Analysis of Analog and Digital MRC in Massive MU-MIMO Systems over Correlated Channels

I. INTRODUCTION

MASSIVE multiple user multiple input multiple output (MU-MIMO) is a key technique for future broadband networks as it provides many benefits. Large-scale antenna systems can increase system capacity and improve energy efficiency on the order of 10 times or more simultaneously [1]. They not only reduce transmit power but also average out small-scale fading as random matrices start to look deterministic, which also offers benefits in robustness and reliability [1]. However, when the system is able to serve more users due to the number of antennas increasing at the base station (BS), the resulting interference among users can have a negative impact on the overall system performance. Zeroforcing (ZF) can suppress the inter-user interference at the cost of more computational complexity. For example, the matrix inverse required by ZF has a complexity order of [TeX:] $$O\left(K^{3}\right)$$ where K is the number of user data streams. Thus, taking communication overheads and complexity into consideration, simplified processing is required.

Among linear processing techniques, digital maximal ratio combining (MRC) and analog MRC¹, due to their simplicity and efficiency, are more practical in massive MU-MIMO systems. The two methods passively reduce interference by taking advantage of favourable propagation². An additional benefit of MRC is its suitability to distributed systems - the processing can be performed independently at each antenna cluster, without additional information exchange [2]. Analog MRC, by utilizing only one radio frequency chain, further reduces hardware cost and power consumption compared with digital MRC. Due to the presence of only a single RF-chain, the average spectral efficiency of massive MIMO systems with pure analog processing is less than the digital counterpart due to the inability to weight the amplitude of the incoming signal. However, despite this disadvantage, most of the current commercial 5G-NR street-macro and micro-cellular solutions within the FR2 bands are based on analog processing [3]. Hence, it is important to analyze analog processing. We note that most analytical work concerning analog MRC is from the perspective of modulation, outage probability and bit error probability [4]-[11].

¹ Analog MRC is also known as equal gain combining (EGC) in classical literature. We chose the term analog MRC in order to align the work with hybrid processing literature.

² 2Favourable propagation refers to the scenario where the number of BS antennas becomes large, causing the user channels to become orthogonal automatically. This enables simple, linear processing techniques, such as MRC, to maximize the system capacity.

Since there are more antennas which are closely-spaced in one physical location, massive MIMO suffers more from spatial correlation than conventional MIMO systems [12]. Although some research shows that system performance is critically dependent on the type of correlation model, they demonstrate this using simulations rather than mathematical analysis.

Although the asymptotic behaviour of large systems with correlation (equal for all users) has been well studied (see [2], [13], [14]), the effect on performance is not straightforward. Some studies demonstrate a negative impact [15], [16] and others suggest a positive impact [17], [18]. The above studies assume equal correlation among users. Recently, the impact of correlation on system performance has been shown to be highly dependent on the correlation model [19], [20]. Furthermore, results and measurements in [20] demonstrate that correlation variability among users enhances the performance of digital MRC. It is intuitively clear that equal correlation structures across users hinders performance as similar statistics implies more similar channels and increased interference. However, this understanding is relatively recent. Many papers still model a multi-user system with equal correlation matrices per user (see the example references [21]–[24]). It is also well known that the level of correlation can impact on system performance both positively and negatively, depending on the processing. However, the joint effect of correlation level and variability across users is little studied. Hence, we were motivated to further explore the impact of both correlation variability and the joint effects of correlation variability and level.

Hence, in this paper we provide new insights into the impact of correlation (including its heterogeneity) on the SINR and spectral efficiency (SE) behaviour of digital MRC and the first such results for analog MRC. Specifically,

· We derive closed form expressions for the expected signal and interference power, and use these to show that the signal-to-interference ratio (SIR) behaviour is highly dependent on the correlation heterogeneity across users.

· We derive a closed-form SIR expression based on the exponential correlation model. The 3D surface plots based on this expression give great insight into the circumstances where high correlation can improve system performance. We also show similar behaviour under the one-ring correlation model.

· We derive new asymptotic results for analog and digital MRC SINR in Rayleigh fading for the benchmark scenarios of uncorrelated and perfectly correlated channels, when both the number of users and antennas go to infinity. For the latter, the system performance suggests that digital and analog MRC would have the same asymptotic behaviour. For general correlated channels, the limits are shown to be critically dependent on the correlation heterogeneity, specifically, correlation is detrimental with equal correlation, but beneficial with variation across users. Depending on the assumptions, the resulting SINR can either vanish in the limit or converge to a constant. For uncorrelated channels, from the derived expression of SINR for i.i.d. Rayleigh fading, around 21.5% performance loss occurs with analog MRC compared with digital MRC.

