A Modal Description of Multiport Antennas

Abstract

This paper presents a modal description of multiport antennas that leads directly to a rigorous network representation and simple quadratic expressions for gain, efficiency, and effective area. The analysis shows that the transmitting and receiving properties of an 𝑁 element antenna array are exactly described by a 2𝑁×2𝑁 element scattering matrix together with a set of 𝑁 orthonormal mode functions and accounts for effects such as mutual coupling, scattering, reflection, and losses. The approach is quite general, only requiring that the antenna be finite and reciprocal. The scattering network description simplifies accounting of power flow while retaining a close connection to the physical antenna characteristics. The orthonormal mode functions provide a complete basis for radiated and received fields, facilitating beamforming. The theory provides rigorous definitions of input-output signals and links them to the underlying electromagnetics in a straightforward manner.

1. Introduction

Many communications, radar, and sensing systems rely on arrays of antenna elements, each with individual receivers, to transmit or receive energy with radiation patterns that may be manipulated in real time by digitally modifying the coherent combination of received signals. The flexibility afforded by such digital beamforming allows, for example, a radar system to electronically track multiple targets with the sensitivity afforded by high gain beams, a high data rate receiving system to communicate simultaneously with multiple users on channels with high spatial selectivity, or a direction finding system capable of locating many emitters with high accuracy. Multiple input, multiple output (MIMO) communication systems in particular have demonstrated robust data transfer in the presence of multipath interference as a result of the redundancy provided by multiple independent spatial channels [1].

The development of the underlying signal processing of such systems typically begins with abstract and often idealized constructs where the signals transmitted and received by individual antenna elements are represented as time-(or phase-) shifted replicas that depend in a straightforward way on the antenna geometry. Often neglected from such analyses are the effects of mutual scattering and coupling between elements as well as scattering from objects in the vicinity of the antenna. A perturbation approach is often utilized in system design to provide first order corrections due to such coupling and scattering with mixed and sometimes contradictory results reported in the literature [2]. A scattering description of MIMO systems has been presented in [3, 4] that takes into account coupling between antenna elements. However, this analysis defines the scattering parameters with respect to a fixed location in the far field and must be modified for different incident/radiation angles. This approach does include structural scattering, although this is generally not important for MIMO arrays.

The analytical description of a multiport antenna has typically been provided by the terminal port impedance, admittance, or scattering matrix together with radiation patterns obtained by successively exciting each port with the others terminated in known loads. Linear combinations of the resulting field patterns are used to obtain desired beam shapes. For the receiving case, equivalent circuits have been developed that allow calculation of the received signals for a multiport antenna under plane wave incidence, including mutual coupling and scattering. For example, [5] provides a rigorous derivation of such an equivalent circuit in the frequency domain and [6] provides the same in the time domain. Alternatively, [7] describes an 𝑁 element multiport antenna by including a test antenna and considering the resulting 𝑁+1 element antenna system. This approach leads to expressions for antenna gain in terms of the mutual impedance (or admittance) between the multiport and test antennas.

This paper presents a modal approach to multiport antennas that provides an exact description of the transmitting and receiving properties of the antenna and leads to quadratic expressions for antenna parameters of interest, such as gain, efficiency, and effective area. The theory shows that a finite multiport antenna with 𝑁 pairs of feed terminals (i.e., ports) will produce exactly 𝑁 orthogonal “radiation” modes whose linear combinations may describe any possible transmit or receive pattern for that array (assuming the terminal port excitations produce linearly independent far zone fields, as is nearly always the case in practice). The scattering network description together with the mode functions completely characterize the transmitting and receiving properties of an antenna, although these modes are not sufficient to completely describe scattering by the antenna due to incident waves.

The analysis presented here is quite general and applies to a large class of antennas. The fundamental assumptions are that the antenna is comprised of linear materials and is finite so that far fields may be asymptotically defined in empty space at a suitably large distance. Because the antenna is assumed finite, this analysis does not apply to (for example) infinite periodic structures, although one could represent each of the Floquet modes as a radiation port applying the same techniques. We also assume the antenna elements and nearby materials are reciprocal to facilitate the analysis of the receive case, although this assumption is not necessary to define the radiation modes. Unlike other antenna modal analyses, this theory allows for dissipation within and around the antenna.

A different but related modal description was developed in the 1960s by Garbacz and Turpin [8] and Harrington and Mautz [9] first for scattering bodies and later for antenna arrays [10]. Their approach was to determine a set of orthogonal excitations that effectively eliminate mutual coupling between elements. We distinguish these as “backscatter” modes. In the lossless case, the radiation and backscatter modes are identical, a fact that limited the previous modal analyses to the lossless case [11]. For antennas with finite conductivity and dielectric losses, the backscatter modes do not produce orthogonal radiated fields and therefore may not be as useful for beamforming as the radiation modes. Backscatter modes and their associated decoupling networks are useful for other purposes and various groups have explored their application [12]. We include a short discussion of backscatter modes in this paper to place them in context with radiation modes and show how both mode sets diagonalize different portions of the complete 2𝑁×2𝑁 antenna scattering matrix.

Following the Introduction (Section 1), this paper is organized into five additional sections. In Section 2, we show how to derive the radiation mode set from antenna measurements. These modes provide a complete orthonormal basis for all possible fields radiated from and received by the antenna. We again note that our modal description is not sufficient to describe the so-called structural mode scattering by the antenna resulting from external energy sources. Problems such as low RCS antenna design require a fuller basis, such as spherical harmonics, to support general antenna scattering. In Section 3, the radiation modes are used to define an antenna scattering matrix, providing a complete network description of the antenna. In addition to the usual terminal port scattering parameters, this matrix also includes parameters describing transmission, reception, and antenna-mode (as opposed to structural mode) scattering. Here, we show that an ideal transformation network that produces the radiation mode excitations diagonalizes the scattering submatrices for transmission (𝑆₂₁), and reception (𝑆₁₂), and show the close connection between 𝑆₂₁ and antenna efficiency. In Section 4, we apply the formalism to produce simple quadratic expressions for antenna efficiency, directivity, and gain and derive the relation between gain and effective area for an arbitrary multiport antenna. In Section 5, we define the backscatter modes as the solutions to a conjugate eigenvalue problem and show how a transformation (i.e., decoupling) network diagonalizes the reflection coefficient submatrix 𝑆₁₁. These two different diagonalizations distinguish the modes sets: one leads to orthonormal-radiated fields and an equivalent network description, the other to decoupled terminal ports. Finally, in Section 6, we present two examples illustrating the application of the theory.

