Performance Analysis of Irregular Repetition Slotted Aloha with Multi-Cell Interference

Massive machine-type communications (mMTC) require random access protocols that serve large numbers of users [1, 2]. One such protocol is irregular repetition slotted aloha (IRSA), a successive interference cancellation (SIC) aided protocol in which users transmit multiple packet replicas in different resource blocks (RBs) [3]. Channel estimation in IRSA is accomplished using training or pilot sequences transmitted by the users at the start of their packets. Assigning mutually orthogonal pilots to users avoids pilot contamination, but is prohibitive in mMTC, since the pilot overhead would be proportional to the total number of users [4]. Thus, pilot contamination (PC), which reduces the accuracy of channel estimation and makes the estimates correlated [5], is unavoidable in mMTC, and significantly degrades the throughput of IRSA. PC is caused by both within-cell and out-of-cell users, termed intra-cell PC and inter-cell PC, respectively. The goal of this paper is to analyze the performance of IRSA, accounting for both intra-cell PC and inter-cell PC.

Initial studies on IRSA with focused on MAC [3] and path loss channels [6]. IRSA has been analyzed in a single-cell (SC) setup, accounting for intra-cell PC, estimation errors, path loss, and MIMO fading [7, 8]. Multi-user interference from users within the same cell is termed intra-cell interference and from users across cells is termed inter-cell interference. In the SC setup, only intra-cell interference affects the decoding of users since users do not face inter-cell interference. In practice, multiple base stations (BSs) are deployed to cover a large region, and thus inter-cell interference is inevitable [9]. Furthermore, MC processing (e.g., MC MMSE combining of signals) schemes can achieve better performance compared to SC processing, since it accounts for inter-cell interference [10].

Our main contributions in this paper are as follows:

1. 1.

   We derive the channel estimates in MC IRSA accounting for path loss, MIMO fading, intra-cell PC, and inter-cell PC.

2. 2.

   We analyze the SINR achieved in MC IRSA, accounting for PC, channel estimation errors, intra-cell interference, and inter-cell interference.

3. 3.

   We provide insights into the effect of system parameters such as number of antennas, pilot length, and SNR on the throughput performance of MC IRSA.

To the best of our knowledge, no existing work has analyzed the effect of MC interference on IRSA. Through numerical simulations, we show that inter-cell PC and inter-cell interference result in up to 70% loss in throughput compared to the SC setup. This loss can be overcome by using about $4-5$x longer pilot sequences. Thus, it is vital to account for the effects of MC interference, in order to obtain realistic insights into the performance of IRSA.

Notation: The symbols $a$, $\mathbf{a}$, $\mathbf{A}$, $[\mathbf{A}]_{i,\mathrel{\mathop{\mathchar 58\relax}}}$, $[\mathbf{A}]_{\mathrel{\mathop{\mathchar 58\relax}},j}$, $\mathbf{0}_{N}$, $\mathbf{1}_{N},$ and $\mathbf{I}_{N}$ denote a scalar, a vector, a matrix, the $i$th row of $\mathbf{A}$, the $j$th column of $\mathbf{A}$, all-zero vector of length $N$, all ones vector of length $N$, and an identity matrix of size $N\times N$, respectively. $[\mathbf{a}]_{\mathcal{S}}$ and $[\mathbf{A}]_{\mathrel{\mathop{\mathchar 58\relax}},\mathcal{S}}$ denote the elements of $\mathbf{a}$ and the columns of $\mathbf{A}$ indexed by the set $\mathcal{S}$, respectively. $\text{diag}(\mathbf{a})$ is a diagonal matrix with diagonal entries given by $\mathbf{a}$. $[N]$ denotes the set $\{1,2,\ldots,N\}$. $|\cdot|$, $\|\cdot\|$, and $[\cdot]^{H}$ denote the magnitude (or cardinality of a set), $\ell_{2}$ norm, and Hermitian operators.

