Decoupling and Matching Strategies for Compact Antenna Arrays

Antenna arrays have been used in various applications and have become an important tool to achieve high spectral efficiency in wireless communications. Its use brings to the communications system an increase in performance in terms of capacity and reliability. Recently one of the main communications architectures that makes use of antenna arrays is the Multiple-Input Multiple-Output (MIMO) technology. MIMO technology has been applied in antenna arrays composed of elements in the order of tens to hundreds. In this way, it is necessary to use the structure of compact antennas that offer all the necessary robustness to the applied project. In this context, it is important to revisit the concepts of mutual coupling and impedance matching among antenna elements in an array. This paper proposes and evaluates three strategies of joint decoupling and impedance matching networks(DMN) for antenna arrays. The first method called DMN with Lumped Elements (DMN-LE) performs the decoupling and impedance matching steps with capacitors and inductors. The second method is called the Ring Hybrid (DMN-RH). It is utilizes a microstrip line in the ring form. With this approach is achieved first the decoupling followed by impedance matching steps. The third method is called Networkless Decoupling and Matching (NDM). It brings a concept of decoupling without the presence of a network itself. A comparison of the methods is performed both analytically and via computer simulations. We conclude that the third method, networkless one, is an interesting new alternative approach.

analyze and revisit two important concepts in an antenna array: mutual coupling and impedance matching among antenna elements in the array. When utilizing antenna arrays it is important to model these arrays in a physically consistent way. This is best done by a matrix description of the array multiport, which of course depends on the type of antenna elements, their orientation and spacing [5].
The topic of designing multiports for antenna arrays, which provide decoupled and matched ports for the amplifiers (either high power amplifiers (HPA) for transmission or low noise amplifiers (LNAs) for reception), has been treated in the literature by several authors [6]- [7], [8], [9]- [10], [11]- [12]. Some of them are using lumped elements, some are using strip lines, and some use the so-called parasitic antenna.
This paper proposes and evaluates three strategies of joint Decoupling and Impedance Matching Network (DMN) for an array of dipole antennas. From the multiport communication theory presented in [3], the appropriate equations for the proposed DMN are taken. The main contribution of this paper is to offer three possible DMN implementations and compare their performance.
The first method, called DMN with Lumped Elements (DMN-LE), performs the decoupling and impedance matching steps by a lumped circuit with capacitors and inductors. This method can easily generalized to any number N, but practically limited to reasonably low N. The second method, called DMN with Ring Hybrid (DMN-RH), is modeled with a microstrip line in a ring format. The DMN-RH is limited to two antenna elements. The third method, called Networkless Decoupling and Matching (NDM), brings a concept of decoupling and impedance matching without the presence of a multiport network itself. This enables the design of an antenna array that performs DMN operations in a simplified and compact manner. For a fair comparison of the three approaches we restrict the detailed analyses and simulation to an array of only two antennas elements. This is justified by the fact that the complexity of all DMN designs is unfortunately growing quadratically with the number of antenna elements N (it grows at least with N 2 + N as shown in [13]), the implementation of DMNs is limited to a low number of antenna elements, say 2, 3 or at most 4. Nevertheless these small arrays are still useful for massive MIMO, because they pave the way to implement a massively large array with a reasonable large number of such small subarrays, each of them with very few antennas, say 2 or 3. In this way,within the subarrays, we use very small spacing between antenna elements and decouple them with realizable DMNs, while the spacing between these subarrays should not be that small, such that the residual mutual coupling could be ignored.
A case study will be conducted in order to compare the performance results and obtain a DMN structure capable of providing high efficiency to the antenna arrays and communication systems.
The rest of paper is organized as follows. Section II brings the general definition and concept of DMN. Section III presents the DMN-LE along with an analysis of the system. Section IV presents the DMN-RH, explaining the process of its design. Section V demonstrates the networkless decoupling technique along with the impedance matching step. Section VI shows computer simulation results and discusses them. Finally, Section VII summarizes the findings and gives an outlook to future works.

