And Far-Field Communications with Large Intelligent Surfaces

Home , Near and far field

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MIMO, law, Massive scaling power metasurface, surface, telligent MEotefrsM o nLSo rcial ag size. large practically of van LIS not an does for interference MR the outperforms and MMSE near-field geometric the in Abstract ag nelgn ufc LS eest raswt a with arrays to refers (LIS) Surface Intelligent Large ne Terms Index ∗ iatmnod nenradl’nomzoe Universit dell’Informazione, Ingegneria di Dipartimento Ti ae tde h pikseta efficiency spectral uplink the studies paper —This er n a-il omnctoswith Communications Far-Field and Near- Itlietrflcigsrae eofiual in- reconfigurable surface, reflecting —Intelligent .I I. † T oa nttt fTcnlg n ikpn University Linköping and Technology of Institute Royal KTH NTRODUCTION N nensta ahhsarea has each that antennas nrad eu Torres Jesus de Andrea L ag nelgn Surfaces Intelligent Large rw,teUsaeeventually are UEs the grows, L = √ NA propaga- fPs,Iay(nradjssorspduiii,luc ([email protected], Italy Pisa, of y ∗ A uaSanguinetti Luca , rthe er fthe of [7], , das ed The . MR) med ous ish. nce ion the act ate an he he he of i- r s - - , p r pc hne gain channel space .Canlmodel Channel A. s channel computing 1. of Lemma way the general of a each provide to to gains [7] in work nthe in n r qal pcdo a on spaced equally are and elydeg-oeg,tu h oa rao h I is LIS the of area total the thus edge-to-edge, deployed i.1 orea nabtaylocation arbitrary an at source A 1: Fig. fpatclylresz.Ti ae ttepeerdcombi preferred the it makes scheme. LIS This an size. for large MR practically to that of superior shown is be that i will performance in considered provides it MMSE be Moreover, conventionally literature. eventually is vanis MIMO what Massive not will the from will UEs different interference the is the grows, which and show near-field size also geometric will LIS We the the distances. UE as to that, size comparable has LIS oteLSlctdi the in located LIS the to a htec a area has each that nas ubrteatna rmlf orgt o yrw according row, by the row 1, right, Fig. to left to from antennas the number h I scnee rudteoii nthe in origin the around centered is LIS The at ( n x [ = h olwn em oe rm[]adetnsprior extends and [6] from comes lemma following The ecnie h I hw nFg ossigof consisting 1 Fig. in shown LIS the consider We n r [ = n − x x [ = Y x s x s y X y , n ) s n n 2 s,n ieto hntaeigi the in traveling when direction x y , [ = s r +( = = n z , n n ∗ Distance Antenna y , osdralsls stoi nen oae at located antenna isotropic lossless a Consider = , mlBjörnson Emil , y − [ = x s ( n 0] n n ] √ wdn([email protected]) Sweden , s T n ( hrcieatnafor antenna receive th − , y , T √ N x 0] √ sgvnby given is , httasisasga hthspolarization has that signal a transmits that y n s N s T A/ z , − y , ) ζ 2 2 I S II. where s N n − s + n , 1) 2 + 2 r ] : , T n z A 1) √ 0] nen lmnso h LIS. the of elements antenna s 2 tthe at h nenshv size have antennas The . √ XY T A SE MODEL YSTEM x n A √ Z − − -plane. (3) N + † √ n x × hrcieatna oae at located antenna, receive th √ A s a o fnx ae where page) next of top (at , √ A √ N n mod( A/ √ n [email protected]) Y − N rd h nensare antennas The grid. 1 = Z 2 ( 1 x − n ieto.Tefree- The direction. s N , . . . , y , x . − XY n s 1 z , + , pae fwe If -plane. √ X s x √ ) N s slocated is N transmits o A ) × anten- NA ning √ (4) (2) (1) A h, n . xy xy 2 2 1 zs 2 −1 zs ζs,rn = + tan (3) 4π  y2 x2 y2 3  x2 y2  x∈Xs,n y∈Ys,n 3 z2 +1 z2 + z2 +1 z2 + z2 +1 X X s s s s s  q q  LIS s,n = √A/2+ yn ys, √A/2 yn + ys . (5) Y − − X n o Lemma 1 is important when quantifying the channel gain d2 in the so-called geometric near-field of the array [4], [8],1 θ2 because it takes into account the three fundamental properties d1 θ1 that makes it different from the far-field: 1) the distance to the UE 2 elements varies over the array; 2) the effective antenna areas vary since the element