A Hybrid Scheme for Reconfigurable Intelligent Surfaces: How Many Elements Should be Estimated?

In this work, a low-complexity hybrid scheme is presented for a wireless network assisted by a reconfigurable intelligent surface (RIS), where channel estimation is required for only a subset of the elements. Specifically, in order to reduce the channel training overhead and boost the performance of the RIS-aided network, the RIS is partitioned in two sub-surfaces, which are sequentially activated to assist the communication. The elements of the first sub-surface align their phase shifts, based on the acquired channel state information (CSI) from a channel training period, whereas the elements of the second sub-surface randomly rotate the phase of the incident signals. The performance of the proposed scheme is investigated under the effect of imperfect CSI acquisition at the RIS. Analytical expressions for the outage probability are derived and useful insights on the optimal configuration of the RIS are provided. We show that, by optimizing the number of elements that need to be estimated, the proposed scheme provides significant performance gains and overcomes the limitations caused by the imperfect CSI acquisition.


I. INTRODUCTION
Reconfigurable intelligent surfaces (RISs) have emerged as an appealing technology for the deployment of future 6G communications, which can support massive connectivity networks with high data rates, enhanced reliability and low latency [1].An RIS is equipped with a large number of passive reflecting elements, embedded on a planar surface, which can be smartly adjusted through a dedicated controller to efficiently configure the wireless propagation environment.This technology is therefore a cost effective solution, which can significantly enhance energy efficiency, due to the passive operation of the elements, and improve the spectral efficiency, since the RIS operates in an ideal full-duplex mode [2].
Driven by the controllable nature of this technology, several works have recently investigated RIS-aided networks for This work was co-funded by the European Regional Development Fund and the Republic of Cyprus through the Research and Innovation Foundation, under the project ENTERPRISES/0521/0068 (AGRILORA).It has also received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 819819) and from the Agence Universitaire de la Francophonie (AUF) under an Inter-university Scientific Cooperation project (PCSI).various communication scenarios [3]- [5].In particular, in [3] the authors study an RIS-assisted multiple-input single-output (MISO) system with multiple users and show that, when the active transmit beamforming at the access point (AP) and the passive reflect beamforming at the RIS are jointly optimized, the transmit power at the AP can be minimized.Furthermore, in [4] the RIS has been considered for assisting nonorthogonal multiple access (NOMA) communications, and it is demonstrated that RIS-assisted NOMA improves spectral efficiency and can be used to serve additional users.In [5], the employment of RISs has been also studied for jointly achieving passive beamforming and information transfer through a reflection pattern modulation scheme, which enhances the system's performance through passive beamforming at the RIS and simultaneously conveys information from the RIS to the user by using an appropriate mapping method.
Most of the existing works base their proposed solutions on the assumption that perfect knowledge of the channel state information (CSI) is available.However, channel estimation in RIS-assisted systems is not a trivial task, due to the passive nature of the reflecting elements and the fact that the RIS does not have signal processing capabilities [1].Towards this direction, several channel estimation methods for RIS-aided networks have been suggested.In [6], the cascaded transmitter-RIS-receiver channels are estimated by individually activating one-by-one the elements at the RIS.Moreover, in [7], a discrete Fourier transform (DFT)-based method has been proposed for estimating the cascaded channels.In this case, the RIS training states are given as the columns of a DFT matrix, resulting in higher estimation accuracy [8].Nevertheless, for a large number of elements, performing channel estimation for all the RIS elements becomes impractical, since the incurred training overhead is prohibitively large and causes significant performance degradation, while the resources are usually limited.To avoid the above limitations, some efforts have been made to reduce the training requirements of such systems.Specifically, in [9] the authors suggest to reduce channel estimation overhead through an element-grouping method.Moreover, a low-overhead RIS reconfiguration scheme is proposed in [10] which, instead of acquiring full CSI for configuring the RIS, requires only the estimation of the mobile user's position.Finally, various low-complexity RIS schemes are proposed in [11], which induce random phase shifts to the incident signal in order to enhance the performance of RISaided systems, resulting in low-to-zero CSI requirements.
Motivated by this, in this work we study the performance of a low-complexity hybrid scheme for an RIS-aided network, where only a subset of the elements are used for channel estimation, and we investigate how the performance is affected under an imperfect CSI acquisition scenario at the RIS.In particular, this scheme partitions the RIS into two sub-surfaces, where the elements of the first sub-surface optimize their phase shifts through a coherent beamforming (CB) design, based on the estimated CSI, while the remaining elements induce random rotations (RR).By sequentially activating each subsurface, a time-varying channel is produced, which can boost the performance of the considered network.The proposed scheme can significantly reduce the channel training overhead for the RIS configuration and can be easily adapted to several channel estimation methods that have been proposed for RISaided systems.Through this approach, we aim to provide a tradeoff between the number of RIS elements that need to be considered for channel estimation in order to optimize the performance of the considered system.Therefore, we present a complete analytical framework in terms of outage probability and we provide useful insights on the effect of the number of elements used for channel estimation and the consideration of imperfect CSI.Our results show that the proposed scheme can provide higher coding gain, compared to the case where all the elements induce RR, and overcomes the performance limitations caused by the imperfect CSI acquisition in the full channel estimation scenario.