2. Radiation Modes

A multiport antenna consisting of 𝑁 elements, each with a pair of feed terminals (or a waveguide port), is capable of transmitting or receiving a variety of field patterns. The so-called element patterns of the antenna are produced by exciting each of the feed terminals separately with the ports terminated in known impedances. The element patterns are similar to translated versions of the pattern produced by exciting a single element in isolation but are altered by the coupling between, and scattering by, neighboring elements. Because the materials comprising the antenna are assumed linear, superposition holds and the 𝑁 complex incident wave amplitudes provide 𝑁 (complex) degrees of freedom that span the space of all possible radiated fields from the array. For a reciprocal antenna, the transmitting and receiving element patterns are identical so the receiving antenna possesses the same 𝑁 (complex) degrees of freedom. It is important to note that we distinguish “radiated” field patterns, which are due to terminal port excitations, from “scattered” field patterns, which may result (for example) from plane wave incidence. The space of complex fields spanned by all linear combinations of incident terminal port wave amplitudes represents radiated fields but is not sufficient to describe the scattered fields. The portion of scattered fields that lies in this space is called “antenna mode” scattering, and the portion outside this space is called “structural mode” scattering.

To facilitate power flow computations, we define a 2𝑁×2𝑁 scattering matrix that represents an 𝑁 element multiport antenna. Such a network description of antennas was first proposed in [13], with more detail provided in [14], which utilized spherical modes to define the far field “ports”. Because the infinite set of spherical modes is sufficient to describe arbitrary field distributions, this approach describes not only the transmitting and receiving properties of the antenna, but also the structural scattering due to incident fields. When knowledge of these structural scattering properties is not needed, as is often the case in practice, one may reduce the set to exactly 𝑁 modes (generally equal to the number of terminal ports) that provide a complete basis for radiated and received fields.

First, some remarks must be made concerning antenna ports. At the terminal side of the array, we use the definitions of ports and associated voltages, currents, and incident/reflected waves from microwave circuit theory. Rigorous definitions of equivalent terminal ports have been presented in detail elsewhere (e.g., see [15–17]) and need not be repeated here. On the radiation side of the array, we must begin by defining equivalent circuit “radiation” ports and we do this by requiring that the power flowing to and from each port be independent of power flowing in the other ports. This condition is implicitly assumed in any set of circuit ports and will be met if and only if the set of port fields is orthogonal. We note that on the terminal side of the array this is often satisfied through isolation. For example, coaxial cables have (assumed) perfect field confinement. If a physical connection supports multiple modes (e.g., overmoded waveguide), each of the (orthogonal) modes may be represented by a separate port in the equivalent network. On the radiation side of the array, we must determine an orthogonal set of fields that supports all possible radiated patterns in order to properly define equivalent radiation ports.

2.1. Transmitting

Consider waves incident upon the antenna input terminals and radiated by the antenna elements (i.e., the transmitting case). Exciting the 𝑛th terminal port with one Watt of time average incident power at zero phase with all other ports terminated in their reference impedances produces a far zone electric field 𝐸_𝑛(𝑟,𝜃,𝜙) (we use RMS field amplitudes throughout this paper), described with respect to a spherical coordinate system with the origin of some fixed location within or near the antenna. We seek linear combinations of these outward traveling waves𝐸⁻_𝑛=^𝑁−1_𝑚=0𝑈_𝑚,𝑛𝐸_𝑚 (1) that form an orthogonal set in the far field:1𝜂_𝑜𝐸⁻_𝑝^∗⋅𝐸⁻_𝑞𝑑𝑆=_𝑚𝑛𝑈^∗_𝑚,𝑝1𝜂_𝑜𝐸^∗_𝑚⋅𝐸_𝑛𝑑𝑆𝑈_𝑛,𝑞=_𝑚𝑛𝑈^∗_𝑚,𝑝𝑀_𝑚,𝑛𝑈_𝑛,𝑞=𝜆_𝑝²𝛿_𝑝,𝑞, (2) where 𝜂_𝑜=𝜇_𝑜/𝜀_𝑜 is the intrinsic impedance of free space, 𝛿 is the Kronicker delta function, and the asterisk denotes complex conjugate. The negative superscript indicates outgoing waves, and a positive superscript will indicate incoming waves. The differential area is 𝑑𝑆=𝑟²sin𝜃𝑑𝜃𝑑𝜙=𝑟²𝑑Ω, and the integration extends over the entire sphere 0<𝜃<𝜋, 0<𝜙<2𝜋 as 𝑟→∞.

Each element of the matrix𝑀_𝑚,𝑛=1𝜂_𝑜𝐸^∗_𝑚⋅𝐸_𝑛𝑑𝑆 (3) is proportional to the overlap integral of the fields produced by two terminal-port excitations. Expressing last equality of (2) in matrix notation, we have𝑈^𝐻𝑀𝑈=Λ^𝐻Λ, (4) where the superscript indicates the Hermitian conjugate transpose. The orthogonal fields 𝐸⁻_𝑛 and 𝐻⁻_𝑛=(1/𝜂_𝑜)𝑢_𝑟×𝐸⁻_𝑛 will be referred to as radiation mode fields, with each mode representing a radiation port in a manner similar to port definitions of uniform waveguides [17]. The diagonal matrix Λ contains complex parameters 𝜆_𝑛, whose magnitude squared turns out to be the total efficiency for the 𝑛th mode. The coefficient matrix 𝑈 contains the eigenvectors of 𝑀 as its columns, each column providing the incident wave amplitudes for the corresponding mode excitation. We note that the eigenvector matrix 𝑈 is not unique since each column may be multiplied by a phase factor while still satisfying (4). Once the matrix 𝑈 is specified, one may compute the modal fields from the terminal port excited radiation fields using (1).

Because 𝑀 is Hermitian, we are guaranteed a set of 𝑁 orthogonal modes, one for each set of terminal pairs, for nearly all practical cases of interest (we ignore pathological cases where two terminal ports produce linearly dependent radiated fields). We assume for simplicity that the number of radiating elements is equal to the number of terminal ports, but this is not necessary. The theory applies equally well when there are fewer terminal ports than radiating elements. For this case, the number of radiation modes is equal to the number of terminal ports, not the number of antenna elements.