II. DECOUPLING AND MATCHING NETWORK (DMN)
The DMN is a device capable of performing the decoupling operations between the elements of an antenna array and the impedance matching of the connected amplifier ports. The DMN ports are designed to provide an impedance of 50Ω on the source side. Fig. 1 shows the representation of the 2N-Port DMN providing decoupled ports matched to the sources, each source with impedance R = 50Ω. In this representation the source comprised of ports T 1 to T N represents the insertion of the signal that will be transmitted in each port. On the other side of the DMN, on the ports A 1 to A N , the antenna array with its impedance matrix (Z AT ) is connected.
Equations (1) and (2) show the definition of the voltage and current vectors at the aforementioned ports: The purpose of this method is to obtain the matrix Z M T such that the sources achieve power matching and decoupling by the DMN as, where I is the identity matrix. This can be achieved with the following equations: where, from [3] and [14]. The antenna array that will be discussed in this paper is presented in Fig. 2. This array consists of two half-wavelength dipole antennas where the distance between them is given by ρ. The feed is also represented in Fig. 2, located at the center between the λ/4 segments with its respective voltage u A and current i A for each antenna element.

III. DMN WITH LUMPED ELEMENTS (DMN-LE)
The DMN-LE performs the decoupling between the elements of the antenna array and the impedance matching between the array and the source by means of lumped elements. Lumped elements are widely used in microwave systems to perform impedance transformation [6], [7], [15]- [17].

A. Design of DMN Lumped Elements
For the detailed design of DMN-LE it is convenient to work with the admittance matrix Y M T . Its representation will be referred to as a general 4-port implemented with the reactive elements Y 1 to Y 10 according to Fig. 3, when the number of elements of the antenna array is defined as N = 2. For this diagram, the ports u T1 and u T2 represent the input ports of the DMN-LE, while u A1 and u A2 represent the output ports connecting the antenna array. The DMN-LE is designed to operate at a center reference frequency f r .
The DMN-LE is initially characterized by Y M T and in order to perform the modeling of each of its elements, it is necessary to know the impedance matrix Z AT of the antenna array. The matrix Y M T is obtained from Z M T by inversion as, where, and The impedance matrix of the array in Fig. 2 has been computed according to [18,Chapter 13] and results in (12).
where each of its elements is defined as Y 1 = y 11 + y 12 + y 13 + y 14 , Y 2 = y 22 + y 12 + y 23 + y 24 , Y 3 = y 33 + y 13 + y 23 + y 34 , Y 4 = y 44 + y 14 + y 24 + y 34 , The conversion of each element of the matrix Y M T into capacitive or inductive elements is performed based on Y i = jB i , as shown in (15) and (16). Thus, if B i > 0, and if B i < 0, we have, where ω = 2πf r . For the DMN-LE modeling with the addition of losses, the equivalent circuit model employs the concept of Q-factor. In this way, using the Q-factor obtained from the manufacturer of the electronic component, [19] and [20], it is possible to take losses into account.

IV. DMN WITH RING HYBRID (DMN-RH)
The Ring Hybrid (RH) is a linear microwave device which is generally defined as a 4-port [8]. It is composed of the branch-line coupler design with a 90 • hybrid junction, with spacing of λ/4 between ports T 1 and A 2 , T 1 and A 1 , and A 1 and T 2 . The spacing between ports T 2 and A 2 is 3λ/4. Such a RH is readily implemented using microstrip lines as shown in Fig. 4. In relation to obtaining the impedance matching of the ports T 1 and T 2 , a transmission line segment is added to each of these ports.
In this way, the RH is a useful solution for the DMN allowing the operation of mutual decoupling between the ports and the impedance matching, only with its use. Fig. 4 shows the standard design of the RH with impedance characteristic Z 0 .

A. Design of the DMN-RH
The RH modeling is based on formatting the microstrip configuration parameters that will be used. To the design of DMN-RH, it is necessary to define the parameters of the transmission line segments both in the ring itself and at the ports T 1 and T 2 . Thus, to define the line widths of the ring, the impedance Z 0 , and the port impedances Z 1 (T 1 ) and Z 2 (T 2 ) are required. The derivation of these parameters is based on the circuit analysis of the RH microstrip according to [8]. For the derivation of parameter Z 0 (RH characteristic impedance) the equation is used based on the impedance parameters of the dipole antenna array, the Z AT matrix, defined previously, where For the derivation of Z 1 and Z 2 of the impedances at T 1 and T 2 , which still have to be matched to R, the following equations are used Section VI-C shows how this can be achieved.

