# Optimal SNR Analysis for Single-user RIS Systems in Ricean and Rayleigh Environments

We present an analysis of the optimal uplink (UL) SNR of a SIMO Reconfigurable Intelligent Surface (RIS)-aided wireless link. We assume that the channel between base station (BS) and RIS is a rank-1 LOS channel while the user (UE)-RIS and UE-BS channels are correlated Ricean. For the optimal RIS matrix, we derive an exact closed form expression for the mean SNR and an approximation for the SNR variance leading to an accurate gamma approximation to the distribution of the UL SNR. Furthermore, we analytically characterise the effects of correlation and the Ricean K-factor on SNR, showing that increasing the K-factor and correlation in the UE-BS channel can have negative effects on the mean SNR, while increasing the K-factor and correlation in the UE-RIS channel improves system performance. We also present favourable and unfavourable channel scenarios which provide insight into the sort of environments that improve or degrade the mean SNR. We also show that the relative gain in the mean SNR when transitioning from an unfavourable to a favourable environment saturates to (4-π)/π as N →∞

## Authors

• 3 publications
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08/21/2020

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## I Introduction

Reconfigurable Intelligent Surface (RIS) aided wireless networks are currently the subject of considerable research attention due to their ability to manipulate the channel between users (UEs) and base station (BS) via the RIS. Assuming that channel state information (CSI) is known at the RIS, one can intelligently alter the RIS phases, essentially changing the channel to improve system performance. Here, we focus on a single user system and assume a common system scenario where a RIS is carefully located near the BS such that a rank-1 line-of-sight (LOS) channel is formed between the BS and RIS. System scenarios with a LOS channel between the BS-RIS and a single-user are also considered in [22, 15, 14, 21] with motivation for the LOS assumption given in [14]. All of these existing works aim to enhance the system to achieve some optimal system performance (sum rate, SINR, etc.) by tuning the RIS phases. In particular, [14] gives a closed form RIS phase solution without the presence of a direct UE-BS channel for a single user setting, while [21] gives a closed form phase solution with the presence of a direct channel. We note, however, once the optimal RIS has been defined there is no exact analysis of the mean SNR and no analysis of correlation impact on the mean SNR in [22, 15, 14, 21].

For the UE to RIS and the direct UE to BS links, the presence of scattering is a more reasonable assumption as is spatial correlation in the channels, especially at the RIS where small inter-element spacing may be envisaged. Several papers do consider spatial correlation in the small-scale fading channels [14, 24, 23, 12, 25, 4, 13, 26], however, these papers are simulation based and no analysis is given on the impact of correlation on the mean SNR.

Statistical properties of the RIS-aided channel have been investigated in existing literature. For example, [8, 3] provide a closed form expression for the mean SNR in the absence of a UE-BS channel with [3]

additionally providing a probability density function (PDF) and a cumulative distribution function (CDF) for the distribution of the mean SNR. In

[8, 18], an upper bound is given for the ergodic capacity and in [7] a lower bound is given for the ergodic capacity. However, there are no closed form expressions for the mean SNR and SNR variance for an optimum RIS-aided wireless system, in the presence of correlated Ricean UE-BS, UE-RIS fading channels. We further note that the work presented in this paper is an extension for the work done in [17] which considered correlated Rayleigh UE-BS, UE-RIS fading channels.

In this paper, we focus on an analysis of the optimal uplink (UL) SNR for a single user RIS aided link with a rank-1 LOS RIS-BS channel and correlated Ricean fading for the UE-BS and UE-RIS channels. The contributions of this paper are as follows:

• An exact closed-form result for mean SNR and an approximate closed form expression for SNR variance are derived. These are used to show that a gamma distribution provides a good approximation of the UL optimal SNR distribution. Furthermore, using the analysis presented, we are able to reduce the mean SNR and SNR variance expressions when the UE-BS and UE-RIS links are correlated Rayleigh to agree with those in

[17].

• The analysis is leveraged to gain insight into the impact of the K-factor and spatial correlation on the mean SNR. We show that increasing the K-factor and correlation in the UE-BS channel has negative effects, while increasing the K-factor and correlation in the UE-RIS channel improves the mean SNR.

