In this paper, the considered system comprises a modified version of Wyner's linear cellular array [4, 12, 28], which has been used extensively as a tractable model for studying MJD scenarios [29]. In the modified model studied herein, MJD is possible for clusters of *M* adjacent BSs while the focus is on the uplink. Unlike [23, 24], IA is employed herein to mitigate intercluster interference between cluster-edge cells. Let us assume that *K* UTs are positioned between each pair of neighboring BSs with path loss coefficients 1 and α, respectively (Figure 1). All BSs and UTs are equipped with *n* = *K* + 1 antennas ^{b}[10]. to enable IA over the multiple spatial dimensions for the clustered UTs. In this setting, four scenarios of intercluster interference are considered, namely global MJD, IA, RDMA and CI. It should be noted that only cluster-edge UTs employ interference mitigation techniques, while UTs in the interior of the cluster use the optimal wideband transmission scheme with superposition coding as in [5]. Successive interference cancellation is employed in each cluster processor in order to recover the UT signals. Furthermore, each cluster processor has full CSI for all the wireless links in its coverage area. The following subsections explain the mode of operation for each approach and describe the analytical derivation of the per-cell sum-rate throughput.

### 3.1 Global multicell joint decoding

In global MJD, a central processor is able to jointly decode the signals received by neighboring clusters and, therefore, no intercluster interference takes place. In other words, the entire cellular system can be assumed to be comprised of a single extensive cluster. As it can be seen, this case serves as an upper bound to the IA case. The received *n* × 1 symbol vector y_{
i
}at any random BS can be expressed as follows:

{\mathbf{y}}_{i}\left(t\right)={\mathbf{G}}_{i,i}\left(t\right){\mathbf{x}}_{i}\left(t\right)+\alpha {\mathbf{G}}_{i,i+1}\left(t\right){\mathbf{x}}_{i+1}\left(t\right)+{\mathbf{z}}_{i}\left(t\right),

(1)

where the *n* × 1 vector **z** denotes AWGN with E\left[{\mathbf{z}}_{i}\right]=0 and E\left[{\mathbf{z}}_{i}{\mathbf{z}}_{i}^{H}\right]=\mathbf{I}. The *K* *n* × 1 vector x_{
i
}denotes the transmitted symbol vector of the *i* th UT group with E\left[{\mathbf{x}}_{i}{\mathbf{x}}_{i}^{H}\right]=\gamma \mathbf{I} where γ is the transmit Signal to Noise ratio per UT antenna. The *n* × *Kn* channel matrix {\mathbf{G}}_{i,i}~\mathcal{C}\mathcal{N}\left(0,{\mathbf{I}}_{n}\right) includes the flat fading coefficients of the *i* th UT group towards the *i* th BS modelled as independent identically distributed (i.i.d.) complex circularly symmetric (c.c.s.) random variables. Similarly, the term α**G**_{i, i+1}(*t*)**x**_{i+1}(*t*) represents the received signal at the *i* th BS originating from the UTs of the neighboring cell indexed *i + 1*. The scaling factor *α* < 1 models the amount of received intercell interference which depends on the path loss model and the density of the cellular system^{c}. Another intuitive description of the *α* factor is that it models the power imbalance between intra-cell and inter-cell signals.

Assuming a memoryless channel, the system channel model can be written in a vectorial form as follows:

\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{z},

(2)

where the aggregate channel matrix has dimensions *Mn* × (*M + 1*)*Kn* and can be modelled as:

\mathbf{H}=\mathbf{\Sigma}\odot \mathbf{G}

(3)

with \mathbf{\Sigma}=\stackrel{\u0303}{\mathbf{\Sigma}}\otimes {\mathbf{I}}_{n\times kn} being a block-Toeplitz matrix and \mathbf{G}~\mathcal{C}\mathcal{N}\left(0,{{\mathbf{I}}_{M}}_{n}\right). In addition, \stackrel{\u0303}{\mathbf{\Sigma}} is a *M* × *M +* 1 Toeplitz matrix structured as follows:

\stackrel{\u0303}{\mathbf{\Sigma}}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill \alpha \hfill & \hfill 0\hfill & \hfill \cdots \hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill \alpha \hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill \\ \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill \vdots \hfill \\ \hfill \alpha \hfill \end{array}\right]

(4)

Assuming no CSI at the UTs, the per-cell capacity is given by the MIMO multiple access (MAC) channel capacity:

\begin{array}{ll}\hfill {\mathsf{\text{C}}}_{\mathsf{\text{MJD}}}& =\frac{1}{M}E\left[\mathcal{I}\mathsf{\text{(x;y\u2014}}\mathbf{H}\mathsf{\text{)}}\right]\phantom{\rule{2em}{0ex}}\\ =\frac{1}{M}E\left[log\phantom{\rule{0.3em}{0ex}}det\left({\mathsf{\text{I}}}_{Mn}+\gamma \mathbf{H}{\mathbf{H}}^{H}\right)\right].\phantom{\rule{2em}{0ex}}\end{array}

(5)

