Including Antenna Effects Into Capacity Formulations of Line-of-Sight MIMO Channels

Analytical expressions for the capacity of multiple-input multiple-output (MIMO) line-of-sight (LOS) channels that include antenna effects in both the near- and far-field regions are provided. They extend previous formulations for LOS capacity in that they provide insights into capacity results across the full range of antenna element separations. This range includes element separations tending to zero, and more generally, we show that MIMO LOS capacity at antenna element separations of less than five wavelengths differs from previous results without antenna effects. In addition, the results reveal an optimum antenna element separation to maximize the capacity for single-rank MIMO LOS channels. A novelty of the approach is that all results are obtained with analytical expressions and can thus be easily incorporated into system formulations. The results have applications in MIMO LOS channels, including those involving unmanned aerial vehicles (UAVs) and wireless communication systems in millimetre-wave and terahertz bands.

Including Antenna Effects Into Capacity Formulations of Line-of-Sight MIMO Channels Jun Qian , Member, IEEE, Shanpu Shen , Senior Member, IEEE, and Ross Murch , Fellow, IEEE Abstract-Analytical expressions for the capacity of multipleinput multiple-output (MIMO) line-of-sight (LOS) channels that include antenna effects in both the near-and far-field regions are provided.They extend previous formulations for LOS capacity in that they provide insights into capacity results across the full range of antenna element separations.This range includes element separations tending to zero, and more generally, we show that MIMO LOS capacity at antenna element separations of less than five wavelengths differs from previous results without antenna effects.In addition, the results reveal an optimum antenna element separation to maximize the capacity for singlerank MIMO LOS channels.A novelty of the approach is that all results are obtained with analytical expressions and can thus be easily incorporated into system formulations.The results have applications in MIMO LOS channels, including those involving unmanned aerial vehicles (UAVs) and wireless communication systems in millimetre-wave and terahertz bands.Index Terms-Capacity, linear arrays, MIMO LOS channel, mutual coupling, radiated power, terahertz.

I. INTRODUCTION
M ULTIPLE-INPUT multiple-output (MIMO) line-of- sight (LOS) systems have become important due to their application in wireless systems such as unmanned aerial vehicles (UAV) and millimetre-wave (mmWave) wireless communications [1], [2].They also have potential for use in future sixth-generation (6G) wireless communications involving near-field LOS channels where Terahertz (THz) bands (from 0.1 to 10 THz) have been advocated [3], [4].At these frequencies, MIMO communication will be important to realize the multiple wavelength antenna apertures necessary for compensating the effects of the small wavelengths [1], [2], [5].
Previous MIMO LOS capacity formulations have used models that do not include antenna effects [6], [7], [8].While these have yielded valuable results for capacity trade-offs, the effects of the antennas on these channels are also important [2], [6].For example, in previous formulations [6], [7], there is no lower limit to the antenna element separations that achieve capacity gains.If all the antenna element separations tend to zero, the MIMO antenna performance will also tend to that of a single antenna.This type of effect is not modelled by existing approaches [6], [8].Whenever antenna element separations become less than around five wavelengths, the effects of the antennas should be included to provide a complete picture of the MIMO LOS channel.Mutual coupling has also been considered in holographic MIMO and also extra-large MIMO systems [9], [10] where the LOS scenario is particularly relevant.This letter extends existing LOS capacity models by incorporating antenna effects, and the results are valid for both near-and far-field regions.The results are provided entirely in analytical form and can be carried over to other applications straightforwardly.We also formulate the general MIMO LOS channel and its capacity for all antenna element separations to provide unified capacity results for LOS channels [6], [7].The results also predict an optimum separation that achieves maximum capacity in the single-rank channel regime.
The specific contributions can be listed as follows: 1) We provide analytical expressions that include antenna effects in the formulation of the capacity of MIMO LOS channels across the full range of antenna element separations.
2) We provide analytical expressions for the single-rank MIMO LOS capacity when antenna effects are considered and show that array directivity and mutual coupling manifest as the same antenna effect.Simulation results for the capacity of general MIMO LOS channels are also provided.
3) We provide approximations for the optimum element separation for maximum capacity in the single-rank regime.
Organization: Section II describes the MIMO LOS channel configuration and channel model.Sections III and IV study the antenna effects and the proposed MIMO LOS capacity with these antenna effects.

