Average Rate and Error Probability Analysis in Short Packet Communications over RIS-aided URLLC Systems
Ramin Hashemi, Samad Ali, Nurul Huda Mahmood, Matti Latva-aho
11 Average Rate and Error Probability Analysis inShort Packet Communications over RIS-aidedURLLC Systems
Ramin Hashemi,
Student Member, IEEE,
Samad Ali
Member, IEEE , Nurul Huda Mahmood
Member, IEEE , andMatti Latva-aho,
Senior Member, IEEE . Abstract —In this paper, the average achievable rate and errorprobability of a reconfigurable intelligent surface (RIS) aidedsystems is investigated for the finite blocklength (FBL) regime.The performance loss due to the presence of phase errors arisingfrom limited quantization levels as well as hardware impairmentsat the RIS elements is also discussed. First, the composite channelcontaining the direct path plus the product of reflected channelsthrough the RIS is characterized. Then, the distribution of thereceived signal-to-noise ratio (SNR) is matched to a Gammarandom variable whose parameters depend on the total numberof RIS elements, phase errors and the channels’ path loss. Next,by considering the FBL regime, the achievable rate expressionand error probability are identified and the correspondingaverage rate and average error probability are elaborated basedon the proposed SNR distribution. Furthermore, the impact ofthe presence of phase error due to either limited quantizationlevels or hardware impairments on the average rate and errorprobability is discussed. The numerical results show that MonteCarlo simulations conform to matched Gamma distribution toreceived SNR for sufficiently large number of RIS elements. Inaddition, the system reliability indicated by the tightness of theSNR distribution increases when RIS is leveraged particularlywhen only the reflected channel exists. This highlights theadvantages of RIS-aided communications for ultra-reliable andlow-latency systems. The difference between Shannon capacityand achievable rate in FBL regime is also discussed. Additionally,the required number of RIS elements to achieve a desired errorprobability in the FBL regime will be significantly reduced whenthe phase shifts are performed without error.
Index Terms —Average achievable rate, block error probability,finite blocklength (FBL), factory automation, reconfigurable in-telligent surface (RIS), ultra-reliable low-latency communications(URLLC).
I. I
NTRODUCTION
The fourth industrial evolution or Industry 4.0 is aimed atdigitizing industrial technology towards decentralized manu-facturing of products and automation of tasks with reducedhuman involvement in various industrial processes. Industry4.0 [1] is powered by Industrial Internet of Things (IIoT) [2]which interconnects various elements like sensors and otherinstruments with industrial management applications throughindustrial control networks (ICN) [3] to enable real-timecontrolling of ubiquitous actuators (AC) and machines acrossthe smart factory. To this end, traditional wired connections
R. Hashemi, S. Ali, NH. Mahmood, and M. Latva-aho are with theCentre for Wireless Communications (CWC), University of Oulu, 90014Oulu, Finland. e-mails: (ramin.hashemi@oulu.fi, samad.ali@oulu.fi, nurul-huda.mahmood@oulu.fi, matti.latva-aho@oulu.fi). are being replaced with wireless networks [4] to minimizethe infrastructure expenditure, and achieve higher flexibility.However, this requires guaranteeing wired connectivity per-formance with wireless links. Hence, having an ultra-reliableand high-precision physical layer communication link is ofparamount importance for many industrial applications.In 5G new radio (NR), ultra-reliable and low-latency com-munication (URLLC) [5], [6] is one of the three main servicecategories that address the requirements of Industry 4.0 [7].In URLLC, the high-reliability interprets error probabilitiesof less than − , and low-latency targets to 1 ms end-to-end delay. The main attributes of URLLC are realized e.g.by leveraging the file contents caching [8], utilization ofshorter transmission time interval (TTI) [9], grant free access[10], and multi-connectivity. URLLC messages usually carrycontrol information, hence the packet lengths are generallyultra-short. As a result, the blocklength of the channel isfinite which necessitates a thorough analysis of achievablerate and decoding error probability as investigated in [11],[12]. However, the URLLC transmission demands are not metentirely by the above solutions as the main challenge to ensurehigh reliability is the random nature of the propagation channelmainly due to multipath fading.Recently, the reconfigurable intelligent surface (RIS) tech-nology [13], [14] is introduced as a means to improve thespectral efficiency and coverage of wireless communicationsystems by influencing the propagation environment. Theuse of RIS brings intelligence to the physical channel. Thestructure of an RIS is composed of a metasurface wherea programmable controller configures and adjusts the phaseand/or amplitude response of the metasurface to modify thebehaviour of the reflection of an incident wave. The aim of thisoperation is that the received signals at a particular receiverlocation are constructively added so that the system perfor-mance enhances in terms of increasing e.g. the signal-to-noiseratio (SNR). Based on leveraging passive or active elementsat each phase shifter, the RISs are classified into passive andactive devices, respectively. Therefore, the RIS technology canbe effectively utilized in URLLC short packet transmissionsunder finite blocklength (FBL) regime in order to improve theIIoT networks’ performance in terms of enhancing the receivedsignal quality and ensuring high reliability. In this paper, ouraim is to shed some light on the average achievable rate,and error probability analysis of RIS-aided IIoT networks inFBL regime that relies on only statistical measures of channel a r X i v : . [ c s . I T ] F e b response. A. Related Work1) RIS Related Studies on Channel Characterization:
Anumber of studies investigate either the ergodic capacity oroutage probability analysis of RIS-aided systems by identify-ing the characteristics of the channel response and receivedSNR [15]–[26]. In [15] the distribution of the absolute valueof the composite channel containing direct link is consideredto be a Gaussian random variable (RV) for large RIS elementsaccording to the central limit theorem (CLT) and then theergodic capacity is studied. The authors in [16] considereda phase shift error in RIS elements which is distributed asvon Mises or uniform RV, following which the distributionof the SNR is approximated to a Gamma RV. Then, theaverage error probability in an infinite blocklength channelis analyzed. In [17] the authors express that the probabilitydensity function (PDF) of the reflected channel response in anRIS-aided non-orthogonal multiple access (NOMA) networkis a Gaussian RV for a large number of RIS elements. Then,the diversity order analysis was studied for fully constructiveadding of signals in the presence of phase error at RIS. Thecomposite channel considered in this paper does