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Communication Through a Large Reflecting Surface with Phase Errors

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Communication Through a Large Reflecting Surface with Phase Errors 1 Communication Through a Large Reflecting Surface With Phase Errors Mihai-Alin Badiu and Justin P. Coon, Senior Member, IEEE Abstract—Assume the communication between a source and The latter have particularly attractive properties and are ex- a destination is supported by a large reflecting surface (LRS), pected to be a key enabler of smart wireless environments [2], which consists of an array of reflector elements with adjustable [3]. reflection phases. By knowing the phase shifts induced by the composite propagation channels through the LRS, the phases of The focus of this paper is on the concept of a large reflective the reflectors can be configured such that the signals combine surface (LRS) which, irrespective of the building technology, coherently at the destination, which improves the communica- can be abstracted into an array of passive reflector elements tion performance. However, perfect phase estimation or high- that induce adjustable phase shifts on the reflected signals. An precision configuration of the reflection phases is unfeasible. In LRS placed somewhere between a source and a destination can this paper, we study the transmission through an LRS with phase errors that have a generic distribution. We show that greatly enhance the communication between the two when the the LRS-based composite channel is equivalent to a point-to- phase shifts are appropriately configured such that the reflected point Nakagami fading channel. This equivalent representation waves combine coherently at the destination [1], [7], [8], [9]. allows for theoretical analysis of the performance and can help Most of the studies assume the phase shifts induced by the the system designer study the interplay between performance, composite propagation channel through the LRS are perfectly the distribution of phase errors, and the number of reflectors. Numerical evaluation of the error probability for a limited known and the reflector phases are set to the ideal values, with number of reflectors confirms the theoretical prediction and no errors. However, perfect phase estimation or high-precision shows that the performance is remarkably robust against the configuration of the reflection phases is unfeasible. In [10], phase errors. [11], the LRS-based system is optimized over low-resolution Index Terms—Reflect-array, intelligent reflecting surface, phase shifts. meta-surface, phase errors, large system analysis, Nakagami In this paper, we study the communication through an LRS fading, error probability. whose induced phase shifts deviate from the ideal values ac- cording to a generic probability distribution. This set-up could, for example, represent imperfect phase estimation, quantized I. INTRODUCTION reflection phases, or both. We show that the transmission The wireless propagation environment is traditionally through a large number n of imperfect reflectors is equivalent viewed as a detrimental medium of communication that dis- to a point-to-point communication over a Nakagami fading torts the transmitted signals in an uncontrollable manner. Wire- channel with the following characteristics: the diversity order 2 less systems have improved consistently over the years through grows with n, the average SNR scales with n relative to the development of increasingly sophisticated transmission and the average single-reflector SNR, and both parameters are reception techniques that counteract the undesirable effects attenuated by the phase uncertainty. Numerical results for a of propagation. An alternative to this conventional reactive limited number of reflectors validate the theoretical prediction approach is currently gaining interest, as several works pro- and show that accurate knowledge or representation of the pose to achieve a level of control over the propagation of ideal phase shifts is not necessary to closely approach the arXiv:1906.10751v1 [eess.SP] 25 Jun 2019 electromagnetic waves by placing in the environment ‘smart’ ideal performance. passive devices that manipulate the impinging waves according to a desired functionality, such as reflection in an intended II. SYSTEM MODEL direction, full absorption, polarization, or filtering [1], [2], [3]. Furthermore, by endowing such devices with interconnec- The considered system comprises a single-antenna source S, tion capability, they can be introduced in control loops that a single-antenna destination D and an LRS with n reflector ele- enable adaptation to the changing conditions of the wireless ments R1,..., Rn, which assist the communication between S network, thus creating a smart radio environment. Examples of and D. Since we want to study the effect of the phase errors at reconfigurable devices that can manipulate EM waves include the LRS, we restrict our focus on the transmission through the tunable reflect-arrays [4], [1], frequency-selective surfaces [5], LRS only and assume there is no direct link between S and D. or meta-surfaces consisting of a thin layer of metamaterials [6]. Assuming slow and flat fading, let Hi1 and Hi2 be the complex fading coefficients of the S-to-Ri and, respectively, Ri-to-D The authors are with the Department of Engineering Science, Uni- channels. The reflectors are sufficiently spaced apart such that versity of Oxford, Parks Road, Oxford, OX1 3PJ, UK, (email: mi- all the 2n coefficients are mutually independent. Furthermore, [email protected]; [email protected]). This material is based upon work supported by, or in part by, the U. S. Army they have unit power and average magnitudes Research Laboratory and the U. S. Army Research Office under contract/grant number W911NF-19-1-0048. a = E[ H ] < 1 and a = E[ H ] < 1, (1) 1 | i1| 2 | i2| 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.
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