The mutual coupling between antenna elements affects the antenna parameters like terminal impedances, reflection coefficients and hence the antenna array performance in terms of radiation characteristics, output signal-to-interference noise ratio (SINR), and radar cross section (RCS). This coupling effect is also known to directly or indirectly influence the steady state and transient response, the resolution capability, interference rejection, and direction-of-arrival (DOA) estimation competence of the array. Researchers have proposed several techniques and designs for optimal performance of phased array in a given signal environment, counteracting the coupling effect. This paper presents a comprehensive review of the methods that model and mitigate the mutual coupling effect for different types of arrays. The parameters that get affected due to the presence of coupling thereby degrading the array performance are discussed. The techniques for optimization of the antenna characteristics in the presence of coupling are also included.

A phased array comprises of definitely arranged, finite sized antenna elements, which are fed by an appropriate feed network. In such an array, the fields radiated from one antenna might be received by the other elements. Furthermore, this signal might get reflected, reradiated, or scattered. The properties of these signals depend on the power of the signal, reflection coefficients, and the additional electrical phase introduced due to the propagation delay from one element to the other. This kind of interaction between the antenna elements will lead to coupling effect and hence can alter the array characteristics.

In other words, in a phased array, the electromagnetic (EM) characteristics of a particular antenna element influence the other elements and are themselves influenced by the elements in their proximity. This interelement influence or mutual coupling between the antennas is dependent on various factors, namely, number and type of antenna elements (A), interelement spacing, relative orientation of elements, radiation characteristics of the radiators, scan angle, bandwidth, direction of arrival (DOA) of the incident signals, and the components of the feed network (Figure

Schematic of adaptive antenna array.

The presence of coupling in an array changes the terminal impedances of the antennas, reflection coefficients, and the array gain. These being the fundamental properties of the array have a greater influence on their radiation characteristics, output signal-to-interference plus noise ratio (SINR) and radar cross section (RCS). Furthermore, it affects the steady state response, transient response, speed of response, resolution capability, interference rejection ability, and DOA estimation competence of the array.

Several researchers have studied the effect of mutual coupling on different types of adaptive arrays. These include Yagi array [

The parameters governing the array performance are obtained using various techniques like method of moments (MoM), multiple signal classification (MUSIC), estimation of signal parameters via rotational invariance techniques (ESPRIT), scheme for spatial multiplexing of local elements (SMILE), and direct data domain (DDD) algorithms. The algorithms used to estimate the coupling can be extended towards the compensation of its effect. Moreover optimization techniques such as genetic algorithm (GA), particle swarm optimization (PSO), and linear programming (LP) can be used in conjuncture with these techniques towards the enhancement of array efficiency. This paper presents a unified review of the techniques and the designs proposed for mitigating mutual coupling effect in phased arrays. The performance parameters that get affected due to the coupling effect in antenna array are discussed. The work reported in open domain towards efficient array design mitigating mutual coupling effect is reviewed and compared.

The subsequent sections describe the analysis and compensation of mutual coupling effect in phased arrays using these techniques. The effect of mutual coupling on the array parameters such as antenna impedance and steering vector, which further affects the radiation pattern, resolution and interference suppression ability, DOA estimation, output SINR, response speed, and RCS is described in Section

The antenna radiation pattern depends mainly on the impedance at the antenna terminals. However, the antenna impedance of phased array is significantly different in comparison to that of an isolated element. This variation in impedance is due to the presence of coupling between the array elements. In general, for an array of

In general, the mutual impedance matrix is a square matrix with the order corresponding to the array size. This implies that the computations involved in arriving at the matrix coefficients increases with the size of array. However, it is possible to exploit certain properties of coupling for reducing the computation complexity. One such property is the inverse dependence of coupling coefficients on the distance between the array elements [

Variation of mutual impedance between two half-wavelength, center-fed dipoles with interelement spacing.