We consider an uplink MC system with $Q$ cells, where each cell has an $N$-antenna BS located at its center. We refer to the BS at the center of the $q$th cell as the $q$th BS. Every cell has $M$ single antenna users arbitrarily deployed within the cell who wish to communicate with their own BS. The time-frequency resource is divided into $T$ RBs. These $T$ RBs are common to all the cells, and thus, a total of $QM$ users contend over the $T$ RBs. Each user randomly accesses a subset of the available RBs according to the IRSA protocol, and transmit packet replicas in the chosen RBs. Each replica comprises of a header containing pilot symbols for channel estimation, and a payload containing data and error correction symbols.

The access of the RBs by the users can be represented by an access pattern matrix $\mathbf{G}=[\mathbf{G}_{1},\mathbf{G}_{2},\ldots,\mathbf{G}_{Q}]\in\{0,1\}^{T\times QM}$. Here $\mathbf{G}_{j}\in\{0,1\}^{T\times M}$ represents the access pattern matrix of the users in the $j$th cell, and $g_{tji}=[\mathbf{G}_{j}]_{ti}$ is the access coefficient such that $g_{tji}=1$ if the $i$th user in the $j$th cell transmits in the $t$th RB, and $g_{tji}=0$ otherwise. The $i$th user in the $j$th cell samples its repetition factor $d_{ji}$ from a preset probability distribution. It then chooses $d_{ji}$ RBs from the $T$ RBs uniformly at random for transmission. The access pattern matrix is known at the BS, which is made possible by using pseudo-random matrices generated from a seed that is available at the BS and the users [8]. This can be done in an offline fashion.

The received signal at any BS in the $t$th RB is a superposition of the packets transmitted by the users who choose to transmit in the $t$th RB, across all cells. In the pilot phase, the $i$th user in the $j$th cell transmits a pilot $\mathbf{p}_{ji}\in\mathbb{C}^{\tau}$ in all the RBs that it has chosen to transmit in, where $\tau$ denotes the length of the pilot sequence. The received pilot signal at the $q$th BS in the $t$th RB, denoted by $\mathbf{Y}_{tq}^{p}\in\mathbb{C}^{N\times\tau}$, is

where $\mathbf{N}_{tq}^{p}\in\mathbb{C}^{N\times\tau}$ is the additive complex white Gaussian noise at the $q$th BS with $[{\mathbf{N}_{tq}^{p}}]_{nr}\mathrel{\overset{\text{i.i.d.}}{\scalebox{1.5}[1.0]{$\sim$}}}\mathcal{CN}(0,N_{0})$ $\forall\ n\in[N],\ r\in[\tau]$ and $t\in[T]$, and $N_{0}$ is the noise variance. Here, $\mathbf{h}_{tji}^{q}\in\mathbb{C}^{N}$ is the uplink channel vector between the $i$th user in the $j$th cell and the $q$th BS on the $t$th RB. The fading is modeled as block-fading, quasi-static and Rayleigh distributed. The uplink channel is distributed as $\mathbf{h}_{tji}^{q}\mathrel{\overset{\text{i.i.d.}}{\scalebox{1.5}[1.0]{$\sim$}}}\mathcal{CN}(\mathbf{0}_{N},\beta_{ji}^{q}\sigma_{h}^{2}\mathbf{I}_{N}),\ \forall\ t\in[T],\ i\in[M]$ and $j\in[Q]$, where $\sigma_{h}^{2}$ is the fading variance, and $\beta_{ji}^{q}$ is the path loss coefficient between the $i$th user in the $j$th cell and the $q$th BS.

In the data phase, the received data signal at the $q$th BS in the $t$th RB is denoted by $\mathbf{y}_{tq}\in\mathbb{C}^{N}$ and is given by

where $x_{ji}$ is a data symbol with $\mathbb{E}[x_{ji}]=0$ and $\mathbb{E}[|x_{ji}|^{2}]=p_{ji}$, i.e., with transmit power $p_{ji}$, and $\mathbf{n}_{tq}\in\mathbb{C}^{N}$ is the complex additive white Gaussian noise at the BS, with $[\mathbf{n}_{tq}]_{n}\mathrel{\overset{\text{i.i.d.}}{\scalebox{1.5}[1.0]{$\sim$}}}\mathcal{CN}(0,N_{0}),$ $\forall\ n\in[N]$ and $t\in[T]$.

