not need to be included. Indeed, first, the real parts of those terminations do not exactly correspond to physical resistors, since they correspond to the input resistances of LNAs; second, the contributions from noisy amplifiers have already been accounted for in the noise-coupling budget recalled above. Factor α0 = 1 − ζ corresponds to the ratio between the power dissipated in the array and the power delivered to it, while considering excitation at the element of interest. For infinite arrays, the ASM can be used to determine the current distribution of the infinite lossy array when only one element is excited, while the ASM-MBF may be used when large finite arrays are analyzed.

[63] In a first instance, one may consider that the thermal noise contributions received at each port are uncorrelated, i.e., the corresponding powers are simply added, taking into account the beamforming weights of each antenna. However, in dense arrays, it makes sense to consider that, when ports are very close to each other, the thermal noise contributions at different ports may be correlated. The corresponding result may be computed with the help of the theory of noise correlation in lossy networks [Warnick et al., 2010; Twiss, 1955]. In this approach, to obtain the contribution from noise generated by the lossy antennas, contributions from external noise sources (momentarily assuming an isotropic noise temperature equal to the physical temperature of the array), computed with the help of beam coupling factors (see section 3), is withdrawn from the general result for noise correlation in a N-port network [Twiss, 1955]. One may also obtain the contributions from antennas themselves directly from the current distributions obtained over the array for excitations at the ports of interest. The correlation coefficient between noise contributions is then proportional to a scalar product between those currents over the array surface [Craeye, 2005]. In the case where all front-end amplifiers have the same input impedance, the total equivalent noise temperature after beamforming is then proportional to the losses integrated over the total array surface, assuming that the array is used on transmit with the beamforming weights equal to those considered on receive. For infinite arrays, this calculation can be limited to the unit cell. In view of those correlations, the corresponding noise power resulting from lossy antennas, as seen at the output of the beamformer, will also be dependent on the scan angle.

[64] The intricate effects of antenna mutual coupling on both signal and noise call for new definitions that allow the representation of the whole beamforming system. Such definitions, fully consistent with classical antenna-theory definitions, have been presented first by Ivashina et al. [2008], then upgraded by Warnick et al. [2010]. Two key components of the latter extended theoretical framework are the isotropic noise response, as well as the link with noise theory for linear N-port systems.

11. Conclusion

[65] Array mutual coupling has been a subject of research for more than half a century. The impressive list of references already found in early compilations, like chapters 2–4 of Hansen [1966], gives an idea of how soon this fascinating subject has attracted attention. Indeed, in dense arrays, i.e., when the average element spacing is smaller than about half a wavelength (which is generally the case for regular arrays), mutual coupling cannot be omitted. If this phenomenon is neglected, important errors can arise, among which major discrepancies in terms of power budget. Besides, mutual coupling often plays a major role in the functioning of some antennas and of some arrays, like Yagi antennas and self-complementary arrays. Regarding the latter, as is the case for most ultrawideband arrays, mutual coupling plays a vital role to allow the realization of ultrawideband performance, which is primarily measured by the scan-dependent active input impedance. The main quantities that characterize finite antenna arrays are the embedded element patterns of all antennas, which are different from each other, as well as the impedance matrix of the array, seen as a N-port circuit. It is important to notice that, in general, it is not sufficient to know the latter to be able to correct for the effects of mutual coupling in the former. This point is known by most of the electromagnetic community dealing with arrays, but seems not to have fully percolated among the array signal processing community. A priori, such a simple link is only possible for a certain class of (“minimum-scattering” or “single-mode”) antennas. However, there is a relationship that holds for any type of lossless antenna between impedance matrix and beam coupling factors, increasingly used in the radiometry community and also of interest to evaluate signal correlations in MIMO applications. We reviewed several different formulations, starting from first principles. We also showed that many methods developed for the analysis of finite regular phased arrays can actually be explained with the help of the array scanning method developed by Munk and Burrell [1979]. In a method of moments scheme, important savings in terms of number of unknowns can be achieved by aggregating elementary basis functions in the form of “macro basis functions” (MBFs). We showed that the array scanning method provides an excellent way of constructing such MBFs. Finally, in the framework of the Square Kilometer Array project for radio astronomy, noise generated by front-end amplifiers and coupled back through neighboring elements received increasing attention. It has been shown that the active reflection coefficient is, again, a very useful quantity to analyze this phenomenon. New theoretical tools have also been developed for the analysis of noise generated by the lossy antennas. These observations called for an extension of traditional figures of merit for antennas and antenna arrays, as provided by Warnick et al. [2010].