II. SYSTEM MODEL
We consider an RIS-aided communication system shown in Fig. 1, where the communication between a single-antenna transmitter (Tx) and a single-antenna receiver (Rx) is assisted by the employment of an RIS equipped with Q reflecting elements.We assume a scenario where a direct link between the Tx and the Rx does not exist (e.g.due to path blockages or heavy shadowing); hence the communication is achieved only through the RIS [11], [12].
the channel vectors from the Tx to the RIS and from the RIS to the Rx, respectively.We consider that all wireless links exhibit fading which follows a flat quasi-static Rayleigh block fading model1 [12]; the channel coefficients remain invariant during a coherence interval of T channel uses, but change independently between different coherence intervals, by following a circularly symmetric complex Gaussian distribution with zero mean and unit variance i.e., h i and g i ∼ CN (0, 1).
At an arbitrary time instance t, the reflection amplitude and the phase shift of each RIS element are modified, according to the proposed scheme that will be presented in Section III.We denote by the reflection vector of the RIS at the t-th time instance, where a i,t ∈ [0, 1] is the reflection amplitude of the i-th RIS element, φ i,t ∈ [0, 2π) is the induced phase shift and  = √ −1 is the imaginary unit.In this setup, we consider an RIS hardware implementation where, at every time instance, each reflecting element can be found in two possible states, either to be turned ON or OFF [5], [6].Once an element is at the ON state, its reflection amplitude is set to 1 (full reflection) and we can only control its phase shift; conversely, at the OFF state the reflection amplitude is set to 0 (no reflection2 ) and the element can not assist the communication.Finally, we assume that at the beginning of each coherence interval we have no prior knowledge of the CSI at the RIS.The received signal at the Rx at the t-th time instance is expressed as where n t ∼ CN (0, σ 2 ) is the additive white Gaussian noise (AWGN) with variance σ 2 , x t is the signal transmitted by the Tx with transmit power P t and v i h i g i , 1 ≤ i ≤ Q, denotes the product channel corresponding to the i-th RIS element.

III. HYBRID COHERENT BEAMFORMING/RANDOM
ROTATIONS RIS SCHEME We now proceed to the description of the proposed hybrid CB/RR RIS scheme, which implements a temporal processing strategy for the reconfiguration of the RIS elements, in order to increase the achieved rate of the considered system.For the implementation of the proposed scheme, we assume that the RIS is partitioned into two non-overlapping sub-surfaces S 1 and S 2 , consisting of Q e and Q r = Q − Q e elements, respectively.The elements of S 1 are configured based on a CB design i.e., the phase shifts are aligned to the phases of the corresponding channel coefficients, while the remaining elements follow an RR approach by inducing random phase shifts to the incident signals.An example of this approach is shown in Fig. 1, where the two sub-surfaces S 1 and S 2 (containing Q e = 10 and Q r = 20 elements, respectively) are indicated by the black solid line.
In order to optimize the phase shift parameters of the elements that belong to S 1 , the knowledge of the related product channels v i is required.Since in the considered scenario prior CSI knowledge at the RIS is not available, a training period needs to be allocated for the required CSI acquisition.Therefore, each coherence interval is divided into two phases, as shown in Fig. 2: the channel training phase of duration τ ≤ T , where the product channels v i corresponding to the elements of S 1 are estimated, and the data transmission phase occuring for the remaining period T − τ , where the two subsurfaces are sequentially activated to assist the transmission of the desired signals from the Tx to the Rx.In the following, we provide a more detailed description of each phase.
1) Channel training phase: The first phase is dedicated for the estimation of the channels associated with the elements of S 1 .During this phase, the elements that belong to S 2 are set to the OFF state.The remaining elements switch their reflection coefficients, based on the RIS training states of the adopted channel training method.Without loss of generality, we consider a DFT-based training method for estimating the product channels [7].Specifically, during the training period τ a pilot signal x p = 1 is transmitted with fixed power P o .This period is further divided into Q e sub-phases of equal duration.During the j-th sub-phase, all the elements of S 1 are turned ON and their phase shifts are set equal to the values of the j-th column of a Q e ×Q e DFT matrix.Note that the columns of the DFT matrix are mutually orthogonal.The cascaded channels vi , i ∈ S 1 , are estimated by using a minimum mean square error (MMSE) approach [8].In the considered scenario we assume imperfect CSI acquisition, where the variance of the channel estimation error is given by [8], [14] σ 2) Data transmission phase: In the second phase, the desired data signals are sent with transmit power P from the Tx to the Rx.During this phase, the RIS controller activates S 1 and S 2 sequentially to assist the communication procedure.In particular, the data transmission phase is divided into two subphases of equal duration3 i.e., (T −τ )/2.In the first sub-phase, the elements of S 1 are ON and the elements of S 2 remain at the OFF state.The phase shifts of the activated elements are adjusted to align the phases of the product channels, based on the estimated CSI from the channel training phase, in order to yield the CB gains and maximize the achieved rate [8].The design parameters for the RIS elements during the first sub-phase are summarized below Finally, in the second sub-phase of data transmission, the communication is assisted only by S 2 .Specifically, in this sub-phase, the elements of S 2 are now activated, while the elements of S 1 are turned OFF.Each activated element randomly rotates the phase of the incident signal, by following a uniform distribution in [0, 2π).The reflection coefficients are therefore given by a i,RR = 1, i-th element ∈ S 2 ; 0, otherwise, As a result, the end-to-end channel over a coherence interval is composed of two independent parallel sub-channels in the time domain, where for each sub-channel only the activated elements in each sub-phase need to be considered.It is worth noting that with the proposed scheme, we are able to reduce the channel training overhead, since channel estimation is required for only a subset of elements.In the following section, we evaluate the performance of the proposed scheme in terms of outage probability.Moreover, we compare our scheme with two benchmark scenarios: the RR-only scenario, where all the elements randomly rotate the phase of the incident signals (Q e = 0 and τ = 0), and the conventional full-CB scheme, where all the elements align their phase shifts based on the estimated CSI (Q e = Q) [8].Note that, in both cases, all the elements of the RIS are activated for the whole data transmission phase.