In the far field of the antenna, the radiation mode fields are spherical waves proportional to equivalent circuit voltage amplitudes:𝐸⁻_𝑛𝑟,𝜃,𝜙=𝑉⁻_𝑛𝑒^{−𝑗𝑘𝑟}𝑟𝑒_𝑛̂𝜃,𝜙, (5) where 𝑉⁻_𝑛 is the outward traveling (complex constant) voltage amplitude and 𝑘=𝜔√𝜇_𝑜𝜀_𝑜. The complex vector mode functions 𝑒_𝑛̂=𝑢_𝑛𝑒_𝑛 (with unit polarization vector 𝑢_𝑛) are scaled to form an orthonormal set:𝑒_𝑛̂⋅𝑒^∗_𝑚̂𝑑Ω=𝛿_𝑛,𝑚. (6) The field representation (5) with orthonormal mode functions is the same as that used for uniform waveguides. We note that (6) allows the phase of each mode function 𝑒_𝑛̂ to be arbitrarily chosen. Inserting (5) into (2) and using (6), we find that |𝑉⁻_𝑛|²=𝜂_𝑜|𝜆_𝑛|², so that the radiation mode fields are related to the normalized mode functions as𝐸⁻_𝑛𝑟,𝜃,𝜙=√𝜂_𝑜𝜆_𝑛𝑒_𝑛̂𝑒𝜃,𝜙^{−𝑗𝑘𝑟}𝑟. (7) Equating (7) with (1), we may determine the complex 𝜆_𝑛 directly from the measured fields:𝜆_𝑛=1√𝜂_𝑜𝑒^𝑗𝑘𝑟_𝑚𝑈_𝑚𝑛𝑟𝐸_𝑚⋅𝑒^∗_𝑛̂𝑑Ω. (8)

Generally, spherical waves from an arbitrary source can be expanded in terms of spherical modes that are products of spherical Bessel functions and Tesseral harmonics, with associated mode propagation constants and mode impedances. In the far field, the radial dependences of all the modes asymptotically approach the form (5), the propagation constants approach the value 𝑘, and the wave impedances approach 𝜂_𝑜. Thus, the far field spherical modes are degenerate so that any linear combination is also a mode. The relation between the modal fields and voltages given in (5) is therefore consistent with (but different from) the microwave circuit formalism of [16]. Our modal representation takes advantage of the high degree of degeneracy to represent the fields in a manner convenient for antenna engineering and measurement.

As a result, we may describe the radiated mode fields in terms of an equivalent circuit, as indicated in Figure 1. Each of the modal field amplitudes is represented by a traveling wave voltage on a radial transmission line of characteristic impedance 𝜂_𝑜, one for each mode. The orthogonality of the modal fields ensures that the total radiated power is the sum of the powers traveling on each radial line.

Figure 1 

Radiated fields are represented by radial waveguides, one per mode, whose amplitudes are given by the traveling wave voltages and currents. Orthogonality of the modal fields ensures that power flows independently along the radial guides.

An arbitrary field 𝐸 radiated from the antenna may be expressed as a superposition of mode functions𝐸=^𝑁−1_𝑛=0𝑉⁻_𝑛𝑒^{−𝑗𝑘𝑟}𝑟𝑒_𝑛̂𝜃,𝜙, (9) and we may determine the equivalent voltage amplitudes using orthogonality (6):𝑉⁻_𝑛𝑒^{−𝑗𝑘𝑟}𝑟=𝐸𝑟,𝜃,𝜙⋅𝑒̂^∗_𝑛𝜃,𝜙𝑑Ω. (10)

The directive gain pattern 𝐷_𝑛 for the 𝑛th radiation port is related to the mode functions using (5) and (6):𝐷_𝑛𝜃,𝜙=4𝜋𝐸⁻_𝑛²∬𝐸⁻_𝑛²𝑒̂𝑑Ω=4𝜋_𝑛². (11) The gain in the 𝜃,𝜙 direction for an arbitrary radiated field (9) may be computed using (6):𝐷𝜃,𝜙=4𝜋𝐸𝜃,𝜙²∬𝐸𝜃,𝜙²𝑉𝑑Ω=4𝜋^−𝐻𝑒^∗̂⋅𝑒^𝑇̂𝑉⁻𝑉^−𝐻𝑉⁻, (12) where 𝑉⁻ (without subscript) is an 𝑁 element vector representing the voltages of the outgoing waves, and the matrix [𝑒^∗̂⋅𝑒^𝑇̂]_𝑚,𝑛≡𝑒^∗_𝑚̂⋅𝑒_𝑛̂ expresses the outer product. We will find this matrix notation convenient in later sections.

2.2. Receiving

The receiving properties of the antenna may be determined by application of Lorentz’s reciprocity relation, as was done in [15]. We take a different approach, noting that the time reversal properties of Maxwell’s equations in free space suggest that the antenna receives energy in modes that are the conjugate of the transmission mode functions,𝐸⁺_𝑛𝑟,𝜃,𝜙=𝑉⁺_𝑛𝑒^𝑗𝑘𝑟𝑟𝑒^∗_𝑛̂𝜃,𝜙. (13) One may verify that the conjugate ensures the correct polarization matching as well as collocated phase centers for transmit and receive. Because the transmitting and receiving mode functions are different, the circuit formalism is not the same as for standard circuits. Incoming and outgoing waves travel on equivalent transmission lines with different mode functions, although we represent them together pictorially (see Figure 1) for convenience. When the mode functions are real, the standard waveguide network formalism applies, and this occurs when the coordinate system is located at the phase center of the mode. Allowing the incoming and outgoing mode functions to be different departs from conventional circuit theory, but this eliminates the need to determine modal phase centers, something that is difficult to accomplish in practice.

We may use the receiving mode functions to compute the available power for a plane wave incident from the 𝜃_𝑜,𝜙_𝑜 direction. Assuming an 𝐸 field polarization unit vector 𝑢_pw and amplitude 𝐸_𝑜, the incident field is𝐸_pw=𝑢_pw𝐸_𝑜𝑒^{𝑗𝑘⋅𝑟}=𝑢_pw𝐸_𝑜𝑒^{𝑗𝑘𝑟[sin𝜃𝑜sin𝜃cos(𝜙−𝜙𝑜)+cos𝜃𝑜cos𝜃]}. (14) We represent this plane wave as a linear combination of incoming receive modes:𝐸_pw≈^𝑁−1_𝑚=0𝑉⁺_𝑚𝑒^𝑗𝑘𝑟𝑟𝑒^∗_𝑚̂𝜃,𝜙. (15) Note that the superposition (15) is only an approximation to the incident plane wave field since the mode functions are not a sufficient basis for arbitrary field distributions. Structural scattering is scattered field energy not supported by the mode function basis.

The incident voltage amplitudes are determined from (15) using orthogonality,𝑉⁺_𝑛𝑒^𝑗𝑘𝑟𝑟=𝐸_pw⋅𝑒_𝑛̂𝜃,𝜙𝑑Ω=𝐸_𝑜𝑢_pw⋅𝑒_𝑛̂×𝑒^{𝑗𝑘𝑟[sin𝜃𝑜sin𝜃cos(𝜙−𝜙𝑜)+cos𝜃𝑜cos𝜃]}𝑑Ω. (16) This integral may be evaluated asymptotically as 𝑟→∞ using the method of steepest descents, with details given in the appendix A. We find the equivalent voltage amplitude for the incident wave is𝑉⁺_𝑛=−𝑗𝜆𝐸_𝑜𝑢_pw⋅𝑒_𝑛̂𝜃_𝑜,𝜙_𝑜=−𝑗𝜆𝐸_pw⋅𝑒_𝑛̂, (17) where 𝜆 is the wavelength of the incident radiation. This expression relates the equivalent voltage to the amplitude, direction, and polarization of the incident plane wave. The network model for plane wave incidence is shown in Figure 2. The expressions for the equivalent voltage sources in Figure 2 are similar to those derived in [5]. One must keep in mind that each radiation port in Figure 2 represents a single mode and is comprised of contributions from all of the antenna elements.