Furthermore, the self- and mutual impedances of (

Carter [

Ehrlich and Short [

The mutual impedance between the microstrip dipoles of arbitrary configuration, printed on a grounded substrate was determined using integral equation-based method [

Inami et al. [

Similar integral equation-based method was employed by Eleftheriades and Rebeiz [

Pozar [

Hansen and Patzold [

The reaction theorem was used to calculate the mutual impedance between two printed antennas of a grounded isotropic dielectric substrate [

The mutual coupling in arbitrarily located parallel cylindrical dipoles [

The fuzzy modelling technique can be employed to calculate the input impedance of two coupled monopole antennas [

The mutual coupling in a nonlinearly loaded antenna array [

The response of an array towards the incident signal is expressed mathematically as steering vector. This vector is dependent on the antenna element positions, inter-element spacing, radiation characteristics of each element, and the polarization of the incident wave. The steering vector of an array can be obtained by direct measurement of complex array element patterns; however, it proves to be difficult. Moreover, it demands for a huge memory to save the measured data, and hence does not present a practical approach [

The presence of coupling between the array elements affects the steering vector, and hence the array response [

As the CSV does not account for coupling effects, the compensation of received voltages becomes mandatory for the analysis of a practical array. To overcome this difficulty, Yuan et al. [

The CSV and USV of an array are related to each other by the following relation:

The presence of coupling in between the elements changes the impedance and hence the radiation pattern of the phased array. This indicates that the accurate calculation of the radiation pattern of an array is feasible only if (i) CSV with an appropriate compensation technique or (ii) USV is used for the calculation. This is apparent from Figure

Synthesized pattern of a 2-element dipole array. Desired signal (green arrow): 0°, 40 dB; 1 jammer (red arrow): 60°, 0 dB.

Zhang et al. [

In order to form a desired receiving array pattern, proper weights with and without mutual coupling are needed. For an array of omnidirectional antenna elements, the steering vector in the absence of mutual coupling is given by

However, when mutual coupling is considered, the steering vector is modified as

Liao and Chan [

The effect of coupling on the pattern synthesis of an array can be analyzed using classical techniques (like pattern multiplication), numerical techniques (like MoM), and active element patterns (AEP) method [

An adaptive antenna is expected to produce a pattern, which has its main beam towards the desired signal, and nulls towards the undesired signals. It is also required that patterns have sufficiently low sidelobe level (SLL). Although the standard pattern synthesis methods [

The principle of pattern multiplication uses the knowledge of currents at the feed terminals of individual elements to arrive at the complete array pattern. This classical technique is applicable only to the array with similar elements as it assumes identical element patterns for all individual array elements. The radiation pattern of an array is expressed as the product of an element and array factor. However, in a practical antenna array, the presence of coupling results in the variation of individual element patterns. Moreover for an array with electrically large elements, which differ in size, shape and/or orientation, the patterns of the individual elements differ considerably. This introduces a noise floor in the array pattern, thereby increasing SLL and degrading the quality of the result.

Such practical situations for which classical approaches become unsuitable can be analyzed using the numerical techniques such as MoM or FDTD. These numerical techniques directly estimate the current distributions over the antenna elements. The voltage and current expansion coefficients are related in terms of self- and mutual impedance matrices. These numerical techniques yield accurate results; however the size of the matrices increases with the array size, and beam scanning. Moreover, these numerical techniques cannot be used for the arrays that are located in highly complex inhomogeneous media.

Kelley and Stutzman [

In unit excitation AEP method, the individual elemental patterns are computed assuming the elements excited by a feed voltage of unit magnitude. These active element patterns represent the pattern of entire array, considering direct excitation of a single element and parasitic excitation of others. Furthermore, these individual patterns are superimposed/summed-up and scaled by a factor of complex-valued feed voltage applied at the terminals to arrive at the complete array pattern. This method is advantageous, as it needs to compute the pattern of individual array elements only once. Moreover, this method is valid for arrays of both similar and/or dissimilar elements, located in inhomogeneous linear media.

