Communication Through a Large Reflecting Surface with Phase Errors

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R eirMme,IEEE Member, Senior 2 a n 1 1 , . . . , ofcet r uulyidpnet Furthermore, independent. mutually are coefficients E[ = n h vrg N clswith scales SNR average the , R | H n hc sittecmuiainbetween communication the assist which , i 1 I S II. | ] < n 1 YSTEM fipretrflcosi equivalent is reflectors imperfect of S D -to- and n nLSwith LRS an and R i M H a n,respectively, and, ODEL i 2 1 E[ = and H | H i 2 i 2 etecomplex the be n | n ] 2 eetrele- reflector < eaieto relative 1 S , R and i could, -to- order ction An . cted sion zed ex- hat (1) gy, 9]. D S D S h y n a e 1 . , , , 2 for all i =1,...,n. In the following, we use III. EQUIVALENT SCALAR FADING CHANNEL In this section, we find an equivalent representation of a = √a1a2. (2) the transmission through an LRS as a point-to-point fading channel. First, we provide the following result. Denote by φi the phase shift induced by Ri. The signal received by the destination over the equivalent baseband Lemma 1. For large n, the coefficient H given by (5) has channel is1 a (non circularly symmetric) complex normal distribution. Its real and imaginary parts, U = (H) and V = (H), are n 2 ℜ 2ℑ jφi independent, and U (µ, σU ) and V (0, σV ) with Y = √γ0 Hi2 e Hi1X + W, (3) ∼ N ∼ N 2 Xi=1 µ = ϕ1a (8) where X is the transmitted symbol with mean zero and 2 1 2 4 σU = (1 + ϕ2 2ϕ1a ) (9) E[X2]=1, W (0, 1) models the normalized receiver 2n − noise and γ is the∼ CN average SNR when only one reflector is 2 1 0 σV = (1 ϕ2). (10) present and accounts for path loss, shadowing and reflection 2n − loss. Proof: See Appendix A.

Ideally, given the phases arg(Hi1) and arg(Hi2), the phase The ideal case where the phase shifts induced by the LRS- φi is set so as to cancel the overall phase shift arg(Hi1)+ augmented channel are perfectly estimated and the reﬂector arg(Hi2), which maximizes the SNR at the receiver [9]. phases can be set precisely is instantiated as follows. Assuming the phase shifts induced by the channels are not Corollary 1. When there are no phase shift errors, i.e., Θi estimated perfectly and/or the desired phases cannot be set 0, for all i = 1,...,n, the coefﬁcient H in (5) is real and≡ precisely (e.g., when only a discrete set of phases is techno- distributed as a2, (1 a4)/n . logically possible), we model the deviation of from the ideal N − φi setting by the phase noise Θi, which is randomly distributed Proof: Since the phase errors are zero, we have ϕ1 = on [ π, π) according to a certain circular distribution [12]. ϕ2 = 1. This implies that (H) 0 in Lemma 1 and the − ℑ ≡ Now, we express the received signal as result follows. The ideal case of no phase errors and Rayleigh fading (i.e.,