• Given the analyses, we present favourable and unfavourable channel scenarios, which provide insight into the sort of environments that would improve and degrade the mean SNR. For systems with a large number of RIS elements, we show that when changing from favourable to unfavourable channel scenarios, improvements in the mean SNR saturate at a relative gain of as .

Notation: represents statistical expectation. is the Real operator. denotes the

norm. Upper and lower boldface letters represent matrices and vectors, respectively.

denotes a complex Gaussian distribution with mean

and covariance matrix .

denotes a uniform random variable taking on values between

and .

denotes a chi-squared distribution with

degrees of freedom. represents an matrix with unit entries. The transpose, Hermitian transpose and complex conjugate operators are denoted as , respectively. The trace and diagonal operators are denoted by and , respectively. The angle of a vector of length is defined as and the exponent of a vector is defined as . denotes the Kronecker product. denotes a Laguerre function of non-integer degree .

## Ii System Model

As shown in Fig. 1, we examine a RIS aided single user single input multiple output (SIMO) system where a RIS with reflective elements is located close to a BS with antennas such that a rank-1 LOS condition is achieved between the RIS and BS.

### Ii-a Channel Model

Let , , be the UE-BS, UE-RIS and RIS-BS channels, respectively. The diagonal matrix , where for , contains the reflection coefficients for each RIS element. The global UL channel is thus represented by

 h=hd+HbrΦhru, (1)

where we consider the correlated Ricean channels:

 hd hru =√βru(ηruaru+ζruR1/2ruuru)≜√βru~hru

with

 ηd=√κd1+κd,ζd=√11+κd, ηru=√κru1+κru,ζru=√11+κru,

where and are the link gains between UE-BS and UE-RIS respectively, and are the correlation matrices for UE-BS and UE-RIS links, respectively, and . Here, and are the Ricean K-factors for the UE-BS and UE-RIS channels, respectively. The LOS paths and are topology specific steering vectors at the BS and RIS respectively. Particular examples of steering vectors for a vertical uniform rectangular array (VURA) are given in Sec. V.

Note, that the correlation matrices and can represent any correlation model. For simulation purposes, we will use the well-known exponential decay model,

 (Rru)ik=ρdi,kd% rru,(Rd)ik=ρdi,kdbd, (2)

where , . is the distance between the and antenna/element at the BS/RIS. and are the nearest-neighbour BS antenna separation and RIS element separation, respectively, which are measured in wavelength units. and are the nearest neighbour BS antenna and RIS element correlations, respectively.

The rank-1 LOS channel from RIS to BS has link gain and is given by where and are topology specific steering vectors at the BS and RIS, respectively.

### Ii-B Optimal SNR

Using (1), the received signal at the BS is, where is the transmitted signal with power and . For a single user, Matched Filtering (MF) is the optimal combining method so the optimal UL SNR is, where . The optimal RIS phase matrix to maximize the SNR can be computed using the main steps outlined in [21, Sec. III-B] but using an UL channel model instead of downlink. Substituting the channel vectors and through some matrix algebraic manipulation, the optimal RIS phase matrix is,

 (3)

Substituting (3) into , the optimal UL global channel is,

 h=hd+√βbrβruψN∑n=1∣∣~hru,n∣∣ab, (4)

where , giving the optimal UL SNR as

 SNR=(hHd+α∗aHb)(hd+αab)¯τ, (5)

where and . The variable is given its own notation as it arises in many of the derivations for the SNR.

## Iii Mean SNR and SNR Variance

### Iii-a Correlated Ricean Channels

Here, we provide an exact closed form result for and an exact expression for when and are correlated Ricean channels. Utilising these expressions, we then show that they can be reduced to the expressions given in [17] when and are correlated Rayleigh channels.

###### Theorem 1.

The mean SNR is given by

with given by

 FR=N∑i=1N∑k=1i≠k(1−|ρik|2)21+κruexp{−2κru−2μcκru1−|ρik|2} ×Γ2(m+32)1F21(m+32,n+1,κru(1+|ρik|2−2μc)1−|ρik|2) (7)

and , where , is the confluent hypergeometric function, , , , and is given by and for [9, Eq. (26b)].

