Systematic Design Method for Mutual Coupling Reduction in Closely Spaced Patch Antennas

In this article, a systematic design approach is presented to reduce the mutual coupling between a pair of closely spaced microstrip patch antennas (MPAs). This method can effectively overcome the main drawback of previous two-path cancellation methods, i.e., lack a systematic design guideline and heavily rely on the time-expensive trial-and-error procedure to mitigate mutual coupling in a best-effort manner. To this end, we first propose the concept and workflow of the systematic design method, and introduce the circuit-level modeling upon the proposed two coupling routes. Then, prototyped on a practical and simple physical structure, we perform certain accurate extractions and derive the rigorous conditions for both of isolation between two patches and impedance matching of a single patch based on these two paths. Afterwards, according to the circuit-level discussion on lumped elements, the corresponding decoupling structure could be implemented substantially and designed quantitatively at the antenna level for particularized closely spaced MPAs. Finally, a prototype is simulated, fabricated, and measured to validate the proposed method. The results show that the circuit-model calculation, full-wave simulation, and measurement are in good agreement. By virtue of such a simple decoupling structure, the poor 7-dB isolation is dramatically improved to 58 dB, and the isolation is enhanced beyond 20 dB in the entire impedance matching bandwidth range of 4.38 ~ 4.60 GHz (4.9%), with the closely center-to-center spacing of $0.37~{\lambda }_{0}$ .


I. INTRODUCTION
I N RECENT years, multiple-input and multiple-output (MIMO) antenna systems have become more and more prevalent in various modern wireless communication systems due to the benefits of improved spectrum utilization and enhanced channel capacity. With the growing demands for flexibility and high throughput in wireless systems, it makes sense to reduce the size of the antennas arrays used in MIMO systems [1], [2]. However, severe correlation and mutual coupling will inevitably occur between the close spacing of antenna elements, which unfortunately compromises the performance of MIMO systems [3]. On the other hand, since the intrinsic advantages of low profile, lightweight, and ease of integration, microstrip patch antennas (MPAs) have been regarded as an increasingly popular form of the antenna element in MIMO systems. In light of this, it is highly urgent and significant to find a simple, effective, and design-friendly method to reduce the mutual coupling between the closely spaced patch antennas.
Another category of technique is introducing a decoupling and matching network (DMN) between the antenna elements and the actual input ports to achieve isolation and matching in circuit characteristics [14], [15], [16]. The separated networks facilitate the design analytically and systematically, but yield ohmic losses as demonstrated and require occupying additional spaces, e.g., multiple layers and overall footprint [17]. Besides, superimposing different modes or electric fields to eliminate the total electric fields at the input ports could also be used to reduce mutual coupling. In [18], [19], [20], [21], a pair of hybrid modes, eigenmodes, and specific in-set feeding ports of the MPAs were utilized skillfully, respectively, so that the mutual coupling can be suppressed without additional structures. Most of them tend to modify the patch elements heavily and lack systematic guidance. This hurdle has been settled in [22] based on the even-odd characteristic modes method, but it is only applicable under a single patch so far. In [23], a distributed inductance was introduced between the antenna elements and designed based on the common mode and differential mode (CM/DM) cancellation theory, but the limited level of decoupling prevents its application to some highly isolated scenarios [24], [25].
Very recently, a decoupling structure composed of a microstrip line and a shorting pin is presented in [26] to reduce the mutual coupling between closely coupled MPAs in H-plane. Compared with other works that introduce an extra coupling path to eliminate the original coupling path, such as the microstrip U-section [27], array-antenna decoupling surface (ADS) [28], and neutralization lines (NL) [29], the skillfully sided arrangement makes this scheme more appropriate for closely spaced MPAs. Unfortunately, constrained by the fact that the two paths cannot be separated independently and invisible, the only qualitative design guideline could be presented under the assumption that the two coupling paths are equally amplitude but out of phase. As such, these existing two-path cancellation efforts are unfortunately faced with the design-unfriendly dilemma, that is, impedance matching and mutual coupling reduction can be only achieved in a best-effort manner via trial and error during the design process, which would waste a lot of time on simulation or optimization.
In this article, a systematic design approach is proposed to reduce the mutual coupling between two closely spaced MPAs. This concept is inspired by the classical filtering antenna work [30] and introduced to address two major challenges, i.e., 1. A quantitative design guideline is presented to significantly reduce the design complexity under twopath decoupling; 2. The isolation performance is analyzed and predicted theoretically to exploit the potential of the decoupling structure more effectively. In the remainder if this article, Section II describes the concept of the systematic design method, perform the circuit-level accurate modeling, and derive the conditions of high isolation between two closely spaced patches and good impedance matching of each patch. According to the theoretical investigation at the circuit level, the design and implementation at the antenna level are illustrated in detail in Section III. Section IV gives the experimental validation. The measured results demonstrate that excellent isolation performance, sufficient operating bandwidth, and improved radiation performance are all achieved solely with a very simple decoupling structure. Finally, the conclusion is given in Section V.