not have adirect path between the access point (AP) and the users. In[18] the distribution of SNR is approximated as a GammaRV by employing the moment matching technique and theergodic capacity as well as the outage probability is studiedin an infinite blocklength model i.e. conventional Shannoncapacity formula is considered. Note that the assumed compos-ite channel contains the direct link plus the reflected channelfrom RIS with arbitrary phase shifts so that only the statisticalbehavior of the phase shifts was taken into account. In [19]the analytical PDF of the received SNR was found in terms ofcascade channel characteristics in an RIS-NOMA network anda transmission design method was presented based on spatialdivision multiple access (SDMA). The impact of RIS phaseerror is also studied in outage probability analysis. In [20]the upperbound of the ergodic rate is maximized under Ricianfading channel between the multi-antenna AP and the user (orRIS). The optimal phase shifts are derived based on the ergodicrate depending on the phase configuration matrix. The authorsin [21] derived a closed-form approximation for the ergodicrate of cell-edge users based on Taylor series expansion oflogarithmic function. The perfect phase shift assignment isassumed and the cascade channel distribution was consideredas a normal RV according to CLT. In [22] the upperbound forthe ergodic capacity is derived when perfect phase shift at theRIS is performed. The upperbound is based on the Jensen’sinequality for Shannon capacity formula. The authors showedthat employing a decode and forward relay will enhance theupperbound capacity significantly.Furthermore, the study in [23] analyzed the impact offinite quantization levels on the performance of an RIS-aidedtransmission by using a tight approximation for the ergodiccapacity without assuming a specific distribution for receivedSNR. The best case and worst case channel characteristicsare formulated as a Gamma RV with separate shape and rate parameters for each case in [24]. In [25] the absolutevalue of the reflected channel is considered as a GammaRV then, the ergodic rate and outage probability analysisis investigated in terms of total RIS elements and havingdiscrete phase shifts. Note that the analysis is performed overinfinite blocklength regime without the presence of the directchannel. Finally, the authors in [26] considered the optimalSNR derived in [27] and then, they proposed that the SNRdistribution is composed of the product of three independentGamma RVs and sum of two scaled non-central chi-squareRVs based on the eigenvalues of the channel matrices ofRIS-AP and RIS-user. The authors compare the proposedanalytical distributions with the case that the SNR is onlyapproximated with one Gamma RV. The numerical resultsshowed that there is negligible difference in consideringGamma distribution for the SNR compared with preciseanalytical distributions. Furthermore, to evaluate the averageachievable rate, it is intractable to perform computation of theexpectations concerning SNR distribution when a complexexpression is taken into account. Therefore, assuming thereceived SNR as a Gamma RV is tractable and sufficientlyaccurate.
2) URLLC Studies on Average Achievable Rate and AverageError Probability:
Several papers investigate the performanceanalysis of URLLC systems [28]–[35] in finite blocklength(FBL) channel model. In [28] the authors proposed to employmassive multiple-input multiple-output (MIMO) systems toleverage in IIoT networks to reduce the latency. The lowerbound achievable uplink ergodic rates of massive MIMO sys-tem with finite blocklength codes is analyzed by convexifyingthe rate formula which holds under specific conditions. Thesame authors analyzed secure URLLC in IoT applications [29]and presented resource allocation problems. The authors in[30], [31] studied the analysis of ergodic achievable data ratein FBL regime. In [30] a MISO network is considered andthe uplink channel training was studied instead of downlinktraining by deriving the lower bound of the ergodic ratewhereas in [31] channel state information (CSI) is acquiredby downlink channel training. Then, the optimal number oftraining symbols is proposed based on the average data rateexpression. In [32] the downlink MIMO NOMA systems’average error probability under Nakagami-m fading modelis investigated in FBL regime. It should be noted that in[32] the ergodic capacity analysis is not addressed and theaverage error probability is studied based on a well-knownlinear approximation for the Q-function. In [33] the analysis oflower bound achievable ergodic rate in URLLC transmission ispresented based on convexifying the achievable rate functionwhere the function is convex on a specific interval. Grant-freeuplink access for an RIS-assisted industrial MIMO networkis studied in [34] as well as outage probability is done basedon numerical Monte Carlo simulations. The effective capacitywhich is the maximum transmission rate under certain delayconstraints was studied in [35]. The authors introduced aclosed-form expression for the effective capacity in a Rayleighfading channel.
B. Contributions
Even though the aforementioned studies cover the topicsof RIS and short-packet communication, to the best of ourknowledge, there is no previous reports on the the performanceanalysis of an RIS-aided transmission with/without the pres-ence of phase noise in an FBL regime for URLLC applications.Motivated by the above works we aim to elaborate on theanalysis of average achievable rate and block error probabilityof RIS-aided factory automation wireless transmissions underFBL model. We extend our results for the case with errors inthe RIS phase shift adjustments arising due to, e.g., limitationof quantization bits or hardware imperfections. The contribu-tions of our work are summarized in the following ● The downlink received signal containing the direct linkplus a reflected signal from the RIS to the AC is identi-fied. Then, the received SNR is statistically matched to aGamma RV with/without the presence of phase noise atthe RIS elements which is due to the quantization erroror hardware impairments at the RIS phase controller. ● The average achievable rate and error probability assum-ing FBL regime is mathematically elaborated in terms ofthe characteristics of Gamma RV for the SNR. ● Since the results involves computing high-complexityfunctions, a tractable and closed-form lower bound for-mula for the average achievable rate and error probabilityunder FBL regime is presented. ● The impact of quantization error and hardware impair-ments are modeled as a uniform RV in the phase shiftargument. Then, the corresponding modifications due tothese effects on the SNR distributions are studied in detailand mathematical equations for the mean and the varianceof presented Gamma RV are identified. Furthermore, aclosed-form formula which is useful in design consider-ations to find total channel blocklength value is studiedin terms of the average rate and dispersion functions aswell as the effect of the phase error on total channelblocklength.