Channel estimation is performed based on the received pilot signal in each cell. The signals and the channel estimates are indexed by the decoding iteration $k$, since they are recomputed in every iteration. We denote the set of users in the $j$th cell who have not yet been decoded up to the $k$th decoding iteration by $\mathcal{S}_{kj}$. For some $m\in\mathcal{S}_{kj}$, let $\mathcal{S}_{kj}^{m}\triangleq\mathcal{S}_{kj}\setminus\{m\},$ with $\mathcal{S}_{1j}=[M]$. Let the set of all cell indices be denoted by $\mathcal{Q}\triangleq\{1,2,\ldots,Q\}$, and let $\mathcal{Q}^{q}\triangleq\mathcal{Q}\setminus\{q\}$. The received pilot signal at the $q$th BS in the $t$th RB in the $k$th decoding iteration is given by

where the first term contains signals from users within the $q$th cell who have not yet been decoded up to the $k$th decoding iteration, i.e., $\forall i\in\mathcal{S}_{kq}$. The second term contains signals from all users outside the $q$th cell, i.e., from every $i\in\mathcal{S}_{1j},\forall j\in\mathcal{Q}^{q}$. We note that there is no coordination among BSs, and thus, all the users outside the $q$th cell do not get decoded by the $q$th BS, and they permanently interfere with the decoding of users in other cells, across all the decoding iterations.

Let $\mathcal{G}_{tq}\triangleq\{i\in\mathcal{S}_{1q}|g_{tqi}=1\}$ denote the set of users within the $q$th cell who have transmitted in the $t$th RB, with $M_{tq}=|\mathcal{G}_{tq}|$. We denote the set of users in the $q$th cell who have transmitted on the $t$th RB but have not yet been decoded up to the $k$th decoding iteration by $\mathcal{M}_{tq}^{qk}\triangleq\mathcal{G}_{tq}\cap\mathcal{S}_{kq},$ with ${M}_{tq}^{qk}\triangleq|\mathcal{M}_{tq}^{qk}|$. Let $\mathbf{H}_{tj}^{q}\triangleq[\mathbf{h}_{tj1}^{q},\mathbf{h}_{tj2}^{q},\ldots\mathbf{h}_{tjM}^{q}]$ contain the uplink channels between all the users in the $j$th cell and the $q$th BS, with $\mathbf{H}_{tq}^{qk}\triangleq[\mathbf{H}_{tq}^{q}]_{\mathrel{\mathop{\mathchar 58\relax}},\mathcal{M}_{tq}^{qk}}$ and $\mathbf{H}_{tj}^{qk}\triangleq[\mathbf{H}_{tj}^{q}]_{\mathrel{\mathop{\mathchar 58\relax}},\mathcal{G}_{tj}},\forall j\in\mathcal{Q}^{q}$. Let $\mathbf{P}_{j}\triangleq[\mathbf{p}_{j1},\mathbf{p}_{j2},\ldots,\mathbf{p}_{jM}]$ contain the pilots of all users within the $j$th cell, with $\mathbf{P}_{tq}^{qk}\triangleq[\mathbf{P}_{q}]_{\mathrel{\mathop{\mathchar 58\relax}},\mathcal{M}_{tq}^{qk}}$ and $\mathbf{P}_{tj}^{qk}\triangleq[\mathbf{P}_{j}]_{\mathrel{\mathop{\mathchar 58\relax}},\mathcal{G}_{tj}},\forall j\in\mathcal{Q}^{q}$. Let $\mathbf{B}_{j}^{q}\triangleq\sigma_{h}^{2}\text{diag}(\beta_{j1}^{q},\beta_{j2}^{q},\ldots,\beta_{jM}^{q})$ contain the path loss coefficients between the users within the $j$th cell and the $q$th BS, with $\mathbf{B}_{tq}^{qk}\triangleq[\mathbf{B}_{q}^{q}]_{\mathrel{\mathop{\mathchar 58\relax}},\mathcal{M}_{tq}^{qk}}$ and $\mathbf{B}_{tj}^{qk}\triangleq[\mathbf{B}_{j}^{q}]_{\mathrel{\mathop{\mathchar 58\relax}},\mathcal{G}_{tj}},\forall j\in\mathcal{Q}^{q}$. Thus, the received pilot signal from (3) can be written as