II. MIMO LOS CONFIGURATION
We consider a narrow band N × M MIMO LOS antenna system where there are M antennas at the transmitter and N antennas at the receiver as shown in Fig. 1 [2], [11].For convenience, the first element of the transmit array is located at the coordinate origin, and the first element of the receive array is separated from the coordinate origin by D along the x-axis coordinate as shown.We define d n,m (n = 1, . . ., N , m = 1, . . ., M ) as the distance between the mth transmit antenna and nth receive antenna and is written where (x m , y m , z m ) and (x n , y n , z n ) are the coordinates of the mth transmit antenna and nth receive antenna respectively.In the remainder of this letter, we will always use the exact form  for d n,m , as in ( 1), without near-or far-field approximations unless stated otherwise.
For the transmitter and receiver, we consider uniform linear arrays (ULA) where the elements are separated by respective d t and d r .For example, the transmit ULA oriented along the z-axis the element coordinates would be (x n , y n , z n ) = (0, 0, (n − 1)d t ).The work can be extended to uniform planar arrays (UPA), but it is not considered here for expediency of explanation.
We consider waves with wavelength λ and wavenumber k = 2π/λ.We consider point source arrays, so that an individual channel link formed by the ULA MIMO LOS channel model H LOS ∈ C N ×M can be given by [3], [12] where the individual channel link has been normalized by the channel formed between the transmit and receive elements at their respective origins.It should also be noted that the channel model relates the current input at the transmit antenna elements to the open circuit voltage at the receive antenna elements.
For reference later, we take the Rayleigh distance [3], the boundary between near-and far-field regions, as since there are two arrays.For convenience, we set N = M and d t = d r = d in the following.

III. ANTENNA EFFECT ANALYSIS A. Total Power Radiated
To determine the total power radiated from the transmit array, we can integrate the radiated power density over any surface enclosing the array (since by power conservation, the same power passes through every enclosing surface in both near-and far-field).For convenience, with no loss of accuracy, we therefore use an enclosing spherical surface with a radius in the far-field region.In addition, since the transmit array is located at the origin, we can orient the enclosing surface so that the array aligns with the z-axis without loss of generality.
Using these considerations allows us to use the far-field array pattern for an N-element ULA spaced along the z-axis to find the total normalized radiated power.Assuming all element excitation's are set to unity and identical, this is written [13] (4) Integrating over the enclosing sphere, the total radiated power normalized to that of a single element is given by where sinc(x) = sin(x)/x is the unnormalized sinc function.Plots of ( 5) are shown in Fig. 2, as a function of element separation for N = 2 and 8.The total radiation varies significantly with element separations for up to 5λ.The ratio of the power radiated at very small element separations compared to those at large separations is approximately 1/N.This is because, at very small element separations, the patterns add coherently for all radiation directions, while at larger separations, the patterns do not add coherently for all directions.We can find that with larger N, the peak is shifted to the left since the end-fire direction becomes the dominant radiating component, and therefore the peaks tend to whole-number wavelength separations.The effect of this power variation has not been incorporated into previous LOS MIMO capacity results [6], [8].
Given that the squared sum of the excitation's should be related to the input power, it might be thought that the variation in the radiated power in (5) violates the principle of power conservation.This is resolved by realizing that although the current excitation's of the array are constant, the input voltages or antenna element impedance varies as a function of element separation.This issue has manifested itself in (5) even though antenna impedance (including mutual coupling) is not explicitly considered.A fundamental relation exists between the element patterns and antenna impedance, as noted previously [14].While the effect of mutual coupling could be reduced by using special antenna designs or a decoupling network, the resultant antenna patterns would also change and, in effect, prevent their use for LOS applications.Finding the radiated power by using mutual coupling is described next.

B. Mutual Coupling
A different but equivalent viewpoint for the results shown in Fig. 2  antenna impedance defined by [15] where Z can also be written in terms of antenna resistance and reactance as Z = R + jX.In ( 6), Z n represents the mutual impedance between antenna elements with separation distance nd when n ∈ 1, 2, . . ., N − 1. Z 0 = R 0 + jX 0 is the antenna self-impedance.
There are two limiting configurations of the circuit model (6).In the first, when element separations are large, the mutual impedance Z n (n = 1, . . ., N − 1) is zero, and all elements can be considered isolated with self impedance Z 0 ; In the second, when the separations tend to zero, the mutual coupling Z n → Z 0 , the elements in Z will all be identical and it will have rank one.If all the excitation's are also identical, the N antennas will appear to be connected in parallel, and this forms an equivalent single antenna (see ( 6)) with overall parallel impedance Z 0 .
The total power radiated by the array using ( 6) for the general current excitation case can be written as where R(•) refers to the real part.If all excitation's are unity and identical, and the antenna elements are also identical, then the total radiated power, normalized to that of a single element, can be written as Equating this with (5) reveals that the normalized mutual resistance is R n /R 0 = sinc(knd ) for the point array [14].It shows that total radiated power and mutual coupling manifest as the same antenna effect.As a side note, utilizing the normalized mutual resistances, we can determine the total radiated power for any current excitation by using (7).For example, if there are an even number of elements with alternating −1 and 1 excitation's, the total radiated power for element separations tending to zero would also tend to zero.Note that the equal weight configuration is a specific example considered for illustration, and general element weightings are used for finding capacity with water-filling in later simulations.