II. SYSTEM MODEL

We consider an uplink massive MIMO system with N antennas serving K single antenna users. The [TeX:] $$N \times 1$$ channel vector for user i can be written as [TeX:] $$\mathbf{h}_{i}=\mathbf{R}_{i}^{1 / 2} \mathbf{u}_{i},$$ where [TeX:] $$\mathbf{u}_{i} \sim \mathcal{C} \mathcal{N}(\mathbf{0}, \mathbf{I}), \mathbf{R}_{i}=\beta_{i} \Sigma_{i}, \Sigma_{\mathbf{i}}$$ is the [TeX:] $$N \times N$$ spatial correlation matrix and [TeX:] $$\beta_{i}$$ is the large-scale link gain. As we are analysing convergence issues under three scenarios, we are not modeling the link gain based on classic path-loss and lognormal shadowing because the substantial variation caused by

shadowing will make it inconvenient to see the limiting effects. We adopt a similar approach to that in [14] to counter this problem. We have two models for [TeX:] $$\beta_{i},$$ equal and unequal power for each user. We use the equal link gain case as a reference and we assume [TeX:] $$\beta=1$$ for all users. For the unequal case, we set [TeX:] $$\beta_{1}=1$$ for the desired user and for the interfering users, we select [TeX:] $$\beta_{l}$$ from the exponential decay function [TeX:] $$A e^{-\lambda x}$$ such that the average interference power is equal to [TeX:] $$\beta_{1}.$$ As there is no zero link gain in practice, we guarantee a minimum link gain by cutting off the least 10% values of [TeX:] $$A e^{-\lambda x},$$ which means [TeX:] $$\beta_{l}$$ has a range of [TeX:] $$[(1 / 10) A, A].$$ Hence, [TeX:] $$\beta_{l}$$ is given by [TeX:] $$\beta_{l}=A e^{-(l-2) \lambda}.$$ This model gives K - 1 values of [TeX:] $$\beta_{l}$$ decaying exponentially over [TeX:] $$[(1 / 10) A, A].$$ The channel matrix is [TeX:] $$\mathbf{H}=\left[\mathbf{h}_{1} \mathbf{h}_{2} \cdots \mathbf{h}_{K}\right].$$ We assume perfect channel knowledge at the BS and equal transmit power, [TeX:] $$P_{\mathrm{t}},$$ for each user. Thus, the received signal at the BS can be expressed as

where [TeX:] $$\mathbf{n} \sim \mathcal{C N}\left(\mathbf{0}, \sigma_{\mathrm{n}}^{2} \mathbf{I}\right)$$ is white Gaussian noise, s is the data symbol vector from the K users and [TeX:] $$\mathbb{E}\left[\mathbf{s s}^{H}\right]=\mathbf{I}.$$ Without loss of generality, [TeX:] $$\sigma_{\mathrm{n}}^{2}$$ is assumed to be 1. The signal after combining at the BS to detect the ith user is given by

where [TeX:] $$\mathbf{g}_{i}=\mathbf{h}_{i}$$ for digital MRC and [TeX:] $$\mathbf{g}_{i}=\hat{\mathbf{h}}_{i}$$ for analog MRC, where [TeX:] $$\hat{\mathbf{h}}_{i}=\exp \left(j \angle \mathbf{h}_{i}\right) \text { and } \angle \mathbf{h}_{i}$$ indicates the vector of angles of [TeX:] $$\mathbf{h}_{i}.$$ The corresponding SINR is given by [25]

To enable the analysis, we apply the following commonly used approximation: if [TeX:] $$X=\sum X_{i} \text { and } Y= \sum Y_{i}$$ are both sums of non-negative random variables, then [TeX:] $$\mathbb{E}\left[\log _{2}(1+X / Y)\right] \approx \log _{2}(1+\mathbb{E}[X] / E[Y])$$ [26]. Independence between X and Y is not required and the result becomes more accurate when the number of the summation terms in X and Y is large [26]. This behaviour is due to the law of large numbers [26] where the numerator and denominator approach their mean values and their variances become small. On the uplink, using (3), this approximation gives the per-user spectral efficiency of

The expression in (4) allows for the analysis of achievable rates for linear processing schemes, such as matched filter (MF) ZF, minimum mean squared error (MMSE) and to gain new insights into their performance, in particular the effects of correlation.