II. SYSTEMATIC DESIGN METHODOLOGY
The topology of the proposed systematic design method for the mutual coupling reduction in closely spaced patch antennas is depicted in Fig. 1. Traditional decoupling design approaches mainly focus on the antenna level, with the concept of introducing and designing an additional decoupling structure (ADS) to mitigate the isolation between the two patch antennas in a best-effort manner. A few works have also tried to use equivalent circuits to help better explain the decoupling principle [31], [32], [33]. However, it is known that the antenna-level design method only establishes the relationship between the additional decoupling structure and the scattering parameters of the input ports, while the original coupling between the antennas becomes a neglected black box and its performance is hardly analyzed. In this context, only qualitative analysis and design can be realized on the antenna level, but not a systematic quantitative design. To address this technical challenge, we additionally introduce a circuit-level modeling during the analysis and design procedure so that the two paths (Path A and Path B) become intuitively visible and they can be analyzed independently, and then map the circuit-level investigation back to the antenna level to achieve a systematic design. The detailed workflow is illustrated as: Step 1: Accurately model the antenna coupling path (Path A). The original coupling between the radiating patches needs to be extracted precisely; Step 2: The path of the antenna through the additional decoupling structure (Path B) is accurately modeled; (The above two steps correspond to Stage 1 in Fig. 1

)
Step 3: Analyze the relationship between the two paths at the circuit level to obtain the isolation conditions and the quantitative relationship between the circuit components of the two paths; Step 4: Analyze the impedance matching conditions at the circuit level to obtain the constraints and design goals; (These two steps correspond to Stage 2 in Fig. 1

)
Step 5: Complete the physical design at the antenna level by using the mapping relationship between the circuit models and the dimensional parameters, as well as the above isolation and impedance matching conditions. (This step corresponds to Stage 3 in Fig. 3 and will be illustrated in detail in Section III.) Fig. 2 depicts a simple and practical example for further illustrating the concept of the proposed systematic design method, which consists of two identical rectangular patches arranged in an E-plane coupled manner and two identical side-by-side arrangement half-wavelength microstrip lines. The two closely spaced patches are fed by two coaxial probes placed along the central axis, respectively. The corresponding equivalent model at the circuit level of the entire structure is shown in Fig. 3, and its detailed modeling procedure is described as below.

A. ACCURATE MODELING FOR TWO PATHS: PATH A
Firstly, the equivalent circuit model of the original coupling path (Path A) is depicted in Fig. 4, which is modeled as follows. The two parallel GLC circuits are used to equivalently model the two identical patch radiators, where the values (G a , L a , and C a ) can be calculated analytically under a single patch based on the classical cavity model theory [34] or the extraction approach reported in work [35]. The admittanceinverter (J 0 ) is employed to represent the external coupling between the input port and the patch antenna, which is replaced here by a parasitic series inductance (L p ) due to the probe feeding. As for the mutual coupling between the two radiated patches, it is modeled as a capacitive π -network with the value of C m . Obviously, C m is the crucial parameter of this equivalent circuit, which can be calculated by [36] where C a = C b , and CC is the coupling coefficient between the two radiating patches as can be calculated by (13) and (14) in the Appendix, where details of the extracted method and their derivations are also provided. According to the dimensions of the patch radiators listed in the caption of Fig. 2, the corresponding lumped elements are calculated to be G a = 7.41 mS, L a = 0.48 nH, C a = 3.28 pF, and L p = 2.59 nH. C m is easily determined to be 0.28 pF based on (1), (14), and the two frequencies (f 1 and f 2 ) from the S 21 magnitude response are shown in Fig. 5. As such, the calculated S-parameters of Path A can be derived as depicted in Fig. 6, which are in general agreement with the full-wave simulation results within the impedance matching bandwidth of the single patch antenna (about 5% theoretically). The slight out-of-band inconsistency is mainly caused by minor VOLUME 4, 2023 351 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  resistive coupling, but it hardly affects the actual accuracy in analysis of in-band decoupling and thus it can be fully neglected in the modeling [37].

