Holographic MIMO Communications: What is the benefit of closely spaced antennas?

Holographic MIMO refers to a (possibly large) array with a large number of individually controlled and densely deployed antennas. The objective of this paper is to provide further insight into the use of closely spaced antennas in the uplink and downlink of a multi-user Holographic MIMO system. To this end, we utilize multiport communication theory, which ensures physically consistent uplink and downlink models. We first consider a simple uplink scenario with two side-by-side half-wavelength dipoles, two users, and single-path line-of-sight propagation, and show both analytically and numerically that the array gain and average spectral efficiency strongly depend on the directions from which the signals are received and on the array matching network used. The numerical results are then used to extend the analysis to more practical scenarios involving larger arrays of dipoles (arranged in a uniform linear array) and a larger number of users. The case where the antennas are densely packed in a space-constrained factor form is also considered. It is found that the spectral efficiency increases with decreasing antenna spacing only for arrays of moderate size, e.g. in the order of a few wavelengths. In comparison, larger arrays with closely spaced antennas show only marginal improvements in spectral efficiency compared to half-wavelength arrays.


I. INTRODUCTION
Communication theorists are always on the lookout for new technologies to improve the speed and reliability of wireless communications. Chief among the technologies that blossomed into major advances is the multiple antenna technology, whose latest implementation is Massive MIMO (multiple-input multiple-output) [1], [2]. Inspired by its potential benefits [3], new research directions are taking place under different names [4], e.g., Holographic MIMO [5] and large intelligent surfaces [6]. Particularly, the former concept refers to an array (possibly electromagnetically large, i.e., compared to the wavelength) with a massive number of closely spaced antennas whose electromagnetic interactions inevitably results into mutual coupling [7]. Although few exceptions exist, e.g., [8]- [12], the vast majority of the MIMO literature has entirely neglected mutual coupling since A. A. D'Amico and L. Sanguinetti are with the Dipartimento di Ingegneria dell'Informazione, University of Pisa, Pisa, Italy (email: luca.sanguinetti@unipi.it, antonio.damico@unipi.it). This work is supported by Huawei Technologies Sweden AB. signals are received and on the array matching network used at the BS. Advantages can be obtained only with impedance matching (e.g., [24]) and under certain conditions, which may not be met in practical systems. In these cases, the gains may be marginal or even non-existent. The internal losses within the dipole antennas are also shown to significantly impact the spectral efficiency as the spacing reduces. Numerical results are then used to show that similar conclusions hold true in more practical scenarios with an arbitrary number of UEs and an arbitrary number of dipole antennas at the BS. Particularly, the analysis is conducted in the following two cases: i) the number of dipoles is fixed as we vary their spacing; i) the array size is fixed as we vary the dipole spacing.
In the latter case, it turns out that the spectral efficiency increases as the antenna distance reduces.
However, this comes from the larger energy that is collected by the larger number of dipoles, not from the mutual coupling. Interestingly, the spectral efficiency tends to increase less and less as the size of the antenna array increases (compared to the wavelength).
Although most of the analysis focuses on the uplink, we also investigate the downlink. Particular attention is given to the uplink and downlink duality in the presence of different matching networks. Specifically, we show that the downlink and uplink channels are reciprocal up to a linear transformation. In line with [22], the ordinary channel reciprocity (i.e., no linear transformation) holds true only if full matching networks (that are hard to implement in arrays with many antennas) are employed at both sides. Numerical results are used to quantify the spectral efficiency loss when the linear transformation is not applied.
The remainder of this paper is organized as follows. In Section II, we review the Multiport Communication Theory from [17]. In Section III, we show how to compute the mutual coupling impedance matrix when a uniform linear array made of half-wavelength dipoles is used at both sides. In Section IV, the uplink and downlink signal models for Holographic MIMO communications are derived on the basis of the multiport communication model. The concept of uplink and downlink duality is also discussed. To showcase what is the impact of mutual coupling, a simple case study with two dipole antennas and two UEs is considered in Section V. The analysis is then extended in Section VI to more realistic scenarios with multiple antennas, multiple UEs and arrays of varying or fixed aperture. Conclusions are drawn in Section VI.
Reproducible research: The Matlab code used to obtain the simulation results will be made available upon completion of the review process.  [17].