Y = n√γ0HX + W, (4) a = √π/2) is studied in [9], which Corollary 1 recovers. Corollary 2. When the phase shift errors are uniformly where distributed on [ π, π), which corresponds to a complete lack 1 n − jΘi of knowledge about the phases of Hi1 and Hi2, i =1,...,n, H = Hi1 Hi2 e C. (5) n | || | ∈ the coefficient H in (5) has a circularly-symmetric complex Xi=1 normal distribution with mean zero and variance 1/n, which All the variables in the r.h.s. of (4) and (5) are mutually means the equivalent channel resembles Rayleigh fading. independent. We assume Θi, i = 1,...,n, are i.i.d. with common characteristic function expressed as a sequence of Proof: For the uniform circular distribution, ϕ1 = ϕ2 =0 and the result follows from Lemma 1. complex numbers ϕp p Z, { } ∈ When ϕ1 > 0, even though U and V are Gaussians with jpΘ non-zero and, respectively, zero means, the magnitude H = ϕp = E[e ], (6) 2 2 | 2| √U + V does not have a Rice distribution because σU = 2 6 which are also called trigonometric (or circular) moments [12]. σV in general. We obtain the following result instead. Note that ϕ 1, for all p Z; for p 1 equality | p| ≤ ∈ | | ≥ Theorem 1. For large n and ϕ1 > 0, the magnitude of H holds if, and only if, the distribution is degenerate (Dirac in (5) has a Nakagami distribution with pdf delta). A distribution that is broad on the circle (signifying m high uncertainty) gives small magnitudes of the trigonometric 2m 2m 1 m 2 f H (x)= x − exp x , (11) moments (for the uniform distribution, ϕ =0 for p 1). It is | | Γ(m)µ2m −µ2 p ≥ reasonable to assume that the distribution of the error Θi has where the fading parameter m is given by mean direction zero (arg E[ejΘi ]=0) and its pdf is symmetric n ϕ2a4 around zero. Due to the symmetry, the characteristic function 1 (12) m = 2 4 . takes real values, i.e., ϕ R, for all p Z. 2 1+ ϕ2 2ϕ1a p ∈ ∈ − The receiver performs ideal coherent detection based on the Proof: See Appendix B. sufficient statistic Thus, the transmission through the LRS is statistically equivalent to a direct transmission over a Nakagami fading j arg(H) 2 2 e− Y = n√γ0 H X + W ′, (7) channel. The equivalent channel has n larger receive power | | j arg(H) 2 where W ′ = e− W (0, 1). The magnitude of the fading coefficient has a Nakagami distribution, ∼ CN although its phase is not uniformly distributed as is sometimes assumed for Nakagami fading. However, this is irrelevant, since we consider phase coherent 1The signal is normalized by the standard deviation of the receiver noise. detection. 3 relative to the transmission through one reflector. The average 8 single-reflector SNR γ0 can be very low because of the attenuation of the constituent channels and reflection. Next, 7 we make a few remarks. 6 n = 256 Remark 1. The full distributions of the 2n coefficients Hi1, Hi2, i = 1,...,n, are irrelevant to the distribution of H , 5 as this depends only on the average magnitudes (1).| The|