###### Proof.

See App. A for the derivation of (1). ∎

The formula for the mean SNR in (1) is compact and intuitive except for the term which is easier to understand from its definition in App. A as

 (8)

Note that evaluation of

requires the product moment of two correlated Ricean variables. While compact expressions are given in

[27] for the PDF of two correlated Ricean variables, it is important to note that these PDFs are for the special case where the LOS components are the same. Here, there is a phase shift that occurs between any two RIS elements and this assumption cannot be made. Hence, we have used the more general bivariate Ricean distribution in [9]. When the UE-RIS channel is non-LOS (correlated Rayleigh) then the complexity in all the expressions reduces considerably.

###### Theorem 2.

The SNR Variance is given by

with

 (10) (11) +Ld2⎡⎣Bκd∣∣aHdab∣∣2A2(1+κd)+√π(M−B)1+κd⎤⎦ (12)
 (13) T23=η2dM√π2Ld (14) I=−3A√π4√κd(1+κd)exp⎧⎨⎩j∠aHdab−κd∣∣aHdab∣∣22A2⎫⎬⎭ (15)

where is given in Theorem 1, is given by (1), , , is the Whittaker M function, and .

###### Proof.

See App. B and App. C for the derivation of (2). ∎

In (2), all terms are known except for and . To the best of our knowledge, these moments are intractable and approximations are required for these moments. Note that is positive, unimodal, and the sum of random variables. A gamma distribution is commonly used to approximate such distributions, eg., [8]. Since the first and second moments of are known exactly, we are able to fit a gamma approximation to with the correct first and second moments. Using this gamma approximation, we obtain approximations for and as presented in the following Corollary,

###### Corollary.

To obtain an approximation of the SNR variance in closed form, we approximate the 3 and 4 moments of by,

 E{Y3} =b3a2∏k=1(k+a),E{Y4}=b4a3∏k=1(k+a) (16)

with

 a =N2πL2ru4(N+FR)−N2πL2ru b =2N√πLru(N+FR−N2π24L2ru)

where is given by (1) and is given by (10).

###### Proof.

See App. D for the derivation of (16). ∎

Note that for high values of correlation and the K-factor, (1) is computationally expensive as in such scenarios the number of terms required in the double summation over and grows large. In these situations, an alternative approach is to replace the double summation with its integral equivalent, derived in App. F, and presented in the following cases where and . The case of perfect correlation, , provides a useful benchmark to evaluate the SNR trends as the level of correlation increases.

### Iii-B Case 1: Fr for |ρik|<1

 FR =N∑i=1N∑k=1i≠kζ2ru√1−|ρik|22√π∫2π0∫∞0re−r2 ×√|ai|2+r2+2r|ai|cos(θ−θi) ×L1/2(−b(|ak|2+r2+2r|ak|cos(θ−θk))) dθdr, (17)

where , , and .

### Iii-C Benchmark Case: Fr when |ρik|=1

For the benchmark case of maximum correlation, , both (1) and (III-B) result in an indeterminate answer. However, we can use the integral form of and find its result as which, using the derivation in App. F, gives

 FR =ζ2ruπN∑i=1N∑k=1i≠k∫2π0∫∞0re−r2 (18)

where and are defined in Sec. III-B. Note that the double numeric integrations required in (III-B) and (III-C) are computationally convenient as: one integral is finite range and the integrand is smooth and rapidly decaying as .

### Iii-D Special Case 1: Uncorrelated Ricean Case

For independent Ricean fading, we provide exact closed form expressions for both and . The mean SNR is given by

 E{SNR}= +βdM+βbrβruM(N+FR))¯τ, (19)

with given by

 FR=πN(N−1)4(ζruL1/2(−κru))2. (20)

This can be easily derived from Theorem 1 by noting that is the sum of products of iid Ricean random variables over all for and since and .

The SNR Variance is given by

 Var{SNR}= (β2dζ2d[2η2dM+ζ2dM] +β3/2d√βbrβru√MNLruπ[2√π(T21+T22+T23)−MLd] +βdβbrβru[4(η2d∣∣aHbad∣∣2+ζ2dM)(N+FR) −N2Mπ2L2ruL2d4] +√βd(βbrβru)3/2M3/2Ld[2√πC1u−N(N+FR)πLru] +(Mβbrβru)2[C2u−(N+FR)2])¯τ2, (21)

with

 T22=2ηd√MR{aHdRdab[I−ηdaHdab√π2√MLd]} (25) T23=η2dM√π2Ld (26) (27) C1u=3√πN4Lru1+π3/2N!8(N−3)!L3ru+3√πN!2(N−2)!Lru (28) C2u=N(2ζ2ru+η2ru(4ζru+η2ru))+π2N!16(N−4)!L4ru +3N!(N−2)!(1+√πLru1)+3πN!2(N−3)!L2ru. (29)

Equations (III-D)-(29) follow from Theorem 2 using the substitutions , , and the version of given in (20). A consequence of having uncorrelated UE-BS and UE-RIS channels, is that the expectations and are known exactly and these are given by (28) and (29) respectively.

