1 Introduction
A holographic MIMO (multipleinput multipleoutput) system consists of a physically large antenna surface with (approximately) continuous aperture [3]. This concept is also known as extremely large aperture Massive MIMO [4], large intelligent surfaces [5, 6], and holographic beamforming [7].
In multipleantenna channels, the channel capacity grows linearly with the number of spatial degrees of freedom (DoF) [8, 9, 10], which is therefore a key performance measure. Holographic MIMO represents the ultimate form of a spatiallyconstrained multipleantenna channel where the number of antennas goes to infinity. A fundamental question thus arises: given an area limitation on the aperture, what are the available DoF of a holographic MIMO system? The statical models typically used in the MIMO literature are insufficient to answer this question [2]. Consider for example the uncorrelated Rayleigh fading model where the channel is statistically uncorrelated across all antenna pairs. In this case, increases unboundedly as [9]. However, as grows large in a given area, spatial correlation among antenna pairs naturally arises and therefore cannot increase indefinitely. This is analogous to a bandlimited waveform (timedomain) channel, given the bandwidth constraint and transmission interval , increasing the number of time samples will not increase the capacity indefinitely. The available DoF are fundamentally limited to [11].
To demonstrate an analogous result for multipleantenna channels, statistical models driven by electromagnetic theory considerations are required [12]. To this end, a continuous formulation is somewhat preferable [13]. Results in this direction can be found in [5, 14, 15]. In [14], the authors study the reception of a monochromatic complexvalued electromagnetic field over a sphericallysymmetric aperture (e.g., disk, ball) under a random nonlineofsight (NLoS) propagation. The DoF are computed by means of a signal space approach, which follows directly from the reasoning involved to derive the Shannon formula. Particularly, a continuous series expansion of the electromagnetic random channel is obtained over an orthonormal basis with statistically uncorrelated coefficients, as for the famous KarhunenLoève expansion [16]. The DoF are thus simply obtained by counting the average number of nonzero coefficients, and are found to grow with the surface area of the aperture rather than with its volume, measured in units of wavelengthsquared. The extension to a more general nonmonochromatic electromagnetic field is provided in [15]
and it is based on the celebrated Landau’s eigenvalue theorem
[17]. However, an explicit form for the basis is hardly available in general and often leads to nontrivial representations [14, 18, 15], which rely on special functions or unfamiliar subsidiary mathematical results.A similar result was recently obtained in [5] by considering a monochromatic electromagnetic field over a rectangular aperture under a deterministic LoS propagation. Unlike [14, 15], this leads to a simple deterministic channel model, which is first used in [5] to evaluate the capacity normalized by the deployed area of the aperture, and then to compute the DoF. A more realistic random NLoS propagation environment is considered in [1, 2]
and leads to a Fourier planewave spectral representation of the random electromagnetic channel in Cartesian coordinates. Suitably discretized, this provides a Fourier planewave series representation of the channel of the form of an orthonormal series expansion with statisticallyuncorrelated Gaussiandistributed coefficients. This can be regarded as the asymptotic version of the multivariate KarhunenLoève series expansion
[16]. In this paper, we use the Fourier planewave series representation in [1, 2] to compute the average DoF of a holographic MIMO system, i.e, a large, but finite, spatiallycontinuous aperture.(1) 
(4) 
2 Preliminaries
We begin by reviewing the basics of the analytical framework developed in [1, 2]. Consider electromagnetic waves propagation in every direction^{1}^{1}1The nonisotropic case is handled by observing that it is obtained from the isotropic case through a linear timeinvariant filtering operation [1, 2]. through a homogeneous, isotropic, and infinite random scattered medium [1]. Under these settings, each of the three Cartesian components of the electrical field can be described independently by the scalar wave equation [19] and electromagnetic waves qualitatively behave as acoustic waves [1]. Hence, the channel over an infinitely large spatiallycontinuous aperture can be modeled as a Gaussian spacefrequency scalar random field: , which is a function of frequency and spatial coordinates . We treat only monochromatic waves, i.e., propagating at the same frequency , which can thus be omitted. We assume that can be modeled as a zeromean, spatiallystationary and Gaussian random field. This is a scenario of primary interest in wireless communications.