snr 4 individual coefficients may even have different distributions, as f long as they are mutually independent and have the respective 3 means. 2 Furthermore, the phase errors manifest in (11) through 1 the first two trigonometric moments (6) of their common n = 16 distribution, which are subunitary reals for the considered 0 zero-mean symmetric distributions. 0 0.5 1 1.5 2 2.5 3 γ/γ¯ 2 2 Remark 2. The average SNR, γ¯ = n γ0 E[ H ] = 2 2 4 2 2 | | n γ0ϕ1a gamma distribution for n large, see (20), and the scaling| | factor 2 IV. PERFORMANCE ANALYSIS n γ0 is O(1), as pointed out in Remark 3. As an example, the pdf (13) is displayed in Fig. 1 for the We investigate how the performance of the system is im- following scenario: the phase errors have a zero-mean von pacted by the system size and phase errors. We consider the Mises distribution with concentration ; the -to- chan- following types of errors: κ =8 S Ri nels exhibit Rician fading with unit power and a Rice factor phase estimation errors: Θ is modelled as a zero-mean • i of K = 1, which gives a = π F ( 1/2; 1; K), von Mises variable whose concentration parameter κ 1 4(K+1) 1 1 q − captures the accuracy of the estimation. In this case, the while the fading over the Ri-to-D channels is Rayleigh with characteristic function is ϕ = Ip(κ) , where I is the unit power, for which a2 = √π/2. We observe a very p I0(κ) p modified Bessel function of the first kind and order p [12]. good agreement between the theoretical approximation and the quantization errors: when only a discrete set of 2q phases Monte Carlo estimate even for a moderate value of n. Also, • when n becomes large, the pdf concentrates around the mean can be configured, q 1, the error Θi is assumed to be ≥ q q γ¯ and approaches a Gaussian pdf. uniformly distributed over [ 2− π, 2− π]. For this distri- − − sin(2− q π) sin(2 q+1π) bution, we obtain ϕ1 = 2−q π and ϕ2 = 2−q+1π . We study these types of errors separately, although the case B. Average error probability with both estimation and quantization errors can be easily We evaluate through simulations the average error proba- treated because, assuming they are independent, the resulting bility, Pe, of the LRS system for BPSK transmission, n = 32 ϕ1 and ϕ2 are given by the product of the respective trigono- and the same fading models as in the previous example. In metric moments. Fig. 2, the results for estimation and quantization errors are plotted, together with the theoretical performance [13] of the A. Distribution of the SNR Nakagami fading channel given by Theorem 1. We observe a good agreement between the simulations and the theoretical The performance of the system is fundamentally determined prediction even for a limited . Other results not included here by the distribution of the instantaneous SNR, which equals n show that the match is closer for n = 64, whereas for n = 16 n2γ H 2. 0| | the theoretical approximation is less accurate, although fairly Corollary 3. For large n, the instantaneous SNR, n2γ H 2, good. As illustrated in Fig. 2a, even a vague knowledge of 0| | is gamma distributed. Its pdf can be expressed as the ideal phase shifts can be beneficial: e.g., when κ =2 (for m which the distribution is broad) the gap to ideal performance m m 1 m fsnr(γ)= γ − exp γ , (13) is of about dB, while for (for which the distribution Γ(m)¯γm − γ¯ 5 κ =8 is still not very concentrated) Pe is within 1 dB from the ideal 2 2 4 where m is given by (12) and γ¯ = n ϕ1a γ0 is the average case. It is also remarkable to observe in Fig. 2b that phase SNR. quantization of just one bit is already at about 5 dB from the 3The upper bound would correspond to the contrived case when the emitted ideal Pe, while with two bits the error probability becomes waves all arrive at the receiver without any absorption. close to the ideal. 4

100 100

10-2 10-2 e e P P

10-4 10-4

10-6 10-6 -35 -30 -25 -20 -15 -10 -35 -30 -25 -20 -15 -10 γ0 (dB) γ0 (dB) (a) (b) Fig. 2. Average probability of error for n = 32 under phase estimation errors (left) and quantization errors (right). The curves correspond to the theoretical performance for the Nakagami channel in Theorem 1, whereas the marked points are obtained via Monte Carlo simulations of the LRS system with 107 trials. The dashed curves represent the ideal performance with zero errors. Left: the concentration parameter of the von Mises distribution is κ = 2 and κ = 8, in increasing order of performance. Right: the number of quantization bits is q = 1, 2 and 3 bits, in increasing order of performance.

The average error probability for the Nakagami channel at APPENDIX A high average SNR γ¯ is [14] PROOF OF LEMMA 1

m 1 1 The complex variables jΘi , , are m − Γ(m + 2 ) m n Hi1 Hi2 e i =1,...,n Pe γ¯− . (14) | 2 || | 4 2 ≈ 2√πΓ(m) i.i.d. with common mean µ = a ϕ1, variance ν =1 a ϕ1 and pseudo-variance ρ = ϕ a4ϕ2 (see [15] for the− second-| | 2 − 1 2 2 4 Gd order characterization of complex variables). According to Using γ¯ = n ϕ1a γ0, we express Pe(γ0) = (Gcγ0)− , such that the diversity and coding gains w.r.t. the aver- the assumptions on the distribution of Θi, the trigonometric moments ϕ1, ϕ2 R and, consequently, µ,ρ R. By invok- age single-reflector SNR γ0 are Gd = m and Gc = ∈ ∈ −1 1/m m Γ( + 1 ) ing the central limit theorem, we approximate the distribution 2 2 4 m m 2 − . These relationships may be use- n ϕ1a 2√πΓ(m) of H in (5) for n large by a complex normal distribution, h i ful to find the number of reflector elements required to achieve (µ,ν/n,ρ/n). Let U = (H) and V = (H). We CN ℜ ℑ a certain Gd or Gc, given the distribution of the phase have [15] errors (e.g., κ for estimation errors). While n can be easily established in the case of the diversity gain based on (12), 1 ν ρ Cov[U, V ]= + =0. (15) the relationship is more complicated in the case of the coding 2ℑ −n n gain. Being jointly Gaussian with zero covariance, it follows that 2 U and V are independent. Furthermore, U (µ, σU ) and 2 2 ∼ N V (0, σ ) with µ = a ϕ1, V. CONCLUSION ∼ N V