### Iii-E Special Case 2: Correlated Rayleigh Case

We obtain mean SNR and SNR variance expressions for correlated Rayleigh channels and by setting .

#### Iii-E1 Mean SNR

Note that when , and . Similarly when , and . As such, the second term in (1) reduces down to the second term in [17, Eq. (8)]. The third term can be reduced to its correlated Rayleigh form as shown in [17, Eq. (8)] by reducing to (31) (see App. E). The final form after reducing all of the terms in (1) results in the mean SNR expression in [17, Eq. (8)]; given below for completeness:

###### Theorem 3.

The mean SNR when and are correlated Rayleigh channels is given by

 E{SNR}= (βdM+NAπ√βdβbrβru2+βbrβruM(N+F))¯τ, (30)

with given by

 (31)

and , where is the Gaussian hypergeometric function and .

#### Iii-E2 SNR Variance

In addition to the steps undertaken in III-E1, we further note that . This eliminates many of the terms in (2) of Theorem 2. Using App. E to collapse down to its correlated Rayleigh form, we obtain the final SNR variance expression for correlated Rayleigh channels, consistent with that given in [17, Eq. (10)] and given below for completeness:

###### Theorem 4.

The SNR variance when and are correlated Rayleigh channels is given by

 Var{SNR} =(β2dtr{R2d}+β3/2d√βbrβruNπ(B−MA) +βdβbrβruA2(4(N+F)−N2π24) +MA√βd(βbrβru)3/2(2√πC1−N(N+F)π) +(Mβbrβru)2(C2−(N+F)2))¯τ2, (32)

where , is given by (31), is given in Theorem 1 and .

###### Corollary.

To obtain an SNR variance approximation in closed form, we approximate the 3, 4 moments of by,

 E{Y3} =b3a2∏k=1(k+a),E{Y4}=b4a3∏k=1(k+a) (33)

with

 a=N2π4(N+F)−N2π,b=2N√π(N+F−N2π4).

where is given by (31).

## Iv Performance Insights based on E{SNR}

Note that the first term in the mean SNR in (1) is influenced by variants of the function for , , where the factor of arises from the and terms. Hence, we study the behaviour of before developing performance insights based on the mean SNR.

### Iv-a Behaviour of L1/2(−αx)/√1+x

Using [1, Eq. (13.6.9)] the Laguerre function can be rewritten in terms of the confluent hypergeometric function, , as

 L1/2(−αx)=1F1(−12;1;αx). (34)

Using the series expansion for [1, Eq. (13.1.2)] and the Maclaurin series expansion of , we have the following expansion for ,

 q(α,x) =(1+αx2−α2x216+…)(1−x2+3x28−…) =1+α−12x+6−4α−α216x2+…

First, we look at the behaviour near the origin. Noting that , then for , is increasing in and for , is decreasing in .

Next, we look at the behaviour as . Using the asymptotic behaviour of [1, Eq. (13.1.5)], we have

 q(α,x) ∼2√αxπ(1+x)(1+14αx) (35) →2√απ as x→∞. (36)

From (35), we see that as , for and for . Differentiating (35) and simplifying gives

 dq(α,x)dx≈12αx3/2√1+x(αx+11+x−12), (37)

for large values of . From (37), we make the following observations. As , for and for .

Collating these properties, we obtain the following result:

###### Result 1.

The main characteristics of are given by

 q(α,x)⟹⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩increases near the origin,%forα≥1decreases near the origin,for α<1positive gradient as x→∞,for α≥1/2negative gradient as x→∞,for α<1/2≥1 as x→∞,for α≥π/4<1 as x→∞,for α<π/4.

