Floquet spectral almost periodic modulation of massive finite and infinite strongly coupled arrays: Dense massive MIMO, intelligent surfaces, 5G and 6G applications

In this paper, we introduce a new formulation based on Floquet (Fourier) spectral 1 analysis combined with a spectral modulation technique (and its spatial form) to study strongly 2 coupled sublattices predefined in the infinite and large finite extent of almost periodic antenna 3 arrays (e.g metasurfaces). This analysis is very relevant for dense massive MIMO, intelligent 4 surfaces, 5G, and 6G applications (used for very small areas with a large number of elements 5 such as millimeter and terahertz waves applications). The numerical method that is adopted to 6 model the structure is the method of moments simplified by equivalent circuits MoM GEC. Other 7 numerical methods (as the ASM array scanning method and windowing Fourier method) used 8 this analysis in their kernel that to treat periodic and pseudo-periodic (or quasi-periodic) arrays. 9


Introduction
Antenna arrays, and in particular dense (or massive) coupled almost-periodic 13 antenna arrays, have been of great interest in telecommunications and RF electronic 14 applications (such as dense massive MIMO applications, smart surfaces, 5G and 6G) [11] 15 - [14], including those used for very small surfaces with large numbers of elements such 16 as millimeter and terahertz array applications. Therefore, the spectrum analysis based 17 on a Fourier transformation (in the Floquet domain) is proposed to simplify the EM 18 calculation on an elementary cell surrounded by periodic walls, as explained in [9]- [15] 19 (in other research, they use periodic Green's functions) [17]- [28]. In the bibliography 20 and recent studies, only spatial modulation techniques have been proposed to study 21 periodic systems with large sizes [33]- [36]. Except in our case, a Fourier spectral analysis 22 is presented to introduce a spectral modulation technique and its spatial equivalent 23 (Fourier and Fourier inverse) to study strongly coupled sub arrays in an infinite and large 24 finite almost periodic support [6], [30]- [32] . In this context, several numerical methods 25 such as FDTD and FEM and other integral methods like the method of moments and 26 full-wave methods [16] are proposed to resolve the given problem. In our work, we 27 are interested only in the method of moments combined with equivalent circuits and 28 Floquet analysis to study the suggested structure with the principle of modulation. This 29 work is divided into four parts: we start with an explication of the almost-periodic 30 modal (or spectral) modulation and its spatial equivalent to examining strongly coupled 31 cells [1]- [5]. Then, we applied MoM-GEC as a numerical method to solve the proposed 32 problem [7]- [15], [29] . Next, several numerical results are presented to confirm the 33 validity of the approach. Finally, some conclusions are established. 34 35 strongly coupled arrays 36 The concept is a signal processing concept for a filter with a periodic spectral response, as shown in [2]- [4] . His response is described as an impulse response function that is given by:

Almost periodic spectral (Floquet) modulation and its spatial equivalent to study
Its Fourier representation of equation (1) (in the spatial domain) yields to From the transformations by way of analogy , that we take into account, we note that x 37 is a spatial coordinate and α ∈ [− π d , π d ] is a spectral coordinate in the Brillouin domain. 38 Note that also Ω is a spectral coordinate, as α. where H(x) is the Fourier transform of 39 K(Ω) and is defined as the optical (or optoelectronic [2] , [3] ) transfer function, U(x) 40 and V(x) are the Fourier transforms of u(α) and V(Ω), respectively. More details are 41 provided in [2]. Let us consider f α (x) a spectral periodic response for infinite array that is written [5], [14] : α is a continuous Floquet modes α ∈ [− π d , π d ] and x ( or x n = nd x ) represent the position in the spatial domain (a spatial distribution). with and f n (x) = f 0 (x − nd) , n ∈ Z . f n (x) is a periodic function. Now, let's put the given spectral modulation [2]: Next , we are considering : Possible to take TF −1 (U ∞ mod (α)) = U ∞ mod (x)(it depends the Fourier notation) hence, A simple demonstration from (5) to (6) is provided : = TF −1 (u(α) The equation (7) is established refering to the theorem of aliasing given in [41] Notice that my old published work [7] on rectangular pulse functions can be a special 45 case of this process of analysis and development. Where are the Fourier series coefficients of the periodic pulse train (in the presence of Floquet 47 modes) , as described in [39].