B. ACCURATE MODELING FOR TWO PATHS: PATH B
Subsequently, the equivalent circuit of the additional coupling path (Path B) is modeled and depicted in Fig. 7. The modeling of the two radiating patches is the same as that in Path A. A parallel LC circuit is used to model the opencircuit half-wavelength microstrip line, whose values (L s and C s ) can be calculated based on the closed-form expressions in [38]. Specifically, the characteristic impedance in the formulas also can be readily obtained with the help of some transmission line tools. The admittance-inverters (J c1 and J c2 ) represent the coupling extent between the two radiating patches and the non-radiative half-wave microstrip line, respectively. Their values can be determined as follows: using the method reported in work [39] to extract the coupling coefficient and then employing the relationship between J-inverter and the coupling coefficient to calculate and determine [40].
Since the overall structure of the two closely spaced patch antennas is geometrically symmetric with respect to the yoz plane, the absolute values of the extracted coupling coefficients are the same but with the opposite signs, that is, The minus sign is caused by the microstrip line operating in the half-wave mode with opposite electric fields on two sides, and it can be deduced from the coupling coefficient as defined in [41]. Furthermore, the additional decoupling section (framed by the blue dashed line) can be regarded as a unity and is derived as When substituting J c1 = −J c2 = J c into (2) and simplifying, it is unexpectedly found that this A-matrix has the same form as a series inductance, such that This means that the mutual coupling in Path B and Path A constitutes a duality. Herein, the ω = 2π f in (3) is actually the desired frequency for realization of excellent decoupling, and not the resonant frequency (f s ) of the half-wave microstrip line. In contrast, if f = f s , there is L m = 0, implying that no coupling is present in Path B or Path B fully disappears. This is a completely different conclusion as derived from the qualitative analysis that a parasitic halfwave resonator operates at the isolated frequency in other relevant works.

C. CONDITION FOR ISOLATION
Till now, the two coupling paths are successfully separated and independently visible based on the accurate equivalent circuit model mentioned above. Path A and Path B could be regarded as two second-order lossy bandpass filter with electric and magnetic coupling, respectively. Thus, it can be assumed that the equivalent coupling lumped elements in the combined paths satisfy [41] When substituting (3) into (4) and simplifying, it is yielded where C s and L s can be further mapped to the physical decoupling structure and predetermined based on the quasistatic theory of microstrip line structure, thus yielding where ε e is the effective permittivity, c is the speed of light in free space, and Z s is the characteristic impedance depending on W d and H, and L d is the physical length of the microstrip line. As demonstrated in (5) and (6), the coupling coefficient between the radiating patch and the decoupling structure could be quantitatively calculated at the operating frequency after the mutual coupling between the specified closely spaced MPAs is determined. Thus, a set of lumped-element values (L s = 1.5 nH, C s = 1.65 pF, and J c = 0.01 S) in Path B are obtained accordingly based on the extracted and calculated values of Path A in Section II-A.
Then, the transmission characteristics of the two paths and the total path are calculated using the prescribed variables and eventually depicted in Fig. 8. Herein, although the resonator loss (insertion loss) denoted by G a does not affect the mutual coupling as proved in Fig. 5, it does have any significant effect on the port impedance matching (external quality factor), which will indirectly distort the transmission response. Hence, the transmission responses discussed below are initially in the lossless case (G a = 0). As shown in Fig. 8(a), both paths present a typical response of the secondorder bandpass filter with a staggering frequency (seeing in the inset). Within the band of interest, the amplitudes of the two paths partially overlap, as can be intuitively seen from the magnitude ratio (S a 21 /S b 21 ). Meanwhile, the phase difference (φ b 21 − φ a 21 ) of the two paths is determined by the staggering passband frequency, which shows a 180 • out of phase at f z . As such, the transmission response of the total path is plotted in Fig. 8(c), which shows a corresponding typical zero-point response at f z . When the radiation loss is considered, the total transmission response also produces a zero point at f z as previously exhibited, and the amplitude decreases in the nearby frequency band. Therefore, these results demonstrate indeed that, at the proposed circuit level, the transmission coefficient (mutual coupling) between the two ports could be reduced effectively over a certain range of bandwidth with our well-performed modeling investigations.