C. Channel Model With Mutual Coupling
Using (6), we can extend (2) to include antenna effects.We can use the general mutual coupling channel model first provided in [16].We assume that both the receiver and transmitter have identical elements and conjugate matching is utilized such that (6), Z L = Z H (where Z L is the matched load).Single-point matching is usually considered a more practical scenario but for ease of analysis we use conjugate matching and the result trend will be similar [16].Then, the MIMO LOS channel model with mutual coupling can be written as where R r and R t are the respective receiver and transmitter antenna resistances with R R = R R H /2 R R 1/2 .For large element separations without mutual coupling, R T 1/2 and R R 1/2 become identity matrices with respective √ R t , √ R r scaling.

IV. MIMO LOS CAPACITY WITH ANTENNA EFFECTS
Using the channel model in (9), the channel capacity for the general N × N MIMO LOS channel with mutual coupling can be written as where ρ = P /σ 2 0 is the average SNR of each receiver antenna [8], in which P is the total transmit power, σ 2 0 represents the additive Gaussian noise power [2].Furthermore, Q is the covariance matrix representing the power allocation with trace(Q) = N.In (10), the term H mc is found from ( 9) by using the resistances obtained from the text below (8).
The selection of the covariance matrix Q has an impact on capacity.For equal power allocation, Q is set to the identity matrix so that all spatial sub-channels are allocated the same power.If the channels are known to the transmitter, then optimum water-filling power allocation can be performed to achieve the optimum channel capacity.
A special case of water-filling to consider is when the channel model is in the far-field zone.This occurs when the element separations are small or the transmitter-receiver distance D is large (greater than the Rayleigh distance following (3)), so that the communication system is operating purely in the far-field zone.In this case, the channel becomes single-rank, with all channel elements tending to unity.In this situation, beamforming or, more generally, water-filling can be utilised to increase capacity.
To gain insight into the far-field zone we incorporate beamforming with combining weights w T ∈ C N ×1 and w R ∈ C N ×1 at the transmitter and receiver with ||w T || = ||w R || = 1.The MIMO channel can then be written as a SISO channel, where h LOS ∈ C N ×1 is defined by the single-rank channel matrix decomposition where D T is the transmitter directivity since T w T | 2 can be thought of as the radiated power density and the input power is unity.
for the receiver is defined similarly.For illustration purposes we assume that w R = w T = w have equal antenna weighting and We can also write directivity, D T = D R = D, using ( 4) and ( 5), since directivity is defined as the maximum Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
in pattern power density normalized to the total isotropic radiated power density [13].By noting that the maximum power density (4) of the array occurs when θ = 90 • and is |F (φ, θ)| 2 = N 2 and using (5), we obtain since P radiated in ( 5) is already normalized to the radiated power of an isotropic element.The single-rank capacity can then be written in terms of directivity following (13) as Moreover, for the N = 2 configuration, water-filling and beamforming with equal antenna weighting are the same.More generally, for N > 2, water-filling capacity is different from that with equal antenna weightings and results in superdirectivity at low antenna element separations [17] as shown later.
Capacity results that include antenna effects for both nearand far-fields are illustrated with the examples that follow.Capacity results are provided first as a function of element separation and then as a function of array separation.

