Electromagnetic Lensfocusing Antenna Enabled Massive MIMO
Abstract
Massive multipleinput multipleoutput (MIMO) techniques have been recently advanced to tremendously improve the performance of wireless networks. However, the use of very large antenna arrays brings new issues, such as the significantly increased hardware cost and signal processing cost and complexity. In order to reap the enormous gain of massive MIMO and yet reduce its cost to an affordable level, this paper proposes a novel system design by integrating an electromagnetic (EM) lens with the large antenna array, termed electromagnetic lens antenna (ELA). An ELA has the capability of focusing the power of any incident plane wave passing through the EM lens to a small subset of the antenna array, while the location of focal area is dependent on the angle of arrival (AoA) of the wave. As compared to conventional antenna arrays without the EM lens, the proposed system can substantially reduce the number of required radio frequency (RF) chains at the receiver and hence, the implementation costs. In this paper, we investigate the proposed system under a simplified singleuser uplink transmission setup, by characterizing the power distribution of the ELA as well as the resulting channel model. Furthermore, by assuming antenna selection used at the receiver, we show the throughput gains of the proposed system over conventional antenna arrays given the same number of selected antennas.
I Introduction
Multiantenna or multipleinput multipleoutput (MIMO) systems have been shown to offer great advantages over conventional singleantenna systems in pointtopoint, singlecell multiuser MIMO, as well as multicell MIMO transmissions [36, 377, 130]. Recently, an even more advanced multiantenna technique known as massive MIMO has been proposed, where antenna arrays with large or ultralarge number of elements are deployed at the base stations (BSs) to reap the MIMO transmission benefits on a greater scale (see [374] and references therein). For example, given a massive MIMO system of antennas serving about terminals with the same timefrequency resource, a simultaneous increase of both the spectral efficiency (in bits/sec/Hz) by times and the energy efficiency (in bits/Joule) by times can be achieved [375], as compared to the reference system with one single antenna serving a single terminal. Other benefits of massive MIMO include the asymptotic optimality of simple linear processing schemes such as maximalratio combining (MRC), the resilience against failures of individual antenna elements, and the possibility to simplify the multipleaccess techniques, etc.
Despite many promising benefits, massive MIMO systems are faced with new challenges, which, if not tackled successfully, could roadblock their widely deployment in practice. Firstly, the use of large antenna array increases the hardware cost considerably. Even with inexpensive antenna elements, the cost associated with the radio frequency (RF) elements, which include mixers, D/A and A/D converters, and amplifiers etc., grows up significantly with the increasing number of antennas used. Secondly, the complexity of signal processing increases drastically due to the large number of branch signals that need to be processed from all antennas, as well as the increased number of channel parameters that need to be estimated for coherent communications. Thirdly, the total energy consumption including that for the RF chains may be greatly increased due to the use of large number of antennas, which can even negate the power saving with massive MIMO transmissions.
In order to capture the promising gains of massive MIMO and yet reduce its cost to an affordable level, we propose in this paper a novel system design as shown in Fig. 1, where a new component called electromagnetic (EM) lens is integrated with the large antenna array, termed electromagnetic lens antenna (ELA). An EM lens is usually built with dielectric material with curved front and/or rear surfaces. With the geometry carefully designed, the EM lens is able to change the paths of the incident EM waves in a desired manner so that the arrival signal energy is focused to a much smaller region of the antenna array. Furthermore, for a given EM lens, the spatial power distribution of any uniform plane wave passing through it is determined by the angle of arrival (AoA) of the incident wave. This is demonstrated in Fig. 2, where the Efield distribution of a practical EM lens with the refractive index of two is shown. The aperture diameter and thickness of the EM lens is and , respectively, where is the wavelength in free space. It is observed that as the incident angle changes from to , the location of the strongest Efield distribution sweeps accordingly.