The expectation of the signal and interference terms in (4) for digital MRC are, respectively, given by [27]

The relationship between correlation and performance is seen in (5) and (6) via the terms [TeX:] $$\operatorname{tr}\left(\Sigma_{i}^{2}\right) \text { and } \operatorname{tr}\left(\Sigma_{i} \Sigma_{l}\right).$$ To explore this relationship, consider the exponential correlation model in [28] where [TeX:] $$\left(\Sigma_{i}\right)_{r s}=\rho^{|s-r|} \exp \left(j(s-r) \phi_{i}\right),$$ so that users have the same amplitude correlation parameter, [TeX:] $$\rho,$$ but user specific phases, [TeX:] $$\phi_{i} \sim U[0,2 \pi] .$$ Using this model, straightforward algebra allows [TeX:] $$\operatorname{tr}\left(\Sigma_{i} \Sigma_{l}\right)$$ to be written as (7), where [TeX:] $$\Delta=\phi_{i}-\phi_{j}.$$ Obviously, [TeX:] $$\operatorname{tr}\left(\Sigma_{i} \Sigma_{l}\right)$$ is a function of [TeX:] $$\rho$$ and [TeX:] $$\Delta .$$ Less obviously, the relationship between interference and [TeX:] $$\rho$$ is different for different values of [TeX:] $$\Delta.$$ To see this, consider the two special cases of (7):

where (9) is for the case of even N. When the phase parameters are aligned [TeX:] $$(\Delta=0),$$ the correlation matrices are identical, the first numerator term of (8) dominates and the interference increases with [TeX:] $$\rho.$$ When the two phases are orthogonal [TeX:] $$(\Delta=\pi / 2),$$ the first numerator term in (9) dominates and the interference decreases with [TeX:] $$\rho.$$ Hence, increasing the size of the correlation can have opposite effects when users experience the same correlation matrix [TeX:] $$\left(\phi_{i}=\phi_{j}\right)$$ and when the correlation matrices differ [TeX:] $$\left(\phi_{i}-\phi_{j}=\pi / 2\right).$$ Equivalent results to (5) and (6) for analog MRC are available in [25] but are given in terms of Gaussian hypergeometric functions which make interpretation difficult. Hence, we look at the extreme cases of zero and perfect correlation in Section III.

III. ASYMPTOTIC SINR ANALYSIS

We derive the asymptotic SINR for analog and digital MRC under the two benchmark scenarios of i.i.d. and perfectly correlated Rayleigh fading. We assume N and K grow at the same rate, so that [TeX:] $$\alpha=N / K$$ is fixed and this asymptotic regime is described by [TeX:] $$\lim _{N \rightarrow \infty}.$$ The derivations require the strong law of large numbers (S.L.L.N), the version given in [29, Th. 5.4.3] (Result 1) being adequate for these proofs.

[TeX:] $$\text { 1: If } X_{1}, X_{2}, \cdots$$ re independent with finite means [TeX:] $$\mu_{1}, \mu_{2}, \cdots$$ and variances [TeX:] $$\sigma_{1}^{2}, \sigma_{2}^{2}, \cdots,$$ then [TeX:] $$(1 / L) \sum_{n=1}^{L} X_{n} \stackrel{\text { as. }}{\longrightarrow} \mu,$$ a.s. [TeX:] $$L \rightarrow \infty,$$ where [TeX:] $$\mu= \lim _{L \rightarrow \infty}\left((1 / L) \sum_{n=1}^{L} \mu_{n}\right)$$ if [TeX:] $$\sum_{n=1}^{L}\left(1 / L^{2}\right) \operatorname{Var}\left(X_{n}\right)<\infty$$ [29, Th. 5.4.3].