IV. OUTAGE PROBABILITY ANALYSIS
In this section, we present our main analytical results for the performance of the considered scheme.Specifically, we derive the achieved outage probability, which is defined as the probability that the achieved rate over a coherence interval is 2023 IEEE Global Communications Conference: Wireless Communications not above a predefined threshold r.Based on the presented RIS scheme, the outage probability can be written as where is the sum-rate achieved over the CB sub-phase of the data transmission phase with ρ = P (σ 2 + Q e P σ 2 e ) −1 , and is the achieved sum-rate of the RR sub-phase.In what follows, we provide an approximation that can sufficiently describe the outage performance of the proposed scheme, by using the moment matching method and the central limit theorem (CLT).
Theorem 1.The outage probability of the proposed hybrid CB/RR scheme is approximated by where Γ(•) is the complete gamma function, γ(•, •) denotes the lower incomplete gamma function [15], The proof can be found in the Appendix.It is clear that under the RR-only scenario i.e., Q e = 0, a channel training phase is not required (τ = 0), and so the whole coherence interval is used for data transmission.Therefore, we need to consider only (8) for the achieved rate of the RIS-aided system and the approximated outage expression is simplified to On the other hand, by considering the benchmark scenario of the conventional full-CB scheme, where Q e = Q, the outage probability can be expressed as It can be easily observed that the outage performance of the considered RIS-aided system is significantly affected by the size of each RIS sub-surface.Although the employment of the CB design at the RIS can provide high performance gains, estimating the channels for a large number of elements incurs high training overhead, which inevitably degrades the system's performance.Thus, in this work, we aim to find the optimal number of elements in each sub-surface in order to minimize the outage probability i.e., By using the derived theoretical expressions for Π(Q e , T ), the above optimal value can be easily obtained through an exhaustive search method on the number of elements that should be estimated.
We now turn our attention to the asymptotic outage performance of the system as P increases.When P → ∞ then the instantaneous signal-to-noise ratio (SNR) over the CB subphase converges to Then, by using that exp(−x) ≈ 1 − x for x ≈ 0 and by taking the smallest order term of ( 9) as the dominant term, we observe that the outage probability decreases with the rate of 1/P i.e., lim (18) Note that, under the RR-only scenario (Q e = 0) the achieved outage probability follows a similar behavior, since asymptotically it converges to On the other hand, when employing the full-CB scheme, due to the channel estimation error imposed by the imperfect CSI acquisition the outage probability converges to a constant floor given by lim It is therefore deduced that, at high SNR the CB/RR scheme overcomes the outage floor that limits the performance of the full-CB case due to imperfect CSI acquisition, since in the proposed scheme the two sub-surfaces are sequentially activated and the channel estimation error affects only the elements of S 1 .In particular, when P → ∞ both the RR-only scenario and the CB/RR scheme outperform the full-CB case, while the optimal performance is obtained by the partition size Q e that achieves the highest coding gain.