II. REVIEW OF MULTIPORT COMMUNICATION THEORY
Consider a narrowband communication system equipped with M antennas at the receiver and N antennas at the source. This is described by the following discrete-time input-output relation [25]: where y ∈ C M and x ∈ C N denote the output and input vectors, respectively. The vector x must satisfy E{x H x} ≤ P T to constrain the total transmit power. Also, n ∼ N C (0, R n ) is the additive Gaussian noise and H ∈ C M ×N is the MIMO channel matrix. The input-output relation in (1) can be used to model a great variety of multiple antenna communication systems. In order to successfully model a particular one, there is the need to encode the physical context of the system into it. This is exactly the point where the circuit theoretic concept of linear multiports from [17] comes into play. The physical model, based on the circuit theoretic approach, is shown in Fig. 1.
It consists of four basic parts: signal generation, impedance matching, antenna mutual coupling, and noise. The meaning of each part is briefly reviewed next. More details can be found in [17].

A. Signal generation and power
The generation of the nth physical signal that is to be transmitted is modeled by a voltage source, with complex envelope v G,n , in series with the impedance Z G = R G + jX G . The average available power of the voltage generator is P a,n = where the expectation accounts for signal randomness.
Letting v G = [v G,1 , v G,2 , . . . , v G,N ] T , the total average available power is thus The transmit/receive matching networks are multiport systems described by the impedance matrices Z MT and Z MR . In particular, Z MT ∈ C 2N ×2N and Z MR ∈ C 2M ×2M are given by with {Z MT,ij ∈ C N ×N ; i = 1, 2, j = 1, 2} and {Z MR,ij ∈ C M ×M ; i = 1, 2, j = 1, 2}. We assume that the impedance matching networks are lossless, reciprocal [16], and noiseless [26].
The impedance matrix Z A ∈ C (N +M )×(N +M ) accounts for the mutual coupling between antennas and can be partitioned as: Particularly, Z AT ∈ C N ×N and Z AR ∈ C M ×M quantify the mutual coupling at the transmit and receive sides (intra-array coupling), respectively, while Z ATR ∈ C N ×M and Z ART ∈ C M ×N model the mutual coupling between the transmit and the receive arrays (inter-array coupling). Because antennas are reciprocal, e.g. [7], we have Z AT = Z T AT , Z AR = Z T AR , and Z ATR = Z T ART . A common approximation for Z A follows from the unilateral assumption, according to which Z ATR ≈ 0 N ×M . This basically implies that the currents at the receiver do not produce effects on the transmitter. From a mathematical standpoint, it requires that ||Z AT i AT || ≫ ||Z ATR i AR || where i AT and i AR are the vectors of currents at the transmitting and receiving arrays, respectively (see Fig. 1). In practice, it implies that the transmitting antennas are not affected by the presence of the receiving antennas. This is true as long as the transmit and receive arrays are sufficiently separated in space as it happens in any practical communication network. 1

C. Losses in the antennas
Notice that even though antennas may possess minimal loss, this can become significant when substantial electric currents are required to transmit a specific power. Particularly when dealing with close antenna spacing, the internal losses within the antenna can significantly impact the 1 It may not hold true if different short-range applications are considered, e.g., near-field communications or short-range simultaneous wireless information and power transfer systems. 6 performance. A common way to account for this, it is to include a dissipation resistance that is connected in series. This implies that the impedance matrices Z AT and Z AR must be replaced with where k B is the Boltzmann constant, T A is the noise temperature of the antennas, while ∆f is the equivalent noise bandwidth that depends on the bandwidth of the desired signal.
The intrinsic noise is produced by the subsystems that follow the receive matching network such as LNAs, mixers, and ADCs. Most of the noise originates from the LNAs, and thus can be modelled by using the voltage and current vectors [16], [27] given by v LNA and i LNA , respectively.
Both v LNA and i LNA are zero-mean random vectors, with the following statistics [16,Eq. (10)]: where R N is the so-called noise resistance of the LNAs, usually indicated in the manufacturer data sheets. The complex parameter ρ = E{v LNA,m i * LNA,m } E{|v LNA,m | 2 }E{|i LNA,m | 2 } accounts for the correlation between voltage and current noise generators at each port.