A. Asymptotic Analysis for i.i.d. Rayleigh Fading

First, for digital MRC, from (3), the interference power is,

In (10), [TeX:] $$u_{l 1}$$ is the first element of [TeX:] $$\mathbf{u}_{l}$$ and the second equality follows from the rank-1 eigen-decomposition of [TeX:] $$\mathbf{h}_{i} \mathbf{h}_{i}^{\mathrm{H}}.$$ The desired signal power is [TeX:] $$P_{\mathrm{t}}\left(\mathbf{h}_{i}^{\mathrm{H}} \mathbf{h}_{i}\right)^{2}=P_{\mathrm{t}} \beta_{i}^{2}\left(\mathbf{u}_{i}^{\mathrm{H}} \mathbf{u}_{i}\right)^{2}$$ and the noise power is [TeX:] $$\mathbf{h}_{i}^{\mathrm{H}} \mathbf{h}_{i}=\beta_{i} \mathbf{u}_{i}^{\mathrm{H}} \mathbf{u}_{i}.$$ Substituting into (3) and simplifying gives

where D denotes digital MRC. Using the S.L.L.N (Result 1), the numerator of (11) converges almost surely (a.s.), giving

Similarly, the denominator of (11) converges almost surely,

Thus, finally we get

where [TeX:] $$\bar{\beta}=\lim _{N \rightarrow \infty} \sum_{i=1}^{K} \beta_{i} / K$$ is the asymptotic mean of the link gains. For the equal [TeX:] $$\beta$$ case, (13) can be written as

For analog MRC, similar steps lead to the result

using [TeX:] $$\hat{\mathbf{h}}_{i}=\sqrt{\beta_{i}} \hat{\mathbf{u}}_{i}.$$ For the numerator, the S.L.L.N gives

For the interference term, we note that

which follows from the rank-1 eigen-decomposition of [TeX:] $$\hat{\mathbf{u}}_{i} \hat{\mathbf{u}}_{i}^{\mathrm{H}}.$$ Thus, from the S.L.L.N, the interference term in (15) is

Substituting (16) and (19) into (15), we obtain

Compared with digital MRC, analog MRC suffers a [TeX:] $$(1- \pi / 4) \times 100 \approx 21.5 \%$$ performance loss in the asymptotic SINR.

B. Asymptotic Analysis, Perfect Correlation, Equal Matrices

Consider the perfectly correlated channel where [TeX:] $$\mathbf{R}_{i}=\beta_{i} \Sigma_{i}$$ and all the elements of [TeX:] $$\Sigma_{i}$$ equal one. Here, [TeX:] $$\mathbf{h}_{i}=\mathbf{h}_{i 1}[11 \cdots 1]^{T}$$ and [TeX:] $$\hat{\mathbf{h}}_{i}=\exp \left(j \angle \mathbf{h}_{i 1}\right)[11 \cdots 1]^{T}.$$ Substituting the perfectly correlated channels and associated combiners into (3) gives:

and

Thus, [TeX:] $$\operatorname{SINR}_{i}^{\mathrm{A}}=\operatorname{SINR}_{i}^{\mathrm{D}}$$ and dividing numerator and denominator by K, the S.L.L.N gives

The SINRs vanish here due to the interference growth which occurs when all the user channels are aligned.

C. Asymptotic Analysis, Perfect Correlation, Unequal Matrices

Here we investigate two types of correlation structures for uniform linear arrays. The exponential correlation model in [28] has a user specific phase, [TeX:] $$\phi_{i} \sim U[0,2 \pi],$$ for the correlation parameter and for perfect correlation (amplitude 1). The correlation matrix is defined by [TeX:] $$\left(\Sigma_{i}\right)_{r s}=\exp \left(j(s-r) \phi_{i}\right).$$ The second model is the classic one-ring model [30], defined by,

where AS is the angle spread, [TeX:] $$\theta_{i}$$ is the central angle for user i and d is the antenna spacing. For this model, taking the limit as the angle spread vanishes gives a perfectly correlated correlation matrix also defined by [TeX:] $$\left(\Sigma_{i}\right)_{r s}=\exp \left(j(s-r) \phi_{i}\right)$$ but here, [TeX:] $$\phi_{i}=2 \pi d \sin \left(\theta_{i}\right).$$ For both models, defining [TeX:] $$\mathbf{a}_{i}= \left[\begin{array}{llll} 1 & \exp \left(-j \phi_{i}\right) & \exp \left(-j 2 \phi_{i}\right) & \cdots & \left.\exp \left(-j(N-1) \phi_{i}\right)\right]^{T} \end{array}\right.$$ allows the channel vectors to be written as [TeX:] $$\mathbf{h}_{i}=\mathbf{h}_{i 1} \mathbf{a}_{i}.$$ Following the steps in Section III-B the corresponding SINRs are

From (25), we see that the SINR depends on the limiting behaviour of the interference component denoted by I. We explore this limiting behaviour by deriving [TeX:] $$\mathbb{E}(I)$$:

which can be written as

Next, we derive [TeX:] $$\left|\mathbb{E}\left[e^{j r \phi_{i}}\right]\right|$$ for the two correlation models.