In this paper, for an initial investigation we apply the proposed ELA system to a simplified singleuser uplink communication setup. We first characterize the power distribution of the EM lens along the line where the antenna array is placed. The channel model with the receiver ELA by incorporating the power distribution of the EM lens is then established. Furthermore, we evaluate the throughput gains of the proposed system over conventional antenna arrays with the antenna selection (AS) scheme applied at the receiver. We show that the proposed ELA design can achieve the same capacity as the conventional antenna array without the EM lens, but requires significantly reduced number of active antennas, thus yielding much lower energy cost and signal processing cost and complexity at the receiver.
Ii System Model
In this paper, we consider a simplified narrowband uplink communication as shown in Fig. 3, where a single mobile user with one omnidirectional antenna transmits to the BS with a large uniform linear array (ULA) consisting of antenna elements deployed along the yaxis and separated by distance of . Without loss of generality, we assume that the antenna array is centered at , so that the location of the th element is
(1) 
We assume that the transmitted signal from the user arrives at the BS antenna array via paths, where the th path, , impinges as a plane wave with AoA of , as shown in Fig. 4. With plane wave assumption, the surfaces of constant phase of an incident wave are parallel planes normal to the direction of incidence, and this is justified when the antenna aperture of the ULA is much smaller than the transmission distance between the user and BS. We further denote the angular spread of the incident waves by , where ; as a result, we have between the user and the th antenna element at the BS is then represented as (without the EM lens applied yet) . The channel coefficient
(2) 
where denotes the wavelength, is the imaginary unit with , represents the power gain of the th component, and denotes the arrival signal phase of the th component. We assume that ’s are independent and uniformly distributed random variables between and , i.e., , . Note that the power gain in general is determined by the distancedependent signal attenuation and shadowing. It is also generally a function of the AoA since the effective aperture of the ULA varies with .
Note that by setting , (2) reduces to the case with singlepath transmission only, e.g., the lineofsight (LOS) scenario. In this case, all antenna elements at the receiver have the same received signal power. In another special case when , becomes a circularly symmetric complex Gaussian (CSCG) random variable, which leads to the wellknown Rayleigh fading channel model [goldsmith2005wireless].
The received baseband signal at the BS is given by
(3) 
where is the transmitted signal power; is the informationbearing signal of the user with unit power; denotes the vector consisting of the complexvalued coefficients of the singleinput multipleoutput (SIMO) channel, i.e., ; and stands for the additive noise vector with components modeled by independent and identically distributed (i.i.d.) zeromean CSCG random variables with variance , i.e., .
Iii EM Lens Embedded Antenna Array
In this section, we investigate further our proposed ELA design for antenna arrays equipped with a front layer of dielectric lens, as shown in Fig. 1. First, we characterize the power distribution at receiving antennas after the EM lens filtering as a function of the AoA. Then we model the resulting SIMO channel with an ELA at the receiver.
Iiia Power Distribution of EM Lens
As illustrated in Fig. 1, an EM lens has two main functions, which are energy focusing and path separation in space, respectively. To specify these functions, we denote as the power density function along the yaxis at the plane of the receiving antenna array, which is activated by an incident plane wave with AoA of is defined over is the aperture diameter of the EM lens, whose center is placed at as shown in Fig. 1. Also note that for a plane wave with AoA , the total power collected by the EM lens, denoted by , is given by , where upon the EM lens. Note that
(4) 
A practical example of the power distribution function of an EM lens is shown in Fig. 5, from which we can observe that: 1) For each incident plane wave, a bell shape power density function is resulted, which demonstrates the energy focusing capability of the EM lens; 2) As increases, the peak power locations shift to the right along yaxis. This implies that the arriving multipath signals with different AoA values can be spatially separated after passing through the EM lens; and 3) The total power collected by the EM lens decreases with increasing .^{1}^{1}1This is due to the fact that the effective aperture of the EM lens is in general proportional to , which decreases with increasing .
For convenience, we define the normalized power density function as
(5) 
thus we have , 5, it is observed that the power density for a given can be coarsely approximated by a Gaussian distribution. Therefore, in the rest of this paper, we assume the normalized power density as a Gaussian distribution function with mean and variance , which specify the peak power location and average power spread, respectively, for an incident wave with AoA of . Thus, we have . From the results shown in Fig.