A. Capacity as a Function of Element Separation
Using (10), we can obtain capacity results for any element separation and across near-and far-field regimes with and without water-filling.In all the figures that follow, a comparison benchmark is used, which consists of previous LOS results without antenna effects [6], [8].
For element separations between 0.1 − 20λ, capacity results are shown in Fig. 3 and Fig. 4 for N = 2 (for D = 100λ) and N = 8 (D = 800λ) respectively on a log distance scale with ρ = 10 dB.For N = 2 (Fig. 3), the Rayleigh distance occurs when the element separation is d ≈ 3.5λ (for D = 100λ) and for N = 8 (Fig. 4) the element separation is d ≈ 1.1λ (for D = 800λ).We have provided results for equal and water-filling power allocation in both figures.We also provide capacity results for the single-rank channel with antenna directivity (14).Comparisons with full-rank MIMO (using Nlog(1 + SNR)), single-rank SISO (using log(1 + SNR)) and the benchmark LOS without antenna effects [6], [8] are included.Note that for separations less than 0.1λ, results are not provided.This is because R becomes singular, implying that some antenna currents are becoming impractically high.In practice, for this case, a constraint on the current also needs to be applied and is not considered further here [18].Fig. 3 and Fig. 4 show that antenna effects cause deviations in the capacity predictions as compared to the benchmark previous results [6], [8].The results with antenna effects oscillate above and below that of the benchmark results, and this is directly due to the effects of radiated power or equivalently mutual coupling.The maximum capacity in the far-field region (d < 3.5λ for N = 2 and d < 1.1λ for N = 8) occurs at antenna element separations of around d ≈ 0.7λ for N = 2 and d ≈ 0.9λ for N = 8.
In Fig. 3, it can be observed that the channel capacity with equal power allocation and mutual coupling is less than that without antenna effects for small element separations.This is because mutual coupling is becoming high.It is also less than the SISO channel because power is being equally allocated,  but there is effectively only one channel.For the situation with water-filling, it can be observed that in the far-field region (d < 3.5λ), the capacity results are the same as that of equal weight beamforming as predicted by (14).However, in the near-field, d > 3.5λ, the rank of the channel becomes full, and the capacity with or without mutual coupling or water-filling all become approximately the same as the antenna element separations are greater than a wavelength.The capacity with equal weight beamforming remains as a single-rank channel, as expected.
Fig. 4 shows the capacity results for N = 8.Again, in the far-field d < 1.1λ, there is a maximum in capacity that occurs at less than λ separation.For small element separations, we can observe that water-filling performs better than beamforming with equal weights.This is due to super-directivity [17], where water-filling finds antenna weights that are different from equal-weight beamforming.Otherwise, the observations follow the same trends as for the N = 2 configuration.
One difference between the previous LOS results without antenna effects [6], [8] and those with antenna effects is the optimum in capacity that is reached for separations slightly less than λ.Using the analytical expression (14), we can find the element separation (d ) for maximum LOS capacity in the far-field or single-rank regime.This can be obtained by taking derivatives of ( 14) with respect to d based on finding D using (5) and setting it to zero Solving for d we find that when N = 2, kd ≈ 4.493 or d = 0.715λ.Moreover, d for increasing N tends to one Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

B. Capacity as a Function of Array Separation
Capacity results, as a function of D, for element separations of d = 0.5λ and d = λ are shown in Fig. 5 and Fig. 6.The Rayleigh distance is shown in the results as vertical lines so that the near-and far-field results can be observed.In the far-field the capacity results tend to the single-rank capacity as expected.In the near-field at certain separations where the channel is full-rank the capacity results tend to full-rank capacity.Moreover, d = λ can achieve higher maximum capacity than d = 0.5λ due to the lower mutual coupling effect.At very short distances, D, the results all tend to single-rank capacities because all the transmit-receive element distances all tend to a multiple of λ/2.Also note that at these short distances, the potential mutual coupling between the transmit and receive antennas is not included.The difference between the capacities with or without water-filling is also demonstrated in the results.In the far-field, capacity with water-filling approaches single-rank capacity (14), while capacity without water-filling approaches capacity without antenna effects.
V. CONCLUSION By utilizing analytical formulations that capture antenna effects, expressions for MIMO LOS capacity in both the near-and far-field regions are developed.These show that at antenna element separations of less than five wavelengths, there are important differences between previous MIMO LOS capacity results and our results that take account of antenna effects.These differences are particularly important for antenna element separations of less than λ and when the number of antennas is low.For the two-element example, the maximum capacity occurs at a separation of d = 0.715λ and tends to a separation of λ for higher numbers of elements.Furthermore, all the results are based on analytical expressions for mutual coupling, and these results can be utilized in other applications, including Rician channels.

Fig. 2 .
Fig. 2. Total radiated power of point array as a function of element separation for N = 2 and 8.Both results have been normalized to unity for easy visualization.
can be obtained directly from the circuit model of the antenna array.The equivalent circuit model relating the input voltages and currents of the array can be written as v = Zi, where i ∈ C N ×1 and v ∈ C N ×1 are the vectors of input currents and voltages at the array and Z ∈ C N ×N is the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 3 .
Fig. 3. Capacity with mutual coupling for N = 2, D = 100λ as a function of element separation.The far-field region occurs when d < 3.5λ.The upper and lower horizontal black lines refer to full-rank and single-rank MIMO capacity without water-filling respectively (MC: mutual coupling, EP: Equal power allocation, WF: water-filling).

Fig. 4 .
Fig. 4. Capacity with mutual coupling for N = 8, D = 800λ as a function of element.The far-field region occurs when d < 1.1λ.The upper and lower horizontal black lines refer to full-rank and single-rank MIMO capacity without water-filling respectively (MC: mutual coupling, EP: Equal power allocation, WF: water-filling).

Fig. 5 .Fig. 6 .
Fig. 5.Capacity of MIMO LOS channel vs. array separation for N = 2.The upper and lower horizontal black lines refer to the full-rank and single-rank MIMO capacity without water-filling respectively.The vertical lines represent the Rayleigh distance for λ/2 and λ element separations respectively.