[66] With the fast development of microwave front-ends, of fast real-time processing architectures and imaging algorithms, and with the increasing importance of civilian applications of phased arrays, it is expected that array theory and techniques will remain a rapidly developing and exciting research area. In this endeavor, the understanding of mutual coupling, as well as its fast and accurate prediction, will remain important components.

Appendix A:: Beam Coupling Factors

[67] The radiated power can be written in terms of embedded element patterns (EEPs) as:

equation image

where the v°,n are the voltage sources of the Thevenin equivalent circuits of the generators. Integrating over the unit sphere, one obtains:

equation image

where B is a matrix containing all the BCFs: B(k, l) = bk,l.

[68] The power leaving the volume formed by all generators can also be written as:

equation image

Defining Y = (Z + ZL)−1, and knowing that i = Y v° and v = Z i, one obtains:

equation image

Taking into account the symmetry of Z, the comparison between (A2) and (A4) leads to (10) in section 3.

[69] The radiated power can also be written in terms of open-circuit patterns:

equation image

where in is the current flowing into the nth element. After integration over the unit sphere, (A5) becomes

equation image

where Bo.c. is a matrix containing all the open-circuit beam coupling factors Bo.c.(k, l) = bk,lo.c.. From (A3), the radiated power can also be written as:

equation image

Hence one arrives to (11).

[70] Finally, the power leaving the volume formed by all generators can also be written in terms of power waves a and b as:

equation image

with b = S a and a = v°/(2equation image) in the matched case, i.e., when ZL,n = ZL = Z°, the (complex) reference impedance used to define the scattering matrix. From (A8), we readily obtain:

equation image

where I is a N × N unit matrix. Still under the assumption of identical loads, one hence finds (12) [Stjernman, 2005].

Appendix B:: Proof of Some Fourier Relationships

[71] Equation (22) can be proven by inverting the order of integration and summation:

equation image

where the integrand is zero for x = n a and n ≠ 0. Equation (23) can be obtained by observing that it corresponds to an inverse Fourier transform. We use the following convention for the Fourier transform pair: F(k) = ∫−∞f(x)ejkxdx and f(x) = 1/(2π)∫−∞F(k)ejkxdk. Writing the forward transform, we obtain:

equation image
equation image

The equality between right-hand sides of (B2) and (B3) is a standard result. For ψ = 0, a proof based on formulas for geometrical series can be found in the work of Papoulis [1962, section 3.2]. The phase factor involving ψ in the expression of C(x, ψ) leads to a ψ shift in spectral domain, as per the basic properties of the Fourier transform.

[72] Result (23), which says that the Fourier transform of a comb function is another comb function, can then be exploited to justify the Fourier series formulas, and their generalization, the Poisson sum formula:

equation image

where F(k) is the Fourier transform of f(x). The Poisson sum formula is used in several papers relating to infinite and finite regular arrays, among which are Munk and Burrell [1979], Ishimaru et al. [1985], and Skrivervik and Mosig [1993].


[73] We thank Andrea Neto, from the Technical University of Delft, for his insightful comments about fields in planar ultrawideband arrays. We also thank the Centre de Calcul Intensif (CISM-UCL-FNRS) for free access to computing facilities.