Figure 2 

For the receiving case, an incident plane wave may be represented by a set of equivalent voltage sources, one for each mode.

The time average incident power at the 𝑛th radiation port may be expressed in terms of the radiation pattern using (17) with (11):𝑃_inc,𝑛=𝑉⁺_𝑛²𝜂_𝑜=𝐸_𝑜²𝜂_𝑜𝜆²𝑢_pw⋅𝑢_𝑛²𝑒_𝑛²=𝑆_pw𝜆²𝐷4𝜋^𝑛𝑢_pw⋅𝑢_𝑛², (18) where 𝑆_pw is the average power density in the incident plane wave and the last factor accounts for polarization mismatch. We note that the polarization mismatch factor |𝑢_pw⋅𝑢_𝑛|² differs from that given in [18] by a complex conjugate. This is due to differences in our definitions of the plane wave polarization vector.

Equation (18) confirms the well-known expression for available receive power in terms of the maximum effective area of the antenna and shows how it arises from an overlap integral of the plane wave with the mode function. The maximum power available to the antenna is the sum of the incident powers from all the modes,𝑃_avail=_𝑛𝑃_inc,𝑛=𝑆_pw𝜆²4𝜋_𝑛𝐷_𝑛𝑢_pw⋅𝑢_𝑛², (19) showing that the maximum available directive gain is the sum of the radiation mode gains. The power collected by the terminal ports depends on the scattering parameters of the antenna array and will be considered in the next section.

We now have a network description of the incident and radiated fields for an 𝑁 element multiport antenna in terms of equivalent voltages and currents on radial transmission lines that represent the network port terminations and have related the voltages and currents to the radiating/received fields through a set of 𝑁 normalized field functions. In the next section, we relate the incident, reflected, radiated, and received fields through a scattering matrix.

3. Scattering Matrix Representation

Incident and scattered waves may be defined in the usual way with respect to the (RMS) traveling wave voltages: 𝑎_𝑛=𝑉⁺_𝑛/√𝜂_𝑜 and 𝑏_𝑛=𝑉⁻_𝑛/√𝜂_𝑜, where the port voltage is 𝑉_𝑛=𝑉⁺_𝑛+𝑉⁻_𝑛 and the current entering the port is 𝐼_𝑛=𝐼⁺_𝑛−𝐼⁻_𝑛. The scattering matrix describes the interactions within the antenna between the incoming and outgoing waves using 𝑏=𝑆𝑎:𝑏₁𝑏₂=𝑆₁₁𝑆₁₂𝑆₂₁𝑆₂₂𝑎₁𝑎₂, (20) where 𝑆₁₁, and so forth, are 𝑁×𝑁 submatrices and 𝑎₁, and so forth, are 𝑁 element vectors. The subscript “1” denotes the terminal side of the array, and the subscript “2” denotes the radiation side. Thus, 𝑆₁₁ is the scattering matrix at the array terminal ports that is typically measured with a vector network analyzer. 𝑆₂₁ describes the transmitted waves for terminal port excitation, 𝑆₁₂ describes received waves at the terminal ports due to incident fields, and 𝑆₂₂ describes waves scattered into the outgoing port fields due to incident fields. As mentioned above, the far field scattering properties of a multiport antenna are not completely described by 𝑆₂₂. In general, a portion of the total scattered energy will be orthogonal to all of the port fields. Our analysis does not represent such structural mode scattering.

The radiation submatrix 𝑆₂₁ may be determined from antenna measurements in the following way. One applies a continuous wave signal (scaled to unity power and zero phase) to each of the elements in turn with the other elements terminated in their respective port reference impedances and measures the radiated far zone complex vector electric fields 𝐸_𝑛(𝑟,𝜃,𝜙). The port field functions are computed using (1)–(6) and the amplitudes using (10) (with 𝑉⁺_𝑛=0 at the radiation ports). The 𝑛th radiated port wave at radial distance 𝑟 due to “unit” excitation of the 𝑚th antenna terminal is𝑆_21,𝑛𝑚=𝑏_{2,𝑛𝑎1,𝑚=1}=1√𝜂_𝑜𝑟𝐸_𝑚⋅𝑒̂^∗_𝑛𝑑Ω=𝑈^∗_𝑚𝑛𝜆_𝑛𝑒^{−𝑗𝑘𝑟}. (21) To arrive at the last expression, we used 𝑈^𝐻𝑈=𝐼 together with (8), noting 𝐸_𝑛=0 for 𝑚≠𝑛. In (21), the radiation port reference planes are all located at the radial distance 𝑟, but these may be moved to any desired location by selecting the value of 𝑟 for each port. It is often convenient to refer the radiation port phases to 𝑟=0.

Expressing 𝑆₂₁ in matrix form (with 𝑟=0), 𝑆₂₁=Λ𝑈^𝐻, (22) where Λ is a diagonal matrix whose diagonal elements are 𝜆_𝑛. Using (22), we may now identify the matrix in (3) as 𝑀=𝑆^𝐻₂₁𝑆₂₁ and the columns of 𝑈 as the incident wave vectors 𝑎₁ that produce the corresponding mode fields. Although there are many orthonormal sets of radiated fields for an antenna (any unitary transformation of the mode field functions gives another orthonormal set), our particular choice is the one that results in an orthonormal set of incident terminal port wave vectors.

The matrix 𝑆₁₂ describes the receiving properties of the antenna. Using (17), we may determine the incident port waves (at reference plane 𝑟=0) corresponding to the plane wave in (16):𝑎_2,𝑛1=−𝑗𝜆√𝜂_𝑜𝐸_pw⋅𝑒_𝑛̂. (23) In principle, we may determine the scattering parameters 𝑆₁₂ by measuring the terminal port outputs due to incident plane wave fields. However, for reciprocal antennas, 𝑆₁₂=𝑆^𝑇₂₁, so these measurements are unnecessary.

The matrix 𝑆₂₂ describes energy backscattered into the radiation port fields due to incident plane waves (or superpositions of plane waves), often referred to as antenna mode scattering. In principle, one may determine 𝑆₂₂ by measuring or simulating the scattered fields due to plane waves incident from various directions, but we leave this analysis beyond the scope of this paper. In practice, the backscattered waves are often unimportant. An exception is low RCS antenna design where both the antenna mode and structural mode scattered fields are important.