E. Input-Output Relation
Under the unilateral approximation, the input-output relation is [16,Eq. (16)] where D and η are given by [16,Eq. (17)] with [16,Eq. (19)] and [16,Eq. (20)] The input-output relation (8) can be written in a slightly different form (which will turn useful with Notice that v OC is the open circuit voltage vector as induced by i AT when i AR = 0, as it follows from (a) in (17). In general, i AR = 0 does not imply that the elements of v OC are the same as if the receive antennas were isolated. This holds true only if the receive antennas are canonical minimum scattering (CMS) antennas. 2 This is the case of half-wavelength dipoles [20]. We finally notice that Dv G = QF R v OC so that, using (17), we get Remark 1. (Input-output relation without matching networks) In the absence of a transmit matching network, the input-output relation can simply be obtained by setting Z T = Z AT and F T = I N in (9) and (13), respectively. Analogously, with no receive matching network the input-output relation can be obtained by replacing Z R with Z AR and F R with I M .
2 According to [28], a canonical minimum-scattering antenna is "invisible" when the accessible waveguide terminals are opencircuited. This means that a vanishing electric current in the antenna does not alter the electromagnetic field.

F. Transmit power and noise covariance matrix
The transmit power is defined as the average active power at the output of the transmit matching network, or equivalently, at the input of the transmit antenna array, i.e., P T = E{Re(v H AT i AT )}. Assuming a lossless transmit matching network, we have with Notice that P T coincides with the radiated power P rad only if the transmit antennas are lossless.
From the statistics of the extrinsic and intrinsic noise, the covariance matrix R η of η in (10) is: where Q is given in (16) and U = U IN + U EN is the correlation matrix ofη with and

G. Matching network optimization
The transmit matching network Z MT can be designed to maximize the power delivered to antennas (power matching or maximum power transfer) [16]. This yields B = I N in (21), and By taking (12) and (15) into account, this can be obtained by setting [16] which yields The receive matching network Z MR can be designed to ensure that the signal-to-noise ratio (SNR) is as large as it can be (noise matching or SNR maximization) [16]. This is achieved with Plugging (27) into (11) and (14) yields with Z opt = R N 1 − (Im{ρ}) 2 + jIm{ρ} and Also, notice that becomes diagonal. The covariance matrix R η = |Z L | 2 |Z L + Z opt | 2 σ 2 I M becomes diagonal with The design of coupled matching networks is very challenging for arrays with a large number of antennas [16], [17]. A practical approach is to make use of a self-impedance matching network [29, Sect. III.B], instead of a full multiport matching network. This approach neglects the mutual coupling among antennas and replaces, in the design of the matching networks, the impedance matrices Z AT and Z AR with the diagonal matrices diag(Z AT ) and diag(Z AR ) that contain only their diagonal elements. It becomes possible to substitute these matrices for the actual ones in (25) and (27) and specify uncoupled matching networks, as described above.

III. COMPUTATION OF THE MUTUAL COUPLING IMPEDANCE MATRIX
Next, we show how to compute the mutual coupling impedance matrix Z A . In particular, we assume that the antennas at both sides are half-wavelength dipoles of length l d = λ/2 and radius a d ≪ l d . Moreover, we assume that the receiver is equipped with a uniform linear array.  A. Impedance matrix Z AR We consider Z AR but the same analysis follows for Z AT . The mutual impedance between dipole p and dipole q is computed as [30,Eq. (25.4.14)] where e qp (s) is the component (along the direction of dipole q) of the electric field produced by a current I p (s ′ ) flowing in dipole p, I q (s) is the current flowing in dipole q, and finally I p and I q are the currents at the input terminals of dipoles p and q, respectively. The current distributions where k = 2π/λ is the wavenumber. Based on (33)  and is approximately 2L H /λ + 1. This means that for d H < λ/2 mutual coupling introduces a significant correlation between the different array elements, as expected. Fig. 2b shows the normalized eigenvalues of U in (22), obtained without a matching network (see Remark 1) and with the parameter values reported in Table I. The behavior of these eigenvalues is quite different from that of Fig. 2a, because noise correlation not only depends on Z AR but also on the LNA parameters, and on the presence (and type) of matching networks, as shown in (23).
In particular, the curve corresponding to d H = λ/2 seems to indicate that a significant correlation exists between the elements ofη even with a half-wavelength spacing between the antennas.
Hence, models based on this assumption have no physical meaning. In the case of Hertzian dipoles, a uniform current distribution is typically assumed, which approximates well the current distribution