**Theorem 3.1**. *In the global MJD case, the per-cell capacity for asymptotically large n converges almost surely (a.s.) to the Marcenko-Pastur (MP) law with appropriate scaling*[6, 10]:

\begin{array}{c}{\mathsf{\text{C}}}_{\mathsf{\text{MJD}}}\to a.s.Kn{\mathcal{V}}_{\mathsf{\text{MP}}}\left(\frac{M}{M+1}n\gamma \left(1+{\alpha}^{2}\right),K\frac{M+1}{M}\right),\\ where\phantom{\rule{1em}{0ex}}{\mathcal{V}}_{\mathsf{\text{MP}}}\left(\gamma ,\beta \right)=log\left(1+\gamma -\frac{1}{4}\varphi \left(\gamma ,\beta \right)\right)+\frac{1}{\beta}log\left(1+\gamma \beta -\frac{1}{4}\varphi \left(\gamma ,\beta \right)\right)-\frac{1}{4\beta \gamma}\varphi \left(\gamma ,\beta \right)\\ and\phantom{\rule{1em}{0ex}}\varphi \left(\gamma ,\beta \right)={\left(\sqrt{\gamma {\left(1+\sqrt{\beta}\right)}^{2}+1}-\sqrt{\gamma {\left(1-\sqrt{\beta}\right)}^{2}+1}\right)}^{2}.\end{array}

(6)

*Proof*. For the sake of completeness and to facilitate latter derivations, an outline of the proof in [6, 10] is provided here. The derivation of this expression is based on an asymptotic analysis in the number of antennas *n* → ∞:

\begin{array}{ll}\hfill \frac{1}{n}{C}_{\mathsf{\text{MJD}}}& =\underset{x\to \infty}{lim}\frac{1}{Mn}E\left[log\phantom{\rule{0.3em}{0ex}}det\left({\mathbf{I}}_{Mn}+\gamma \mathbf{H}{\mathbf{H}}^{H}\right)\right]\phantom{\rule{2em}{0ex}}\\ =\underset{x\to \infty}{lim}E\left[\frac{1}{Mn}\sum _{i=1}^{Mn}log\left(1+M\stackrel{\u0303}{\gamma}{\lambda}_{i}\left(\frac{1}{Mn}\mathbf{H}{\mathbf{H}}^{H}\right)\right)\right]\phantom{\rule{2em}{0ex}}\\ ={\int}_{0}^{\infty}log\left(1+M\stackrel{\u0303}{\gamma}x\right){f}_{\frac{1}{Mn}H{H}^{H}}^{\infty}\left(x\right)dx\phantom{\rule{2em}{0ex}}\\ =K{\int}_{0}^{\infty}log\left(1+M\stackrel{\u0303}{\gamma}x\right){f}_{\frac{1}{Mn}{H}^{H}H}^{\infty}\left(x\right)dx\phantom{\rule{2em}{0ex}}\\ =K{\mathcal{V}}_{\frac{1}{Mn}{\mathbf{H}}^{H}\mathbf{H}}\left(M\stackrel{\u0303}{\gamma}\right)\phantom{\rule{2em}{0ex}}\\ \to a.s.K{\mathcal{V}}_{\mathsf{\text{MP}}}\left(q\left(\mathbf{\Sigma}\right)M\stackrel{\u0303}{\gamma},K\right),\phantom{\rule{2em}{0ex}}\end{array}

(7)

where λ_{i} (**X**) and {f}_{\mathbf{X}}^{\infty} denote the eigenvalues and the asymptotic eigenvalue probability distribution function (a.e.p.d.f.) of matrix **X** respectively and {\mathcal{V}}_{\mathbf{X}}\left(x\right)=E\left[log\left(1+x\mathbf{X}\right)\right] denotes the Shannon transform of **X** with scalar parameter *x*. It should be noted that \stackrel{\u0303}{\gamma}=n\gamma denotes the total UT transmit power normalized by the receiver noise power^{d}. The last step of the derivation is based on unit rank matrices decomposition and analysis on the R-transform domain, as presented in [6, 10]. The scaling factor

q\left(\mathbf{\Sigma}\right)\triangleq \parallel \mathbf{\Sigma}{\parallel}^{2}\u2215\left(Mn\times \left(M+1\right)Kn\right)

(8)

is the Frobenius norm of the **Σ** matrix \u2225\mathbf{\Sigma}\u2225\triangleq \sqrt{tr\left\{{\mathbf{\Sigma}}^{H}\mathbf{\Sigma}\right\}} normalized by the matrix dimensions and

q\left(\mathbf{\Sigma}\right)\stackrel{\left(a\right)}{=}q\left(\stackrel{\u0303}{\mathbf{\Sigma}}\right)=\frac{1+{\alpha}^{2}}{M+1}

(9)

where step (*a*) follows from [10, Eq.(34)]. □

### 3.2 Interference alignment

In order to evaluate the effect of IA as an intercluster interference mitigation technique, a simple precoding scheme is assumed for the cluster-edge UT groups, inspired by [24]. Let us assume a *n* × 1 unit norm reference vector **v** with ||**v**||^{2} = *n* and

{\mathbf{y}}_{1}={\mathbf{G}}_{1,1}{\mathbf{x}}_{1}+\alpha {\mathbf{G}}_{1,2}{\mathbf{x}}_{2}+{\mathbf{z}}_{1}

(10)