C. Notations and Structure of the Paper
In this paper, h ∼ CN ( N × , C N × N ) denotes circularly-symmetric (central) complex normal distribution vector withzero mean N × and covariance matrix C . The operators E [ . ] and V [ . ] denote the statistical expectation and variance,respectively. Also, X ∼ Γ ( a, b ) denotes gamma random vari-able with shape and rate parameters a and b , respectively. Auniform distributed random variable with range [ a, b ] is shownas Y ∼ U( a, b ) . The operation [ . ] H denotes the conjugatetranspose of a matrix or vector.The structure of this paper is organized as follows. InSection II, the system model and mathematical identificationof the received SNR and its distribution is presented. SectionIII presents the derivation of average rate and average errorprobability. In Section IV we extend our derivations to the casewhere phase error occurs. The numerical results are presentedin Section V. Finally, Section VI concludes the paper. Control link 𝒉 RISAP 𝒉 ACRIS
RIS 𝑁 𝑁 ℎ ACAP AC AP AP: Access PointAC: ActuatorRIS: Reconfigurable Intelligent Surface
Fig. 1: The system model.II. S
YSTEM M ODEL
Consider the downlink of an RIS-aided network consist-ing of a single antenna AP and AC where the RIS has N = N × N elements. The channel response between theAP and AC has a direct path component plus a reflectedchannel from the RIS. Let us denote the direct channelas h APAC ∼ CN ( , η AP → AC ) where η AP → AC denotes the pathloss attenuation due to large scale fading. h APRIS ∈ C N × and h RISAC ∈ C N × represent the vector channels from theAP to the RIS and from the RIS to the AC, respectively.The channel vector h APRIS is distributed as
CN ( N × , η AP → RIS N × N ) where η AP → RIS = diag ( η AP → RIS , ..., η AP → RIS N ) is a diagonalmatrix including the path loss coefficients from the AP tothe RIS elements. Similarly, the channel between the RISand AC is distributed as h RISAC ∼ CN ( N × , η RIS → AC N × N ) where η RIS → AC = diag ( η RIS → AC , ..., η RIS → AC N ) denotes the covariancematrix in this case. In factory automation environments eachactuator is almost in a fixed location and there is approximatelylow velocity in different modules. Therefore, the quasi-staticchannel fading model can be applied here. We assume thatthere is no interference for simplicity and leave the analysisin an interference network as future work.The received signal at the AC is given by y ( t ) = ( h APAC + h RISAC H Θ h APRIS ) s ( t ) + n ( t ) , (1)where s ( t ) is the transmitted symbol from the AP with E [∣ s ( t )∣ ] = p in which p is the transmit power, and n ( t ) is the additive white Gaussian noise with E [∣ n ( t )∣ ] = N W where N , W are the noise spectral density and the systembandwidth, respectively. The complex reconfiguration matrix Θ N × N indicates the phase shift and the amplitude attenuationof RIS which is defined as Θ N × N = diag ( β e jθ , β e jθ , ..., β N e jθ N ) ,β n ∈[ , ] , ∀ n ∈ N θ n ∈[− π, π ) , ∀ n ∈ N (2)where N = { , , ..., N } . Note that in our model we haveassumed that the RIS elements have no coupling or thereis no joint processing among elements [13]. Hence, thephase shifts and amplitude control are done independently. The phase alignment error of the RIS is defined as φ n = ∠ h APAC − ∠ [ h RISAC ] n + ∠ [ h APRIS ] n + θ n which occurs due to hardwarelimitations and/or finite number of quantization levels availableat RIS phase shifters. More precisely, without consideringhardware impairments the phase discrete set is selected fromthe following set [36] θ n ∈ Θ = {− π, − π + ∆ , − π + , ..., − π + ( Q − ) ∆ } , ∀ n ∈ N (3)where Q = b is the number of quantization levels, b denotesthe number of bits assigned to a discrete and quantized phaseand ∆ = π b − is the quantization step.Based on the received signal at AC and denoting the numberof information bits L that can be transmitted with targeterror probability ε in r channel uses ( r ≥ ) the maximalachievable rate over a quasi-static additive white gaussianchannel (AWGN) is given by [11] R ∗ ( γ, L, ε ) = Lr = C ( γ ) − Q − ( ε )√ V ( γ ) r + O ( log ( r ) r ) , (4)where C ( γ ) = log ( + γ ) is the Shannon capacity formulaunder infinite blocklength assumption. The dispersion of thechannel is defined as V ( γ ) = ( log ( e )) ( − ( + γ ) ) . Notethat Q − ( . ) is the inverse of Q-function which is defined as Q ( x ) = √ π ∫ ∞ x e − ν / dν and γ = ρ ∣ h APAC + h RISAC H Θ h APRIS ∣ , (5)where ρ = pN W denotes the instantaneous SNR. The approx-imate of r in terms of ε can be expressed as [37] r ≈ L C ( γ ) + ( Q − ( ε )) V ( γ ) ( C ( γ )) (6) + ( Q − ( ε )) V ( γ ) ( C ( γ )) ¿``(cid:192) + L C ( γ )( Q − ( ε )) V ( γ ) , Note that the term O( log ( r ) r ) in (4) is neglected throughoutthis paper as it is approximately zero for r ≥ channel uses.The decoding error probability at the AC for a packet of size L transmitted via r symbols is written as ε = Q ( f ( γ, r, L )) , (7)where f ( γ, r, L ) = √ rV ( γ ) ( log ( + γ ) − Lr ) . It is observedfrom (4) when the blocklength approaches infinity the ratewill be lim r →∞ R ∗ ( γ, L, ε ) = log ( + ρ ∣ h APAC + h RISAC H Θ h APRIS ∣ ) , (8)which is the conventional Shannon capacity formula.To determine the achievable average rate, we need toidentify the distribution of γ . In the following, we presentthe related theorems and derive a closed-form and tractable It should be noted that L that is the size of packets is assumed the samefor actuator and access point. approximation for the average rate. Theorem 1 (SNR distribution) . Let X = ∣ h APAC + h RISAC H Θ h APRIS ∣ and given N >> , the distribution of X is approximatelymatched to a Gamma random variable with the followingparameters [18], [24] X ∼ Γ ( α ′ , β ′ ) , (9) where α and β are given in terms of first and second ordermoment of X as α ′ = ( E [ X ]) E [ X ] − ( E [ X ]) , (10) β ′ = E [ X ] E [ X ] − ( E [ X ]) , (11) where E [ X ] and E [ X ] are given in (43) and (48) , respec-tively when phase error occurs ( φ n ≠ , ∀ n ∈ N ) and aregiven in (49) and (50) when φ n = , ∀ n ∈ N . For SNRdistribution we have γ = ρX . Therefore, E [ γ ] = ρ E [ X ] and E [ γ ] = ρ E [ X ] which implies that γ ∼ Γ ( α, β ) with thesame α as in (10) and β = β ′ ρ .Proof. The detailed proof is given in Appendix A in thepresence of phase error i.e. φ n ≠ , ∀ n ∈ N and AppendixB when φ n = , ∀ n ∈ N . ∎ III. A
VERAGE R ATE AND A VERAGE E RROR P ROBABILITY
A. Average Achievable Rate
In the previous section, we have modeled the SNR distribu-tion, and the related Gamma distribution parameters α and β are obtained for two cases with/without phase errors at RIS.Next, to compute the average rate, the instantaneous achievablerate should be averaged over the SNR distribution which weinvestigate in the next theorem. Theorem 2.