where $\bar{\mathbf{H}}_{tq}^{qk}\triangleq[\mathbf{H}_{tq}^{qk},\mathbf{H}_{t1}^{qk},\ldots,\mathbf{H}_{tq-1}^{qk},\mathbf{H}_{tq+1}^{qk},\ldots,\mathbf{H}_{tQ}^{qk}]\in\mathbb{C}^{N\times\bar{M}_{tq}^{qk}}$, with $\bar{M}_{tq}^{qk}\triangleq{M}_{tq}^{qk}+\sum_{j\in\mathcal{Q}^{q}}{M}_{tj}$, and $\bar{\mathbf{P}}_{tq}^{k}\triangleq[\mathbf{P}_{tq}^{qk},\mathbf{P}_{t1}^{qk},\ldots,\mathbf{P}_{tq-1}^{qk},\mathbf{P}_{tq+1}^{qk},\ldots,\mathbf{P}_{tQ}^{qk}]\in\mathbb{C}^{\tau\times\bar{M}_{tq}^{qk}}$. We define $\bar{\mathbf{B}}_{tq}^{qk}\triangleq[\mathbf{B}_{tq}^{qk},\mathbf{B}_{t1}^{qk},\ldots,\mathbf{B}_{tq-1}^{qk},\mathbf{B}_{tq+1}^{qk},\ldots,\mathbf{B}_{tQ}^{qk}]\in\mathbb{C}^{\bar{M}_{tq}^{qk}\times\bar{M}_{tq}^{qk}}$ to derive the channel estimate. Let $\bar{\mathbf{C}}_{t}^{qk}\!\triangleq\!\bar{\mathbf{P}}_{tq}^{k}\bar{\mathbf{B}}_{tq}^{qk}(\bar{\mathbf{P}}_{tq}^{kH}\bar{\mathbf{P}}_{tq}^{k}\bar{\mathbf{B}}_{tq}^{qk}+N_{0}\mathbf{I}_{\bar{M}_{tq}^{qk}})^{-1}\!\!,$ be split as $\bar{\mathbf{C}}_{t}^{qk}=[\mathbf{C}_{tq}^{qk},\mathbf{C}_{t1}^{qk},\ldots,\mathbf{C}_{tq-1}^{qk},\mathbf{C}_{tq+1}^{qk},\ldots,\mathbf{C}_{tQ}^{qk}]$, and $\mathbf{c}_{tji}^{qk}\triangleq[{\mathbf{C}}_{tj}^{qk}]_{\mathrel{\mathop{\mathchar 58\relax}},i}$.

Let $\rho_{tqm}^{k}$ denote the SINR of the $m$th user in the $q$th cell at the $q$th BS in the $t$th RB in the $k$th decoding iteration. Similar to (2), the received data signal at the $q$th BS in the $t$th RB in the $k$th decoding iteration is given by