are seen from different angles; 3) the UE 1 Z loss from polarization mismatch varies since the signals are Fig. 2: The two UEs are located in the XZ-plane at distances received from different angles. We will use Lemma 1 in the dk for k =1, 2 and have angles θk for k =1, 2. remainder. and n CM×N is thermal noise with i.i.d. elements dis- ∈ B. Signal model tributed as C(0, σ2). We define the average received signal- N SNR 2 We consider two single-antenna UEs that communicate with to-noise ratio (SNR) of UE i as i = pi/σ . The channels the LIS in Fig. 1 under the following assumption, shown in are deterministic and thus can be estimated arbitrarily well Fig. 2. This setup is sufficient to demonstrate a few key results. from pilot signals. Hence, perfect channel state information is assumed. The impact of imperfect knowledge of the interfering Assumption 1. UE k for k =1, 2 is located in the XZ-plane UE channel will be investigated in Section III-D. at distance dk from the center of the array with angle θk ∈ To detect s1 from y, the LIS uses the combining vector v1 [ π/2,π/2]. Both UEs send a signal that has polarization in CM ∈ − , multiplied by the vector y. By treating the interference the Y direction when traveling in the Z direction. SE as noise, the SE for UE 1 is 1 = log2 (1 + γ1) where T CN H We denote by hk = [hk1,...,hkN ] for k = 1, 2 SNR v h 2 ∈ γ = 1| 1 1| (10) the channel between UE k and the LIS. Particularly, hkn = 1 SNR vHh 2 v 2 −jφ 2 1 2 + 1 hkn e kn is the channel from the source to the nth receive | | || || | | 2 is the signal-to-interference-and-noise ratio (SINR). We begin antenna with hkn [0, 1] being the channel gain and | | ∈ by considering MR combining, defined as v = h / h , φkn [0, 2π) the phase shift. Following the geometry stated 1 1 || 1|| in Assumption∈ 1, the two UEs are located at (see Fig. 2) leading to SNR h 2 T T MR 1 1 SNR 2 MR s1 = [xs1 ,ys1 ,zs1 ] = [d1 sin(θ1), 0, d1 cos(θ1)] (6) γ1 = H|| 2|| = 1 h1 1 α (11) |h1 h2| T T SNR 2 || || − 2 ||h1|| +1 s2 = [xs2 ,ys2 ,zs2 ] = [d2 sin(θ2), 0, d2 cos(θ2)] . (7) H 2 |h1 h2| SNR2 2 From Lemma 1, the following corollaries are found. MR ||h1|| with α = H 2 . The term |h1 h2| 1+SNR2 2 Corollary 1 (Exact model). Under Assumption 1, the channel ||h1|| −jφkn 2 hkn = hkn e to the nth receive antenna is obtained as N j(φ1n−φ2n) | | H 2 h1n h2n e h1 h2 n=1 | || | SNR2 | | = SNR2 (12) 2 sk rn h 2 h 2 h = ζs r , φ =2π mod || − ||, 1 . (8) 1 P 1 kn k , n kn || || || || | | λ ! accounts for the interference generated by UE 2 whereas SNR h 2 in (11) represents the received SNR in the ab- Corollary 2 (Far-field approximation). Under Assumption 1, 1|| 1|| if UE k is in the geometric far-field of the array, in the sense sence of interference. Since MR does not do anything against MR −jφkn the interference, the term α in (11) must be interpreted as that dk cos(θk) √NA, then hkn = hkn e is obtained as ≫ | | the performance loss due to the presence of UE 2. Instead of using the suboptimal MR combining, we note cos(θ ) d x sin(θ ) 2 k k n k that γ in (10) is a generalized Rayleigh quotient with respect hkn =A 2 , φkn =2π mod − , 1 . (9) 1 | | 4πdk λ to v and thus is maximized by MMSE combining: 1 M×1 −1 The received signal y C at the LIS is y = s1h1 + 2 ∈ SNR H s2h2 + n where si C(0,pi) is the data signal from UE i v1 = ihihi + IM h1 (13) ∼ N i=1 ! 1 s r X Note that we assume throughout this paper that || − n|| ≫ λ, so the leading to system does not operate in the reactive near-field of the transmit antenna (even if it is in the geometric near-field of the array). In fact, this assumption must γMMSE = SNR h 2 1 αMMSE (14) be made to derive the expression in Lemma 1; see [7] for details. 1 1|| 1|| − 2 i 2 Bk + ( 1) √Bk tan(θk) hk = ξd ,θ ,N = − || || k k 2 i i=1 6π(Bk + 1) 2Bk + tan (θk)+1+2( 1) √Bk tan(θk) X − i 1 p Bk + ( 1) √Bk tan(θk) + tan−1 − (15) 2 i 3π 2Bk + tan (θk)+1+2( 1) √Bk tan(θk)!! − p 30 -20 20 -25 10 -30 0 Upper limit 1/3 -10 -35 -20 -40 -30