The exact average achievable rate of the actuatorin the RIS-aided URLLC transmission given the distributionof SNR γ ∼ Γ ( α, β ) is expressed as ¯ R ( L, ε ) =C − Q − ( ε )√ r C (12) = β α Γ ( α ) ln 2 ∞ ∑ k = k Γ ( k + α ) U ( k + α, + α, β )− Q − ( ε ) β α √ r ln 2 ∞ ∑ k = ( k )(− ) k U ( α, − k + α, β ) , where C and C are given in (19) and (24) , respectivelyand U ( a, b, z ) = ∫ ∞ ( + u ) b − a − u a − e − zu du Γ ( a ) denotes the confluenthypergeometric Kummer U function [38, Eq. (9.211)], and theGamma function is denoted by Γ ( α ) = ∫ ∞ y α − e − y dy .Proof. The instantaneous rate is given by R ∗ ( γ, L, ε ) ≈ C ( γ ) − Q − ( ε )√ V ( γ ) r , (13) where O( log ( r ) r ) is ignored. To calculate the expected valueof (13) in terms of the distribution of γ we should computethe following ¯ R ( L, ε ) = C ‡„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„•„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„(cid:181) E [ log ( + γ )] − Q − ( ε )√ r C ‡„„„„„„„„„„„„„„„„„„„„„„•„„„„„„„„„„„„„„„„„„„„„„(cid:181) E [√ V ( γ )] , (14)where we have used the linearity rule of the expectation. Weinvestigate the two terms C and C involved in (14) separately.According to (14), C is defined as C = E [ log ( + γ )] = ∫ ∞ log ( + u ) f γ ( u ) du, (15)where f γ ( u ) = β α u α − e − βu Γ ( α ) . Therefore, by evaluating theintegral in (15) and after mathematical manipulations it yields C = E [ log ( + γ )] = ∫ ∞ log ( + u ) β α u α − e − βu Γ ( α ) du = β α (− β ) − α Γ ( α ) ln 2 {(− β ) α [ β Γ ( α − ) F ( ,
1; 2 , − α ; β ) + Γ ( α ) ( ψ ( α ) − log ( β ))] + πβ α csc ( πα ) [ Γ ( α ) − Γ ( α, − β )]} , (16)where p F q ( a ; b ; z ) is the generalized hypergeometric function[38, Eq. (9.1)] and ψ ( α ) = Γ ′ ( α ) Γ ( α ) gives the digamma functionwhere they are standard built-in functions in most of thewell-known mathematical software packages. However, weaim to write a more tractable solution for the C whichinvolves simple computational complexity in terms of fewdiverse functions. To do so, first consider the following seriesrepresentation for natural logarithm [38] ln ( + x ) = ∞ ∑ k = k ( xx + ) k , x ≥ (17)then, by substituting in (15) we will have ∫ ∞ log ( + u ) β α u α − e − βu Γ ( α ) du (18) a = ∞ ∑ k = k ∫ ∞ ( uu + ) k β α u α − e − βu Γ ( α ) du b = β α Γ ( α ) ln 2 ∞ ∑ k = k Γ ( k + α ) ∫ ∞ ( + u ) − k u α + k − e − βu Γ ( k + α ) du, where a is done by exchanging the summation and integral andin b a constant factor of Γ ( k + α ) is multiplied in numerator anddenominator. We observe that the integral inside the summa-tion of the last step is defined as the confluent hypergeometricKummer U function U ( a, b, z ) [38, Eq. (9.211)] therefore C = β α Γ ( α ) ln 2 ∞ ∑ k = k Γ ( k + α ) U ( k + α, + α, β ) , (19)where the summation can be truncated to a finite numberin practical numerical situations, e.g. with negligiblemismatch. It should be noted that (19) computes only theconfluent hypergeometric function and, well-known gammafunction at each iteration of the summation, henceforth hasless computational complexity in comparison with (16). In the following we evaluate the expectation associated with C which is given by C = E [√ V ( γ )] = ∫ ∞ √ − ( + u ) f γ ( u ) du, (20)to compute the above integral, we adopt the binomial expan-sion of the channel dispersion which is given by √ V ( γ ) = ( − ( + γ ) ) = ∞ ∑ n = (− ) n ( n )( + γ ) − n , (21)where ( n ) = . ( . − ) ... ( . − n + ) n ! for n ≠ and ( ) = .Plugging (21) in (20) yields C = ∫ ∞ ∞ ∑ n = (− ) n ( n )( + u ) − n f γ ( u ) du, (22)where by replacing the integral and summation as well assubstituting the definition of f γ ( u ) , C can be rewritten as C = β α ∞ ∑ n = (− ) n ( n ) ∫ ∞ ( α ) ( + u ) − n u α − e − βu du, (23)we observe that the integral inside the summation can bereformulated in terms of confluent hypergeometric KummerU function U ( a, b, z ) defined earlier, henceforth C = β α ∞ ∑ n = ( n )(− ) n U ( α, − n + α, β ) . (24)consequently, if we substitute the mathematical expressionsobtained in (19) and (24) in the average rate formula in (14),the final result will be obtained which completes the proof. ∎ It is worth mentioning that the average rate given inTheorem 2 involves evaluating high-computational complexityfunctions as well as infinite summations which may notbe beneficial in practical situations and resource allocationalgorithms. Therefore, we propose a tractable lower boundapproximation for the average rate in the following corollary.
Corollary 1.
A tractable and closed-form approximate lower-bound expression for the average rate ¯ R ( L, ε ) is given by ¯ R LB ( L, ε ) ≈ ˜ C − Q − ( ε )√ r ˜ C = log ( + α β ( α + ) ) (25) − Q − ( ε ) √ r ( − β + e β β ( α + β − ) E α ( β )) , where E n ( z ) is the exponential integral function [38, Eq.(8.211)], ˜ C and ˜ C are given in (28) and (31) , respectively.Proof. First, we study the term involving Shannon capacity,i.e. C = E [ log ( + γ )] . Invoking Jensen’s inequality [39] wehave C = E [ log ( + γ )] ≥ log ( + E [ γ − ] ) , (26)where to arrive in a lower-bound we should elaborate the right-hand of (26). To continue, we apply Taylor series expansion of γ then, take average from both sides which yields E [ γ − ] ≈ E [ γ ] + V [ γ ] E [ γ ] = E [ γ ] E [ γ ] , (27)substituting (27) in (26) yields C ≥ ˜ C = log ( + E [ γ ] E [ γ ] ) = log ( + α β ( α + ) ) , (28)where E [ γ ] and E [ γ ] can be evaluated straightforward fromthe SNR distribution γ ∼ Γ ( α, β ) such that E [ γ ] = αβ and E [ γ ] = α ( α + ) β where α and β are investigated in Theorem1. It is worth mentioning that another approximation for C can be written based on truncated Taylor series expansion of ln ( + γ ) as given by ln ( + γ ) ≈ ln ( + γ ) + + γ ( γ − γ ) − ( + γ ) ( γ − γ ) , (29)then, by replacing γ = E [ γ ] and taking average from bothsides which yields C = E [ log ( + γ )] ≈ log ( + E [ γ ]) − V [ γ ] ( + E [ γ ]) ln 2 , (30)however, using the above expression does not give eitherlowerbound or upperbound for C therefore, for comparisonpurposes the lowerbound given in (28) is advantageous.Next, we investigate C in (20). To do so, we apply thetruncated binomial approximation ( + x ) ϑ ≈ + ϑx for ∣ x ∣ < , ∣ ϑx ∣ < to approximate the channel dispersion as √ V ( u ) = √ − ( + u ) ≈ ( − ( + u ) ) . By substitutingthe approximate dispersion expression in (20) we will have ˜ C = E [√ V ( γ )] ≈ ∫ ∞ ( − ( + u ) ) f γ ( u ) du = ∫ ∞ ( − ( + u ) ) β α u α − e − βu Γ ( α ) du = ( − β α ∫ ∞ ( + u ) u α − e − βu Γ ( α ) du ) a =
12 ln 2 ( − β + e β β ( α + β − ) E α ( β )) . (31)where a is obtained through integrating by part and somemanipulations. Finally, by replacing (28) and (31) in (14) theapproximate result will be obtained. Note that since C is lowerbounded by ˜ C and C ≤ ˜ C the achievable rate will attain itslower bound because of negative sign in C in (25). ∎ B. Average Decoding Error Probability