Using (22), the full antenna scattering matrix may be expressed as𝑆𝑆=¹¹𝑈^∗ΛΛ𝑈^𝐻𝑆₂₂. (24) The sets of incident waves that produce the radiation modes may (in theory) be generated using a lossless, reciprocal transformation network, illustrated in Figure 3, whose scattering matrix is 𝑆_𝑈=0𝑈^𝑇𝑈0. (25) The scattering matrix for the combined transformation network and antenna is𝑈𝑆′=^𝑇𝑆₁₁𝑈ΛΛ𝑆₂₂ (26) showing that this mode transformation diagonalizes the radiation and reception submatrices.

Figure 3 

A transformation network placed on the terminal side of the array produces the mode excitations. For example, a wave incident upon a port on the left side of 𝑆𝑈 produces the corresponding set of modal incident waves at the antenna terminals. This transformation network diagonalizes the antenna S parameter submatrices 𝑆21 and𝑆12.

To illustrate the formalism, we will compute the waves received by one array (antenna II) due to terminal port incident waves radiated by another (antenna I). This may be useful for MIMO antenna analysis, although here we assume a perfect channel (empty space) for simplicity. The waves received by antenna II are 𝑏^II₁=𝑆^II₁₂𝑎^II₂. The waves incident upon antenna II may be related to incident electric field due to antenna I 𝐸^Iusing (23):𝑎^II₂1=−𝑗𝜆√𝜂_𝑜𝐸^I⋅𝑒̂^II. (27) But this electric field may be expressed in terms of outgoing waves from antenna I using (9):𝐸^I=_𝑛√𝜂_𝑜𝑏^I_2,𝑛𝑒̂_𝑛^I𝑒^{−𝑗𝑘𝑟}𝑟. (28) Inserting (28) into (27) and using 𝑏^I₂=𝑆^I₂₁𝑎^I₁, we have𝑏₁^𝐼𝐼𝑒=−𝑗𝜆^{−𝑗𝑘𝑟}𝑟𝑆^II₁₂𝑒̂^II⋅𝑒̂^I𝑇𝑆^I₂₁𝑎^I₁, (29) where the mode overlap matrix uses the notation described just below (12), namely [𝑒̂^II⋅(𝑒^Î)^𝑇]_𝑚,𝑛=𝑒̂_𝑚^II⋅𝑒̂_𝑛^I. This result expresses the scattering parameter 𝑆^total₂₁ between the two sets of antenna terminals in terms of the antenna 𝑆 parameters and mode functions. Following the same analysis with antenna II transmitting and antenna I receiving produces 𝑆^total₁₂=(𝑆^total₂₁)^𝑇, confirming reciprocity.

We see that the result of cascading antenna scattering networks is different than for microwave circuits. The reason is that each antenna mode set depends on the physical location of that antenna. When two identical antennas are separated, their mode patterns are different and this is taken into account by a mode overlap matrix, which in our present case is [𝑒̂^II⋅(𝑒^Î)^𝑇]. A realistic communications channel contains multipath scattering from objects such as the ground or buildings that further mixes the modes between the two antennas. However, as long as these scattering events are in the far fields of the antennas, our analysis correctly accounts for mutual coupling and scattering between antenna elements.

4. Efficiency, Gain, and Effective Area

The multiport antenna scattering parameters and mode functions allow us to compute various antenna parameters, such as efficiency, gain, and effective area, in a straightforward manner. The total efficiency (including reflection and dissipative losses) for a particular excitation is defined as the ratio of the total radiated power to the incident power 𝜀_tot=𝑏^𝐻₂𝑏₂𝑎^𝐻₁𝑎₁=𝑎^𝐻₁𝑆^𝐻₂₁𝑆₂₁𝑎₁𝑎^𝐻₁𝑎₁. (30) It is useful to define mode efficiencies, the total efficiency when the terminal ports are excited by a mode vector (i.e., 𝑎₁ is equal to a column of 𝑈). Using (22), these are given by the diagonal elements of𝑈^𝐻𝑆^𝐻₂₁𝑆₂₁𝑈𝑈^𝐻𝑈=Λ^𝐻Λ, (31) which are just the eigenvalues |𝜆_𝑛|² of 𝑀=𝑆^𝐻₂₁𝑆₂₁. Thus, we identify Λ^𝐻Λ as the (diagonal) mode efficiency matrix.

The directive gain may be expressed in terms of the outgoing waves 𝑏₂=𝑉⁻₂/√𝜂_𝑜 using (12) and the incident terminal port waves using 𝑏₂=𝑆₂₁𝑎₁: 𝐷𝑎𝜃,𝜙=4𝜋^𝐻1𝑆^𝐻₂₁𝑒^∗̂⋅𝑒^𝑇̂𝑆₂₁𝑎₁𝑎^𝐻₁𝑆^𝐻₂₁𝑆₂₁𝑎₁, (32) showing how the directive gain pattern depends on the excitations 𝑎₁. The power gain relates the radiated power density to the power accepted by the antenna terminals. In terms of the incident wave vector, the power gain is𝐺_𝑝𝑎𝜃,𝜙=4𝜋^𝐻1𝑆^𝐻₂₁𝑒^∗̂⋅𝑒^𝑇̂𝑆₂₁𝑎₁𝑎^𝐻₁𝐼−𝑆^𝐻₁₁𝑆₁₁𝑎₁. (33) Finally, the realized gain (that we indicate without subscript) relates the radiated power density to the incident power𝐺𝑎𝜃,𝜙=4𝜋^𝐻1𝑆^𝐻₂₁𝑒^∗̂⋅𝑒^𝑇̂𝑆₂₁𝑎₁𝑎^𝐻₁𝑎₁. (34)

For the receiving case, we may again relate the effective area to the transmitting gain, but we must specify exactly what gain we mean since (32), (33), and (34) supply infinite possibilities, each depending on the choice of incident waves 𝑎₁. Considering the plane wave given in (14), the incident wave amplitudes 𝑎₂ may be computed using (17). The result is 𝑒_pw,𝑛𝜃_𝑜,𝜙_𝑜=𝑢_pw⋅𝑒_𝑛̂𝜃_𝑜,𝜙_𝑜, (35) which is the scalar components of the vector field functions in the polarization direction of the incident plane wave, so that 𝑎_2,𝑛=−𝑗𝜆(𝐸_𝑜/√𝜂_𝑜)𝑒_pw. Using this notation with (22), 𝑏₁=𝑆₁₂𝑎₂, and 𝑆₁₂=𝑆^𝑇₂₁, we find that the total power collected by the antenna terminals is𝑏^𝐻₁𝑏₁=𝜆²𝐸_𝑜²𝜂_𝑜𝑒^𝐻_pw𝑆^𝐻₁₂𝑆₁₂𝑒_pw=𝜆²𝐸_𝑜²𝜂_𝑜𝑒^𝐻_pwΛ^𝐻Λ𝑒_pw, (36) so that the effective area is 𝐴_𝑒𝜃_𝑜,𝜙_𝑜=𝜆²𝑒^𝐻_pwΛ^𝐻Λ𝑒_pw. (37) This expression explicitly shows how the effective area depends on the mode efficiencies.