-40 -1 0 1 2 -45 10 10 10 10 10-1 100 101 102

2 H 2 Fig. 3: Behavior of the channel gain h1 using either the |h1 h2| || || Fig. 4: Behavior of the interference gain h 2 using either exact model or the far-field approximation. The desired UE 1 || 1|| ◦ the exact model or the far-field approximation. The desired has θ1 =2 and is at a distance d1 = 25λ =2.5 m. ◦ ◦ UE 1 is at θ1 =2 while the interfering UE 2 is at θ2 = 2 . H 2 − |h1 h2| Both UEs are at a distance d1 = d2 = 2.5 m. Note that the SNR2 2 MMSE ||h1|| with α = 2 . valleys not always reaching their correct value, dB, due 1+SNR2||h2|| −∞ The SINR expression in (14) contains some of the same to the limited numerical precision. terms as in (11), but has a different structure which makes it behave differently. Unlike αMR in (11), the term αMMSE in LIS larger than L = 1 m is already enough to notice the (14) must be interpreted as the performance loss encountered approximation gap, whereas L 10 m is needed to approach ≥ with MMSE when cancelling the interference from UE 2. the upper limit of 1/3 (this limit comes from considering the polarization mismatch loss [6] and differs from the 1/2 limit III. NEAR- AND FAR-FIELD ANALYSIS in [1]). This shows the importance of properly modeling the We now analyze the system in the near- and far-field cases. near-field. We begin by reviewing the behavior of the channel gain [6], and then we will look into the interference gain and spectral A. Interference gain H 2 |h1 h2| efficiency. From [6, Prop. 1], we have the following result. We now analyze the normalized interference gain 2 . ||h1|| 2 We notice that the computation of a closed-form expression Proposition 1. Under Assumption 1, we have that hk = || || with the exact channel model is challenging while it takes the ξd ,θ ,N where ξd ,θ ,N is given by (15) with k k k k simple form with the far-field approximation [9, Eq. (12)] NA (16) Bk = 2 2 . H 2 2 4d cos (θk) h h A cos(θ ) sin(πLΩ/λ) k | 1 2| = 2 (18) h 2 4πd2 √ In the far-field case, it becomes 1 2 sin(π AΩ/λ) || || cos(θk) h 2 = NA . (17) with L = √NA and Ω = sin(θ2 ) sin(θ1). Fig. 4 plots the k 2 hHh 2 || || 4πd | −1 2| k the normalized interference gain 2 as a function of , ||h1|| L Note that both (15) and (17) depend on the total LIS area, using either the exact model and the far-field approximation NA. Hence, the channel gain is independent of the wave- from Corollaries 1 and 2, respectively. We consider a setup length. Since practical elements are sub-wavelength-sized, the with λ = 0.1 m in which the two UEs have different angles ◦ ◦ number of elements N that is needed to achieve a given θ1 = 2 and θ2 = 2 , but have the same distance from 2 − channel gain is inversely proportional to λ . the LIS, given by d1 = d2 = 2.5 m. In line with the results 2 Fig. 3 shows the channel gain h1 as a function of of Fig. 3, the gap between the exact model and the far-field the size of the LIS, i.e., L = √NA||, using|| either the exact approximation is noticeable when L 1 m. We notice that expression in (15) or the far-field approximation in (17). We L > 100 m is needed with the exact≥ model to approach its ◦ consider a setup with λ =0.1 m in which UE 1 has θ1 =2 upper limit. This is more than 3 dB higher than the upper and d1 = 25λ = 2.5 m. The results of Fig. 3 show that an limit with the far-field approximation. 20