(6) 
where is given by
(7) 
We further assume for simplicity that , 6), we have neglected the boundary effect by letting in order to have . Furthermore, we define as the power beamwidth, i.e., with a distance away from the center of the power distribution , the power level drops by off the peak value at the center. A simple calculation with Gaussian distribution reveals that . Practically, and both increase with the aperture of the EM lens, since a largersize EM lens generally has a wider range of the focal area. . Note that in (
IiiB Channel Model of ELA
With the normalized power density given in (6), we are now ready to derive the new channel coefficients for the SIMO system introduced in Section II with an ELA at the receiver. We assume the EM lens and the antenna array are appropriately designed and installed so that for the th path component with AoA , we have: 1) the power collected by the EM lens is equal to that received by the antenna array without EM lens, i.e., ; and 2) the power received by the th antenna element with EM lens due to the th component, denoted as , is obtained by integrating the power density function as ,
(8a)  
(8b) 
where , , and denotes the location of the th antenna element, which is given by (1). Note that represents the fraction of the power received by the th antenna element due to the th component. We thus have , . With defined in (IIIB), the channel coefficient in (2) for the th antenna with the EM lens is modified as
(9) 
where is the phase change due to the EM lens, which is assumed to be independently distributed over both and as . Let . Then the received signal at the BS with an ELA is given by
(10) 
where , and have been defined in (3).
Iv Antenna Selection and Performance Analysis
In this section, we compare the performance for the SIMO system with versus without the EM lens under the antenna selection framework.
Iva Antenna Selection
As the number of receiving antennas becomes too large, it is costly in terms of both hardware implementation and energy consumption to make all antennas operate at the same time. A practical lowcost solution is thus antenna selection (AS) [369], where the “best” subset of out of , , receiving antennas are selected for processing the received signal, as shown in Fig. 6. AS reduces the number of required RF chains significantly from to if is much smaller than . For SIMO systems, the optimal combining scheme for the output signals of the selected antennas is known to be maximalratio combining (MRC), with the optimal weights given as the conjugated complex coefficients of the corresponding channels [goldsmith2005wireless]. As a result, the combiner output SNR is equal to the sum of individual branches’ SNRs. For example, for the SIMO system given in (3) without the EM lens, we denote the received SNR of the th antenna as
(11) 
Then, for AS with a given , it follows that the maximum combiner output SNR, denoted as , is achieved by combining the branches with the highest SNRs via MRC, which yields
(12) 
where denotes a permutation such that . The achievable rate in bits/sec/Hz (bps/Hz) for the SIMO channel without the EM lens is then given by [goldsmith2005wireless]
(13) 
Similarly, for the SIMO system given in (10) with the EM lens, we denote the received SNR of the th antenna as , the output SNR by combining the strongest antenna signals via MRC as , and the achievable rate as , which are given by
(14)  
(15)  
(16) 
Note that if , for both SIMO systems with or without the EM lens, MRC is indeed capacity optimal [goldsmith2005wireless], i.e., , , where and denote the capacity of the SIMO channels given in (3) and (10), respectively.
IvB Rate Comparison
Next, we compare the achievable rates and by AS with given for the SIMO systems with versus without the EM lens. For simplicity, we consider the scenario with one single propagation path only (i.e., ), say the th path, which may be the LOS path in an outdoor environment for example. From (2) and (9), we then have , , and , with defined in (IIIB). In the following, we omit the path index for brevity. Then the SNRs received by each antenna given in (11) and (14) for the singlepath case reduce to
(17)  
(18) 
Since , if all the antennas are used, i.e. , from (12) and (15), we have . As a result, with singlepath transmission only, there is no rate/capacity improvement for the system with over without EM lens, i.e., , and . However, if , we show in the following via majorization theory that a strictly positive rate gain can be achieved with the EM lens.
Definition 1
(Majorization[372]) Given , is majorized by , denoted as , if
(19)  
(20) 
Intuitively, indicates that the elements in are “less spread out” than those in . A simple result from majorization theory is that a vector with equal components is majorized by another vector that has the same elementwise sum, as given by the following lemma.