B. Impedance matrix Z ART
The impedance matrix Z ART is needed to obtain the open-circuit voltage array response repre- which requires the computation of D OC . In general, v OC is influenced by various factors, including the type of antennas, array configuration, polarization and transmission medium. Assuming that the receive array is in the far-field region of the transmitter and possible scatterers, v OC is produced by the superposition of a (possibly large) number L of plane waves, each reaching the receiver from a particular azimuth angle φ i ∈ [−π/2, π/2) and elevation angle θ i ∈ [−π/2, π/2). In the Appendix, we assume a single transmitting antenna (i.e., Z ART and D OC reduce to the vectors z ART and d OC , respectively) and show that T is the wave vector that describes the phase variation of the plane wave with respect to the Cartesian coordinates and u m is the center coordinate of dipole m. The model (34) can be generalized to a continuum of plane waves as follows (e.g., [33]) where α(θ, φ) accounts for the small-scale fading and must be modeled stochastically. A common way to model α(θ, φ) is a zero-mean, complex-Gaussian random process with cross-correlation [33] E{α where β(θ, φ) is the average channel gain, and f (θ, φ) is the normalized spatial scattering function [33], such that

IV. HOLOGRAPHIC MIMO COMMUNICATIONS
We consider a communication system where the BS is equipped with M BS antennas and serves networks are used at each UE in uplink (i.e., for power matching) and downlink (i.e., for noise matching). This is reasonable since a single antenna is used at each UE.

A. Uplink data transmission
In the uplink, the vector v L ∈ C M BS of voltages measured at the BS is generated by the superposition of the generator's voltages {v G,i ; i = 1, . . . , K} of the K single-antenna (i.e., N = 1) transmitting UEs. The dimensionless input-output relation can be obtained from (8) as where c is an arbitrary constant, measured in V 2 , needed to obtain a dimensionless relationship.
The vector d ul i ∈ C M BS associated with the single-antenna UE i is obtained from (19) and reads where (a) follows from (17) whereas (b) is because a matching network for maximum power transfer is used by UE i. From (24) and (26), this implies Z ul AT is the transmitting antenna impedance. In (38), we have defined where P a,i = 1 4R G E{|v G,i | 2 } is its available power and is the fraction of available power delivered to the transmitting antenna. Notice that (a) follows because Z ul T = (Z ul G ) * when a matching network for maximum power transfer is used by UE i.
of the multi-user MIMO system in the form (1) follows: The data signal The vector where R η is given by (22). Since c is an arbitrary constant, we assume c = 1 V 2 without loss of generality.
To decode x ul k , the vector y ul is processed with the combining vector u k ∈ C M BS . By treating the interference as noise, the spectral efficiency (SE) for UE k is log 2 is the SINR. We consider both MR and MMSE combining [2]. MR has low computational complexity and maximizes the power of the desired signal, but neglects interference. MMSE has higher complexity but it maximizes the SINR in (43). In the first case,

B. Downlink data transmission
In the downlink, the voltage v dl L,k ∈ C measured at the single antenna of UE k is generated by the voltage vector v G ∈ C M BS at the BS array. From (8) where c is an arbitrary constant measured in V 2 . The vector d dl k ∈ C M BS is obtained from (19): since the receiving UE has a single antenna. In particular, (a) derives from (18) as D OC is a 1×M BS matrix (i.e., a row vector) whose transpose is exactly (28) and (29). Also, we have defined By setting y dl the input-output relation follows in the form Since a noise matching network is used at each UE, we have that n dl k ∼ N C (0, c −1 σ 2 dl ) with σ 2 dl given by (31). The vector x dl is obtained as wherex dl i ∼ N C (0, p i ) is the information-bearing signal and w i is the precoding vector associated with UE i that satisfies E{||w i || 2 } = 1 so that E{||x dl || 2 } = K i=1 p i and P T = c 4R G K i=1 p i . By treating the interference as noise, the downlink SE for UE k is log 2 is the SINR for c = 1 V 2 . We assume that w k = w k /||w k || and consider both MR precoding with

C. Uplink and downlink duality
The concept of uplink and downlink duality in wireless communications refers to the relationship between the uplink and downlink channels in a communication system, with the exception of a scaling factor. The duality principle states that the uplink channel vector is proportional to the transpose of the downlink channel vector, with a scaling factor that depends on various factors (e.g., antenna gains). The significance of uplink and downlink duality lies in its practical implications for system design and optimization [2]. By exploiting this duality, system parameters and algorithms can be jointly designed for both uplink and downlink transmissions, simplifying system complexity and improving overall performance [2,Sec. 4]. For example, channel estimation and combining techniques developed for uplink can be applied to the downlink without modification, leading to significant savings in complexity. Next, we will discuss this duality in three different cases at the BS: 1) when arbitrary matching networks (e.g., self-impedance matching networks) are used, 2) when no matching network is employed, and 3) when full power and noise matching networks are employed. It should be noted that in all these cases, z ul ART,k = z dl ART,k holds, which is a result of the reciprocity principle in electromagnetic propagation.