The dependence of the element pattern on the array geometry can be explicitly mentioned using an exponential term. Such approach in which individual element patterns vary due to the presence of additional spatial phase information factor is called phase-adjusted unit-excitation AEP method. This method considers the spatial translation of the array elements, unlike the unit-excitation element pattern, which refers only to the origin of the array coordinate system. Although the phase-adjusted element patterns differ for different array geometries, the concept is useful in the development of approximate array analysis methods.

The computational complexity of both unit-excitation and phase-adjusted unit-excitation methods increases with the array size due to the need of the active element pattern data for every array element. However, if the uniform array is infinitely large, then the phase-adjusted active element patterns of individual elements will become identical. In such scenarios, the complete array pattern can be expressed in the form of an average active element pattern, which will be the active element pattern of a typical interior element [

Hybrid active element pattern method, proposed by Kelley and Stutzman [

In a large array, almost all the elements experience similar EM environment, unlike small array with prominent edge effects. This causes a considerable difference in the individual element patterns of the array, leading to higher SLL. Moreover, small arrays require exceedingly fine control over both magnitude and phase of each element for accurate beam steering. Darwood et al. [

The mutual coupling compensation can be done by multiplying the inverse of coupling matrix and

This process is simple for an array with single mode elements; the coupling compensation matrix,

The coupling matrix can be estimated [

This solution is based on the assumptions that

The second method for calculating coupling coefficients is based on the scattering matrix

The measurement of the network parameters becomes difficult when the feed lines between the element apertures and output terminals are not matched. Furthermore, this method requires each element to be driven in both transmit and receive modes. The information about the reference plane corresponding to the phase center of each radiating element is required, which is not feasible for a real array, composed of nonideal elements and complex feed network [

Darwood et al. [

The mutual coupling compensation requires the knowledge of coupling coefficients, which can be waived by using the experimental method of applying retrodirective beams [

In general, the coupling matrix-based methods assume that the coupling matrix is an averaged effect of the angle-dependent relationship between the active element patterns and the stand-alone element patterns. In such scenario, the minimum mean-square error (MMSE) matching of the two pattern sets for a few known incident angle yields the coupling matrix [

In certain scenarios, the antenna arrays are loaded with nonlinear devices in view of protection from the external power. The analysis of such arrays is complex due to the nonlinear characteristics of each array element. For such arrays, Lee [

The mutual coupling in a finite array of printed dipoles fed by a corporate feed network was studied by Lee and Chu [

The effects of coupling for a microstrip GSM phased array fed by a Butler feed network were analyzed [

The presence of mutual coupling amongst the array elements affects the array resolution adversely. Manikas and Fistas [

The mutual coupling worsens the array performance further if the signals are wideband. This is because the array loses its ability to match the desired signals or to null the jammers [

The resolution capability of an array is also affected due to array calibration errors, similar to that of its radiation pattern. Such effects in eigenstructure-based method, MUSIC, presented by Friedlander [

An adaptive array is expected to accurately place sufficiently deep nulls towards the impinging unwanted signals. The presence of mutual coupling between the antenna elements affects both the positioning and the depth of the nulls. The interference rejection ability depends on array geometry, direction of arrivals (DOA) of the signals, and weight adaptation. Riegler and Compton [

Adve and Sarkar [

An improvement over the technique of open circuit voltage method so as to include the scattering effect of antenna elements was presented [

The current distribution of a small helical antenna is shown to be independent of azimuth angle of the incident field, if it impinges from horizontal direction [

Some special techniques were proposed for small and ultrawideband arrays. Darwood et al. [

An adaptive array needs to estimate accurately the emitter location (DOA) and other details so as to suppress it effectively. DOA estimation depends on the array parameters determined by various techniques. These techniques are either spectral based or parametric based [

MUSIC and ESPRIT algorithms are among the popular methods for DOA estimation. The sensitivity of the MUSIC algorithm to the system errors in the presence of coupling was studied by Friedlander [

A preprocessing technique for accurate DOA estimation in coherent signal environment was proposed for uniform circular array [

The coupling affects the phase vectors of radiation sources, which in turn varies the signal covariance matrix and its eigenvalues, affecting the array performance [