1) Exponential Correlation: The model in [28] has [TeX:] $$\phi_{i} \sim U[0,2 \pi]$$ for which [TeX:] $$\mathbb{E}\left[e^{j r \phi_{i}}\right]=0 \text { and } \mathbb{E}(I) \rightarrow 1 / \alpha.$$ For the more general case where [TeX:] $$\phi_{i} \sim U[a, b],$$

Substituting (28) into (27) we have,

where [TeX:] $$\sum_{r=1}^{N-1} 1 / r^{2}=\pi^{2} / 6,$$ and

where [TeX:] $$\gamma$$ is Euler’s constant and [TeX:] $$\epsilon_{N-1} \sim 1 / 2 N.$$ Thus, we see that the mean interference is finite for all uniform distributions. For the particular case when [TeX:] $$[a, b]=[0,2 \pi], \mathbb{E}(I) \rightarrow 1 / \alpha.$$

2) One-ring Correlation: Here, we have [TeX:] $$\phi_{i}=2 \pi d \sin \theta_{i},$$ where [TeX:] $$\theta_{i} \sim U[0,2 \pi].$$ Hence,

Substituting (31) into (27) we have [TeX:] $$\mu_{I}^{\infty}=(1 / \alpha)(1+2 S),$$ where

pFrom [31, eq. 10.17.3], for large argument [TeX:] $$J_{0}(z)= \sqrt{2 / \pi z} \cos (z-\pi / 4)+O\left(z^{-3 / 2}\right).$$ Hence, S exists (is finite) if and only if [TeX:] $$S_{1}$$ exists, where

Using the double angle formula, the first summation in (33) can be written as

From [32, p. 43], the second sum in (34) is finite,

and it is well-known that [TeX:] $$(1 / 2) \sum_{r=1}^{N-1} 1 / r$$ diverges logarithmically. Thus, [TeX:] $$S_{1}$$ diverges, causing both S and [TeX:] $$\mu_{I}^{\infty}$$ to diverge. The simulation results in Section IV also support this claim.

D. Exponential Correlation Model Analysis in General

In Section III-B and Section III-C, we have analysed the asymptotic behaviour under extreme high correlation scenarios. We would also like to study the general behaviour of [TeX:] $$T_{i i} / T_{i l},$$ which is the ratio of signal power [TeX:] $$T_{i i}=\operatorname{tr}\left(\Sigma_{i}^{2}\right)$$ and interference power [TeX:] $$T_{i l}=\operatorname{tr}\left(\Sigma_{i} \Sigma_{l}\right).$$ For simplicity, we only consider two users,

For the exponential correlation model, we have

and

where

Substituting (37) and (38) into (36), we obtain the general expression of [TeX:] $$T_{i i} / T_{i l},$$ which allows us to examine the correlation effect with various correlation coefficients.

E. Analog MRC Correlation Analysis

Next, we consider analog MRC and give new versions of the results in [25] which allow an interpretation of correlation effects. From [33, eq.15.3.3, p. 559], we have

Using this transformation formula allows the signal and interference power terms in (3) and [25] to be written as

Using known results on hypergeometric functions, [33, eq.15.2.1, eq.15.1.1, and eq.15.1.20] we see that [TeX:] $${ }_{2} F_{1}\left(-1 / 2,-1 / 2,1 ;\left|\rho_{i j k}\right|^{2}\right) \text { and }{ }_{2} F_{1}\left(1 / 2,1 / 2,2 ;\left|\rho_{i j k}\right|^{2}\right)$$ both increase monotonically from 1 to [TeX:] $$4 / \pi \text { as }\left|\rho_{i j k}\right|$$ increases from 0 to 1, making (43)–(44) easier to interpret. We see that the mean signal power in (43) is an increasing function of [TeX:] $$\left|\rho_{i j k}\right|.$$ The interference behaviour depends on the similarity of the correlation matrices. If [TeX:] $$\Sigma_{i}=\Sigma_{l}, \text { then } \rho_{l k j} \rho_{i j k}=\left|\rho_{i j k}\right|^{2}>0$$ and interference grows with [TeX:] $$\left|\rho_{i j k}\right|.$$ However, if [TeX:] $$\Sigma_{i} \neq \Sigma_{l},$$ then [TeX:] $$\rho_{l k j} \rho_{i j k}$$ is a complex constant and the sum in (44) will not necessarily grow with [TeX:] $$\left|\rho_{i j k}\right|$$ as terms may cancel. Again, we

see the important property that correlation is detrimental with equal correlation but can be beneficial with sufficient variation across users.