In order to compute the average decoding error probabilityone needs to evaluate the following ¯ ε ≈ E [ Q (√ rV ( γ ) ( log ( + γ ) − Lr ))] , (32)where the expected value is taken over the distribution of γ and, still r is large enough to have a better accuracy. It should be noted that directly computing (32) is intractable toachieve a closed-form solution. Therefore, we adopt a linearapproximation to the function inside the the expected value as Q (√ rV ( γ ) ( log ( + γ ) − Lr )) ≈ g ( γ ) (33)where g ( γ ) is given by [40] g ( γ ) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ , γ ∈ [ , κ ) + ξ ( γ − ξ ) , γ ∈ [ κ , κ ] . γ ∈ [ κ , ∞) (34)where ξ = −√ r π ( Lr − ) , ξ = Lr − , κ = ξ + ξ and κ = ξ − ξ . Consequently, the average error probability definedin (32) will be ¯ ε = ∫ ∞ Q (√ rV ( u ) ( log ( + u ) − Lr )) f γ ( u ) du ≈ ∫ κ f γ ( u ) du + ∫ κ κ ( + ξ ( u − ξ )) f γ ( u ) du =( + ξ ξ ) F γ ( κ ) + ( − ξ ξ ) F γ ( κ ) + ξ ∫ κ κ uf γ ( u ) du = ξ β Γ ( α ) ( β ( κ − κ ) Γ ( α ) − βκ Γ ( α, κ β ) + βκ Γ ( α, κ β )+ Γ ( α + , κ β ) − Γ ( α + , κ β )) , (35)where F γ ( γ ) = γ ( α,β ) Γ ( α ) is the cumulative distribution functionof the SNR that is investigated in Theorem 1 and γ ( a, b ) , Γ ( a, b ) are the lower and upper incomplete gamma functions,respectively [38]. Consequently, (35) gives a closed formformula for the average error probability. C. The Required Number of Channel Blocklengths
By solving a quadratic equation in terms of (cid:36) = √ r in (4)and ignoring the higher order terms O( log ( r ) r ) , an estimateof the average number of channel blocklengths ¯ r = E [ r ] interms of number of RIS elements N and phase shifts will beobtained as follows ¯ r ≈ L (√ ∆ − Q − ( ε )C ) , (36)where ∆ = ( Q − ( ε )C ) + L C . Therefore, ¯ r can beevaluated straightforwardly after determining C and C thatwere investigated earlier. To give an insight on this formula,given the channel blocklength which is defined as r = T W (which T is the transmission duration and W denotes theavailable bandwidth) we can expect that when total thenumber of RIS elements increases the required blocklengthto transmit the symbols will be reduced. This is because thedenominator of the (36) is an increasing function in terms oftotal number of RIS elements so that the whole fraction willbe smaller as N gets larger. This results in lower transmissiontime as well as less power consumption at the AP whichhighlights the suitability of RIS at URLLC systems in the FBL regime. IV. I
MPACT OF P HASE E RROR
In this section we investigate the performance loss due tothe presence of phase error in RIS elements which may occurdue to quantization of the RIS elements’ phase according tothe number of bits assigned to each discrete phase by thecontroller or hardware impairments [13], [14]. Consideringeither hardware impairments or quantization error the resultantimpact will be on the distribution of φ n , ∀ n ∈ N . Therefore,the SNR parameters namely α and β are needed to recompute.Henceforth, we propose a general framework to have a clearunderstanding of this effect. Different distributions can beanalyzed as straightforward without loss of generality.Let us assume that there are only Q quantization levels inwhich b = log ( Q ) bits are assigned to each discrete phaseshift. The RIS chooses each phase shift from the followingset [36] θ n ∈ Θ = { − π, − π + ∆ , − π + , ..., − π + ( Q − ) ∆ } , ∀ n ∈ N (37)where ∆ = π b − . It should be noted that a linear quantizer hasan error e which spans uniformly over − ∆2 ≤ e ≤ ∆2 . Let usdenote the phase error as φ n ∼ U(− (cid:15)π, (cid:15)π ) , ∀ n ∈ N where (cid:15) = b ∈ ( , ] . Additionally, the phase errors of each elementare independently distributed. In this case, we should computethe expected values of (43) and (48) in terms of the phase errordistribution. Therefore, the following expressions are neededfor ∀ n ≠ m ≠ n ′ E [ cos ( φ n )] = sinc ( (cid:15) ) , (38a) E [ cos ( φ n )] = + sinc ( (cid:15) ) , (38b) E [ cos ( φ n − φ m )] = sinc ( (cid:15) ) , (38c) E [ cos ( φ n − φ m )] = + sinc ( (cid:15) ) , (38d) E [ cos ( φ n ) cos ( φ n − φ m )] = sinc ( (cid:15) )( + sinc ( (cid:15) ) ) , (38e) E [ cos ( φ n − φ m ) cos ( φ n ′ − φ n )] = ( + sinc ( (cid:15) )) sinc ( (cid:15) ) , (38f)consequently, the average rate and error probability will beachieved after substituting (38a)–(38f) in (43) and (48) whichyields E [ γ ] = ρ ( η AP → AC + N η AP → RIS η RIS → AC + π sinc ( (cid:15) ) N ( N − ) η RIS → AP η AC → RIS + N π sinc ( (cid:15) ) √ πη AP → AC η RIS → AP η AC → RIS ) , (39)and E [ γ ] is given in (40) on top of the next page. Note thatas in Appendix A we have ς = η AP → AC , (cid:37) = η AP → RIS and ϑ = η RIS → AC . By noting that γ ∼ Γ ( α, β ) where α = ( E [ γ ]) E [ γ ]−( E [ γ ]) , β = E [ γ ] E [ γ ]−( E [ γ ]) the average rate and error probability can beevaluated simply using Theorem 2 and (35), respectively. TABLE I: Simulation parameters.Parameter Default valueRIS location in 2D plane ( d ,10) m d ∈ [ , ] AP location (0,0) mAC position (100,0)AP transmit power ( p ) 200 mWReceiver noise figure (NF) 3 dBTarget error probability ( ε ) − φ n , ∀ n ∈ N ∼ U[− π , π ] Noise power density ( N ) -174 dBm/HzChannel blocklength ( r ) The size of packets ( L ) bits ( Lr = . )Number of realizations Bandwidth ( W ) 200 kHzCarrier frequency 1900 MHzAP height 12.5 mPath loss model( D : distance in meter) PL(dB) = . +
38 log ( D ) V. P
ERFORMANCE E VALUATION
In what follows, we evaluate the proposed derivations andmathematical expressions numerically. Table I shows the cho-sen default values for the network parameters and geometry.Since the carrier freq. is 1900 MHz a typical antenna size willbe around 15 cm. Therefore, the far-field assumption holds truein all simulation scenarios for the given network geometry. InFig. 