{\mathbf{y}}_{M}={\mathbf{G}}_{M,M}{\mathbf{x}}_{M}+\alpha {\mathbf{G}}_{M,M+1}{\mathbf{x}}_{M+1}+{\mathbf{z}}_{M},

(11)

where **y**_{1} and **y**_{
M
}represent the received signal vectors at the first and last BS of the cluster, respectively. The first UT group has to align its input **x**_{1} towards the non-intended BS of the cluster on the left (see Figure A), while the *M* th BS has to filter our the aligned interference coming from the *M +* 1th UT group which belongs to the cluster on the right. These two strategies are described in detail in the following subsections:

#### 3.2.1 Aligned interference filtering

The objective is to suppress the term α**G**_{M, M+1}**x**_{M+1}which represents intercluster interference. It should be noted that UTs of the *M +* 1th cell are assumed to have perfect CSI about the channel coefficients **G**_{M, M+1}. Let us also assume that {\mathbf{x}}_{i}^{j} and {\mathbf{G}}_{\u0129,i}^{j} represent the transmitter vector and channel matrix of the *j* th UT in the *i* th group towards the \u0129\mathsf{\text{th}} BS. In this context, the following precoding scheme is employed to align interference:

{\mathbf{x}}_{M+1}^{j}={\left({\mathbf{G}}_{M,M+1}^{j}\right)}^{-1}{\mathbf{v}}_{j}{x}_{M+1}^{j},

(12)

where **v**_{
j
}= **v** *v*_{
j
}is a scaled version of **v** which satisfies the input power constraint \mathbb{E}\left[{\mathbf{x}}_{M+1}^{j}{\mathbf{x}}_{M+1}^{i}{}^{H}\right]=\gamma \mathbf{I}. This precoding results in unit multiplexing gain and is by no means the optimal IA scheme^{e}[22] provide conditions for classifying a scenario as proper or improper, a property which is shown to be connected to feasibility., but it serves as a tractable way of evaluating the IA performance [23, 24]. the feasibility of IA. Following this approach, the intercluster interference can be expressed as:

\alpha {\mathbf{G}}_{M,M+1}{\mathbf{x}}_{M+1}=\alpha \sum _{j=1}^{K}{\mathbf{G}}_{M,M+1}^{j}{\mathbf{x}}_{M+1}^{j}=\alpha \sum _{j=1}^{K}{\mathbf{G}}_{M,M+1}^{j}{\left({\mathbf{G}}_{M,M+1}^{j}\right)}^{-1}\mathbf{v}{v}_{j}{x}_{M+1}^{j}=\alpha \mathbf{v}\sum _{j=1}^{K}{v}_{j}{x}_{M+1}^{j}.

(13)

It can be easily seen that interference has been aligned across the reference vector and it can be removed using a *K*× *n* zero-forcing filter **Q** designed so that **Q** is a truncated unitary matrix [19] and **Qv = 0**. After filtering, the received signal at the *M* th BS can be expressed as:

{\stackrel{\u0303}{\mathbf{y}}}_{M}=\mathbf{Q}{\mathbf{G}}_{M,M}{\mathbf{x}}_{M}+{\stackrel{\u0303}{\mathbf{z}}}_{M},

(14)

Assuming that the system operates in high-SNR regime and is therefore interference limited, the effect of the AWGN noise colouring {\stackrel{\u0303}{\mathbf{z}}}_{M}=\mathbf{Q}{\mathbf{z}}_{M} can be ignored, namely E\left[{\stackrel{\u0303}{\mathbf{z}}}_{M}{\stackrel{\u0303}{\mathbf{z}}}_{M}^{H}\right]={\mathbf{I}}_{K}.

**Lemma 3.1.** *The Shannon transform of the covariance matrix* of **QG**_{
M,M
}*is equivalent to that of a K* × *K Gaussian matrix* **G**_{
K×K
}.

*Proof.* Using the property det(**I** + γ**AB**) = det(**I** + γ**BA**), it can be written that:

\begin{array}{ll}\hfill det\left({\mathbf{I}}_{K}+\gamma \mathbf{Q}{\mathbf{G}}_{M,M}{\left(\mathbf{Q}{\mathbf{G}}_{M,M}\right)}^{H}\right)& =det\left({\mathbf{I}}_{K}+\gamma \mathbf{Q}{\mathbf{G}}_{M,M}{\mathbf{G}}_{M,M}^{H}{\mathbf{Q}}^{H}\right)\phantom{\rule{2em}{0ex}}\\ =det\left({\mathbf{I}}_{n}+\gamma {\mathbf{G}}_{M,M}^{H}{\mathbf{Q}}^{H}\mathbf{Q}{\mathbf{G}}_{M,M}\right).\phantom{\rule{2em}{0ex}}\end{array}

(15)

The *K* × *n* truncated unitary matrix **Q** has *K* unit singular values and therefore the matrix product **Q**^{H}**Q** has *K* unit eigenvalues and a zero eigenvalue. Applying eigenvalue decomposition on **Q**^{H}**Q**, the left and right eigenvectors can be absorbed by the isotropic Gaussian matrices {\mathbf{G}}_{M,M}^{H} and **G**_{
M,M
}respectively, while the zero eigenvalue removes one of the *n* dimensions. Using the definition of Shannon transform [30], Eq. (15) yields

{{\mathcal{V}}_{\mathbf{Q}{\mathbf{G}}_{M,M}}}_{{\left(\mathbf{Q}{\mathbf{G}}_{M,M}\right)}^{H}}\left(\gamma \right)={\mathcal{V}}_{{\mathbf{G}}_{K\times K}}{\mathbf{G}}_{K\times K}^{H}\left(\gamma \right).