2.1 Fourier Spectral Representation
Every zeromean, secondorder, stationary Gaussian random waveform channel can be represented either in time domain or in frequency domain. The mapping between these two representations is the onedimensional (1D)
Fourier spectral representation. Similarly, can be represented either in spatial domain or wavenumber (also known as spatialfrequency) domain, which represent the time and frequency counterparts, respectively. The spatialwavenumber mapping is given by the 3D Fourier spectral representation in (1) where are the realvalued Cartesian coordinates in the wavenumber domain,is a zeromean stationary whitenoise Gaussian random field with unit spectrum, and
is the power spectral density. The latter has a key role between the two domains, as shown next. Notice that (1) can be regarded as a superposition of an uncountablyinfinite number of planewaves each one having statisticallyindependent Gaussiandistributed random amplitude [1, 2]. The condition excludes the socalled evanescent waves, which decay exponentially fast in space and do not contribute to the farfield propagation.2.2 Fourier PlaneWave Spectral Representation
Since
is a Gaussian spatial random field of electromagnetic nature, each of its realizations needs to satisfy, with probability
, the socalled homogeneous Helmholtz equation where is the wavenumber with being the wavelength. As a direct consequence of the Helmholtz equation, we have that [1, 2](2) 
which is an impulsive function with wavenumber support on the surface of a sphere of radius . Its impulsive nature makes it hard the direct computation of (1) since the squareroot of an impulsive function is not defined. This is addressed in [1, 2] and yields a 2D Fourier planewave spectral representation of where are defined in (4) and are two 2D independent, zeromean, complexvalued, whitenoise Gaussian random fields. The “planewave” terminology refers to the fact that and represent a decomposition of the channel in terms of an uncountably infinite number of planewaves that are spatially propagating in the halfspaces and , respectively. Notice that has bandlimited spectrum in the wavenumber domain since the coordinates in have a compact support given by a disk of radius centered on the origin, as illustrated in Fig. 1. In other words, only a subset of planewaves in (1) effectively propagate. These are called propagating waves. This is a direct consequence of the Helmholtz equation, which thus acts as a 2D linear spacetime invariant physical filter. Notice that the planewaves associated with the propagation directions outside the disk would be associated with evanescent waves.
2.3 Fourier PlaneWave Series Representation
Consider a compact spatiallycontinuous rectangular aperture with side lengths . The channel energy collected over the finite spatial volume is contained in a countably infinite number of planewaves; see Fig. 1. Therefore, (4) can be replaced by a 2D Fourier planewave series representation with periods and along the and axes [2]. Since the channel spectra is bandlimited, only a countably finite number of planewaves propagate through space (i.e., propagating waves). Within the fundamental period, the following approximation is found for :
(5) 
where the discrete spectral support is a 2D lattice ellipse of semiaxes and and the Fourier coefficients are
(6) 
where and
are statisticallyindependent Gaussiandistributed random variables with variances
computed in [2]. Notice that the approximation error of (5) becomes negligible as . At carrier frequency GHz (i.e., cm), an aperture length of m already provides , which increases to at GHz (i.e., cm). This means that (5) is a good approximation in practice.3 Channel Degrees of Freedom
We now use (5) to compute the DoF of linear, planar and volumetric continuous apertures. Before this, we review the reasoning involved to derive the formula [11].
Consider a bandlimited waveform channel of bandwidth and time interval . The ShannonNyquist sampling theorem states that we can approximate as a linear combination of a countably finite number of elements of the cardinal basis of functions with coefficients collected inside the time interval and equally spaced by :
(7) 
where the approximation error becomes negligible as . The limit can be seen as going to infinity while going to zero, but has a higher convergence speed such that . This is because physicsbased signals are of limited energy and thus subject to the phenomena of spectral concentration [12] under which as increases the effective bandwidth gets smaller and smaller. As a consequence, the available DoF are limited to a finite nonzero value
(8) 
which is the product between time interval duration and the frequency bandwidth . Notably, (5) and (7) are two bandlimited orthonormal series expansion having a countablyfinite number of coefficients, whose cardinality determines the space dimension, i.e., the available DoF.