2 1 ν ρ 1 4 2 We showed that the composite channel through an LRS σU = + = (1 + ϕ2 2a ϕ1) (16) 2ℜ n n 2n − with phase errors is equivalent to a point-to-point channel 1 ν ρ 1 σ2 = = (1 ϕ ). (17) with Nakagami fading whose parameters are affected by the V 2ℜ n − n 2n − 2 phase uncertainty through the first two trigonometric moments of the phase error distribution. Despite the phase shift errors, the average SNR still grows with n2 (w.r.t. the single- APPENDIX B reflector SNR) and the diversity order grows linearly with PROOF OF THEOREM 1 n; however, both are attenuated by the phase uncertainty. Also, the distributions of the fading coefficients of the two We first consider the squared magnitude H 2 = U 2 + V 2. 2 2 | | constituent channels manifest in the equivalent channel only The normalized variable U /σU has a non-central chi-squared though their average magnitudes. Numerical results for a distribution, while V 2 is gamma distributed with shape 1/2 2 2 limited number of reflectors showed good agreement with the and scale 2σV . Thus, H is the sum of a scaled non-central theoretical prediction and that the performance is remarkably chi-squared variable and| | a gamma variable. We characterize robust against the phase errors. the distribution of H 2 through its cumulant generating func- | | 5