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For more details, following the pulse (or impulse ) trains, we can introduce the Floquet 49 phases as follows [5], [14] : Note that : which explains how to recover (reconstruct) pulses train by means the Fourier series, as shown in figure(1) and explained in [39] and [40] (see subsection periodic pulse and impulse trains in [39] ) . By adding the Floquet contribution e +jαx , we obtain the definition f α (x) = ∑ N=+∞ n=−N rect(x − nd)e +jαx (See figure (1)). Rect is a rectangular function with a width W.
Then, we can develop a series of rectangle functions into a series of Fourier functions, which allows us to write : (See to figure (1) and [7] ) we can note : K x,n = 2nπ d + α as an wavenumbers leads to decompose The equation (12) is proven based on the example (example 3.17) of [39].

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Now let us apply the superposition theorem of equation (10) ( based on Floquet states) 53 to generate the spatial solution [18], [28]: it is also called modulation (see equation 7) The spatial solution: is a periodic pulses series What we get in (13) is similar to equation (14) of my published work [7] (and the 55 equation (4) of WATANABE reference [6]). Also, it is of the same type as equation (6) 56 and the expansion that follows in equation (7). Then, a spatial modulation that has been 57 performed in equation (8) is being followed . As previously explained in [10], [14], the interactions between cells in spectral domain for periodic finite array are governed by a discrete phase law such that which comes from a rule of three math reasoning. For a large period of finite array D −→ 2π ( 2π is the hole interval of phases ). For a local period of finite array pd −→ α p =? ( is the spectral contribution for one cell in position pd) (p is the index position in finite array and d is the local period ). So, =⇒ α p = 2π p N , where N x is the total number of elements in finite array and p is the index position [22]. Figure (2) explains how to discretize the phases from the infinite case to the finite case.

Contribution of continuous Floquet modes α in an infinite array
Contribution of discrete Floquet modes α p in finite array According to the same figure (2), each cell interacts spectrally with its neighbors through the continuous Floquet modes α (or the phase shift e +jαx ) in infinite case and respectively α p (or the phase shift e +jα p x ) in finite case. which allows writing the spectral solution as the sum of discrete Floquet states, [5], [14], and In the same way f n ( we can rewrite the spectral modulation law for a discrete Floquet modes , [2], Thus, the DFT is written as Eventually

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A part of our results was presented in [7], [8], [10] and [14]. Let us now display the 67 other obtained results.

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This approach can be applied to FMCW radar antennas (to scan the radiation beam GHz, respectively. In this model, we will consider the microstrip patch antenna as a 77 phased array radiator. The dielectric substrate is air. According to figures (3) and (5), 78 the patch antenna has its first resonance (parallel resonance) at 23.7 and 76 GHz. It is a 79 common practice to shift this resonance to 24 and 77 GHz by scaling the length of the 80 planar dipole antenna, as described in [8].

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The next stage is to reconfirm the reflection coefficient of the planar antenna dipole, as   Table 1: Directivity versus some Floquet states (considering 100 antenna arrays) and the superposition theorem (or the modulation as explained in formula (13) which transformed to study a finite array ) for ϕ s = 0, θ s = 30 • steering angles (used for 5G application) figure Fig.1. of [8] shows the spatial radiation pattern of the resulting planar antenna array and cosine array is presented in figure Fig.3. of [8]. After validating the FMCW 98 radar, we propose to evaluate the approach for a very large number of elements that 99 uses the same frequency band at 24 and 77 GHz (for example a lattice of 100 elements 100 ). Fig.6 gives an example of the superposition theorem (or a spatial modulation) for a 101 large array to generate a spatial radiation pattern through the addition of the radiation 102 patterns of Floquet states. After that, Fig.4. of [8] presents the variation of the spatial 103 radiation pattern (obtained using Floquet analysis) in the function of steering angles 104 for 100 antenna elements that distributed in uni-dimensional configuration ( for 5G 105 application). In the same way, Fig.7 and Fig.5. of [8] show the variation of 3D radiation 106 pattern against different steering angles that are described with (θ 0 = 45, ϕ 0 = 0) and 107 (θ 0 = 90, ϕ 0 = 0), and respectively in cartesian coordinates and (u,v) space. Following 108 the same study, the tables (1) and (2)

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In this paper, we have illustrated the principle of Floquet spectral modulation based 120 on the Fourier analysis (and its spatial form ) to study almost periodic sub-arrays (with finite size) in the presence of strong mutual coupling interaction, defined on infinite 122 support (or really largely finite size). Knowing that this study is very useful for the new 123 generation of technologies based on millimeter and terahertz waves in phased arrays, 124 for example in dense massive MIMO, smart surfaces, 5G, and 6G. In future work, we are 125 interested to investigate the randomly modulated almost periodic arrays (Also in the 126 presence of strong coupling).