3.1 Linear Aperture
Assume is a line segment of length along the axis. The Fourier series expansion (5) may be rewritten as
(9) 
where is the 1D Fourier basis and are the statisticallyindependent Gaussian Fourier coefficients. The average available DoF are
(10) 
given by the product between the aperture length and the wavenumber . From Fig. 1, it follows that in the 1D case; that is, represents the wavenumber bandwidth. Therefore, (10) is the spatialwavenumber counterpart of (8) where the time interval and angular frequency bandwidth are replaced with and , respectively.
3.2 Planar Aperture
Assume is a rectangle of side lengths and on the plane. The Fourier series representation in (5) reads
(11) 
where is the 2D Fourier basis, and is the th Fourier coefficient defined over the discrete spectral support . The available DoF are simply the measure of the lattice wavenumber support, which is given by with [20]. Therefore, we have that
(12) 
which is proportional to the surface area of the aperture measured in units of wavelengthsquared. From (10), one may expect that the expansion of a 1D aperture into a 2D aperture may yield DoF. However, this is not the case. The DoF are reduced by a factor , which is exactly the ratio between the areas of the disk and the square circumscribing it. This is due to the fact that evanescent waves are not included in our analysis; see Fig. 1.
3.3 Volumetric Aperture
When is a parallelepiped of side lengths , , and , the Fourier series expansion at any is given by (11) by replacing the th Fourier coefficient with [2]. In this case, it is more convenient to proceed as follows [16]. By collecting the samples along , i.e.,
, we obtain a single random vector
given by, for and(13) 
where is obtained from by a linear application matrix
(14) 
The available DoF are given by the product between the space dimension spanned by the 2D Fourier basis – as given in (12) – and the vector space dimension spanned by . This is given by , since the two columns of are linearly independent regardless of . Thus, the DoF are
(15) 
As for the 2D case, the DoF are proportional to , and not to the volume of . Hence, the expansion of a planar aperture into a volume aperture asymptotically yields only a twofold increase in the available DoF. This is because the upper and lower hemispheres of the spherical wavenumber support can be independently parametrized on the disk [2, Fig. 2(a)], which brings us back to the 2D case. Compared to the DoF obtained for a volumetric deployment in [5, Sec. III.C], the only difference is a factor . This is because in LoS propagation the planewaves impinge on the aperture from only one of the two halfspaces where the aperture is located.
4 Numerical results
Numerical results are now used to validate the accuracy of the analytical framework for apertures of relatively small size. We approximate the continuous aperture by discretizing its spatial domain on a grid of points with spacing . The channel samples generated by sampling (9), (11), and (13) for any are collected into . The uncorrelated Rayleigh fading model [9, 10] and the Clarke’s model [21, 22] are also considered. In both cases, is generated through the discrete KarhunenLoève representation; that is, where , and is the spatial correlation matrix. With uncorrelated Rayleigh fading, . With the Clarke’s model, it is a blockToeplitz matrix with entries with that is obtained by sampling the spatial autocorrelation function between antenna locations and [2].
The DoF are numerically evaluated by generating an ensemble of random vectors
and averaging the number of linear independent vectors enclosed within this ensemble. We begin by considering a linear aperture with
. Fig. 2 illustrates the DoF as a function of . As increases, reduces and the array approaches a continuous aperture. The results show that the DoFs are a monotonically nondecreasing function of . With uncorrelated Rayleigh fading, grows linearly with since the channel samples are uncorrelated across all antenna. On the contrary, is limited to (10) for the channel samples obtained through the Fourier planewave series representation. The Clarke’s model shows a similar behavior with a smoother transition region for the channel eigenvalues due to the different numerical generation method for the channel samples. The practical relevance of the results in Fig. 2 is that they tell us the right number of antennas to be deployed on an aperture of size . Similar results are found when considering higher dimensional apertures and their limits (12) and (15). This is illustrated in Figs. 3 and 4 for the 2D and 3D cases, respectively.5 Conclusions
A random electromagnetic isotropic channel generates, over a spatiallycontained continuous aperture, a number of DoF that is proportional to the surface area, measured in units of wavelengthsquared. We obtained this result by using a Fourier planewave series expansion of the channel, which yields the optimal number of antennas to be deployed under isotropic propagation. This treatment can be extended to include transmit side as well as nonisotropic propagation [2].