Lemma 1
Given and with , , then we have
Let and be the two vectors consisting of the branch SNRs given in (17) and (18), respectively. Since , it then follows from Lemma 1 that . Thus, we have the following proposition.
Proposition 1
For the singlepath transmission case with , we have
where the strict inequality holds if in , all the elements ’s given in (18) are nonequal.
V Numerical Results
In this section, we compare the achievable rates for the SIMO systems with versus without the EM lens by simulations. The BS is assumed to be equipped with an element ULA () with halfwavelength separation between adjacent antennas (). The normalized power density of the EM lens is given by (6), with the power beamwidth set as . The AoA values , , are assumed to be independent and uniformly distributed between and is the angular spread. For the th path with given , by taking into account the effective aperture of the ULA, we assume that the power gain is proportional to , i.e., , where the proportional constant depends on the pathloss and shadowing. The power gain ’s are normalized so that , where is a parameter indicating the average received SNR level for both the cases with and without the EM lens at the receiver. Without loss of generality, we assume in the following that , so that . , , where
Va Rate Comparison in SinglePath Environment
First, we consider the case with singlepath transmission, i.e., in (2). In Fig. 7, the achievable rate versus the number of selected antennas for AS, , is shown for dB. It is observed that the SIMO system with the EM lens strictly outperforms that without the EM lens for all values of , while the two systems achieve the same rate (capacity) when . This is in accordance with the analytical result given in Proposition 1. It is also observed that the rate gain is more pronounced for smaller . For example, with or , an or rate gain is achievable. Moreover, in order to achieve the rate within of the SIMO channel capacity, antennas need to be selected for AS without the EM lens, while this number is significantly reduced to in the case with EM lens, based on the results in Fig. 7.
VB Rate Comparison in Multipath Environment
Next, we consider a multipath environment with paths. With or , we plot in Fig. 8 the achievable rates averaged over random channel realizations with dB and (which may correspond to a practical BS antenna array for one of the six equallycovered sectors in a cell). Significant rate gains are observed for both values considered at relatively small for the SIMO system with versus without the EM lens. It is also observed that the achievable rate of the SIMO system with the EM lens is insensitive to ; however, increasing helps improve the rate of the system without EM lens. This can be explained as follows. For the system without EM lens, the received power at each receiving antenna is due to the superposition of independent multipath components that have equal power over all antennas. As a result, larger leads to more significant power fluctuations across antennas, which makes AS more effective. In contrast, for the system with EM lens, the power of each multipath component is mainly distributed over a certain subset of receiving antennas, which are different for multipaths with different AoAs; thus, the diversity effect by increasing is less notable.
Vi Conclusion and Future Work
In this paper, we have proposed a novel antenna system design for massive MIMO, where an EM lens is deployed together with the large antenna array, termed electromagnetic lens antenna (ELA). An ELA has been shown to offer two main benefits, namely spatial multipath separation and energy focusing. Under a simplified singleuser uplink transmission setup, we have characterized the power distribution of the EM lens, and thereby established a new channel model for the SIMO system with a receiver ELA. Furthermore, we have demonstrated the significant rate gains of the proposed ELA system over conventional antenna arrays assuming the same antenna selection scheme applied at the receiver.
Several important issues remain to be addressed for the proposed ELA system in our future work:

Power Distribution. The performance of an ELA critically depends on the energy focusing and path separation capabilities of the EM lens. Therefore, a more accurate characterization for the power density function of the EM lens is desirable.

Channel Knowledge. The performance analysis in this paper assumes perfect channel knowledge. It is thus necessary to study the system for a more practical scenario with imperfect channel estimation.

Multiuser System. Investigating the proposed system in the multiuser setup is promising. In particular, the spatial multipath separation capability of the proposed ELA system can be further exploited for interference rejection, since the arriving signals from different users generally have different AoAs. This provides a possible solution to the “pilot contamination” problem [374] in massive MIMO.

Downlink Transmission. It is interesting to study the downlink transmissions with an ELA at the BS transmitter, where the celebrated uplinkdownlink duality via channel reciprocity is worth revisiting.