t H 2 tion, K H 2 (t) = ln E[e | | ]. Since U and V are independent REFERENCES | | (Lemma 1), [1] X. Tan, Z. Sun, J. M. Jornet, and D. Pados, “Increasing indoor spectrum sharing capacity using smart reflect-array,” in 2016 IEEE International µ2t Conference on Communications (ICC), May 2016, pp. 1–6. K H 2 (t)= KU 2 (t)+ KV 2 (t)= [2] C. Liaskos, S. Nie, A. Tsioliaridou, A. Pitsillides, S. Ioannidis, and | | 1 2σ2 t − U I. Akyildiz, “A new wireless communication paradigm through software- 1 2 1 2 controlled metasurfaces,” IEEE Communications Magazine, vol. 56, ln(1 2σU t) ln(1 2σV t) (18) no. 9, pp. 162–169, Sep. 2018. − 2 − − 2 − [3] M. D. Renzo, M. Debbah, D.-T. Phan-Huy, A. Zappone, M.-S. We now approximate for large n the first term in the r.h.s. Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos, J. Hoydis, H. Gacanin, J. d. Rosny, A. Bounceur, G. Lerosey, and M. Fink, of (18). We develop its Maclaurin series expansion as “Smart radio environments empowered by reconfigurable AI metasurfaces: an idea whose time has come,” EURASIP Journal on Wireless µ2t Communications and Networking, vol. 2019, no. 1, p. 129, May 2019. 2 2 2 2 2 2 2 3 [4] S. V. Hum, M. Okoniewski, and R. J. Davies, “Modeling and design 2 = µ t + µ (2σU )t + µ (2σU ) t + ... 1 2σU t of electronically tunable reflectarrays,” IEEE Transactions on Antennas − and Propagation, vol. 55, no. 8, pp. 2200–2210, Aug 2007. µ2 ∞ k (4σ2 t)k µ2 ∞ (4σ2 t)k = U = U g(t) [5] L. Subrt and P. Pechac, “Controlling propagation environments using 2 k 1 2 intelligent walls,” in 2012 6th European Conference on Antennas and 4σU 2 − k 4σU k − Xk=1 Xk=1 Propagation (EUCAP), March 2012, pp. 1–5. = ln(1 4σ2 t) [6] T. Jun Cui, M. Q. Qi, X. Wan, J. Zhao, and Q. Cheng, “Coding − − U metamaterials, digital metamaterials and programmable metamaterials,” | {z } where Light: Science & Applications, vol. 3, p. e218, 10 2014. [7] C. Huang, A. Zappone, M. Debbah, and C. Yuen, “Achievable rate maximization by passive intelligent mirrors,” in 2018 IEEE International µ2 ∞ k (4σ2 t)k g(t)= 1 U Conference on Acoustics, Speech and Signal Processing (ICASSP), April 2 k 1 2018, pp. 3714–3718. 4σ − 2 − k U kX=3 [8] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless µ2 ∞ (4σ2 t)k ∞ (4σ2 t)k network: Joint active and passive beamforming design,” in 2018 IEEE U 2 2 2 U Global Communications Conference (GLOBECOM), Dec 2018, pp. 1–6. < 2 =4µ σU t 4σ k k +2 [9] E. Basar, “Transmission Through Large Intelligent Surfaces: A U kX=3 kX=1 New Frontier in Wireless Communications,” arXiv e-prints, p. ∞ (4σ2 t)k arXiv:1902.08463, Feb 2019. < 4µ2σ2 t2 U = 4µ2σ2 t2 ln 1 4σ2 t U k − U − U [10] C. Huang, G. C. Alexandropoulos, A. Zappone, M. Debbah, and Xk=1 C. Yuen, “Energy efficient multi-user MISO communication using low resolution large intelligent surfaces,” in 2018 IEEE Globecom Workshops 2 1 2 (GC Wkshps), Dec 2018, pp. 1–6. Given that σU = O(n− ), we have g(t) = O(n− ), such that (18) is well approximated by4 [11] Q. Wu and R. Zhang, “Beamforming optimization for intelligent reflecting surface with discrete phase shifts,” in ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing 2 µ 2 (ICASSP), May 2019, pp. 7830–7833. K H 2 (t)= ln(1 4σU t) [12] K. V. Mardia and P. E. Jupp, Directional Statistics. John Wiley & Sons, | | 2 −4σU − 2000. 1 1 [13] M. K. Simon and M.-S. Alouini, Digital communication over fading ln(1 2σ2 t) ln(1 2σ2 t) (19) − 2 − U − 2 − V channels. John Wiley & Sons, 2000. [14] Zhengdao Wang and G. B. Giannakis, “What determines average and Expression (19) corresponds to the cumulant generating outage performance in fading channels?” in Global Telecommunications Conference, 2002. GLOBECOM ’02. IEEE, vol. 2, Nov 2002, pp. 1192– function of the sum of three independent gamma variables 2 1196 vol.2. µ 2 2 with (shape, scale) parameters ( 2 , 4σ ), (1/2, 2σ ) and [15] B. Picinbono, “Second-order complex random vectors and normal dis- 4σU U U 2 tributions,” IEEE Transactions on Signal Processing, vol. 44, no. 10, respectively (1/2, 2σV ). While the resulting distribution can pp. 2637–2640, Oct 1996. be characterized based on [16], we simplify by making a [16] P. G. Moschopoulos, “The distribution of the sum of independent gamma further approximation, random variables,” Annals of the Institute of Statistical Mathematics, vol. 37, no. 3, pp. 541–544, Dec 1985. 2 µ 2 2 K H (t)= 2 ln(1 4σU t), (20) | | −4σU −

1 2 which has O(n− ) error. Thus, for large n, H has a gamma 2 µ 2 | | distribution with shape 2 and scale 4σU . It follows through 4σU variable transformation that H has a Nakagami distribution with fading parameter (shape)| |

µ2 n ϕ2a4 m = = 1 4σ2 2 1+ ϕ 2ϕ2a4 U 2 − 1 and spread µ2.

4Alternatively, the Maclaurin expansion of (18) can be truncated to second- order, which corresponds to a Gaussian distribution and incurs an error of O(n−2) as well. However, for moderate values of n, the Gaussian approximation is less accurate than the one we developed.