To relate this to gain, we first consider the transmitting case, but only the component of radiated electric field 𝐸 polarized in the direction of the incident plane wave:𝐸⋅𝑢_pw=𝐸_pw=_𝑛𝑏_2,𝑛𝑒_𝑛̂⋅𝑢_pw𝑒^{−𝑗𝑘𝑟}𝑟=_𝑛𝑏_2,𝑛𝑒_pw,𝑛𝑒^{−𝑗𝑘𝑟}𝑟. (38) The realized gain pattern for this component of radiated field is𝐺_pw𝑏=4𝜋^𝐻2𝑒^∗_pw𝑒^𝑇_pw𝑏₂𝑎^𝐻₁𝑎₁𝑎=4𝜋^𝐻1𝑆^𝐻₂₁𝑒^∗_pw𝑒^𝑇_pw𝑆₂₁𝑎₁𝑎^𝐻₁𝑎₁. (39) As mentioned above, this gain pattern is a function of the incident waves 𝑎₁ and may take on a continuum of values. Because the gain has a quadratic form (Rayleigh’s quotient), its maximum value occurs when the incident wave vector is the eigenvector corresponding to the largest eigenvalue [19]. However, the matrix 𝑒^∗_pw𝑒^𝑇_pw in (39) is an outer product of a single vector and therefore possesses only one degree of freedom. Thus, all but one of its eigenvalues are zero. It is straightforward to show that the incident wave vector 𝑎₁=𝑆^𝐻₂₁𝑒^∗_pw(𝜃_𝑜,𝜙_𝑜) produces the maximum gain (in the 𝜃_𝑜,𝜙_𝑜 direction):𝐺_pwmax=4𝜋𝑒^𝑇_pw𝑆₂₁𝑆^𝐻₂₁𝑒^∗_pw=4𝜋𝑒^𝐻_pwΛ^𝐻Λ𝑒_pw. (40) To arrive at the last equality, we used reciprocity (𝑆₂₁=𝑆^𝑇₁₂) and conjugated the (real) result. Thus, we see that the effective area 𝐴_𝑒=𝜆²𝐺4𝜋^pwmax (41) is proportional to the maximum realized gain pattern computed from the component of radiated electric field that lies in the direction of polarization of the incident plane wave.

In general, the mode coupling matrix [𝑒^∗̂⋅𝑒^𝑇̂] can be expressed as the product of an 𝑁×2 matrix with its conjugate transpose and therefore has only two degrees of freedom. This means that for any chosen direction, the matrix has only two nonzero eigenvalues, with the largest corresponding to maximum realized gain. This excitation produces the maximum gain with a particular polarization, and one can show that the excitation corresponding to the other nonzero eigenvalue is the gain for the orthogonal polarization. The remaining 𝑁−2 zero eigenvalues correspond to the set of incident terminal port waves that produce null fields in the chosen direction.

5. Backscatter Modes

It is often useful to determine the incident terminal port waves that produce retrodirectively reflected waves with the same relative values, and we refer to these as backscatter modes to distinguish them from radiation modes. Mathematically, this is a conjugate eigenvalue problem where we determine the set of incident wave vectors 𝑎₁ that satisfy𝑆₁₁𝑎₁=Γ_𝑛𝑎^∗₁ (42) for a complex reflection coefficient Γ_𝑛. If an orthonormal set of such excitations exists, collected in the columns of a matrix 𝑉 with the 𝑛th column corresponding to the eigenvalue Γ_𝑛, it is straightforward to show that 𝑆₁₁ must be symmetric. Thus, we conclude that the ability to diagonalize 𝑆₁₁ implies terminal port reciprocity.

The unitary matrix 𝑉 whose columns satisfy (42) diagonalizes 𝑆₁₁ in the following way:𝑆₁₁𝑉=𝑉^∗Γ⟶𝑉^𝑇𝑆₁₁𝑉=Γ, (43) where Γ the diagonal matrix of eigenvalues. The matrix 𝑉 describes unitary transformations of the original incident and reflected terminal waves 𝑎₁ and 𝑏₁ to a new set 𝑎^′₁ and 𝑏^′₁ according to𝑎₁=𝑉𝑎^′₁,𝑏₁=𝑉^∗𝑏^′₁. (44) As with the radiation modes, this transformation may be represented by a lossless, reciprocal network that has a scattering matrix𝑆_𝑉=0𝑉^𝑇𝑉0. (45) Inserting this network on the terminal side of the antenna (as in Figure 3 with 𝑆_𝑈→𝑆_𝑉) gives a net scattering matrix for the combined diagonalization network and antenna equal to𝑉𝑆′=^𝑇𝑆₁₁𝑉𝑉^𝑇𝑆₁₂𝑆₂₁𝑉𝑆₂₂=Γ𝑆^′₁₂𝑆^′₂₁𝑆₂₂. (46) Thus, we see that this network diagonalizes 𝑆₁₁, eliminating the mutual coupling between input ports. These networks are referred to as decoupling networks and have been studied by a number of groups [11, 12, 20–22]. It is straightforward to show that for a lossless antenna the incident waves that produce the radiation and backscatter modes are identical (i.e.,𝑈=𝑉).

Radiation and backscatter modes are particularly useful for providing physical insight to superdirective antennas. As the element spacing is reduced below 𝜆/2, one finds that the radiation mode patterns overlap and the mode directivities increase in the end-fire direction (when the element factor allows it). Assuming for simplicity that the antenna elements all have identical polarization, using (32), we see that the radiated wave vector 𝑏₂=𝑆₂₁𝑎₁=𝑒^∗ results in the maximum possible directivity for the antenna𝐷_max=4𝜋𝑒^𝐻𝑒=_𝑛𝐷_𝑛 (47) which is the sum of the mode directivities. The incident waves required to produce these modes is 𝑎₁=𝑆⁻¹₂₁𝑒^∗=𝑈Λ^−1/2𝑒^∗. This result shows that higher-order, low-efficiency modes (small 𝜆_𝑛) require substantial incident power to produce the desired output fields. Mode efficiencies may be increased by impedance matching, and this may be greatly facilitated by a decoupling network. Since mutual coupling is eliminated, each mode input may be matched independently from the rest, achieving (in theory) simultaneous matching of all the modes. Even when decoupling network realization is impractical (as is often the case), the backscatter modal analysis is a convenient way to characterize the antenna. Each backscatter mode has a characteristic input reflection coefficient from which the modal quality factor may be computed.