References
 [1] T. L. Marzetta, “Spatiallystationary propagating random field model for Massive MIMO smallscale fading,” in 2018 IEEE Int. Symposium Inf. Theory (ISIT), June 2018, pp. 391–395.
 [2] A. Pizzo, T. L. Marzetta, and L. Sanguinetti, “Spatial characterization of Holographic MIMO channels,” CoRR, vol. abs/1911.04853, 2019.
 [3] E. Björnson, L. Sanguinetti, H. Wymeersch, J. Hoydis, and T. L. Marzetta, “Massive MIMO is a reality—What is next? Five promising research directions for antenna arrays,” Digital Signal Processing, 2019, to appear.
 [4] A. Amiri, M. Angjelichinoski, E. de Carvalho, and R. W. Heath, “Extremely large aperture massive MIMO: Low complexity receiver architectures,” in IEEE Global Communications Conference Workshops (GLOBECOM Workshops), 2018.
 [5] S. Hu, F. Rusek, and O. Edfors, “Beyond Massive MIMO: The potential of data transmission with large intelligent surfaces,” IEEE Transactions on Signal Processing, vol. 66, no. 10, pp. 2746–2758, May 2018.
 [6] M. Di Renzo, M. Debbah, D.T. PhanHuy, A. Zappone, M.S. Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos, J. Hoydis, H. Gacanin, J. de Rosny, A. Bounceu, G. Lerosey, and M. Fink, “Smart radio environments empowered by reconfigurable AI metasurfaces: an idea whose time has come,” EURASIP Journal on Wireless Communications and Networking, vol. 2019:129, 2019.
 [7] E. J. Black, “Holographic beam forming and MIMO,” Pivotal Commware, Tech. Rep., 2017., 2017.
 [8] David Tse and Pramod Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005.
 [9] Emre Telatar, “Capacity of multiantenna gaussian channels,” European Transactions on Telecommunications, vol. 10, no. 6, pp. 585–595, 1999.
 [10] G. J. Foschini and M.J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communications, vol. 6, no. 3, pp. 311–335, Mar 1998.
 [11] C. E. Shannon, “The mathematical theory of communication,” Bell System Technical Journal, vol. 27, no. 3, pp. 379–423, 1948.
 [12] Massimo Franceschetti, Wave Theory of Information, Cambridge University Press, 2017.
 [13] A. B. Baggeroer, “Space/time random processes and optimum array processing,” Tech. Rep. ADA035593, Naval Undersea Center, San Diego (CA), Apr. 1976.
 [14] A. S. Y. Poon, R. W. Brodersen, and D. N. C. Tse, “Degrees of freedom in multipleantenna channels: a signal space approach,” IEEE Trans. Inf. Theory, vol. 51, no. 2, pp. 523–536, Feb 2005.
 [15] M. Franceschetti, “On Landau’s eigenvalue theorem and information cutsets,” IEEE Trans. Inf. Theory, vol. 61, no. 9, pp. 5042–5051, Sept 2015.

[16]
H. L. Van Trees,
Detection Estimation and Modulation Theory, Part I
, Wiley, 1968.  [17] H. J. Landau, “On szegö’s eingenvalue distribution theorem and nonhermitian kernels,” Journal d’Analyse Mathématique, vol. 28, no. 1, pp. 335–357, Dec 1975.
 [18] H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, fourier analysis and uncertainty — ii,” Bell System Technical Journal, vol. 40, no. 1, pp. 65–84, 1961.
 [19] W. C. Chew, Waves and Fields in Inhomogenous Media, WileyIEEE Press, 1995.
 [20] Ekkehard Krätzel, Lattice Points, Springer Netherlands, 1989.
 [21] R. H. Clarke, “A statistical theory of mobileradio reception,” The Bell System Technical Journal, vol. 47, no. 6, pp. 957–1000, July 1968.
 [22] T. Aulin, “A modified model for the fading signal at a mobile radio channel,” IEEE Trans. Veh. Technol., vol. 28, no. 3, pp. 182–203, Aug 1979.
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