(16)

□

Based on this lemma and for the purposes of the analysis, **QG**_{
M,M
}is replaced by **G**_{K × K}in the equivalent channel matrix.

#### 3.2.2 Interference alignment

The Mth BS has filtered out incoming interference from the cluster on the right (Figure 1), but outgoing intercluster interference should be also aligned to complete the analysis. This affects the first UT group which should align its interference towards the Mth BS of the cluster on the left (Figure 1). Following the same precoding scheme and using Eq. (10)

{\mathbf{G}}_{1,1}{\mathbf{x}}_{1}=\sum _{j=1}^{K}{\mathbf{G}}_{1,1}^{j}{\mathbf{x}}_{1}^{j}=\sum _{j=1}^{K}{\mathbf{G}}_{1,1}^{j}{\left({\mathbf{G}}_{0,1}^{j}\right)}^{-1}\mathbf{v}{v}_{j}{x}_{1}^{j},

(17)

where {\mathbf{G}}_{0,1}^{j} represents the fading coefficients of the *j* th UT of the first group towards the *M* th BS of the neighboring cluster on the left. Since the exact eigenvalue distribution of the matrix product {\mathbf{G}}_{1,1}^{j}{\left({\mathbf{G}}_{0,1}^{j}\right)}^{-1}\mathbf{v}{v}_{j} is not straightforward to derive, for the purposes of rate analysis it is approximated by a Gaussian vector with unit variance. This approximation implies that IA precoding does not affect the statistics of the equivalent channel towards the intended BS.

#### 3.2.3 Equivalent channel matrix

To summarize, IA has the following effects on the channel matrix **H** used for the case of global MJD. The intercluster interference originating from the *M* + 1th UT group is filtered out and thus *Kn* vertical dimensions are lost. During this process, one horizontal dimension of the Mth BS is also filtered out, since it contains the aligned interference from the *M* + 1th UT group. Finally, the first UT group has to precode in order to align its interference towards the Mth BS of neighboring cluster and as a result only *K* out of *Kn* dimensions are preserved. The resulting channel matrix can be described as follows:

{\mathbf{H}}_{\mathsf{\text{IA}}}={\mathbf{\Sigma}}_{\mathsf{\text{IA}}}\odot {\mathbf{G}}_{\mathsf{\text{IA}}},

(18)

where {\mathbf{G}}_{\mathsf{\text{IA}}}~\mathcal{C}\mathcal{N}\left(0,{\mathbf{I}}_{Mn-1}\right)and

{\mathbf{\Sigma}}_{\mathsf{\text{IA}}}=\left[\begin{array}{c}\hfill {\mathbf{\Sigma}}_{\mathsf{\text{1}}}\hfill \\ \hfill {\mathbf{\Sigma}}_{\mathsf{\text{2}}}\hfill \\ \hfill {\mathbf{\Sigma}}_{\mathsf{\text{3}}}\hfill \end{array}\right]

(19)

with {\mathbf{\Sigma}}_{1}=\left[{\mathbf{I}}_{n\times K}\phantom{\rule{2.77695pt}{0ex}}\alpha {\mathbf{I}}_{n\times Kn}{0}_{n\times \left(M-2\right)Kn}\right] being a *n* × (*M* - 1)*Kn* + *K* matrix^{f}, {\mathbf{\Sigma}}_{2}=\left[{0}_{\left(M-2\right)n\times K}{\stackrel{\u0303}{\mathbf{\Sigma}}}_{M-2\times M-1}\otimes {\mathbf{I}}_{n\times Kn}\right] being a (*M* - 2)*n* × (*M* - 1)*Kn* + *K* matrix and {\mathbf{\Sigma}}_{3}=\left[{0}_{n-1\times \left(M-2\right)Kn+K}\phantom{\rule{2.77695pt}{0ex}}{\mathbf{I}}_{n-1\times K\phantom{\rule{0.3em}{0ex}}n}\right] being a *n* - 1 × (*M* - 1)*Kn* + *K* matrix^{g}.

Since all intercluster interference has been filtered out and the effect of filter **Q** has been already incorporated in the structure of **H**_{IA}, the per-cell throughput in the IA case is still given by the MIMO MAC expression:

\begin{array}{ll}\hfill {\mathsf{\text{C}}}_{\mathsf{\text{IA}}}& =\frac{1}{M}E\left[\mathcal{I}\left(\mathbf{x};\mathbf{y}|{\mathbf{H}}_{\mathsf{\text{IA}}}\right)\right]\phantom{\rule{2em}{0ex}}\\ =\frac{1}{M}E\left[log\phantom{\rule{0.5em}{0ex}}det\left({\mathbf{I}}_{Mn-1}+\gamma {\mathbf{H}}_{\mathsf{\text{IA}}}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}\right)\right].\phantom{\rule{2em}{0ex}}\end{array}

(20)

**Theorem 3.2**. *In the IA case, the per-cell throughput can be derived from the R-transform of the a.e.p.d.f. of matrix*\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}.