D. CONDITION FOR IMPEDANCE MATCHING
The impedance matching is another issue that needs to be addressed in addition to the concerned isolation, which can characterize a passband transmission response between the input port and the radiation conductance. To remove the interference caused by the transmission responses between the two ports, port 2 is set as an open-circuit. The input admittances and reflection coefficients looking into from port 1 for two separated paths and the total path are calculated and depicted in Fig. 9. As seen, Path A and Path B have the smallest reflection coefficients at f pa and f pb , respectively, while the best matching point for the total path is located between them. This is because when the coupling of Path B is introduced to mitigate Path A, the total input conductance will gradually shift from Y ina to Y inb , as illustrated in Fig. 9(a). Therefore, the frequency (f p ) at the input conductance peak of Path A could be chosen as the desired frequency for decoupling design (i.e., let f z = f p ), and the input ports will be well-matched naturally when the mutual coupling is suppressed effectively. Most importantly, the above design step of pre-determining the impedance matching frequency greatly simplifies the design complexity, so there is no need to perform further impedance matching procedure by modifying the feeding structure or conducting the parameter study of the decoupling structures.

III. IMPLEMENTATION AND ACTUAL DESIGN
The physical implementation of reducing mutual coupling between a pair of closely spaced MPAs at the antenna level is presented in this section. Based on the discussion at the circuit level in Section II, the rigorous relationship of the lumped elements in the equivalent circuit models for decoupling and impedance matching was successfully obtained. Therefore, for closely spaced MPAs with determined parameters, the dimensions of the corresponding decoupling structures can be synthetically designed through parameter extraction and physically mapping to the dimensional layout. The EM simulation throughout this work is done by using ANSYS high-frequency structure simulator.

A. COUPLING COEFFICIENT BETWEEN TWO CLOSELY SPACED MPAS
In this design, the dimensions of the single MPA and the substrates are the same as those in Section II. Using the extraction technique proposed in the Appendix, the coupling coefficients between the two radiating patches at different edge-to-edge spacing can be easily obtained. Substituting them into (1), the relationship between the coupling capacitance (C m ) and the variation of spacing (G) is calculated and plotted in Fig. 10, which shows that the C m gradually increases as the G decreases. Then, the desired decoupling frequency (f p ) corresponding to different spacings is calculated by substituting different C m into the equivalent circuit of Path A. When G = 3 mm (corresponding to 0.04 λ 0 edge-to-edge spacing, 0.37 λ 0 center-to-center spacing, and maximum coupling of −7 dB) is chosen as the target for decoupling, C m = 0.28 pF and f p = 4.45 GHz can be conveniently determined from Fig. 10. In this context, the transmission responses of the total path at different J c are investigated and calculated at the circuit level using (5) in Section II-C. As shown in Fig. 11, as J c increases from 0.004 S to 0.016 S, the in-band isolation near the zero point of this closely spaced patch array gradually increases as well. In particular, when J c ≥ 0.008 S, S 21 is less than −20 dB in the range of 4.33 ∼ 4.7GHz (8.2%), which is larger than the impedance matching bandwidth (5.2%) that is theoretically achieved by this MPA (denoted by grey region) [42].

B. COUPLING BETWEEN RADIATING PATCH AND DECOUPLING PART
According to the extraction method in previous work [39] and (5), (6) in Section II-C, the relationship between the coupling coefficients J c and the physical parameters of the decoupling structure can be conveniently obtained. As shown in Fig. 12,      extraction will be used in the subsequent simulation and fabrication of the prototype.
In addition, the impact of the distance (G c ) between the decoupling structures and the patch antenna on decoupling performance is investigated. Three cases under different G c are depicted in Fig. 13. Compared to case 1, when only G c is changed, it is seen that case 2 cannot achieve decoupling effectively and the operating frequency also deviates from the expectation. Then, when increasing G c along with adjusting other parameters to achieve decoupling within the desired operating band, it is found that the 20-dB isolation bandwidth is degraded. Therefore, G c not only influences the isolation effectiveness but also has impacts on the isolation bandwidth.
Herein, the above cases also demonstrate that the parameters of the decoupling structure are usually dependent on each other and cannot be analyzed independently, which will severely increase the computations of the conventional method. As shown in Table 1, under the same parametric combinations, the proposed method significantly improves the design efficiency, saving nearly half of the computation time. Most importantly, the computations of the proposed method can be reused when the spacing or the desired isolation is changed.