6. Examples

6.1. Half Wave Dipoles

We apply the modal analysis to an ideal, perfectly conducting dipole of length 𝜆/2, not located at the origin, but translated a distance 𝑎 along the 𝑦-axis. This will provide an analytical example using simple closed form expressions for dipole input impedance and radiated fields presented in many texts. Of course, these expressions are approximate, derived assuming a sinusoidal current distribution on the dipole and neglecting the feed discontinuity, but they are reasonably accurate and widely used in the literature. We displace the dipole from the origin to show that the theory accounts for the locations of modal phase centers which are generally not known. The dipole scattering parameters (𝑆₁₁ and 𝑆₂₁) and mode functions will then be used to compute the mutual coupling between two dipoles (in a scattering representation) and show that our result agrees with that given by a mutual coupling formula obtained using a different method, evaluated for large dipole separation.

Assuming a sinusoidal dipole current, a 𝑧-oriented half wave dipole located at 𝑦=𝑎 radiates a far zone electric field [23]𝐸=𝑗𝜂_𝑜𝐼₁𝑒^{−𝑗𝑘𝑟}))2𝜋𝑟cos((𝜋/2cos𝜃𝑒sin𝜃^{𝑗𝑘𝑎sin𝜃cos𝜙}𝑢_𝜃 (48) proportional to the current 𝐼₁ entering its terminals. This current is related to the incident wave amplitude as 𝑍_𝑜𝐼₁=(1−𝑆₁₁)𝑎₁. The reflection coefficient 𝑆₁₁ may be computed from the input impedance of the dipole [24],𝑍₁₁=𝜂_𝑜()()4𝜋𝛾+ln2𝜋−Ci2𝜋+𝑗Si2𝜋, (49) where 𝛾 is Euler’s constant and Ci and Si are the cosine and sine integrals.

We compute the 𝑀 “matrix” (a single element) by substituting (48) in (3):𝜂𝑀=^𝑜𝑍_𝑜1−𝑆₁₁²1(𝑆4𝜋𝛾+ln2𝜋−Ci2𝜋)=1−₁₁². (50) The last equality was obtained using (49). This matrix is already diagonalized, so the eigenvalue equals 𝑀 and the eigenvector matrix can be taken as unity. The mode function is then̂𝑒𝑒=^{𝑗𝑘𝑎sin𝜃cos𝜙}𝜋()))𝛾+ln2𝜋−Ci2𝜋cos((𝜋/2cos𝜃𝑢sin𝜃^𝜃, (51) normalized to satisfy (6). Note that the phase dependence introduced by the offset phase center is included in the mode function. The parameter 𝜆₁ is determined from (8), which, using (21), gives the scattering parameter 𝑆₂₁=𝑗𝜂_𝑜𝑍_𝑜1−𝑆₁₁𝛾+ln2𝜋−Ci(2𝜋)4𝜋, (52) where we have rotated the radiation port reference plane back to the origin 𝑟=0.

Mutual coupling between identical half wave dipoles may be computed using (29). The mode function for a second dipole located at the origin of a new (primed) coordinate system of distance 𝑑 along the 𝑦-axis of the original system is given by 𝑒̂^II=𝑒^{−𝑗𝑘𝑑}𝜋()()𝛾+ln2𝜋−Ci2𝜋cos𝜋/2cos𝜃′𝑢sin𝜃′^′𝜃, (53) where we have defined the radiation port reference plane so that it coincides with the reference plane of the first dipole. Substituting (51), (52), and (53) in (29), we obtain, after some manipulation, the terminal-to-terminal scattering parameter 𝑆^total₂₁=−𝑗𝜆𝑆^𝐼₂₁𝑆^𝐼𝐼₁₂𝑒^𝐼̂⋅𝑒̂^𝐼𝐼𝜂=𝑗𝜆^𝑜𝑍_𝑜1−𝑆₁₁2𝜋²𝑒^{−𝑗𝑘(𝑑−𝑎)}𝑑. (54) For comparison, the expression for mutual coupling between ideal half-wave broadside-coupled dipoles in the limit of a large separation distance 𝑟 is given by [25]:𝑍^total₂₁𝜂=𝑗^𝑜𝜋𝑒^{−𝑗𝑘𝑟}𝑘𝑟, (55) where we have retained the superscript to distinguish this coupling parameter from that of a single dipole. Because 𝑍^total₂₁<<𝑍_𝑜, the associated scattering parameter is𝑆^total₂₁2≅𝑗𝜋𝑍_𝑜𝜂_𝑜𝑍₁₁+𝑍_𝑜²𝑒^{−𝑗𝑘(𝑑−𝑎)}𝜂𝑘𝑑=𝑗𝜆^𝑜𝑍_𝑜1−𝑆₁₁2𝜋²𝑒^{−𝑗𝑘(𝑑−𝑎)}𝑑, (56) agreeing with (54). Of course, for these simple linear antennas, the standard Z matrix approach is much simpler. But the modal method is easily generalized to arbitrary antenna elements arranged in any manner.

6.2. Open-Ended Waveguide Array

To show how the theory may be used in practice, we consider an array of three open ended waveguides, illustrated in Figure 4, radiating in empty space at a frequency of 3 GHz. This arrangement was chosen for its generality, including a nonplanar arrangement of radiating elements and mixed polarization. The waveguide walls are of infinitesimal thickness and their lengths finite with ideal ports located internally, as indicated in the figure.

Figure 4 

EM simulation model of three open-ended waveguides radiating into empty space, operating at 3 GHz. Each waveguide consists of four perfectly conducting walls, an excitation port, and a radiating aperture. The model was simulated using HFSS with perfectly matched layer (PML) boundaries in the far field.

Three sets of far zone electric fields 𝐸_𝑛 were computed using a commercial finite element simulator with 1 W of power incident from each of the three waveguide ports in succession with the unexcited ports terminated in matched loads. The matrix 𝑀 of overlap integrals was computed as in (3) and then diagonalized, giving the following sets of eigenvalues and eigenvectors:Λ^𝐻Λ=,1.000000.940000.88𝑈=.58∠45^∘.38∠−107^∘.72∠95^∘.77∠55^∘.13∠−10^∘.63∠−68^∘.27∠−51^∘.92∠0.6^∘.29∠41^∘. (57) Because the antennas are perfectly conducting, the matrix 𝑈 also diagonalizes the terminal port scattering matrix 𝑆₁₁=.26∠−151^∘.11∠10^∘.03∠−176^∘∘.11∠10^∘.19∠−178^∘.09∠33^∘.03∠−176^∘.09∠33^∘.25∠−10^∘, (58) in the manner of (43) (𝑉=𝑈).

Next, we use (1) to compute the modal fields 𝐸⁻_𝑛 and normalize them to determine the mode functions according to 𝑒_𝑛̂=𝐸⁻_𝑛∬𝐸⁻_𝑛²=𝑟𝑑Ω𝐸⁻_𝑛√𝜂_𝑜𝜆_𝑛. (59) Note that this particular normalization equates the phases of 𝐸⁻_𝑛 and 𝑒_𝑛̂. A different choice would produce a corresponding phase change in the 𝜆_𝑛. How we allocate the phase between 𝜆_𝑛 and 𝑒_𝑛̂ in (7) is arbitrary.