*Proof.* Following an asymptotic analysis where *n*→ ∞:

\begin{array}{ll}\hfill \frac{1}{n}{C}_{\mathsf{\text{IA}}}& =\underset{n\to \infty}{lim}\frac{1}{Mn}E\left[log\phantom{\rule{0.5em}{0ex}}det\left({\mathbf{I}}_{Mn-1}+\gamma {\mathbf{H}}_{\mathsf{\text{IA}}}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}\right)\right]\phantom{\rule{2em}{0ex}}\\ =\frac{Mn-1}{Mn}\underset{n\to \infty}{lim}E\left[\frac{1}{Mn-1}{\sum}_{i=1}^{Mn-1}log\left(1+\stackrel{\u0303}{\gamma}{\lambda}_{i}\left(\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}\right)\right)\right]\phantom{\rule{2em}{0ex}}\\ =\frac{Mn-1}{Mn}{\int}_{0}^{\infty}log\left(1+\stackrel{\u0303}{\gamma}x\right){f}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}}^{\infty}\left(x\right)dx\phantom{\rule{2em}{0ex}}\\ =\frac{\left(M-1\right)Kn+K}{Mn}{\int}_{0}^{\infty}log\left(1+\stackrel{\u0303}{\gamma}x\right){f}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}}^{\infty}\left(x\right)dx\phantom{\rule{2em}{0ex}}\\ \hfill \text{(5)}\end{array}

(21)

The a.e.p.d.f. of \frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}is obtained by determining the imaginary part of the Stieltjes transform \mathcal{S} for real arguments

{f}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}}^{\infty}\left(x\right)=\underset{x\to 0+}{lim}\frac{1}{\pi}\Im \left\{{\mathcal{S}}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}}\left(x+jy\right)\right\}

(22)

considering that the Stieltjes transform is derived from the R-transform [31] as follows

{\mathcal{S}}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}}^{-1}\left(z\right)={\mathcal{R}}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}}\left(-z\right)-\frac{1}{z}.

(23)

□

**Theorem 3.3.**
*The R-transform of the a.e.p.d.f. of matrix*
\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}
*is given by:*

{\mathcal{R}}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}}\left(z\right)=\sum _{i=1}^{3}{\mathcal{R}}_{\frac{1}{n}{\mathbf{H}}_{i}^{H}{\mathbf{H}}_{i}}\left(z,{k}_{i},{\beta}_{i},{q}_{i}\right)

(24)

with *k*, *β*, *q* *parameters given by:*

\begin{array}{c}{\mathbf{H}}_{1}:{k}_{1}=\frac{K+2}{MK+M-K},{\beta}_{1}=\frac{K}{K+1}+K,{q}_{1}=\frac{1+\left(K+1\right){\alpha}^{2}}{K+2}\\ {\mathbf{H}}_{2}:{k}_{2}=\frac{\left(M-1\right)\left(K+1\right)}{MK+M-K},{\beta}_{2}=\frac{M-1}{M-2}K,{q}_{2}=\frac{M-2}{M-1}\left(1+{\alpha}^{2}\right)\\ {\mathbf{H}}_{3}:{k}_{3}=\frac{K+1}{MK+M-K},{\beta}_{3}=K+1,{q}_{3}=1\end{array}

*and*
{\mathcal{R}}_{\frac{1}{n}{\mathbf{H}}_{i}^{H}{\mathbf{H}}_{i}}
*given by theorem A.1.*

*Proof.* Based on Eq.(19), the matrix {\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}can be decomposed as the following sum:

{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}={\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}+{\mathbf{H}}_{2}^{H}{\mathbf{H}}_{2}+{\mathbf{H}}_{3}^{H}{\mathbf{H}}_{3},

(25)

where **H**_{1} = **Σ**_{1} ⊙**G**_{n×(M-1)Kn+K}, **H**_{2} = **Σ**_{2} ⊙ **G**_{(M-2)n×(M-1)Kn+K}and **H**_{3} = **Σ**_{3} ⊙ **G**_{n-1×(M-1)Kn+K}. Using the property of free additive convolution [30] and Theorem A.1 in Appendix A, Eq. (24) holds in the R-transform domain. □

### 3.3 Resource division multiple access

RDMA entails that the available time or frequency resources are divided into two orthogonal parts assigned to cluster-edge cells in order to eliminate intercluster interference [[16], Efficient isolation scheme]. More specifically, for the first part cluster-edge UTs are inactive and the far-right cluster-edge _{BS} is active. For the second part, cluster-edge UTs are active and the far-right cluster-edge _{BS} is inactive. While the available channel resources are cut in half for cluster-edge UTs, double the power can be transmitted during the second part orthogonal part to ensure a fair comparison amongst various mitigation schemes. The channel modelling is similar to the one in global _{MJD} case (Eq. (1)), although in this case the throughput is analyzed separately for each orthogonal part and subsequently averaged. Assuming no _{CSI} at the UTs, the per-cell throughput in the RDMA case is given by:

\begin{array}{ll}\hfill {C}_{\mathsf{\text{RD}}}& =\frac{{C}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{\mathsf{\text{1}}}}+{C}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{\mathsf{\text{2}}}}}{2}\phantom{\rule{2em}{0ex}}\\ =\frac{1}{2M}\left(E\phantom{\rule{2.77695pt}{0ex}}\left[log\phantom{\rule{0.5em}{0ex}}det\phantom{\rule{2.77695pt}{0ex}}\left({\mathbf{I}}_{Mn}+\gamma {\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}^{H}\right)\right]\right)+E\left[log\phantom{\rule{0.5em}{0ex}}det\left({\mathbf{I}}_{\left(M-1\right)n}+\gamma {\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}^{H}\right)\right],\phantom{\rule{2em}{0ex}}\end{array}