IV. EXPERIMENTAL VERIFICATION A. SIMULATED AND MEASURED RESULTS
In order to verify the validity of the proposed systematic design method, a prototype based on the extracted parameters is designed, simulated, and fabricated. The proposed closely spaced MPAs and the decoupling structures are printed on an F 4 B substrate with the relative permittivity of 2.20 and the loss tangent of 0.002. Two rectangular patches are excited directly by 50-SMA probes. Based on the previous predetermined, the actual length of the decoupling structure only needs to make a minor fine-tune to 28.7 mm. The S-parameters and radiation performances are measured by the Rohde & Schwarz ZNB-20 Vector Network Analyzer and SATIMO measurement system, respectively. Fig. 14 shows the S-parameters of the proposed closely spaced MPAs under the circuit model calculation, full-wave simulation, and measurement, respectively. As seen, the S 21 under circuit model agrees very well with that of the full-wave simulation at only about 20 MHz (0.4%) deviation, showing a good prediction. Meanwhile, the measured results are also basically consistent with the simulation, only exhibiting a slight shifting due to the fabricated tolerances, demonstrating an impedance matching bandwidth range of 4.38 ∼ 4.60 GHz (4.9%) for S 11 < −10 dB and an isolation bandwidth in a frequency range from 4.38 ∼ 4.71 GHz (7.33%) for S 21 < −20 dB. Compared with the initial coupling of −7 dB, the maximum isolation is improved to 58 dB. The excellent isolation performance is achieved by such a simple decoupling structure as mainly attributed to the circuit-level prediction during the proposed design procedure.
The simulated and measured normalized radiation patterns of the decoupled MPAs at the center frequency, which are excited from port 1 and port 2, are shown in Fig. 15. The measured results are in good agreement with the simulated ones. In the E-plane, the radiation patterns of the two MPAs exhibit only a slight asymmetry, which is significantly improved compared to the coupled case. Besides, the measured front-to-back ratio (FBR) and the cross-polarization levels are 18 dB and 22 dB, respectively. In the H-plane, the radiation patterns of two elements are symmetric with an 18-dB FBR and a 20-dB cross-polarization level at the boresight direction. As shown in Fig. 16, the simulated and measured gain and total efficiency are basically in agreement, with slight differences fluctuating in the range of 0.1 ∼ 0.9 dBi and 0.05 ∼ 0.1, which comes from the inevitable fabrication and measurement tolerances. The measured total efficiency varies from 81.1% to 93.2% and the  The ECC performances of the proposed antennas before and after using the decoupling structure within the operating bandwidth are depicted in Fig. 17. As seen, the simulated ECCs are significantly reduced from around 0.19 to 0.002 after the introduction of the decoupling structure, which is also in good agreement with the measured result.

B. COMPARISON
To highlight the merits of the proposed systematic decoupling method, a comprehensive comparison in design  methods, physical dimensions, isolation property, and radiation performance with recently relevant research is tabulated in Table 2. As discussed in Section I, a systematic design method is an urgent requirement for decoupling in closely spaced MPAs, with the benefit of offering a clear design guideline to reduce design complexity and provide performance prediction. Among the traditional decoupling methods, only the DMN method in [14] and [16] has the capability of systematic design but suffers from the multilayer complex structures and sacrifices operating bandwidth.
Here it specifically means that these antennas have a higher profile but similar or even lower operating bandwidth. The CM/DM method proposed in [23] provides a design guideline but cannot predict performance and can only minimize the isolation in a best-effort manner by sweeping the parameters, thus bearing limited in-band isolation. In an interesting work [43], exploiting the self-resonance frequency of the CM and DM can improve the prediction and evaluation ability of CM/DM method. Moreover, the traditional two-paths decoupling method used in [26] and [27] lacks design guidance and relies heavily on a try-and-error approach to design, which still suffers from limited operating bandwidth and decoupling capability.
In summary, the proposed systematic design method based on the two-path decoupling technique has demonstrated excellent performance in many aspects, especially in providing quantitative design guidance and isolation performance prediction. Meanwhile, better isolation performances have been achieved with a simpler decoupling structure, and the impedance matching bandwidth and radiation characteristics have been maintained similar to those of a single MPA. Moreover, the proposed design method has the ability applied in other antenna arrays to help improve decoupling design efficiency, and it can also be employed in broadband or multi-band decoupling designs by increasing the zeros [44].