V. NUMERICAL RESULTS
In this section, we validate our analytical framework and main observations with Monte Carlo simulations.For the sake of presentation, we consider r = 1 bps/Hz and σ 2 = 1.Moreover, for the channel training we assume that τ = Q e channel uses and P o = −10 dB.The proposed scheme is compared with the conventional RR-only scenario (Q e = 0) and the full-CB case (Q e = Q).The simulation results have been obtained after 10 8 simulation runs.Unless otherwise stated, analytical results are depicted with lines and simulation results with markers.

Outage Probability
In Fig. 3, we illustrate the achieved outage probability in terms of the transmit power P for an RIS with Q = 40 elements and a coherence interval of T = 50 channel uses.In particular, Fig. 3 shows the achieved outage probability for the conventional cases of Q e = 0 and Q e = Q, and the optimal partition size of the RIS Q e .It is observed that for very low values of P , dedicating part of the coherence interval for channel estimation does not provide any significant performance gains.Therefore the best option is to employ the RR-only scenario at the RIS in order to make full use of the available time for data transmission.On the other hand, in the high SNR regime it is clear that the hybrid CB/RR RIS scheme can significantly outperform the conventional scenarios.Specifically, we observe that for both the RRonly case and the proposed scheme, the outage probability decreases with the rate of 1/P , while in the full-CB scenario the outage probability converges to the floor value calculated in (20).However, the proposed scheme, under optimal partitioning, can achieve considerably higher coding gain compared to the RR-only case.Finally, we note that the theoretical lines approximate the simulation results exceptionally well, validating the accuracy of the presented analysis.The optimal partition size that achieves the best outage performance, as indicated by the red dotted line in Fig. 3, is provided in Fig. 4. It can be seen that in most cases the optimal performance is achieved by using around half of the total RIS elements for CB.In particular, in the high SNR region the optimal partition size becomes constant at Q e = 23 elements, since in this case the achieved coding gain is maximized.Fig. 5 depicts the outage probability with respect to the total number of RIS elements Q for T = 40 channel uses and transmit power P = −5 dB.We observe that, for Q T the RR-only case provides the best outage performance, as the expected gains from the CB design are insufficient for such low number of elements.As the number of RIS elements increases, the full-CB scenario becomes optimal, since the optimization of the RIS phase shifts yields high performance gains.However, when the number of elements approaches the total channel uses i.e., Q → T , the training overhead becomes large under the full-CB case and the outage probability is significantly deteriorated.On the other hand, the proposed scheme overcomes the above limitations by reducing the required training overhead, since the channel training is performed for only a subset of elements; hence it outperforms the conventional scenarios and provides the best outage performance when the partition size is optimized.

VI. CONCLUSIONS
In this paper, we proposed a hybrid CB/RR scheme for an RIS-assisted network, which requires channel estimation for only a subset of the elements.The presented scheme splits the RIS into two sub-surfaces, which are sequentially activated and employ a CB-based and an RR-based design, respectively, resulting in reduced channel training overhead.The performance of the proposed scheme was investigated under imperfect CSI acquisition.We derived analytical expressions for the outage probability and we showed that, by optimizing the number of estimated elements, the proposed scheme can still benefit from the performance gains obtained by the channel training and can improve the performance beyond the limits imposed by the imperfect CSI at the RIS.Future work includes the extension of the presented scheme to multiple partitions at the RIS, in order to obtain (time) diversity gains through a time coding scheme.

APPENDIX
Let W CB = i∈S1 |v i |.Recall that the channel coefficients h i and g i are independent and identically distributed (i.i.d) complex Gaussian random variables with variance one.By taking into account the errors induced at the channel training phase, the mean and variance of |v i | are equal to and By applying the moment matching method, |v i | is approximated as a Gamma random variable [12], where the shape parameter κ and scale parameter ξ are calculated as Var {|v i |} and ξ = Var {|v i |} and are given by (10).Since W CB is the sum of Q e i.i.d.Gamma random variables with the same parameters, it is also Gamma distributed with parameters Q e κ and ξ.We now let It has been proved in [11] that by applying the CLT, the distribution of W RR converges to a complex Gaussian distribution with zero mean and variance Q r .Therefore, the channel gain 2 is exponentially distributed with parameter 1/Q r .We can now evaluate the outage probability as

Fig. 2 .
Fig. 2. The phases of the transmission procedure and the RIS reconfiguration for the proposed scheme.