(26)

where {\mathsf{\text{C}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}} and {\mathsf{\text{C}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}} denote the capacities for the first and second orthogonal part respectively. For the first part, the cluster processor receives signals from (*M* - 1)*K* UTs through all *M* BSs and the resulting *Mn* × *(M* - 1)*Kn* channel matrix is structured as follows:

\begin{array}{ll}\hfill {\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}& ={\mathbf{\Sigma}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}\odot {\mathbf{G}}_{Mn\times \left(M-1\right)Kn}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{with}}\phantom{\rule{2.77695pt}{0ex}}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{\mathsf{\text{1}}}}=\left[\begin{array}{c}\hfill {\stackrel{\u0303}{\mathbf{H}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{\mathsf{\text{1}}\alpha}}\hfill \\ \hfill \stackrel{\u0303}{\mathbf{H}}\hfill \\ \hfill {\stackrel{\u0303}{\mathbf{H}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{\mathsf{\text{1}}\beta}}\hfill \end{array}\right]\phantom{\rule{2em}{0ex}}\\ \hfill {\mathbf{\Sigma}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}& =\left[\begin{array}{c}\hfill {\stackrel{\u0303}{\mathbf{\Sigma}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{\mathsf{\text{1}}\alpha}}\hfill \\ \hfill \stackrel{\u0303}{\mathbf{\Sigma}}\hfill \\ \hfill {\stackrel{\u0303}{\mathbf{\Sigma}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{\mathsf{\text{1}}\beta}}\hfill \end{array}\right]\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}{\stackrel{\u0303}{\mathbf{\Sigma}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1\alpha}}=\left[\alpha {\mathbf{I}}_{n\times Kn}\phantom{\rule{2.77695pt}{0ex}}{0}_{n\times \left(M-2\right)Kn}\right],\phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0303}{\mathbf{\Sigma}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1\beta}}=\left[{0}_{n\times \left(M-2\right)Kn}{\mathbf{I}}_{n\times Kn}\right].\phantom{\rule{2em}{0ex}}\end{array}

(27)

**Theorem 3.4**. *For the first part of the RDMA case, the per-cell throughput* C_{RD1}*can be derived from the R-transform of the a.e.p.d.f of matrix*\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}^{H}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}, *where:*

{\mathcal{R}}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}^{H}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}}\left(z\right)={\mathcal{R}}_{\frac{1}{n}{B}^{H}B}\left(z,\frac{1}{M-1},K,{a}^{2}\right)+\frac{\left(M-2\right)\left(1+{\alpha}^{2}\right)}{M-1-\left(1+{\alpha}^{2}\right)K\left(M-1\right)z}+{\mathcal{R}}_{\frac{1}{n}{B}^{H}B}\left(z,\frac{1}{M-1},K,1\right).

(28)

*Proof*. Following an asymptotic analysis where *n*→ ∞:

\begin{array}{ll}\hfill \frac{1}{n}{C}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}& =\underset{n\to \infty}{lim}\frac{1}{Mn}E\left[log\phantom{\rule{0.5em}{0ex}}det\left({\mathbf{I}}_{Mn}+\gamma {\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}^{H}\right)\right]\phantom{\rule{2em}{0ex}}\\ =K{\int}_{0}^{\infty}log\left(1+\stackrel{\u0303}{\gamma}x\right){\int}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}^{H}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}}^{\infty}\left(x\right)dx.\phantom{\rule{2em}{0ex}}\end{array}

(29)

Using the matrix decomposition of Eq. (27) and free additive convolution [30]:

{\mathcal{R}}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}^{H}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1}}}\left(z\right)={\mathcal{R}}_{\frac{1}{n}{\stackrel{\u0303}{\mathbf{H}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{\mathsf{\text{1}}\alpha}}^{H}{\stackrel{\u0303}{\mathbf{H}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1\alpha}}}\left(z\right)+{\mathcal{R}}_{\frac{1}{n}{\stackrel{\u0303}{\mathbf{H}}}^{H}\stackrel{\u0303}{\mathbf{H}}}\left(z\right)+{\mathcal{R}}_{\frac{1}{n}\phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0303}{\mathbf{H}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{1\beta}}^{H}\phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0303}{\mathbf{H}}}_{R{D}_{1\beta}}}\left(z\right).