1) Arbitrary matching networks:
In general, when arbitrary matching networks are used (e.g., self-impedance matching networks), we have that F dl T = F ul R and Z dl T = Z ul R . From (38) and (45), the physical channels d ul k and d dl k exhibit reciprocity up to a linear transformation. Particularly, d dl where we have defined A dl,ul = (Z dl We observe that z ul ART,k = z dl ART,k and Z dl AT = Z ul AR in (53) and (54). Therefore, d ul This condition is easily satisfied as it involves the load and generator impedances at the BS.
As for h ul k and h dl k , from (21) we observe that, in the absence of a power matching network, B dl is no longer equal to the identity matrix I M BS but is given by The equation above demonstrates that h dl k can be derived from h ul k by multiplying it with the matrix The results for the downlink channel are summarized in the second column of Table II.
3) With power and noise matching networks: When a noise matching network is used at the BS, the impedance matrix Z MR is equal to Z ⋆ MR in (27). From (29) and (30), d ul k in (38) becomes In the downlink, if a power matching network is used by the BS, then Z MT = Z ⋆ MT so that Z dl T and F dl T reduce to (24) and (26). Hence, (45) becomes Notice that z ul ART,k = z dl ART,k . If the BS uses the same array for transmission and reception, then Z dl AT = Z ul AR . Putting together the above results yields which shows that d ul k and d dl k differ only for a scaling factor. This holds also for h ul i and h dl i since, in the presence of a power matching network, B dl reduces to I M BS as it follows from (21). The results are summarized in the third column of Table II.

V. THE EFFECT OF COUPLING: A CASE STUDY WITH TWO ANTENNAS IN A SINGLE PATH LOS SCENARIO
To showcase what is the impact of mutual coupling in multi-user MIMO, next we consider a simple scenario in uplink with K = 2 UEs and M BS = 2 half-wavelength dipoles in side-by-side configuration. Transmission takes place over a single LoS propagation path with {(θ k , φ k ); k = 1, 2} being the directions of UEs in the far-field of the BS array. For convenience, we let where |µ| < 1 accounts for the normalized mutual coupling between the two receiving antennas at the BS. The shape of µ as a function of the normalized antenna spacing d H /λ is reported in Fig. 3 for R r = 73Ω and R d = 10 −3 R r . Also, we call