2a and Fig. 2b the cumulative distribution function (CDF)of the received SNR is illustrated for two cases namely as inthe presence of the direct channel between the AP and the ACand the case where there is no direct link. Besides, the resultsare shown when perfect phase alignment is performed at theRIS which is referred to as φ n = , ∀ n ∈ N and comparisonis made with uniform distributed phase noise at the RIS as abenchmark. As can be observed from Fig. 2a, Monte Carlosimulations conform to the matched Gamma distribution forSNR. Furthermore, the impact of quantizer noise is shownin Fig. 2a. Although there is a performance gap betweenleveraging n -bit quantizer ( n = , , ) and the optimal case, a2-bit quantizer provides a good trade-off between performanceand training overhead as the gap is . dB (compared to . dB gap with 1-bit) [41]. The 3-bit quantizer is morecloser to the perfect case with small SNR degradation. Itcan be inferred that utilizing a lower-bit quantizer leads toreduction in RIS expenditure since higher bits means highercapacity requirement of the control signalling exchange forphase adjustment of the RIS elements, and hence highercontrol overhead. Henceforth, acquiring lower bit quantizersis mandatory.In Fig. 2b a comparison is shown between three scenarios,I) no direct link and the RIS acts as a simple reflector, II)no direct link with phase adjusted RIS and III) with directlink and phase adjusted RIS. As we observe, the only casewhich has a sharp slope and higher tightness is scenario II.It means that the values of SNR that we expect to receive inscenario II are almost in the same range whereas the range offluctuations in cases I and III which are denoted as intervals E [ γ ] = ρ [ ς + N ς(cid:37)ϑ ( + ( (cid:15) ) + π ( N − ) ( (cid:15) )) + √ ς (cid:37)ϑ ( π √ πN ( (cid:15) ))+ N (cid:37) ϑ ( N + + π ( N − )( N + ) sinc ( (cid:15) ) + ( N − ) sinc ( (cid:15) )+ π ( N − )( N − ) sinc ( (cid:15) )( + sinc ( (cid:15) )) + π sinc ( (cid:15) )( N − )( N − )( N − ))+ N √ ς(cid:37) ϑ sinc ( (cid:15) ) π √ π ( N + + ( N − ) ( sinc ( (cid:15) ) + ( N − ) π
16 sinc ( (cid:15) )))] , (40)of ∗ and ∗∗ , respectively, is noticeable that results in erodingthe perception of reliability . Therefore, though the SNR valueis higher with the direct link, the variation in the SNR impliesthat the channel is less deterministic compared to case II.Hence the RIS is beneficial in guaranteeing a high reliabilityeven if the direct link is absent (e.g., blocked). Furthermore,we observe that in the presence of the direct channel there isnegligible difference between leveraging a 1-bit quantizer witha 2-bit in Fig. 2b. This is because the direct channel is severalorders of magnitude stronger than the reflected channel so thatthe effect of reflected channel is not dominant.By evaluating the average achievable rate as well as takinginto account the channel dispersion, a 2-bit quantizer is com-pared with a 1-bit quantizer in Fig. 3 along with a optimalphase setting. The results also confirm that Monte Carlosimulations approximate to the derived analytical expressionsfor the average rate in Theorem 2. As it is observed, whenthe number of bits assigned to each discrete phase at theRIS increments, the average rate curve is very close to theoptimal phase alignment case. Once more, this shows that toachieve satisfactory accuracy, a few numbers of available bitswill be sufficient instead of high precision and high complexityquantizers. Moreover, it is proved that to reach full diversity inRIS-aided communications the number of quantization levelsmust be Q ≥ [42] that holds when b ≥ . Furthermore, theShannon capacity and the gap with FBL regime is illustratedin Fig. 3. We observe that the gap is increased until somesaturation value. This is because of asymptotically convergingthe channel dispersion to its upperbound when the number ofRIS elements increases which results in improving the SNRin other words lim γ →∞ V ( γ ) = ( log ( e )) .Next, we show the results for average error probabilitygiven in Fig. 4 when there is no direct channel. As weobserve, to achieve a desired error probability the requirednumber of RIS elements is much higher when there is phaseerror at the RIS compared with optimal phase setting. Thisshows the importance of phase alignment precision in the RISparticularly in URLLC applications. For instance, when wedesire to reach an error probability of − , the number ofRIS elements satisfying this condition should be at least 190elements in a perfect phase alignment scenario whereas in caseof having phase errors the required number of RIS elementsshould be at least 290 elements in a 1-bit quantizer and about200 with 2-bit quantizer. On the other hand, one may interpretthese curves as a baseline for design considerations of how many RIS elements should be installed to reach sufficientlylow bit error rate.In Fig. 5 the average error probability results are illustratedfor d ∈ [ , ] , where RIS is located at ( d , 10) on the 2D plane(cf. Table I) (close to AP → close to AC). The number ofRIS elements, N = . We observe that the error probabilitybehavior is somewhat symmetric with respect to the distancefrom either the AP or the AC. The error probability degradesas the RIS moves further away from either the AP or the AC,with the worst performance observed for the case when it isequidistant from both. This is because the path loss, which isproportional to the product of the RIS distances from the APand the AC, is the highest when the two distances are equal.In Fig. 6a the average achievable rate is illustrated for d ∈ [ , ] , where RIS is located at ( d , 10) and there isdirect channel between AP and AC for N = . In contrastto the active relaying schemes where locating in the middleof transmitter and the receiver usually is an optimal choiceto maximize the performance, we observe that the averageachievable rate in the FBL regime will be maximized whenthe RIS is either close to transmitter or receiver. Additionally,there is a gap between the lower bound and exact valuefor all curves. Nevertheless, the gap does not change as thenumber of quantizer bits or the RIS location at ( d , 10) changewhich confirms the suitability of the presented lower bound forcomparison purposes and resource allocation algorithms. Theaverage rate performance without direct channel is illustratedin Fig. 6b where as shown in the curves there is a perfectmatch between the lower bound rate and the Monte Carlosimulations. Furthermore, the impact of phase error on theaverage rate improvement is the same. To have a similaranalysis of rate variation we see that when d = is changedto d = or d = the average rate is increased from . ( . )bpcu to . ( . ) bpcu for 2-bit (1-bit) quantizer.In Fig. 7 the required number of channel uses as a functionof RIS elements is illustrated in terms of quantizer bits at theRIS when target error probability is set to − . It is observedthat, when the number of bits increases, the average channelblocklength will be reduced and asymptotically converges tothe lower curve which is the case without noise at the RIS.There is a significant reduction when the number of quanti-zation bits increases from one to two bits. This shows that insystem level design considerations choosing a 2-bit quantizerwill be beneficial and satisfactory than the complicated higherbit quantizers. Furthermore, given the channel blocklength r = SNR [dB] C D F Monte Carlo SimulationMatched Gamma RV (uniform phase noise)Matched Gamma RV ( n = 0) (a) SNR CDFs without direct link. -30 -20 -10 0 10 20 30 40 SNR [dB] C D F With direct linkNo direct linkNo RIS Without direct link * **
Monte Carlo SNR ( n = 0)Matched gamma RV ( n = 0) Monte Carlo SNR (uniform noise)Matched gamma RV (uniform noise) (b) Comparison of SNR CDFs ( N = ). Fig. 2: The CDF curves of SNR and illustration of matched Gamma RV with Monte Carlo simulations.