(30)

Eq. (28) follows from Eq. (42) with q=\left(M-2\right)q\left(\stackrel{\u0303}{\mathbf{\Sigma}}\right)=\left(M-2\right)\left(1+{\alpha}^{2}\right)\u2215\left(M-1\right),\beta =K\left(M-1\right)\u2215\left(M-2\right) and theorem A.1. □

For the second part, the cluster processor receives signals from *MK* UTs through *M* - 1 BSs and the resulting (*M* - 1)*n* ×*MKn* channel matrix is structured as follows:

\begin{array}{ll}\hfill {\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}& ={\mathbf{\Sigma}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}\odot {\mathbf{G}}_{\left(M-1\right)n\times MKn}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{with}}\phantom{\rule{2.77695pt}{0ex}}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}=\left[\begin{array}{c}\hfill {\stackrel{\u0303}{\mathbf{H}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}\hfill \\ \hfill \stackrel{\u0303}{\mathbf{H}}\hfill \end{array}\right]\phantom{\rule{2em}{0ex}}\\ \hfill {\mathbf{\Sigma}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}& =\left[\begin{array}{c}\hfill {\stackrel{\u0303}{\mathbf{H}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}\hfill \\ \hfill \stackrel{\u0303}{\mathbf{\Sigma}}\hfill \end{array}\right]\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0303}{\mathbf{\Sigma}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}=\left[2{\mathbf{I}}_{n\times Kn}\phantom{\rule{2.77695pt}{0ex}}\alpha {\mathbf{I}}_{n\times Kn}{0}_{n\times \left(M-2\right)Kn}\right],\phantom{\rule{2em}{0ex}}\end{array}

(31)

where the factor 2 is due to the doubling of the transmitted power.

**Theorem 3.5.** *For the second part of the RDMA case, the per-cell throughput*{\mathsf{\text{C}}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}*can be derived from the R-transform of the a.e.p.d.f. of matrix*\frac{1}{n}{\mathbf{H}}_{{}_{{\text{RD}}_{2}}}^{H}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}, *where:*

{\mathcal{R}}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}^{H}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}\left(z\right)=}\frac{\left(M-2\right)\left(1+{\alpha}^{2}\right)}{M-1-\left(1+{\alpha}^{2}\right)K\left(M-1\right)z}+{\mathcal{R}}_{\frac{1}{n}{B}^{H}B}\left(z,\frac{2}{M},2K,2+{\alpha}^{2}\right)

(32)

*Proof.* Following an asymptotic analysis where *n* → ∞:

\begin{array}{ll}\hfill \frac{1}{n}{C}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}& =\underset{n\to \infty}{lim}\frac{1}{Mn}E\left[log\phantom{\rule{0.5em}{0ex}}det\left({\mathbf{I}}_{\left(M-1\right)n}+\gamma {\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}^{H}\right)\right]\phantom{\rule{2em}{0ex}}\\ =K\frac{M-1}{M}{\int}_{0}^{\infty}log\left(1+\stackrel{\u0303}{\gamma}x\right){f}_{\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}^{H}{\mathbf{H}}_{\mathsf{\text{R}}{\mathsf{\text{D}}}_{2}}}^{\infty}\left(x\right)dx.\phantom{\rule{2em}{0ex}}\end{array}

(33)

The rest of this proof follows the steps of Theorem 3.4. □

### 3.4 Cochannel interference allowance

CI is considered as a worst case scenario where no signal processing is performed in order to mitigate intercluster interference and thus interference is treated as additional noise [15]. As it can be seen, this case serves as a lower bound to the IA case. The channel modelling is identical with the one in global MJD case (Eq. (1)), although in this case the cluster-edge UT group contribution α**G**_{M, M}_{+1}(*t*)**x**_{
M
}_{+1}(*t*) is considered as interference. As a result, the interference channel matrix can be expressed as:

{\mathbf{H}}_{\mathsf{\text{I}}}=\left[\begin{array}{c}\hfill {0}_{Mn\times Kn}\hfill \\ \hfill \alpha {\mathbf{G}}_{n\times Kn}\hfill \end{array}\right].

(34)

Assuming no CSI at the UTs, the per-cell throughput in the CI case is given by [15, 32–34]:

\begin{array}{ll}\hfill {\mathsf{\text{C}}}_{\mathsf{\text{CI}}}& =\frac{1}{M}E\left[log\phantom{\rule{0.5em}{0ex}}det\left({\mathbf{I}}_{Mn}+\gamma {\mathbf{H}}_{{\text{RD}}_{1}}{\mathbf{H}}_{{\text{RD}}_{1}}^{H}{\left({\mathbf{I}}_{Mn}+\gamma {\mathbf{H}}_{\mathsf{\text{I}}}{\mathbf{H}}_{\mathsf{\text{I}}}^{H}\right)}^{-1}\right)\right]\phantom{\rule{2em}{0ex}}\\ =\frac{1}{M}E\left[log\phantom{\rule{0.5em}{0ex}}det\left({\mathbf{I}}_{Mn}+\gamma \mathbf{H}{\mathbf{H}}^{H}\right)\right]-\frac{1}{M}E\left[log\phantom{\rule{0.5em}{0ex}}det\left({\mathbf{I}}_{Mn}+\gamma {\mathbf{H}}_{\mathsf{\text{I}}}{\mathbf{H}}_{\mathsf{\text{I}}}^{H}\right)\right]\phantom{\rule{2em}{0ex}}\\ ={\mathsf{\text{C}}}_{\mathsf{\text{MJD}}}-{\mathsf{\text{C}}}_{\mathsf{\text{I}}},\phantom{\rule{2em}{0ex}}\end{array}