A. Array gain
The following result is found for the array gain, which is valid assuming a full matching network.
Proof. In the case of full matching networks, the SNR γ ul as it follows from (57). By using z ul ART,k = α(θ k , φ k )a(θ k , φ k ) and computing the inverse of (60) yields The array gain is obtained after normalization with 1 This has an important impact on the direction of arrival (θ k , φ k ) corresponding to the maximum value of the array gain. From (62) it can be observed that for a fixed value of d H /λ, the maximum value of the array gain is achieved when µ > 0 and corresponds to the minimum value of cos ψ k for (θ k , φ k ). On the other hand, if µ < 0 the maximum is achieved for (θ k , φ k ) corresponding to the maximum value of cos ψ k . Assume for example d H /λ < 0.43, which means 0 ≤ 2πd H /λ < 0.86π < π. Since µ > 0, the maximum array gain is attained when cos ψ k is minimum, i.e., when cos(θ k ) sin(φ k ) = ±1. This condition requires θ k = 0 and φ k = ±π/2, which represents the end-fire direction of arrival. The corresponding maximum array gain is given by If 0.43 < d H /λ < 1, then µ < 0 and the maximum array gain is achieved when cos ψ k is maximum, i.e., when cos(θ k ) sin(φ k ) = 0. This requires φ k = 0 or θ k = ±π/2. In particular, θ k = 0 and φ k = 0 corresponds to the front-fire direction of arrival. In this case, we obtain Maximum Array Gain = 2 1 + µ .
(66) Fig. 4a reports the SNR in dB for UE 1 as a function of φ 1 for different values of d H and with a full matching network, i.e., Z MR = Z ⋆ MR . We assume that UE 1 is located at a distance of 50 meters and that the BS array is at an height of 10 meters, which means θ 1 ≈ −11 • . The key parameters of the BS antenna array are reported in Table I. For comparison, the SNR for the single-antenna case (i.e., M BS = 1) is shown together with the line corresponding to an array gain of 3 dB, i.e., the maximum array gain achievable with two uncoupled antennas. In agreement with the discussion above, the results of Fig. 4a show that, in the presence of a noise matching network, the array gain is maximum for φ 1 = ±π/2 (end-fire), when the antenna spacing d H is below λ/4 since µ > 0. On the contrary, it takes the maximum value for φ 1 = 0 (front-fire) when d H = λ/2 since µ < 0. For all the considered values of d H , there exist ranges of φ 1 for which the array gain is above 3 dB. This proves that moving the antennas close to each other may have a positive effect that becomes negligible when d H is further reduced below λ/10. Interestingly, an array gain A noise matching network is used.
greater than 3 dB can also be obtained for d H = λ/2, when the transmitter is in front-fire. This is possible simply because µ = 0 for d H = λ/2, as shown in Fig. 3.
The impact of the choice of the matching network on the performance when moving the antennas close to each other is illustrated in Fig. 4b and Fig. 4c, where we plot the SNR obtained with the self-impedance matching design (see Sect. II.K) and without a matching network. It can be observed that, for a fixed antenna spacing, the maxima and minima occur at the same values of φ 1 , regardless of the matching network design. However, the specific values of these maxima and minima are strongly influenced by the choice of the matching network. For instance, Fig. 4b demonstrates that reducing d H below λ/4 has a negative impact on both SNR and array gain. Furthermore, it is evident that the best performance, whether with a self-impedance matching network or without any matching network, is achieved when d H = λ/2 and φ 1 = 0.
To gain further insights into the effect of coupling as φ 1 varies, Fig. 5 plots the SNR of UE 1 with a full noise matching network for 0.01 ≤ d H /λ ≤ 1. In particular, the black dashed curve has been obtained with φ 1 uniformly distributed between −π/2 and π/2. The other parameters are the same as in Fig. 4a. The results are in agreement with those from Fig. 4a. Specifically,  antenna case is also reported as a benchmark.
Proof: With half-wavelength dipoles in side-by-side configuration, µ can be approximated Plugging these expressions into (62) yields (67) from which the asymptotic value for d H /λ → 0 follows.

B. Interference gain
The mutual coupling between antennas has also an impact on the interference term u H k h ul i 2 in (43). To show this, the following result is given for MR, i.e., u k = h ul k .
Lemma 2. Consider the uplink with MR and assume that M BS = 2. If a full matching network is used at the BS, then in a single path LoS propagation scenario the interference gain (compared to a single antenna BS) between UEs k and i is Proof: With MR and full matching networks, the normalized interference term is as it follows from (57). From (60), a(θ k , φ k ) H Re{Z ul AR } −1 a(θ i , φ i ) is obtained as and a(θ k , φ k ) H Re{Z ul AR } −1 a(θ k , φ k ) can be obtained from (64). The result in (68) follows. Similarly to the array gain, the interference gain is also influenced by the parameters d H , (θ k , φ k ) and d H , (θ i , φ i ) through ψ k and ψ i . The expression for the interference gain is more complex, making it challenging to gain direct insights into the interplay of these parameters. However, the numerical results shown in Fig. 7 reveal that the coupling effects observed with densely spaced antennas can either enhance or hinder the interference rejection capabilities of MR (but the same considerations apply to MMSE), depending on the directions of arrival of the interfering signals.
For example, assuming that UE 1 is in end-fire (as in Fig. 7a) the best performance is observed with φ 2 = 0 • when d H = λ/2 and for φ 2 ≈ 24 • when d H = λ/10. It is worth observing that, when φ 1 = 0 • , moving the antennas close to each other has minimal effects on interference rejection, as shown in Fig. 7b. In this case, the best performance is obtained with d H = λ/2 and φ 2 ± 90 • .