Pasala and Friel [

The coupling effect on the performance of ESPRIT algorithm for a uniform linear array was studied by Himed and Weiner [

Fletcher and Darwood [

The coupling effect in DOA estimation capacity of a smart array of dipoles was studied using Numerical Electromagnetics Code (NEC). The NEC simulation considers the coupling effect by using compensated steering vectors [

Inoue et al. [

Another approach based on the concept of interpolated arrays was proposed [

Accurate DOA estimation in the presence of coupling for a normal mode helical antenna and dipole array was presented [

The mutual impedance terms in (

As already mentioned, an array response towards the incident field is accurately expressed by USV and not by CSV. Thus the accuracy of DOA estimation algorithms can be improved using USV [

Any conventional method assumes a ULA for mutual coupling compensation. Lindmark [

The techniques of coupling compensation proposed by Coetzee and Yu [

The presence of coupling between the antenna elements affects both steady state and transient response of an array. In general, the output SINR represents the steady-state performance of the array, while the transient response is expressed in terms of speed of array response. Figure

Effect of

The output SINR of a least mean square (LMS) adaptive array in the presence of multiple interfering signals is given by [

For narrowband signals, uniformly distributed over

Effect of inter-element spacing on output SINR of a 16-element array of half-wavelength, center-fed dipoles;

Comparison of output SINR in the presence of mutual coupling by varying the number of antenna elements.

: Variation of output SINR of an array of half-wavelength, center-fed dipoles of fixed aperture with array size.

The output SINR is proportional to the gain of adaptive system based on its input SINR. The presence of coupling affects both input and output SINRs, especially if the inter-element spacing is less [

One of the desired characteristics of the adaptive array is its ability to adapt to the changes in signal environment instantly. This requires a quick updating process of the weight vector of an array, which in turn is a function of feedback loop gain, steering vector, and signal covariance matrix. The signal covariance matrix of an adaptive array is expressed as [

Since the covariance matrix depends on the impedance of antenna elements, the mutual coupling affects the transient response of the array. The coupling changes the eigenvalues of the signal covariance matrix. Smaller inter-element spacing causes greater coupling effect, lowering the eigenvalues, and hence longer transients. This reduces the speed of response of an array, resulting in delayed suppression of jammers. The performance analysis of an adaptive array in the presence of coupling by Dinger [

The major focus for strategic applications is towards the reduction of radar cross section (RCS) of antenna array while maintaining an adequate array functionality in terms of gain, beam steering, and interference rejection. This necessitates the analysis and compensation of mutual coupling in array system. The RCS of an array is affected by coupling effects; mutual coupling changes the terminal impedance of the antenna elements and hence the reflection coefficients within the feed network. The coupling effect depends on the type of antenna element, array geometry, scan angle, and the nature of feed network. Figure

Schematic for 30-element series-fed dipole array.

Effect of mutual coupling on RCS of series-fed linear collinear dipole array of

Effect of mutual coupling on RCS of series-fed linear parallel-in-echelon dipole array of

Effect of mutual coupling on RCS of series-fed linear side-by-side dipole array of

Abdelaziz [

Knowing the coupling factor, actual excited voltages at the antenna terminals are obtained as

Zhang et al. [

The array performance can be further improved by optimizing the array parameters. These optimization techniques can be either global [

In the preceding sections, the effects of the coupling on the array performance and their compensation were discussed. It should be noted that the source of errors that hinder the array performance are due to improper antenna designs. In other words, a careful and efficient design of an antenna system can effectively minimize the mutual coupling between the array elements.