500 1000 1500 2000 2500 3000 3500 4000 N A v e r a g e R a t e [ bp c u ] Shannon Rate ( n = 0, 1-bit, 2-bit) No Phase Noise (Analytical LB)No Phase Noise (Monte Carlo)1-bit (Monte Carlo)1-bit (Analytical LB)2-bit (Monte Carlo)2-bit (Analytical LB)
Fig. 3: The average rate in terms of changing the total RISelements without direct channel.
T W when the number of RIS elements increases the requirednumber of channel blocklength to transmit the symbols willbe reduced. This means that as N increases, the transmissionduration T is reduced given a fixed bandwidth W . Thus, RIScan be leveraged to achieve low latency transmissions forURLLC applications, demonstrating the applicability of RISin FBL regime communications.Finally, the asymptotic behavior of the square of the channeldispersion, as well as the binomial approximation accuracy, isinvestigated in Fig. 8. As it is shown, the channel dispersionand its binomial approximation are well-matched when nophase error exists. The situation is different for the case whenphase error exists and in a lower number of RIS elements,the accuracy is lower. Nevertheless, it can also be inferredthat the channel dispersion asymptotically approaches to itsupperbound when we have a sufficient number of RIS elements( N approximately greater than 150) in both cases. This is
120 140 160 180 200 220 240 260 280 300 N -9 -8 -7 -6 -5 -4 -3 -2 -1 E rr o r P r ob a b ilit y Approx. ( n = 0)Monte Carlo SimulationMatched Gamma RVApprox. (1-bit)Approx. (2-bit) Fig. 4: Average error probability versus RIS elements N without direct channel.because the received SNR increases so that the square root ofchannel dispersion will approach log ( e ) .VI. C ONCLUSION
This paper analyzes the applicability of RIS for ensuringURLLC with FBL transmissions in a factory automationscenario. We have presented the analytical derivation of theaverage achievable rate and we have analyzed the averageerror probability based on matching the received SNR to aGamma RV. First, the analytical derivations of matching theSNR to a Gamma RV is presented whose parameters dependon the statistical mean and variance of the instantaneousSNR. Then, the average achievable rate is investigated for theFBL regime based on computing the related expectation withrespect to proposed SNR distribution. The same analysis isperformed to evaluate the average block error probability andchannel blocklength. Next, the impact of phase error resulting
10 15 20 25 30 35 d -9 -8 -7 -6 -5 -4 -3 -2 -1 E rr o r P r ob a b ilit y Matched RV Monte Carlo Approx.
65 70 75 80 85 90 d -9 -8 -7 -6 -5 -4 -3 -2 -1 Optimal
Fig. 5: Average error probability performance in terms ofchanging the RIS location at ( d , 10) for N = 512.from either quantization noise or hardware impairments isinvestigated in modeled SNR distribution parameters. Thenumerical results have shown that the RIS can be effectivelyemployed in factory automation environments to ensure highreliability and reduce the error probability as a measure ofreliability as well as the transmission latency as required bymany URLLC applications. Furthermore, RIS is also found tosignificantly improve the average achievable rate. As a futurework, the analysis results in this paper can be leveraged forresource allocation problems in RIS-assisted URLLC networksto ensure systems reliability and maximizing average raterelying only on statistical measures of the channel.A PPENDIX AC OMPUTING E [ X ] AND E [ X ] WHEN φ n ≠ FOR ∀ n ∈ N Let us rewrite the random variable X as X = ∣∣ h APAC ∣ + N ∑ n = ∣[ h APRIS ] n ∣ ∣[ h RISAC ] n ∣ e j φ n ∣ , (41)where [ . ] n denotes the n th element of a vector and φ n = ∠ h APAC − ∠ [ h RISAC ] n + ∠ [ h APRIS ] n + θ n . In order to obtain the parameters ofmatched Gamma distribution, we should determine the shapeand rate parameters. To do so, we first calculate the expectedmean value of RV X noting that for a RV h ∼ Rayleigh ( σ ) itholds E [ h ] = √ π σ which is given by E [ X ] = η AP → AC + N ∑ n = η AP → RIS n η RIS → AC n + π N ∑ n = N ∑ m = m ≠ n η RIS → AP n η AC → RIS m cos ( φ n − φ m )+ π √ πη AP → AC N ∑ n = √ η RIS → AP n η AC → RIS n cos ( φ n ) , (42)where we assumed the channel responses are mutually inde-pendent. By neglecting the impact of RIS surface dimensions on large scale fading we will have E [ X ] = η AP → AC + N η AP → RIS η RIS → AC + π η RIS → AP η AC → RIS N ∑ n = N ∑ m = m ≠ n cos ( φ n − φ m )+ π √ πη AP → AC η RIS → AP η AC → RIS N ∑ n = cos ( φ n ) , (43)where