C. Spectral efficiency
Both array and interference gains contribute to the overall SINR and ultimately impact the spectral efficiency of the different UEs. To quantify this, Fig. 8 plots the spectral efficiency of UE 1 as a function of φ 2 in the same simulation scenario of Fig. 7. The single antenna case is also reported as a benchmark. It is assumed the same transmit power for the two users. The results in Figs. 8a-8b can easily be explained with those in Fig. 5 and Figs. 7a-7b. In particular, as expected, the points of minimum/maximum in Figs. 8a and 8b correspond to the points of maximum/minimum in Figs. 7a and 7b. When φ 1 = −90 • (as in Fig. 8a), the maximum SE (more than four times larger compared to the single antenna case) is achieved with d H = λ/10 (when  and 'No' matching network (MN). We assume that the angles of arrival φ 1 and φ 2 for the two UEs are within the range [−π/2, π/2] and that both UEs are located at a distance of 50 meters.
The results indicate that decreasing d H /λ has a detrimental impact on the spectral efficiency, regardless of the matching network employed. Additionally, it is evident that the best performance is achieved when d H /λ ≥ 0.5, meaning that there is no significant advantage in using a full matching network compared to the self-impedance matching design. As anticipated, a considerable reduction in spectral efficiency is observed when no noise matching network is utilized.

VI. NUMERICAL ANALYSIS
The analysis presented above highlights that the mutual coupling effects resulting from closely spaced antennas can potentially provide benefits to the uplink spectral efficiency in single-user and multi-user Holographic MIMO systems, depending on the specific propagation conditions. However, it is important to note that these conditions may not be met in practical network scenarios, and therefore the average gains may be marginal or even non-existent. The analysis focused on a simplified uplink case study with two antennas, two UEs, and single-path LoS propagation. Next, the numerical analysis is expanded to more realistic scenarios, including a larger number of antennas, multiple UEs, and different propagation conditions. Additionally, the  Fig. 13a and to K = 24 in Fig. 13b.
analysis considers the case of densely packed antennas in a space-constrained form factor. By exploring these scenarios, a more comprehensive understanding of the benefits of mutual coupling in Holographic MIMO systems can be obtained.
The key parameters of the system are those reported in Table I limitations, our main emphasis is on the uplink but we put a specific focus on addressing the duality issue in the downlink. Moreover, we still focus on line-of-sight propagation but notice that similar results can be obtained with different channel models, e.g., based on stochastic approaches.
The Matlab code that will be made available upon completion of the review process can be used to generate the omitted results.
A. Fixing the Number of Antennas while Varying Array Size   can be clearly seen in Figure 11b, which displays the channel gain per antenna using MMSE, corresponding to the same simulation scenario as in Figure 11a. The results demonstrate that bringing the antennas closer to each other in Holographic MIMO systems can have adverse effects on the channel gain, even when employing a full matching strategy. This is evident from the declining trend of the channel gain as the antenna spacing decreases.

C. Impact of uplink and downlink duality
We now consider the downlink with MMSE precoding, and with either a full or an SI matching network. From Table II it is seen that, with a full matching network, h ul k and h dl k differ only for a scaling factor. Accordingly, it is correct to design the MMSE precoder in downlink by using the measured value of h ul k in uplink. On the other hand, when an SI matching network is employed, the uplink-downlink channel duality requires to apply a linear transformation to h ul k . A performance loss is incurred if this is not done. Fig. 13a shows the average SE per UE in the same setup of Fig. 10b, i.e., with M BS = 32 antennas and K = 8 UEs. We see that, with a full matched network, the performance in uplink and downlink is the same. As for the SI matching design, two different cases have been considered.
In the first case, the MMSE precoder is computed by using h ul  Fig. 13a with those in Fig. 10b, we see that in the latter case the average SE is the same in uplink and downlink, while a considerable loss is observed in the former case (thicker line), especially at low values of d H /λ. The same conclusions can be drawn from Fig. 13b, obtained in the simulation setting of Fig. 12, which shows the average SE per UE for a fixed size L H = 6λ of the array and K = 10.