V. NETWORKLESS DECOUPLING AND MATCHING (NDM)
This method aims to achieve the impedance matching and decoupling between the elements of the antenna array without having a multi-port impedance matching/decoupling structure. For this, a new antenna element is inserted equidistantly between the existing elements. Furthermore, 3 linear voltage sources are employed as shown in Fig. 5.
The overall electromagnetic field is a linear superposition of the contributions of all three antenna elements. The excitation voltage of the third antenna element is controlled by the voltages of the other two. This control is designed to let all antenna elements decoupled and matched.
Each port has its voltage phasor given as, The generators must also be designed to have proper output impedance Z i .
In summary, the method allows the first two generators to have arbitrary values, while the voltage of the third generator u 0,3 is controlled by the other two, according to where g 1 and g 2 are properly chosen complex constants.

A. Networkless Matching Design
For the NDM modeling it is necessary to define the vectors Z is the impedance matrix of the antenna array consisting of three elements and Z 0 is the diagonal matrix of the three source impedances, as and with a = Z 11 , b = Z 12 and c = Z 13 . Requiring power matching as given by (19) and (20) leads to the following derivation [21]: where, Thus, given Z, we solve the above equation for Z 0 , g 1 and g 2 to have Finally we have to transform the impedances Z 1 , Z 2 and Z 3 to the source resistance of 50Ω by impedance matching twoports (see Fig. 11).
To verify the performance of the NDM with a computer tool, it is important to make sure that the third port is always driven by the voltage u 03 as given by [21], because only then all three ports will be power matched to their corresponding sources. This is achieved by inserting Voltage Controlled Voltage Sources (VCVS). The gains G 1 and G 2 are determined by the chosen beamforming and G 3 is generated as a function of G 1 and G 2 .
This method is somewhat similar to what is called parasitic elements based decoupling [22], in which the auxiliary antenna is short circuited at the feed point and the geometry is optimized to achieve decoupling.

VI. NUMERICAL RESULTS
Numerical results were obtained to enable a performance comparison between the three concepts in terms of the S parameters, which are scattering parameters. Regarding the metric for the bandwidth, the reference thresholds of -20dB and -10dB will be considered. The following simulation conditions are assumed: • Two-element antenna array with half-wavelength dipoles L = λ/2. • Distance between main antennas: d = λ/4. • Reference carrier frequency: f r = 3GHz. • Impedance matrix with a classical model computed according to [18,Chapter 13] . • Frequency range of interest (for measuring bandwidth): 2.9 GHz -3.1 GHz. The main objective of this analysis is to verify the behavior of the mutual coupling (S 12 and S 21 ) and impedance matching (S 11 and S 22 ) of the proposed DMN approaches over the frequency range of interest. Circuit simulations employ the ANSOFT DESIGNER SV Software. It is important to note that for this analysis the antenna array used has the beamforming in the end-fire direction. Thus, this research analyzes DMN structures directed to the array of antennas with their elements in parallel.

A. Baseline Performance
The performance of the antenna array without DMN is presented in Fig. 6. It shows that the resonance frequency performance is around 2.7 GHz (i.e., 300 MHz off the reference frequency). Impedance matching (S 11 or S 22 ) has value of around −26 dB, at resonance frequency, while for the reference frequency a value of −6 dB is observed. S 12 and S 21 assume a value of around −12 dB at the reference frequency. A reference level of −10 dB is considered for evaluating the system bandwidth from the S parameters.