We use a combining vector $\mathbf{a}_{tqm}^{k}\in\mathbb{C}^{N}$ to obtain the post-combined data signal $\tilde{y}_{tqm}^{k}=\mathbf{a}_{tqm}^{kH}{\mathbf{y}_{tq}^{k}}$ as in (6), with $\tilde{\mathbf{h}}_{tqm}^{qk}$ as defined in Theorem 1. This combined signal, used to decode the $m$th user in the $q$th cell, is composed of five terms. The first term $g_{tqm}x_{qm}\mathbf{a}_{tqm}^{kH}\hat{\mathbf{h}}_{tqm}^{qk}$ is the useful signal component of the $m$th user; the term $g_{tqm}x_{qm}\mathbf{a}_{tqm}^{kH}\tilde{\mathbf{h}}_{tqm}^{qk}$ arises due to the estimation error $\tilde{\mathbf{h}}_{tqm}^{k}$; the term $\sum\nolimits_{i\in\mathcal{S}_{kq}^{m}}g_{tqi}x_{qi}\mathbf{a}_{tqm}^{kH}\mathbf{h}_{tqi}^{q}$ represents the intra-cell interference from the users within the $q$th cell who have transmitted in the $t$th RB and have not yet been decoded up to the $k$th decoding iteration; the term $\sum\nolimits_{j\in\mathcal{Q}^{q}}\sum\nolimits_{i\in\mathcal{S}_{1j}}g_{tji}x_{ji}\mathbf{a}_{tqm}^{kH}\mathbf{h}_{tji}^{q}$ models the inter-cell interference from users outside the $q$th cell; and the last term $\mathbf{a}_{tqm}^{kH}{\mathbf{n}_{tq}}$ is the additive noise. We now present the SINR for all the users.

We now present simple and interpretable expressions for the SINR in the massive MIMO (large $N$) regime, and with maximal ratio combining, i.e., $\mathbf{a}_{tqm}^{k}=\hat{\mathbf{h}}_{tqm}^{qk}$ [10].

In this section, the derived SINR analysis is used to evaluate the throughput of MC IRSA via Monte Carlo simulations and provide insights into the impact of various system parameters on the performance of the system. In each simulation, we generate independent realizations of the user locations, the access pattern matrix, and the channels. The throughput in each run is calculated as described in Sec. II-1, and the effective system throughput is calculated by averaging over the runs. We consider a set of $Q=9$ square cells, stacked in a $3\times 3$ grid, and report the performance of the center cell [5]. Each cell has $M$ users spread uniformly at random across an area of $250\times 250$ m², with the BS at the center [10].¹¹1Due to path loss inversion, the area of the cell does not significantly affect the throughput, but affects the area spectral efficiency, which we do not analyze here.

The results in this section are for $T=50$ RBs, $N_{s}=10^{3}$ Monte Carlo runs, $\sigma_{h}^{2}=1$, SINR threshold $\gamma_{\text{th}}=10$. The number of users contending for the $T$ RBs in each cell is computed based on the load $L$ as $M=\lfloor LT\rceil$. The path loss is calculated as $\beta_{ji}^{q}$ (dB) $=-37.6\log_{10}(d_{ji}^{q}/10{\text{m}})$, where $d_{ji}^{q}$ is the distance of the $i$th user in the $j$th cell from the $q$th BS [10]. The pilot sequences are chosen as the columns of the $\tau\times\tau$ discrete Fourier transform matrix normalized to have column norm $\sqrt{\tau P_{\tau}}$. The soliton distribution [7] with $d_{\max}=8$ maximum repetitions is used to generate the repetition factor $d_{ji}$, for the $i$th user in the $j$th cell, whose access vector is formed by uniformly randomly choosing $d_{ji}$ RBs from $T$ RBs without replacement [3].²²2The soliton distribution has been shown to achieve 96% of the throughput that can be achieved with the optimal repetition distribution [6]. The access pattern matrix is formed by stacking the access vectors of all the users. The power level is set to $P=P_{\tau}=10$ dBm [10] and $N_{0}$ is chosen such that the data and pilot SNR are $10$ dB, unless otherwise stated.