IV. NUMERICAL RESULTS

The limiting results in Section III depend on the link gains, [TeX:] $$\beta_{i} .$$ Thus, to examine convergence, we adopt a link gain model similar to that in [14], [34] where we consider scenarios of equal and unequal [TeX:] $$\beta_{i}$$ values for each user. The equal link gain case serves as a reference with[TeX:] $$\beta_{i}=1$$ for all users. The unequal case is modeled via a simple exponential decay for the [TeX:] $$\beta_{i} \text { 's }$$ as described in Section II. The decay is chosen so that the average link gain is [TeX:] $$\bar{\beta}=1$$ and the desired user also has unit link gain. Fig. 1 shows the mean SINR vs N for analog and digital MRC with [TeX:] $$P_{\mathrm{t}}=1 \text { and } \alpha=2.$$ We plot the mean rather than the instantaneous SINR to reduce variability and clearly identify the limits. Both the equal and unequal link gain scenarios are shown. The top four curves represent i.i.d. Rayleigh fading for which (14) and (20) give the limits 2 and [TeX:] $$\pi / 2$$ which are verified in the figure. The unequal power case converges slightly more slowly as the power variation gives less averaging and stability compared to the equal power case. The bottom four curves correspond to perfect correlation with equal correlation matrices. As predicted by the analysis in (23), the mean SINR decays to zero.

Fig. 2 shows the mean interference of the exponential (top) and one-ring (bottom) models with perfect correlation and unequal correlation matrices. Equal link gains, [TeX:] $$\alpha=2,,$$ and user specific angles, [TeX:] $$\phi_{i} \sim U[0,2 \pi],$$ are assumed. In the upper figure we see [TeX:] $$\mathbb{E}[I]$$ converging to the derived limit. In the lower figure, we see the interference (obtained by substituting (31) into (26)) agrees well with the simulated interference and grows logarithmically with N as predicted. Hence, the limiting behaviour is entirely different for the two models.

Next, we consider general levels of correlation between the benchmark results of i.i.d. and perfectly correlated channels. From (5) and (6), we see that [TeX:] $$T_{i i}=\operatorname{tr}\left(\Sigma_{i}^{2}\right) \text { and } T_{i l}=\operatorname{tr}\left(\Sigma_{i} \Sigma_{l}\right)$$ control the effect of correlation on digital MRC. For the exponential correlation model, [TeX:] $$T_{i i} \text { and } T_{i l}$$ are given by (7) which is a function of [TeX:] $$\rho, N$$ and [TeX:] $$\Delta=\phi_{i}-\phi_{l}.$$ In Figs. 3

and 4 we plot [TeX:] $$T_{i l} \text { and } T_{i i} / T_{i l}$$ against [TeX:] $$\rho \text { and } \Delta \text { for } N=32 \text {. }$$ The first measures interference while the second gives the size of the signal term relative to interference, a type of SIR. The interaction between the amplitude of the correlation parameter [TeX:] $$(\rho)$$ and the difference in the phases of the correlation [TeX:] $$(\Delta)$$ is clear. Large amplitude correlation helps the signal relative to interference and reduces interference, thus enhancing SINR, unless the phases are very similar [TeX:] $$(\Delta \approx 0 \text { or } \Delta \approx 2 \pi)$$ when the SINR is adversely affected by large interference. Hence, as long as there is phase parameter diversity, increasing correlation is beneficial to performance. In Fig. 5 and Fig. 6 we plot the equivalent results for the one-ring model. We use (24) to compute [TeX:] $$\Sigma_{i}$$ using N = 32 and note that correlation is controlled by the AS parameter where AS= 0 corresponds to perfect correlation and correlation drops as AS increases. User diversity is controlled by [TeX:] $$\Delta_{\theta}=\phi_{i}-\phi_{l},$$ the difference between the central angles of the two users. Figs. 5 and 6 are symmetric about [TeX:] $$t \Delta_{\theta}=\pi$$ as the correlation matrices are the same for [TeX:] $$\Delta_{\theta}=\Delta_{1} \text { and } \Delta_{\theta}=\pi+\Delta_{1}.$$ Otherwise, the trends are identical to the exponential model where reducing angle spread (corresponding to increased correlation) is beneficial as long as there is diversity in the central angles of the users.