B. Results for DMN-LE
The circuit equivalent of the DMN-LE is shown in Fig.  7. The values of the components are described in Table I, in which the Q-factor values are given according to [19] and [20].
The behavior of the DMN-LE versus the baseline is presented in Fig. 8a, for S 11 and S 22 . Fig. 8b shows the performance for S 12 and S 21 .     By having both decoupling and impedance matching better than -20 dB, a bandwidth of 100 MHz is achieved. For a behavior of −10 dB, a bandwidth of more than 300 MHz is reached.
With these results, subarray modeling is well understood in order to achieve a performance increase of around 500% compared to the baseline results at the reference frequency for S 11 . For S 12 , there is an increase of 150% compared to the baseline results at the reference frequency.
Comparing the results obtained in this method with others publications, it is possible to realize that the proposed modeling produces important results, in order to meet the necessary requirements, both in mutual coupling and impedance match-  ing. In [23], for example, a DMN network is presented which shows an increase of 18% in the return loss bandwidth by −10 dB. With DMN-LE, a 40% increase from −9.8 dB to −14 dB in the main frequency and a greater bandwidth range were observed. In [10], another DMN is shown which presents a 10 element DMN circuit. Therefore, a simplification in the DMN-LE can be observed by reducing the number of elements and especially in this case, a single element is eliminated.
In [13] a general method is outlined to design DMNs for arbitrary number of antenna elements. For a simple design approach the number of necessary circuit elements is given by 2N 2 + N , which can be reduced to N 2 + N with a more elaborate technique. The design method, which has been used here is somewhat in between with 3 2 (N 2 + N ). Anyway, unfortunately the number of needed circuit elements grows quadratically with the number of antenna elements restricting the method to a moderately low number of antenna elements.

C. Results for DMN-RH
The values obtained from the equations presented for the DMN-RH are show in Table II. Based on these values, the microstrip parameters are obtained.
To obtain the width and length from the impedance value and the electric length, ANSOFT DESIGN SV's microstrip line calculator is used [24]. The substrate RO3006 material is used [25], which has r = 6.15 and the thickness of the substrate is 1.52mm. The thickness of the strip line and the ground plane is 35µm. Now the ports T 1 and T 2 have to be matched to the source resistance of 50Ω. At port T 1 a strip line is connected with characteristic impedance providing the desired transformed port resistance of 50Ω and electric length θ 1 = 2π λ l 1 with Unfortunately T 2 could not be matched in the same way with a single strip line. First the reactive part of Z 2 is eliminated by a strip line with Z 21 = 50Ω and appropriate length l 21 . Then a strip line with length λ and appropriate characteristic impedance Z 22 is used leading to design values given in Table III. The layout for this RH with the matching strip lines for port T 1 and T 2 is given in Fig. 9.  22 11.0431 mm w 22 7.0753 mm The bandwidth for decoupling (S 21 and S 12 ) better than −20 dB is larger than 400 MHz, for matching at port T 1 (S 11 ) better than −20 dB is 250 MHz, while matching at port T 2 (S 22 ) is the bottleneck. Here a bandwidth of only 100 MHz for being better than −10 dB and less than 50 MHz for better than −20 dB is achieved.
This result shows an increase of around 700% to 800% compared to the baseline results for S 11 and S 12 , respectively, in the reference frequency. Similarly, there is an improvement in bandwidth mainly for the S 11 , with a better result than the defined bandwidth, below 2.9 GHz and above 3.1 GHz, considering a threshold -10 dB. In the same way, S 12 and S 21 shows a behavior with an increase of around 550%, at the main frequency. Therefore, according to the observation of these behaviors, it is possible to notice that this structure, despite having a physical limitation on the number of elements, has positive premises for the modeling of antenna subarrays.
In this analysis, when compared to [11] and [26], it is possible to observe that the proposed structure provides better yields, both in the coupling and impedance matching for the used antenna array. In [26], it can be observed that in both behaviors, S 11 and S 22 have a limited bandwidth. For the DMN-RH, it is observed a wider bandwidth mainly for S 22 and S 12 , which demonstrates better behaviors among the modeling.