(35)

where C_{I} denotes the throughput of the interfering UT group normalized by the cluster size:

\begin{array}{c}{\text{C}}_{\text{I}}=\frac{1}{M}\mathbb{E}\left[\mathcal{I}(\mathbf{x};\mathbf{y}|{\mathbf{H}}_{\text{I}})\right]\\ =\frac{1}{M}\mathbb{E}\left[\mathrm{log}\mathrm{det}\left({\mathbf{I}}_{Mn}+\gamma {\mathbf{H}}_{\text{I}}{\mathbf{H}}_{\text{I}}^{H}\right)\right].\end{array}

(36)

**Theorem 3.6.**
*In the CI case, the per-cell throughput converges almost surely (a.s.) to a difference of two scaled versions of the the MP law:*

{C}_{\mathsf{\text{CI}}}\to a.s.Kn{\mathcal{V}}_{\mathsf{\text{MP}}}\left(\frac{M}{M+1}n\gamma \left(1+{\alpha}^{2}\right),K\frac{M+1}{M}\right)-\frac{Kn}{M}{\mathcal{V}}_{\mathsf{\text{MP}}}\left({\alpha}^{2}n\gamma ,K\right).

(37)

*Proof.* Following an asymptotic analysis in the number of antennas *n* n →∞:

\begin{array}{ll}\hfill \frac{1}{n}{C}_{\mathsf{\text{I}}}& =\underset{x\to \infty}{lim}\frac{1}{Mn}E\left[log\phantom{\rule{0.5em}{0ex}}det\left({\mathbf{I}}_{Mn}+\gamma {\mathbf{H}}_{\mathsf{\text{I}}}{\mathbf{H}}_{\mathsf{\text{I}}}^{H}\right)\right]\phantom{\rule{2em}{0ex}}\\ =\underset{x\to \infty}{lim}\frac{1}{Mn}E\left[log\phantom{\rule{0.5em}{0ex}}det\left({\mathbf{I}}_{n}+\gamma {\alpha}^{2}{\mathbf{G}}_{n\times Kn}{\mathbf{G}}_{n\times Kn}^{H}\right)\right]\phantom{\rule{2em}{0ex}}\\ \to a.s.\frac{K}{M}{\mathcal{V}}_{\mathsf{\text{MP}}}\left({\alpha}^{2}\stackrel{\u0303}{\gamma},K\right).\phantom{\rule{2em}{0ex}}\end{array}

(38)

Eq. (37) follows from Eq. (35), (38) and Theorem 3.1. □

### 3.5 Degrees of freedom

This section focuses on comparing the degrees of freedom for each of the considered cases. The degrees of freedom determine the number of independent signal dimensions in the high SNR regime [35] and it is also known as prelog or multiplexing gain in the literature. It is a useful metric in cases where interference is the main impairment and AWGN can be considered unimportant.

**Theorem 3.7**. *The degrees of freedom per BS antenna for the global MJD, I A, RDM A and CI cases are given by:*

{d}_{\mathsf{\text{MJD}}}=1,{d}_{\mathsf{\text{IA}}}=1-\frac{1}{Mn},{d}_{\mathsf{\text{RN}}}=1-\frac{1}{2M},{d}_{\mathsf{\text{CI}}}=1-\frac{1}{M}.

(39)

*Proof.* Eq. (39) can be derived straightforwardly by counting the receive dimensions of the equivalent channel matrices (Eq. (3) for global MJD, Eq. (18) for IA, Eqs. (27) and (31) for RDMA, Eq. (34) for CI) and normalizing by the number of BS antennas. □

**Lemma 3.2**. *The following inequalities apply for the dofs of eq.* (39):

{d}_{\mathsf{\text{MJD}}}\ge {d}_{\mathsf{\text{IA}}}\ge {d}_{\mathsf{\text{RD}}}>{d}_{\mathsf{\text{CI}}}.

(40)

**Remark 3.1**. *It can be observed that d*_{IA}*= d*_{RD}*only for single UT per cell equipped with two antennas (K* = 1, *n* = 2*). For all other cases, d*_{IA} > *d*_{RD}. *Furthermore, it is worth noting that when the number of UTs K and antennas n grows to infinity*, lim_{
K,n
}→ ∞ *d*_{IA} = *d*_{MJD}*which entails a multiuser ga* in. *However, in practice the number of served UTs is limited by the number of antennas (n = K+* 1*) which can be supported at the BS- and more importantly at UT-side due to size limitations*.

### 3.6 Complexity considerations

This paragraph discusses the complexity of each scheme in terms of decoding processing and required CSI. In general, the complexity of MJD is exponential with the number of users [36] and full CSI is required at the central processor for all users which are to be decoded. This implies that global MJD is highly complex since all system users have to be processed at a single point. On the other hand, clustering approaches reduce the number of jointly-processed users and as a result complexity. Furthermore, CI is the least complex since no action is taken to mitigate intercluster interference. RDMA has an equivalent receiver complexity with CI, but in addition it requires coordination between adjacent clusters in terms of splitting the resources. For example, time division would require inter-cluster synchronization, while frequency division could be even static. Finally, IA is the most complex since CSI towards the non-intended BS is also needed at the transmitter in order to align the interference. Subsequently, additional processing is needed at the receiver side to filter out the aligned interference.