In Fig. 7 we show the simulated cumulative distribution function (CDF) of the mean SE for analog MRC assuming the exponential correlation structure. The randomness is due to the variation of drops including path loss and lognormal shadowing effects. The link gains are given by [TeX:] $$\beta_{i}=A \zeta_{i}\left(d_{0} / d_{i}\right)^{\gamma},$$ where [TeX:] $$d_{i}$$ is the distance to the BS and [TeX:] $$\zeta_{i}$$ is lognormal shadowing. There are four users, each with a single antenna, uniformly located in a cell with the radius of 100 meters. The unit-less constant A = 30 dB, the reference distance [TeX:] $$d_{0}=1$$ meter, N = 32, the pathloss exponent [TeX:] $$\gamma=3.5$$ and the standard deviation of shadowing is 6 dB. The transmit power [TeX:] $$P_{\mathrm{t}}$$ is chosen to guarantee that 95% of the time the SNR exceeds 0 dB. As predicted by the analysis, for a fixed correlation parameter (equal correlation matrices for the users) increasing the correlation decreases SE, whereas for differing correlation parameters (unequal correlation matrices for the

users), correlation improves the SE. The difference in SE can be very large so we next investigate how much of this variation is encountered with more realistic ray-based channels³ based on clusters of scatterers. We adopt the measured angular parameters in [35]. The number of clusters C = 3, and the number of subpaths per cluster L = 16. Each subpath angle of arrival is modeled by a central cluster angle with a Gaussian distribution (zero mean and a standard deviation [TeX:] $$\left.\left(\sigma_{c}\right) \text { of } 14.4^{\circ}\right)$$ plus a subray offset angle which is Laplacian with a standard deviation [TeX:] $$\left(\sigma_{l}\right) \text { of } 6.24^{\circ}.$$ Following [34], the cluster powers decay exponentially from cluster 1 to cluster C such that [TeX:] $$C_{c}=(1 / 10) C_{1}$$ and equal power among subrays is assumed. We also adopt another two sets of parameters, [TeX:] $$C=2, L=20, \sigma_{c}=2^{\circ}, \sigma_{l}=1^{\circ}$$ (narrow spread) and [TeX:] $$C=2, L=20, \sigma_{c}=30^{\circ}, \sigma_{l}=10^{\circ}$$ (wide spread), to compare with the parameters in [35]. From the figure, we can see that wider angular spreads increase system performance, which agrees with the results based on Rayleigh channels with unequal correlations. The SE based on measured parameters is similar to the Rayleigh results with less extreme values of [TeX:] $$\rho .$$ Little change is observed as the angular spread is increased from [TeX:] $$\sigma_{c}=14.4^{\circ}, \sigma_{l}=6.24^{\circ} \text { to } \sigma_{c}=30^{\circ}, \sigma_{l}=10^{\circ}$$ but considerable losses arise for the very narrow angle spread case [TeX:] $$\left(\sigma_{c}=2^{\circ}, \sigma_{l}=1^{\circ}\right).$$ Hence, severe losses due to correlation are likely to be rare, but could exist, for example, in indoor non-line of sight environments where channels have limited numbers of clusters and narrow spread.

³ Ray-based channel models assume that the signal arriving at an antenna is composed of multiple narrow beams (rays), originating from scattering clusters, see, e.g. [35].

V. CONCLUSION

We have presented the first analysis of the effects of correlation on analog processing, compared analog to digital MRC and demonstrated that heterogeneous correlation effects extend to analog MRC. We have derived the expected signal and interference power, demonstrating that SIR decreases when the user correlation matrices are identical, but increases when they are different. We derived asymptotic SINR expressions for both analog and digital MRC for benchmark scenarios of

uncorrelated and fully correlated Rayleigh channels, demonstrating that the performance is critically dependent on the correlation scenario. We have shown that for uncorrelated fading the SINR converges to a constant. For fully correlated channels

SINR converges to zero for equal correlation matrices, whereas for unequal correlation matrices SINR converges to zero or a constant, depending on the correlation model.