V. CONCLUSION
This article presents a systematic design method to suppress the mutual coupling between closely spaced patch antennas. Unlike traditional two-path cancellation methods that rely on trial and error to mitigate mutual coupling in a best-effort manner, the proposed scheme can provide a design guideline and predict the isolation performance. Due to the inability to separate the two paths at the antenna level and the original coupling path is difficult to analyze, the analysis at the circuit level is introduced. Based on a practical and simple structure, the conditions of isolation and impedance matching are then derived by accurate extraction and modeling to obtain the rigorous relationships between the lumped elements of the two paths. As such, the antenna-level implementation and quantitative design can be conveniently realized based on the circuit-level discussion. To validate the proposed method, a prototype is in final simulated, fabricated, and measured. The corresponding circuit-model calculation, full-wave simulation, and measurement of the specific two closely spaced MPAs agree well, thereby evidently demonstrating significant improvements in isolation characteristics, impedance matching, and radiation performance.

APPENDIX
The detailed derivations for extracting and calculating the coupling coefficient (CC) directly from the two radiating patches are presented in this section, which is inspired and based on the previous work [39]. The equivalent circuit model for extracting the CC is depicted in Fig. 18(a), where the parallel GLC circuits and the interstage coupling components are the same as those in Path A shown in Fig. 4, except that the external coupling at the ports is set to weak coupling. The network in Fig. 18(b) is the corresponding lossless network without the radiation conductance (G a and G b ) of the two closely spaced radiating patches, and the rest of the lumped components are the same as in Fig. 18(a). It is well known that the CC can be directly extracted from the two steep peaks of the transmission responses in the lossless network. Therefore, as long as the relationship between the two networks is deduced, the CC can also be directly extracted from the lossy network in the same manner.
Firstly, the transmission response of the lossy network in Fig. 18(a) can be obtained by multiplying the five-part cascaded A-matrices, such that where A R , B R , C R , and D R are given by (7b), shown at the bottom of the page. Then, the transmission matrix of the lossless network without G a and G b can be expressed as where the elements of this matrix are given by It can be seen that the terms in (8b) are exactly all the terms in (7b) of the lossy network that excludes G a and G b . Obviously, the lossy terms must be eliminated if one would like to extract directly from the lossy network. For the lossless parts of the lossy network, A and D in the ABCD matrix are pure real quantities, while B and C are pure imaginary quantities. More specifically, the weak coupling can be simply represented by an admittance inverter, such that where J 1 is the value of the J-inverter. The matrix of the middle part is obtained by multiplying the corresponding cascaded component matrices, which is given by Therefore, the relationship between the lossless and the lossy matrices can be simplified and derived as ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Unfortunately, G a and G b are still present in B R and cannot be ignored. Interestingly, if the A-matrix of the lossless network is converted into a Z-matrix, it can be found that only Z R12 includes B R . Since this network is reciprocal, that is, Z R12 = Z R21 , the term of Z R12 could be replaced by Z R21 , and the Z-matrix can be substituted into (11) and deduced as Finally, the transmission response in the lossy network with no effect from radiation conductance can be expressed as where Z 0 is the port impedance.
As shown in Fig. 5, the transmission response (denoted by the red line) extracted directly from the network including radiation loss by using (13) exhibits two significant steep peaks compared to the original response (denoted by the blue line), which are consistent with the two steep peaks extracted from the lossless network (denoted by the black dashed line). Now, the CC between the two radiating patches can be readily and accurately determined by substituting the two extracted frequencies into the following equation in final.
where f a and f b are the self-resonant frequencies of the two patch antennas, f 1 and f 2 are the characteristic frequencies resulting from the patch coupling.