Lindmark et al. [

Another design technique to mitigate the effects of coupling is to use dummy columns terminated with matched loads on each side of the array. This is effective as it pseudo equalizes the environment around the outer columns of the array to that at its inner columns. Although such an array design shows an improved performance [

A wideband folded dipole array in the presence of mutual coupling was analyzed [

In general, patch antenna is designed on a thick substrate for wideband performance and higher data rates. However, this enhances the coupling effect, as thicker substrate supports higher amount of current flow in the form of surface waves. Fredrick et al. [

Blank and Hutt [

Many attempts have been made to compensate the effect of coupling in microstrip antennas. The finite difference time domain (FDTD) method was used to analyze the array of electromagnetic band gap structures, composing printed antennas on a single isotropic dielectric substrate [

Yousefzadeh et al. [

Yang and Rahmat-Samii [

Bait-Suwailam et al. [

Digital beamforming (DBF) of an array is preferred over analog beamforming, owing to low sidelobe beamforming, adaptive interference cancellation, high-resolution DOA estimation, and easy compensation for coupling and calibration errors [

Demarcke et al. [

In general, beamforming is viewed as a constrained optimization problem. Thus the evolutionary algorithms and related swarm-based techniques, useful for solving unconstrained optimization problems, are not applicable readily for beamforming. The improvement can be achieved by optimizing the system parameters using algorithms like particle swarm optimization (PSO). Basu and Mahanti [

The mutual coupling in conformal arrays is dependent on the curvature of surface on which antennas are mounted. A majority of techniques used to analyze the conformal arrays [

Pathak and Wang [

Wills [

The radiation pattern of a dipole array mounted on a real complex conducting structure [

Obelleiro et al. [

A method of finite element-boundary integral (FE-BI) was used to determine the mutual impedance between conformal cavity-backed patch antennas [

A finite array of microstrip patch antennas loaded with dielectric layers on a cylindrical structure was studied by Vegni and Toscano [

The effect of human coupling on the performance of textile antenna, like a log periodic folded dipole array (LPFDA) antenna, was analyzed [

The mutual coupling between the apertures of dielectric-covered PEC circular cylinders was determined in terms of tangential magnetic current sources of the waveguide-fed aperture antennas/arrays [

The discussion presented so far shows that the conformal arrays can be analyzed using various techniques, each with their own merits and demerits. This indicates that a few of them when carefully chosen and combined [

Raffaelli et al. [

A conformal conical slot array was optimally synthesized taking coupling effect into account [

Thors et al. [

The performance of the phased array deteriorates due to the presence of coupling; however, this cannot be generalized. The presence of coupling is reported to be advantageous in certain cases. The coupling has positive effect on the channel capacity of multiple element antenna (MEA) systems [

The presence of coupling between the array elements is shown to be desirable, if arrays are to be made self-calibrating [

This paper presents an overview of the methods that model mutual coupling effect in terms of impedance matrix for different arrays. Researchers have extended the conventional methods based on self- and mutual impedance matrix to include the effects of calibration errors and near field scatterers. These methods aimed at compensating the effects of mutual coupling by including the coupling matrix in the pattern generation. There are autocalibration methods which mitigate the effects of structural scattering along with the mutual coupling mitigation; such methods facilitate the analysis of conformal phased arrays.

The trend moved towards developing the techniques, which estimate the parameters affecting the performance of real-scenario arrays accurately in extreme conditions including that for coherent signals with minimum number of inputs. This reduced the computational complexity and facilitated easy experimental verification and parametric analysis. It has been shown that the accurate array pattern synthesis is feasible only if the effect of coupling on the array manifold is considered. Thus the USV is used instead of CSV in the coupling analysis of phased arrays mounted over the platform, such as aircraft wings or mobile phones.

Further hybrid techniques, like UTD-MoM, spectral domain method with UTD, were shown to be better in terms of both accuracy and computation. Most of these techniques are suitable only for uniform and infinitely large arrays in narrowband scenarios. Therefore, adaptive array based technique was developed to deal with the small non-uniform planar and circular arrays operating over a wide frequency range.

The effect of mutual coupling on the parameters like terminal impedances and eigenvalues has been discussed considering the radiation pattern, steady state response, transient response, and the RCS of the array. Many compensation techniques are analyzed to mitigate the adverse effects of coupling on array performance issues such as high resolution, DOA estimation, and interference suppression. These have been further simplified due to the optimization of antenna design parameters. It is suggested that a good and efficient design of an antenna system would compensate for the mutual coupling effects. However, in few cases the presence of coupling has been proved advantageous as well.

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