VII. CONCLUSIONS
Building on the multiport communication theory (e.g., [16], [17]), a physically-consistent representation of MIMO channels can be derived and directly used by communication theorists as a baseline for modelling the uplink and downlink of holographic MIMO communications, with closely spaced antennas. Particularly, we used it to study the effects of mutual coupling on the spectral efficiency and to gain insights into interplay between antenna spacing and impedance matching network designs. We focus on half-wavelength dipoles. Numerical and analytical results showed that, for the investigated scenarios, having a fixed number of closely spaced antennas can provide benefits to the spectral efficiency with impedance matching but only under some specific propagation conditions, which may not be met in practical scenarios. In these cases, the average gains may be marginal or even non-existent. We explored a scenario where antennas were closely packed in a space-constrained form, and we demonstrated that reducing the antenna distance led to an increase in spectral efficiency. However, this increase becomes negligible as the antenna aperture size grows (in the order of tens of wavelengths). The uplink and duality duality was also investigated for different matching network designs. We limited our study to uniform linear arrays of side-by-side half-wavelength dipoles operating in far-field LoS conditions. However, we notice that the provided framework can be used to extend the results to different array configurations, antennas and propagation conditions. APPENDIX A Let E inc denote an electric field incident on the receive array, produced by the voltage sources.
We model E inc as a plane wave that reaches the receiver from a particular azimuth angle φ inc ∈ [−π/2, π/2) and elevation angle θ inc ∈ [−π/2, π/2). In this case, assuming that the receive array consists of canonical minimum scattering (CMS) antennas and the incident field is linearly polarized, the mth element of v OC can be written as v OC,m = E inc · l eff,m (θ inc , φ inc ) where l eff,m (θ inc , φ inc ) is the effective length (or effective height) [7, Sect. 2.15] of the isolated mth element of the array, in the (θ inc , φ inc ) direction. The effective length is a far-field parameter given by l eff (θ, φ) = l θ (θ, φ) θ + l φ (θ, φ) φ [7, Eq. (2-91)] where θ and φ are the local orthogonal unit vectors in the directions of θ and φ. For λ/2-dipoles, we have l eff (θ, φ) = λg(θ, φ) θ where g(θ, φ) = cos( π 2 sin θ) π cos θ is the radiation pattern. The effective length is also useful for expressing the field E radiated by an antenna at a distance d, in the far-zone region. Denoting with i AT the current feeding the antenna, we can write [7, Eq. (2-92)] where (θ t , φ t ) is the angle of departure. Based on (71) and (72), we can compute v OC and Z ART in a LoS scenario. Assuming a single transmitting antenna (i.e., Z ART reduces to a vector z ART ) and a receive ULA consisting of CMS antennas with the same effective length, we obtain where α ′ (θ inc , φ inc ) = je jψ 0 l eff (θ t , φ t ) · l eff (θ inc , φ inc ) 2λd (74) ψ 0 is the reference phase at the center of receive array, d is the distance between the centers of transmitting dipole and receive array, a(θ, φ) = [e jk(θ,φ) T u 1 , . . . , e jk(θ,φ) T u M ] T , k(θ, φ) = i AT = F T T (Z G +Z T ) −1 v G we obtain with α(θ inc , φ inc ) = F T T (Z G +Z T ) −1 Z 0 α ′ (θ inc , φ inc ). In general, the open-circuit voltage v OC is produced by the superposition of a (possibly large) number L of incoming waves. Assuming that each wave can be modeled as a plane wave with a particular azimuth angle φ i ∈ [−π/2, π/2) and elevation angle θ i ∈ [−π/2, π/2), we obtain (34).

APPENDIX B CONNECTION WITH THE SCATTERING REPRESENTATION
Instead of working with voltages and currents, incident and reflected power waves can be used to describe multiport systems (e.g., [15]). Particularly, at port n of an arbitrary multiport network, we can define the scattering parameters a n and b n , which represent the complex envelopes of the inward-propagating (incident) and outward-propagating (reflected) power waves, respectively. They are related to the voltage and current, v n and i n , measured at the same port, by the following linear transformations [34], [35,Ch. 4]: a n = v n + Z n i n 2 Re(Z n ) b n = v n − Z * n i n 2 Re(Z n ) where Z n is a chosen reference impedance used for computing the scattering parameters. The physical meaning of a n and b n can be appreciated by computing |a n | 2 − |b n | 2 = Re(v n i * n ) which represents the total power flowing into port n. This is valid for any reference impedance where F and G are diagonal matrices with the nth diagonal elements given by 1/2 Re(Z n ) and Z n , respectively. Based on (77), each impedance matrix can be replaced with the corresponding scattering matrix. In doing so, we obtain a description of the communication system in terms of scattering parameters instead of voltages and currents. The representation of a communication system through the scattering or impedance matrices is only a matter of convenience. Both descriptions are almost equivalent (except for constructed special cases). The authors in [15] found the use of S-matrices more convenient for capacity computation.