Point-of-interest 5 MMSE MR 0 10-1 100 101 102

(a) Spectral efﬁciency versus L = √NA (a) SE with MMSE 20

5 MMSE Point-of-interest

MR 0 10-1 100 101 102 103

(b) Spectral efﬁciency versus z = zs1 = zs2 when L = 6 m

Fig. 5: SE behavior with MR and MMSE. The SE in the ideal (b) SE with MR interference-free case is reported as reference. Fig. 6: SE in bit/s/Hz achieved by UE 1 when the interfering B. Spectral efficiency analysis UE 2 is transmitting at different locations over the XZ plane. Both MMSE and MR are considered. − Fig. 5a plots the SE achieved by UE 1 with MR and MMSE as a function of L, using either the exact model or the far- To gain further insights into the large performance gap field approximation from Corollaries 1 and 2. We consider between MMSE and MR in the near-field, Fig. 6 shows the ◦ ◦ the same setup of Fig. 4; that is, λ = 0.1 m, θ1 = 2 , SE in bit/s/Hz when the desired UE 1 is fixed at θ1 = 2 ◦ θ2 = 2 and d1 = d2 = 2.5 m. We set p1 = p2 = 30 dBm d1 = 2.5 m and the interfering UE 2 is transmitting from −2 2 and σ = 0 dBm. The SE log (1 + SNR1 h1 ) computed different locations over the XZ plane, whose distance from 2 || || with the exact model in the ideal interference-free case is the point-of-interest is measured− in wavelengths. We assume also reported as a reference. Fig. 5a shows that the SE λ = 0.1 m and use an LIS of size L = 6 m. Fig. 6a reveals saturates with both schemes as L increases. However, while that the SE with MMSE is low only in an elliptic region MMSE quickly converges to the interference-free case, the around the point-of-interest, whose semi-major axis (along performance gap with MR is substantial with the exact model, z direction) is roughly half-a-wavelength. This means that while it asymptotically vanishes with the far-field model as in MMSE can efficiently reject any interfering signal that comes [9, Sec. III-D]. from a location that is at least half-a-wavelength away. On the Fig. 5b plots the SE achieved by UE 1 with MR and MMSE contrary, Fig. 6b shows that a low SE is achieved with MR as a function of z = zs1 = zs2 , using either the exact model for most of the locations the interfering UE is transmitting or the far-field approximation. We consider an LIS of size from. This is because MR does not do anything against the 2 L = 6 m, and set p1 = p2 = 30 dBm and σ = 0 dBm. interference. The region where the SE achieves its minimum MMSE provides the same SE as in the interference-free case value is still an ellipse but with a larger area. We see that the for z 10 m, whereas it is lower for larger values. The SE gap SE with both MMSE and MR ranges from 7 to 1 bit/s/Hz between≤ MMSE and MR is relatively large for the values of z (a reduction of 86%), but while in the MMSE≈ case the highest of practical interest, i.e., smaller than tens of meters. MMSE values are located in most of the observed area, that is not the converges to MR only when z 40 m. case when using MR. ≥ 10 7.5