In Fig. 2, we show the effect of the load $L$ on the center cell’s throughput $\mathcal{T}_{C}$. All the curves increase linearly till a peak, which is the desired region of operation, and then drop quickly to zero as the system becomes interference limited. All the users’ packets are successfully decoded in the linear region of increase, and at high $L$, beyond the peak, the throughputs drop to zero. For $N=8,\gamma_{\rm th}=10,$ we see a 70% drop in the peak throughput from $\mathcal{T}_{C}=4$ at $L=4$ for SC to $\mathcal{T}_{C}=1.2$ at $L=1.2$ for MC. This is because users face a high degree of inter-cell interference in the MC setup, unlike the SC setup, especially at high $L$. In the SC setup, the peak throughput reduces from $\mathcal{T}_{C}=4$ for $N=8$ to $\mathcal{T}_{C}=3$ at $L=3$ for $N=4$. This trend is similar to the MC setup for which the peak throughputs are $\mathcal{T}_{C}=4,2.6,1.2$ for $N=32,16,8$, respectively. This is because the system’s interference suppression ability with MMSE combining reduces as we decrease $N$ [10]. This holds true with $\gamma_{\rm th}=6$ also, which corresponds to a lower SINR threshold, and consequently higher $\mathcal{T}_{C}$. To summarize, at high $L$, there is a high degree of inter-cell interference which SC processing does not account for, resulting in a substantial drop in performance.

Fig. 3 studies the impact of the pilot length $\tau$. The performance of SC IRSA at all $L$ is optimal (note that the throughput is upper bounded by $L$) for $\tau>10$. In MC IRSA, nearly optimal throughputs are achieved for $L=1,2,3$ at $\tau=10,30,40$, respectively. The throughput for $L=4$ does not improve much with $\tau$. At high $L$, the impact of inter-cell interference is severe, as expected. Increasing $\tau$ implies that each cell has a higher number of orthogonal pilots, and hence can help in reducing intra-cell PC, but the system is still impacted by inter-cell PC and inter-cell interference. Thus, we see that MC IRSA requires significantly higher $\tau$ (at least $4-5$x) to overcome inter-cell PC and inter-cell interference to achieve the same performance as that of SC IRSA.

In Fig. 4, we study the effect of $N$ for $L=1,2,3,4$, with SNR $=10,-5$ dB. Nearly optimal throughputs can be achieved with $N=8,16,32,32$ for SNR $=10$ dB, and with $N=64$ for SNR $=-5$ dB. The system performance improves because of the array gain and higher interference suppression ability at high $N$. This aids in reducing not only intra-cell interference, but also inter-cell interference. However, as discussed in Remark 3, the SINRs of the users have a coherent interference component that scales with $N$. Thus, while an increase in $N$ helps reducing intra-cell interference and inter-cell interference, and improves the system performance, it does not reduce intra-cell PC and inter-cell PC. Similar observations about $N$ can be made where we study the impact of SNR in Fig. 5. At very low SNR, the system is noise limited, and increasing $N$ does not help increase the throughput, which is at zero. For $N=16$, the throughput is always zero and nearly zero for $L=4$ and $L=3$, respectively. Optimal throughputs are obtained at higher SNRs for $N=32$ and $64$. Since boosting transmit powers of the users scales both the signal and interference components equally, the SINR does not increase, and therefore the system performance saturates with SNR. To summarize, increasing $\tau$, $N$, and the SNR can judiciously help reduce the impact of intra-cell PC and inter-cell PC, as well as intra-cell interference and inter-cell interference.

This paper studied the effect of MC interference, namely inter-cell PC and inter-cell interference, on the performance of IRSA. The users across cells perform path loss inversion with respect to their own BSs and employ a $\tau$-length pilot codebook for channel estimation. Firstly, the channel estimates were derived, accounting for path loss, MIMO fading, intra-cell PC, and intra-cell interference. The corresponding SINR of all the users were derived accounting for channel estimation errors, inter-cell PC, and inter-cell interference. It was seen that MC IRSA had a significant degradation in performance compared to SC IRSA, even resulting in up to 70% loss of throughput in certain regimes. Recuperating this loss requires at least $4-5$x larger pilot length in MC IRSA to yield the same performance as that of SC IRSA. Increasing $\tau,N$, and SNR helped improve the performance of MC IRSA. These results underscore the importance of accounting for multiuser interference in analyzing IRSA in multi-cell settings. Future work could include design of optimal pilot sequences to reduce PC and density evolution [3] to obtain the asymptotic throughput.