D. Results for Networkless Decoupling and Matching
The results obtained from (19) to (27) for the NDM modeling are presented in the Tables IV and V, as well as can be visualized in Fig. 11, in which the circuit analysis for the NDM is presented. It is formed by an AC voltage source with a polar voltage with magnitude of 1V and phase of 0 • , and connected to three VCVSs each instantiated by the respective values of u 0 .
The VCVSs are connected to LC-twoports, which provide the impedance matching from 50Ω to the required impedances Z 1 , Z 2 and Z 3 given in Table IV. The conversion from u 0 to u 0 compensates for the magnitude and phase changes caused by these matching twoports.    Fig. 12 shows that the bandwidth for decoupling being better than −20 dB is quite large (more than 250 MHz), for matching the achievable bandwidth is less than 100 MHz for better than −20 dB and more than 200 MHz for −10 dB.
The results show an increase of around 433% and 466% in S 11 and S 22 , respectively, at the reference frequency, compared to the baseline results and a better bandwidth mainly in the S 22 , reaching good values above 3.1 GHz, at the threshold of −10 dB. Moreover, it is observed that the third antenna included by the method has decoupling values with the other Fig. 11. NDM circuit.

Parameters
Values Obtained (V) antennas around −40 dB, showing a weak influence among the others, and not degrading the general behavior of the subarray. For S 12 and S 21 , it is observed an improvement of around 344% at the main frequency, compared to the baseline results. Therefore, this method shows its ability to perform decouple and impedance matching operations in subarray modeling. Fig. 13 shows the graph of comparison among each one of the three proposed DMN methods. In this way, it is possible to perceive the behavior improvement in each method. The NDM method stands out for having the best improvement at the reference frequency for S 11 and S 12 .

E. Comparison Among DMN approaches
Comparing the results in Fig. 13, it can be seen that, as far as matching (S 11 , S 22 ) is concerned, the DMN-LE has the best performance with a −20 dB bandwidth of 100 MHz, while the DMN-RH is falling short with a −20 dB bandwidth of about 50 MHz, and the NDM is in between with about 70 MHz of bandwidth. As far as decoupling (S 12 , S 21 ) is concerned, the DMN-RH is the best option with very large −20 dB bandwidth of more than 400 MHz, followed by the NDM with more than 300 MHz and finally the DMN-LE with around 90 MHz of bandwidth. Summarizing, the NDM seems to present a good compromise.
Once these results are obtained, it can be understood that the modeling of subarrays by each of these methods leads to very reliable performances even considering their limitations regarding the composition of each subarray.

VII. CONCLUSION
The analysis carried out in this paper described three methods for impedance matching and decoupling for a baseline antenna array in which the scattering parameters (S 11 , S 22 , S 12 and S 21 ) were investigated for the 2.9 GHz to 3.1 GHz frequency band (200 MHz bandwidth) around the reference frequency of 3 GHz.
With the DMN-LE method, it can be observed that, despite the favorable behavior with respect to the baseline, it demands a complex circuit structure which makes its design difficult for compact applications. On the other hand, the DMN-RH method presented a different performance in terms of bandwidth for each of its ports which reduces its overall bandwidth. Therefore, it ends up being discarded in favor of a DMN solution that offers the same behavior in both output ports. Finally the NDM method showed excellent performance, enabling a compact and efficient antenna system for the impedance matching and the minimization of the destructive effects of coupling between the elements of the proposed antenna array.
Therefore, this proposed method has as its main perspective the modeling of multiport array not limited to a 2-element arrays, as in [12]. However when adding more than two antennas the method becomes obviously more complex. With this understanding, modeling and development of subarray structures is the applicable solution which allows the composition of MIMO antenna arrays so that the distance between each subarray is sufficiently capable of guaranteeing the noninteraction between the subarrays, allowing the total composition in a single antenna array.
Therefore, for the overall conclusion, it is possible to notice that from a impedance matching point of view, the DMN-LE provides the highest bandwidth, while, from a decoupling point of view, the DMN-RH is the best solution and the NDM is in between the two aspects. Thus, the choice for the best  method will depend on the specific requirements, in which one of the three best suits. As future works, the analysis of the NDM regarding the gain and beamforming of the antenna array can be investigated; the analysis of a three element antenna array can be performed; and a generalization of this formulation to more elements of an antenna array can be proposed.