8 7

6 6.5 4 MMSE 6 2

MR 5.5 0 -1 0 1 2 10 10 10 10 0.5 1 1.5 2 2.5 3

Fig. 7: Impact of polarization on the SE and interference gain Fig. 8: SE with the exact model when the position of the of UE 1 for the same setup in Fig. 5a. interfering UE is uncertain within a radius r, measured in wavelenghts. We assume , m and an inter-UE C. Impact of polarization mismatch λ = 0.1 L = 6 distance is 17.5 cm=1.75 λ. Unlike [1], the analysis and results above take into account the polarization mismatch, which varies the received power IV. SUMMARY of each antenna element since the signals are received from This paper showed that a realistic assessment of the uplink different angles [6]. This has a double impact on the system SE achievable by two single-antenna UEs communicating with performance. Firstly, the global channel gain is reduced and an LIS requires the use of an exact near-ﬁeld channel model, converges to 1/3 instead of 1/2 as L grows large; see Fig. 3. whenever the distance of transmitting UEs is comparable with Secondly, the mutual interference between UEs changes due the LIS size, L. It also showed that increasing L unboundedly to the varying power footprint induced by the polarization does not guarantee the suppression of interference with MR mismatch. To quantify the joint effect of these two facts, Fig. 7 combining, especially when the UEs are closely located in considers the same setup as in Fig. 5a and compares the SE space. MMSE combining is still needed to efﬁciently suppress obtained with the exact model when ignoring or considering the interference. Particularly, the SE with MMSE quickly the polarization mismatch loss. The two effects have a non- converges to the interference-free case when the UEs are negligible impact on both MMSE and MR when L is large. transmitting from a distance of few meters and values of L of For example, for L = 100 the SE reduction is 6.5% with practical interest are considered, i.e., in the range 1 L 10 MMSE, whereas it is 15% with MR. However, for values of meters. However, the SE with MMSE deteriorates fast≤ in≤ the L of practical interest, i.e., in the range between L = 1 and presence of channel estimation errors. Lastly, we showed that L = 10 meters, the polarization effects loss is limited to 5%. the polarization mismatch should not be ignored since it has D. Imperfect knowledge of interfering UE channel a non-negligible impact on SE, especially when L grows.

The SE analysis above reveals that MMSE performs much REFERENCES better than MR in both the near- and far- fields. However, the [1] S. Hu, F. Rusek, and O. Edfors, “Beyond Massive MIMO: The potential analysis relies on perfect knowledge of the channel vectors of data transmission with large intelligent surfaces,” vol. 66, no. 10, pp. h1, h2 . We now evaluate the robustness of MMSE against 2746–2758, 2018. { } [2] J. Zhao, “A survey of intelligent reflecting surfaces (IRSs): Towards 6G imperfect knowledge of h2 in (13). This is done by assuming wireless communication networks,” CoRR, vol. abs/1907.04789, 2019. that the location of UE 2 for the computation of h2 in (13) is [3] A. Pizzo, T. L. Marzetta, and L. Sanguinetti, “Spatially-stationary model imperfectly known. Particularly, we assume that the estimated for Holographic MIMO small-scale fading,” IEEE J. Sel. Areas Commun., position differs from the true one by an error that is uniformly vol. 38, no. 9, 2020. [4] J. C. B. Garcia, A. Sibille, and M. Kamoun, “Reconfigurable intelligent distributed in a circle of radius r, centered at the true position. surfaces: Bridging the gap between scattering and reflection,” IEEE J. Fig. 8 shows the average SE as a function of r, expressed in Sel. Areas Commun., vol. 38, no. 11, 2020. wavelengths with λ = 0.1 m, in the same setup of Fig. 5a [5] E. Basar, “Reconfigurable intelligent surface-based index modulation: A new beyond MIMO paradigm for 6G,” IEEE Trans. Commun., vol. 68, with L = 6 m. The average is taken with respect to the no. 5, 2020. randomly generated errors. We see that MR is unaffected by [6] E. Björnson and L. Sanguinetti, “Power scaling laws and near-field the imperfect knowledge of the interfering UE position since behaviors of Massive MIMO and intelligent reflecting surfaces,” IEEE Open J. Commun. Society, vol. 1, pp. 1306–1324, 2020. MR does not rely on the knowledge of h2. On the other hand, [7] D. Dardari, “Communicating with large intelligent surfaces: Fundamental the SE with MMSE decreases very fast. In line with Fig. 6a, limits and models,” IEEE J. Sel. Areas Commun., vol. 38, no. 11, 2020. an estimation error of half-a-wavelength is enough for a 65% [8] S. W. Ellingson, “Path loss in reconfigurable intelligent surface-enabled channels,” CoRR, vol. abs/1912.06759, 2019. reduction. This calls for estimation schemes with centimeter [9] E. Björnson and L. Sanguinetti, “Utility-based precoding optimization accuracy (having λ = 10 cm) which is far beyond what we can framework for large intelligent surfaces,” in Asilomar Conference on achieve with state-of-the-art solutions in wireless applications. Signals, Systems and Computers, Nov 2019.