### Iv-B Effect of κru on E{SNR}

Only the first and third terms in (1) are affected by . The first term contains so is increasing with from Result 1. In the third term, affects the mean SNR via the expression. Applying Hölders inequality [5, E.q. (1.1)] to in (8) gives . This upper bound is achieved when . Hence, we conclude that increasing improves the mean SNR.

### Iv-C Effect of κd on E{SNR}

Only the first term in (1) is affected by through the expression

From Result 1, this expression increases with when , decreases when and has a more complex behavior in-between.

This pattern can be explained by the fact that measures the alignment of the RIS-BS channel, , with the LOS component of the direct path, , while measures the alignment of with the scattered component of the direct path. Hence, when the RIS-BS aligns strongly with the direct LOS path then improves the SNR. Conversely, when the RIS-BS aligns strongly with the direct scattered channel then increasing reduces the SNR.

The dominant efect here is the case, where decreases the mean SNR. This is because it is unlikely for two independent steering vectors to strongly align, whereas can align with in several ways as the correlation matrix is usually full-rank (see [17]).

### Iv-D Effect of ρru on E{SNR}

The effect of correlation in the UE-RIS channel, , on the mean SNR is confined to the variable in (1). Identifying the behavior of with respect to is very difficult due to the oscillations in the series expansion given in (1) caused by the term.

However, when we have the correlated Rayleigh case and becomes in (31) which increases with correlation (see [17]). Hence, we conjecture the same broad trend here, so that the mean SNR usually benefits from increasing correlation in .

It is worth noting that the majority of the existing literature does not have the complication of the term in the bivarate Ricean PDF. This is due to the assumption that both of the Ricean variables have identical LOS components (see [27] for example). However, this simplifying assumption is not valid here and the more complex version in [9] is required.

### Iv-E Effect of ρd on E{SNR}

Only variable in the first term of (1) is affected by . It is shown in [17, Sec. IV.B] that as , we can usually expect for large . In contrast, when , we have . Hence, the mean SNR benefits from low correlation in the UE-BS link.

### Iv-F Favourable and Unfavourable Conditions

From Sec. IV-B - Sec. IV-E, low correlation and low K-factor in the UE-BS link along with a high K-factor in the UE-RIS link tend to improve . Hence, we define the favourable channel scenario as an iid Rayleigh channel between UE and BS and pure LOS between UE and RIS. Conversely, the unfavourable channel scenario comprises an iid Rayleigh UE-RIS channel and a LOS UE-BS channel.

Using the analysis in Sec. IV, the mean SNR in the favourable channel scenario is given by,

 E{SNRfav} =(βdM+N√Mπ√βdβbrβru+βbrβruMN2)¯τ, (38)

while the mean SNR in the unfavourable channel scenario is,

using the result for for uncorrelated Rayleigh fading (see [17, Eq. (14)]).

Next we consider the relative difference between these two scenarios for large RIS sizes. Defining the gain

 Gainfav−unfav=E{SNRfav}−E{SNRunfav}E{SNRunfav}, (40)

it is straightforward to show that

 limN→∞Gainfav−unfav=4−ππ≈27.32%.

Hence, for large RIS, the asymptotic relative gain between the channel scenarios is approximately 27.32%.

In order to find the maximum gain, we rewrite (40) as

 Gainfav−unfav≜N2D1+ND2N2D3+ND4+D5. (41)

Differentiating (41) with respect to is elementary and allows the maximum to be obtained. Details are omitted but results are shown in Sec. V.

## V Results

We present numerical results to verify the analysis in Sec. IV. Firstly, note that we do not consider cell-wide averaging as the focus is on the SNR distribution over the fast fading. Furthermore, the relationship between the SNR and the path gains, and , is straightforward, as shown in (1). Hence, we present numerical results for fixed link gains. In particular, since the RIS-BS link is LOS, we assume where m. Next, for simplicity, . This was chosen to give the 95%-ile of the SNR distribution as 25 dB in the baseline case of moderate channel correlation and identical Ricean K-factors for the LOS and scattered paths (defined as , ), with BS antennas and RIS elements.

As stated in Sec. II-A, the steering vectors for are not restricted to any particular formation. However, for simulation purposes, we will use the VURA model as outlined in [10], but in the plane with equal spacing in both dimensions at both the RIS and BS. The and components of the steering vector at the BS are and which are given by

 [1,ej2πdbsin(θA)sin(