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 rank1 lineofsight (LOS) channel is formed between the BS and RIS. System scenarios with a LOS channel between the BSRIS and a singleuser 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 UEBS 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 interelement spacing may be envisaged. Several papers do consider spatial correlation in the smallscale 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 RISaided 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 UEBS 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 RISaided wireless system, in the presence of correlated Ricean UEBS, UERIS 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 UEBS, UERIS 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 rank1 LOS RISBS channel and correlated Ricean fading for the UEBS and UERIS channels. The contributions of this paper are as follows:

An exact closedform 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 UEBS and UERIS links are correlated Rayleigh to agree with those in
[17]. 
The analysis is leveraged to gain insight into the impact of the Kfactor and spatial correlation on the mean SNR. We show that increasing the Kfactor and correlation in the UEBS channel has negative effects, while increasing the Kfactor and correlation in the UERIS 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 chisquared 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 noninteger 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 rank1 LOS condition is achieved between the RIS and BS.
Iia Channel Model
Let , , be the UEBS, UERIS and RISBS channels, respectively. The diagonal matrix , where for , contains the reflection coefficients for each RIS element. The global UL channel is thus represented by
(1) 
where we consider the correlated Ricean channels:
with
where and are the link gains between UEBS and UERIS respectively, and are the correlation matrices for UEBS and UERIS links, respectively, and . Here, and are the Ricean Kfactors for the UEBS and UERIS 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 wellknown exponential decay model,
(2) 
where , . is the distance between the and antenna/element at the BS/RIS. and are the nearestneighbour 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 rank1 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.
IiB 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. IIIB] 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,
(4) 
where , giving the optimal UL SNR as
(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
Iiia 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
(6) 
with given by
(7) 
and , where , is the confluent hypergeometric function, , , , and is given by and for [9, Eq. (26b)].
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 UERIS channel is nonLOS (correlated Rayleigh) then the complexity in all the expressions reduces considerably.Theorem 2.
The SNR Variance is given by
(9) 
with
(10)  
(11)  
(12) 
(13)  
(14)  
(15) 
where is given in Theorem 1, is given by (1), , , is the Whittaker M function, and .
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.
Note that for high values of correlation and the Kfactor, (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.
IiiB Case 1: for
(17) 
where , , and .
IiiC Benchmark Case: when
For the benchmark case of maximum correlation, , both (1) and (IIIB) 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
(18) 
where and are defined in Sec. IIIB. Note that the double numeric integrations required in (IIIB) and (IIIC) are computationally convenient as: one integral is finite range and the integrand is smooth and rapidly decaying as .
IiiD 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
(19) 
with given by
(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
(21) 
with
(22)  
(23)  
(24) 
(25)  
(26)  
(27)  
(28)  
(29) 
Equations (IIID)(29) follow from Theorem 2 using the substitutions , , and the version of given in (20). A consequence of having uncorrelated UEBS and UERIS channels, is that the expectations and are known exactly and these are given by (28) and (29) respectively.
IiiE Special Case 2: Correlated Rayleigh Case
We obtain mean SNR and SNR variance expressions for correlated Rayleigh channels and by setting .
IiiE1 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
(30) 
with given by
(31) 
and , where is the Gaussian hypergeometric function and .
IiiE2 SNR Variance
In addition to the steps undertaken in IIIE1, 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
(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,
(33) 
with
where is given by (31).
Iv Performance Insights based on
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.
Iva Behaviour of
Using [1, Eq. (13.6.9)] the Laguerre function can be rewritten in terms of the confluent hypergeometric function, , as
(34) 
Using the series expansion for [1, Eq. (13.1.2)] and the Maclaurin series expansion of , we have the following expansion for ,
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
(35)  
(36) 
From (35), we see that as , for and for . Differentiating (35) and simplifying gives
(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
IvB Effect of on
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.
IvC Effect of on
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 inbetween.
This pattern can be explained by the fact that measures the alignment of the RISBS 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 RISBS aligns strongly with the direct LOS path then improves the SNR. Conversely, when the RISBS 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 fullrank (see [17]).
IvD Effect of on
The effect of correlation in the UERIS 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.
IvE Effect of on
IvF Favourable and Unfavourable Conditions
From Sec. IVB  Sec. IVE, low correlation and low Kfactor in the UEBS link along with a high Kfactor in the UERIS 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 UERIS channel and a LOS UEBS channel.
Using the analysis in Sec. IV, the mean SNR in the favourable channel scenario is given by,
(38) 
while the mean SNR in the unfavourable channel scenario is,
(39) 
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
(40) 
it is straightforward to show that
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
(41) 
V Results
We present numerical results to verify the analysis in Sec. IV. Firstly, note that we do not consider cellwide 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 RISBS 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 Kfactors for the LOS and scattered paths (defined as , ), with BS antennas and RIS elements.
As stated in Sec. IIA, 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
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