η AP → AC = η AP → AC n , η AP → RIS = η AP → RIS n and η RIS → AC = η RIS → AC n ∀ n ∈ N .In what follows we continue by computing the expectedvalue of X which is defined as E [ X ] = E ⎡⎢⎢⎢⎢⎣∣∣ h APAC ∣ + N ∑ n = ∣[ h APRIS ] n ∣∣[ h RISAC ] n ∣ e j φ n ∣ ⎤⎥⎥⎥⎥⎦ , (44)for notation simplicity we define c ∶= ∣ h APAC ∣ , (45a) a n ∶= ∣[ h APRIS ] n ∣ , (45b) b n ∶= ∣[ h RISAC ] n ∣ . (45c)The binomial expansion of the expression inside expectationyields E [∣ c + N ∑ n = a n b n e j φ n ∣ ] = E ⎡⎢⎢⎢⎢⎣ c + c N ∑ n = a n b n ( + ( φ n ))+ c N ∑ n = N ∑ m = m ≠ n a n b n a m b m ( cos ( φ n − φ m ) + ( φ n ) cos ( φ m ))+ c N ∑ n = a n b n cos ( φ n ) + N ∑ n = N ∑ m = m ≠ n a n b n a m b m + N ∑ n = a n b n + N ∑ n = a n b n N ∑ n = N ∑ m = m ≠ n a n b n a m b m cos ( φ n − φ m )+ c N ∑ n = N ∑ m = m ≠ n a n b n a m b m cos ( φ m ) + c N ∑ n = a n b n cos ( φ n )+ ( N ∑ n = N ∑ m = m ≠ n a n b n a m b m cos ( φ n − φ m )) + c N ∑ n = a n b n cos ( φ n ) N ∑ n = N ∑ m = m ≠ n a n b n a m b m cos ( φ n − φ m )⎤⎥⎥⎥⎥⎦ , (46)since E [ c ] = √ πη AP → AC , E [ a n ] = √ πη AP → RIS and E [ b n ] = √ πη RIS → AC and also noting that for a RV h ∼ Rayleigh ( σ ) wehave E [ h ] =√ π σ, (47a) E [ h ] = σ , (47b) E [ h ] = √ π σ , (47c) E [ h ] = σ . (47d)
10 20 30 40 50 60 70 80 90 d A v e r a g e R a t e [ bp c u ] No Phase Noise (Analytical LB)No Phase Noise (Monte Carlo)Uniform Phase Noise (Analytical LB), 1-bitUniform Phase Noise (Monte Carlo), 1-bit (a) With direct channel for N = .
10 20 30 40 50 60 70 80 90 d A v e r a g e R a t e [ bp c u ] No Phase Noise (Analytical LB)No Phase Noise (Monte Carlo)Uniform Phase Noise (Analytical LB), 2-bitUniform Phase Noise (Monte Carlo), 2-bitUniform Phase Noise (Analytical LB), 1-bitUniform Phase Noise (Monte Carlo),1-bit (b) Without direct channel for N = . Fig. 6: The impact of changing the RIS location at ( d , 10) on the average rate.
50 100 150 200 250 300 350 400 450 500 N A v e r a g e N u m b e r o f C h a nn e l U s e s No Phase NoiseUniform Phase Noise (1-bit)Uniform Phase Noise (2-bit)
Fig. 7: The average number of channel uses versus totalnumber of RIS elements N .Based on above, and after some mathematical manipulationsthe equation (46) will be written as given in Eq. (48) on top ofthe next page. where E [ c ] = η AP → AC = ς , E [ a n ] = η AP → RIS = (cid:37) and E [ b n ] = η RIS → AC = ϑ .A PPENDIX BC OMPUTING E [ X ] AND E [ X ] WHEN φ n = FOR ∀ n ∈ N In a special case where the phase adjustment is perfectlydone at the RIS we have φ n = , ∀ n ∈ N which yields E [ X ] = η AP → AC + N η AP → RIS η RIS → AC + π N ( N − ) η AP → RIS η RIS → AC + πN √ πη AP → AC η AP → RIS η RIS → AC , (49)
50 100 150 200 250 300 350 400 N r = E [ V ] No Phase Noise (Approximation)No Phase Noise (Monte Carlo)No Phase Noise (Exact)Uniform Phase Noise (Approximation), 1bitUniform Phase Noise (Monte Carlo), 1-bitUniform Phase Noise (Exact), 1-bitUniform Phase Noise (Exact), 2-bitUniform Phase Noise (Monte Carlo), 2-bitUniform Phase Noise (Approximation), 2-bit
Fig. 8: Square channel dispersion ( E [√ V ( γ )] ) and its asymp-totic behaviour for large number of N .and E [ X ] = E ⎡⎢⎢⎢⎢⎣∣ c + N ∑ n = a n b n ∣ ⎤⎥⎥⎥⎥⎦ = ς + ς(cid:37)ϑN ( + ( N − ) π )+ N π . √ ς (cid:37)ϑ + (cid:37) ϑ N ( π ( N − )( N − )( N − )+ π ( N − )( N − ) + N + )+ √ ς(cid:37) ϑ N π . ( π ( N − )( N − ) + N − ) . (50)where ς = η AP → AC , (cid:37) = η AP → RIS and ϑ = η RIS → AC . E [∣ c + N ∑ n = a n b n e j φ n ∣ ] = ς + ς(cid:37)ϑ ( N + N ∑ n = cos ( φ n ) + π N ∑ n = N ∑ m = m ≠ n ( cos ( φ n − φ m ) + ( φ n ) cos ( φ m )) ) + π √ πς (cid:37)ϑ N ∑ n = cos ( φ n )+ (cid:37) ϑ ( N ( N + ) + π ( N + ) N ∑ n = N ∑ m = m ≠ n cos ( φ n − φ m ) + N ∑ n = N ∑ m = m ≠ n cos ( φ n − φ m )+ π N ∑ n = N ∑ m = m ≠ n N ∑ n ′ = n ′ ≠ n,m cos ( φ n − φ m ) cos ( φ n ′ − φ n ) + π N ∑ n = N ∑ m = m ≠ n N ∑ n ′ = n ′ ≠ n,m cos ( φ n − φ m ) cos ( φ n ′ − φ m )+ π N ∑ n = N ∑ m = m ≠ n N ∑ n ′ = ,n ′ ≠ n,m N ∑ m ′ = ,m ′ ≠ n,m,n ′ cos ( φ n − φ m ) cos ( φ n ′ − φ m ′ ))+ √ ς(cid:37) ϑ π √ π ( N + N ∑ n = cos ( φ n ) + N ∑ n = N ∑ m = m ≠ n cos ( φ n − φ m ) cos ( φ n )+ N ∑ n = N ∑ m = m ≠ n cos ( φ n − φ m ) cos ( φ m ) + π N ∑ n = N ∑ m = m ≠ n N ∑ n ′ = n ′ ≠ n,m cos ( φ n − φ m ) cos ( φ n ′ )) , (48)R EFERENCES[1] G. Aceto, V. Persico, and A. Pescapé, “A survey on informationand communication technologies for Industry 4.0: State-of-the-art, tax-onomies, perspectives, and challenges,”
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