Figure 5 contains plots of the modal directivity patterns in dB versus polar angle 𝜃, each taken in the plane of 𝜙 containing the maximum total directivity. The 𝜃 and 𝜙 components of directivity are plotted separately to provide more detail. The radiation scattering matrix 𝑆₂₁ is computed using (22): 𝑆₂₁=.59∠−45^∘.77∠−55^∘.27∠51^∘.36∠107^∘.13∠10^∘.89∠−0.6^∘.68∠−95^∘.59∠68^∘.28∠−41^∘. (60)

Figure 5 

Plots of directivity 𝐷 (dB) versus theta for the three radiation modes. Each plot is taken at the phi cut containing maximum total directivity. The three modes are mutually orthogonal over the entire sphere.

Knowledge of the scattering submatrix 𝑆₂₁ and the mode functions 𝑒_𝑛̂ makes many antenna calculations straightforward. As an example, we determine the maximum possible realized gain of this antenna. We first compute the mode overlap matrix [𝑒^∗̂⋅𝑒^𝑇̂]_𝑚𝑛=𝑒^∗_𝑚̂(𝜃,𝜙)⋅𝑒_𝑛̂(𝜃,𝜙) and then the matrix product 𝐶=𝑆^𝐻₂₁[𝑒^∗̂⋅𝑒^𝑇̂]𝑆₂₁ from (34). The realized gain is maximized in the 𝜃,𝜙 direction by a terminal port incident wave vector equal to the eigenvector corresponding to the largest eigenvalue of 𝐶, denoted 𝜆_𝑐,max(𝜃,𝜙). For the antenna considered here, the largest of all such eigenvalues occurs for 𝜃=15.5^∘, 𝜙=82^∘ (within the accuracy of our discretization of 𝜃,𝜙) with a value 𝜆_𝑐,max=0.68. This corresponds to a maximum possible realized gain of 9.3 dB. The incident wave vector that produces this gain is𝑎₁=0.70∠32^∘0.70∠25^∘0.12∠−125^∘ (61) with a gain pattern shown in Figure 6, plotted in the 𝜙 cut containing the maximum. The unit polarization vector for this radiated field is determined using (9) with (61):𝑢₁=1∠71^∘1∠5^∘. (62) An excitation based on the other nonzero eigenvector of 𝐶 in this direction produces the orthogonal polarization𝑢₂=1∠−47^∘1∠67^∘ (63) with a corresponding gain of 6.7 dB.

Figure 6 

Plot of the realized gain (dB) versus theta for the excitation that maximizes this quantity. The phi cut is the plane containing the maximum gain value of 9.3 dB.

7. Conclusion

The formalism presented in this paper provides a network description of multiport antennas, derived directly from measurements, that exactly accounts for mutual coupling and scattering between elements. Since the radiation and reception properties of the antenna are completely described by a scattering matrix and a set of orthonormal mode functions, the approach simplifies many antenna computations and allows for easy accounting of power transfer. We believe this theory gives physical insight to superdirectivity, and will be useful for various antenna applications such as non-Foster matching of antenna arrays, direction finding, phased array beamforming, and compressive sensing.

Appendix

Asymptotic Evaluation of Overlap Integral

We simplify the integral (16) by rotating the coordinate system so that the plane wave is incident along the positive 𝑧-axis, and first asymptotically evaluate the integral over 𝜃 as 𝑟→∞:𝐼=^𝜋₀𝑓(𝜃)𝑒^{𝑗𝑘𝑟cos𝜃}sin𝜃𝑑𝜃, (A.1) where 𝑓(𝜃)=𝑢_𝑝𝑤⋅𝑒_𝑛̂(𝜃) for any 𝜙. The argument of the exponent has saddle points where its derivative vanishes, namely, 𝜃=0 and 𝜃=𝜋. Considering 𝜃 as a complex variable we integrate along a contour 𝐶 on the real axis from 0 to 𝜋 in the complex plane. A well-behaved function 𝑓(𝜃) will produce an integrand that is analytic in and around the integration contour, allowing us to deform the contour along a path for which the imaginary part of the exponent vanishes at the saddle point. Considering the saddle point at 𝜃=0 first, the steepest decent contour satisfies Im𝑗𝑘𝑟cos𝜃−𝑗𝑘𝑟=0. (A.2) Expressing 𝜃 in real (𝜃_𝑟) and imaginary (𝜃_𝑖) parts, we choose the contour with 𝜃_𝑟=𝜃_𝑖 so that (A.2) is satisfied to first order near the saddle point. We then change variables so that the exponent has a quadratic exponential decay away from the saddle point using𝑤²=−2𝑗𝑘𝑟(cos𝜃−1)⟶𝑤𝑑𝑤=𝑗𝑘𝑟sin𝜃𝑑𝜃. (A.3) The contribution to the integral from the 𝜃=0 saddle point is then𝐼₀≈𝑒^𝑗𝑘𝑟𝑗𝑘𝑟^∞₀𝑓(𝑤)𝑒^{−(1/2)𝑤2}𝑤𝑑𝑤≈𝑓_𝜃=0𝑒^𝑗𝑘𝑟𝑗𝑘𝑟. (A.4) The above integral is taken over a real variable because our choice of transformation (A.3) places our integration path on the real axis in the 𝑤-plane. The exponential decay in the exponent allows us to integrate to infinity with little error.

The contribution from the saddle point at 𝜃=𝜋 may be evaluated in a similar manner giving the complete first order asymptotic solution for (A.1) as𝐼≈𝑓_𝜃=0𝑒^𝑗𝑘𝑟𝑗𝑘𝑟+𝑓_𝜃=𝜋𝑒^{−𝑗𝑘𝑟}𝑗𝑘𝑟. (A.5) The evaluation of a function in spherical coordinates at 𝜃=0 and 𝜃=𝜋 eliminates the 𝜙 dependence so that the integral over 𝜙 in (16) simply gives an additional factor of 2𝜋. Rotating the coordinate system back to the original directions, we arrive at the final result𝑢_pw⋅𝑒_𝑛̂𝜃_𝑜,𝜙_𝑜𝑒^{𝑗𝑘𝑟[sin𝜃𝑜sin𝜃cos(𝜙−𝜙𝑜)+cos𝜃𝑜cos𝜃]}𝑢𝑑Ω≈2𝜋_𝑝⋅𝑒_𝑛̂𝜃_𝑜,𝜙_𝑜𝑒^𝑗𝑘𝑟+𝑢𝑗𝑘𝑟^𝑝⋅𝑒_𝑛̂𝜋−𝜃_𝑜,𝜙_𝑜𝑒+𝜋^{−𝑗𝑘𝑟}.𝑗𝑘𝑟 (A.6) The incoming and outgoing waves are both present because the “incident” plane wave exists in all space, incoming on one side of the array and outgoing on the other. It is only the incoming term that is relevant to our analysis of the receiving properties of the array.