Can Massive MIMO Support URLLC?

The received signal at the $m$th antenna of the BS and the $n$th subband is expressed as

where $\bm{\Phi}=[\bm{\phi}_{1}\,\bm{\phi}_{2}\,...\,\bm{\phi}_{K}]\in\mathbb{C}^{\tau\times K}$ is the pilot matrix with $\tau$ denotes the pilot length and $\|\bm{\phi}_{k}\|_{2}=1$ for $k=1,2,...,K$. $\mathbf{A}$ is the indicator matrix defined as $\mathbf{A}=\text{diag}\{[a_{1},a_{2},...,a_{K}]\}$ with $a_{k}=1$ if the $k$th UE is active and $a_{k}=0$ if the $k$th UE is not active. $\bar{\mathbf{g}}_{[m]}^{n}=\sqrt{\tau\rho_{p}}[g_{m,1}^{n},g_{m,2}^{n},...,g_{m,K}^{n}]^{T}$ where $\rho_{p}$ is the normalized signal-to-noise ratio (SNR) of each pilot symbol. $\mathbf{w}_{m}^{n}$ is the noise vector with $\sim i.i.d.\ \mathcal{CN}(0,1)$ entries.

To use orthogonal pilots, we must have $\tau\geq K$. We assume $\tau=K$ to minimize the pilot overhead. Then $\bm{\phi}_{k}^{H}\bm{\phi}_{k^{\prime}}=\delta(k-k^{\prime})$ for $k,k^{\prime}=1,2,...,K$ and the optimal Neyman-Pearson (NP) [6] detector can be used for active UE detection. We denote $\mathbf{f}_{k}$ as the pre-processing operator for the $k$th UE detection. Due to the application of orthogonal pilots, the signal used for the $k$th UE detection, $z_{m,k}^{n}$, can be obtained by a simple correlation operation on $\mathbf{y}_{m}^{n}$:

where set $\mathcal{A}$ is the active UE set. Note that if the $k$th UE is active, $z_{m,k}^{n}=g_{m,k}^{n}+\bm{\phi}_{k}^{H}\mathbf{w}_{m}^{n}\sim\mathcal{CN}(0,\tau\rho_{p}\beta_{k}+1)$ and if the $k$th UE is not active, $z_{m,k}^{n}=\bm{\phi}_{k}^{H}\mathbf{w}_{m}^{n}\sim\mathcal{CN}(0,1)$. Define $\mathbf{z}_{k}=[\hat{\mathbf{z}}_{1,k}^{1},\hat{\mathbf{z}}_{2,k}^{1},...,\hat{\mathbf{z}}_{M,k}^{1},\hat{\mathbf{z}}_{1,k}^{2},...,\hat{\mathbf{z}}_{m,k}^{n},...,\hat{\mathbf{z}}_{M,k}^{N}]^{T}\in\mathbb{C}^{MN\times 1}$ where $\hat{\mathbf{z}}_{m,k}^{n}=[Re\{z_{m,k}^{n}\},Im\{z_{m,k}^{n}\}]$. Based on the $i.i.d.\ \mathcal{CN}$ assumption of $\{h_{m,k}^{n}\}$, we can obtain the following two hypotheses about whether the $k$th UE is active:

where $\sigma_{0,k}^{2}=1$ and $\sigma_{1,k}^{2}=\tau\rho_{p}\beta_{k}+1$. We define the test statistics as $T(\mathbf{z}_{k})=\sum_{n=1}^{N}\sum_{m=1}^{M}|z_{m,k}^{n}|^{2}$. According to (LABEL:eq:hypotheses) and the NP detector procedures given in [6], the $k$th UE is identified as active if

where $Q_{\mathcal{X}_{2MN}^{2}}(\cdot)$ is the complementary cumulative distribution function of $\mathcal{X}_{2MN}^{2}$, which is the Chi-square distribution with $2MN$ degree of freedom and $\text{P}_{\text{FA}}$ is a predetermined probability of false alarm.

The application of short non-orthogonal pilots as configured in the 3GPP standards (e.g., Gold sequence) means $\tau<K$. Due to this fact, the approach of simple correlation pre-processing + NP detector does not work properly for Gold sequence. In this section, we adopt coordinate-wise descend algorithms from [7] for active UE detection. The detailed form of the algorithm is given in Algorithm 1 where ML, MMV, and NNLS are short for Maximum Likelihood, Multiple Measurement Vector, and Non-Negative Least Squares, respectively. In Algorithm 1, $\bar{\mathbf{Y}}=\sqrt{\tau\rho_{p}}\bm{\Phi}\mathbf{A}\mathbf{\bar{G}}^{T}+\mathbf{\bar{W}}$ where $\bar{\mathbf{G}}^{T}=[(\mathbf{G}^{1})^{T},(\mathbf{G}^{2})^{T},...,(\mathbf{G}^{N})^{T}]$ and $(\mathbf{G}^{n})^{T}=[\mathbf{\bar{g}}_{[1]}^{n}\,\mathbf{\bar{g}}_{[2]}^{n}\,...\,\mathbf{\bar{g}}_{[M]}^{n}]\in\mathbb{C}^{K\times M}$ is the channel coefficient matrix at the first subcarrier of the $n$th subband¹¹1Note that active UE detection via Algorithm 1 requires iteration. In addition, given a target $\text{P}_{\text{FA}}$ there is no closed-form expression for the test threshold, $\gamma^{\prime}$, which requires numerical method to determine..

Since each subcarrier has the same pilot length in the coherence interval based pilot setting, without loss of generality we consider one subcarrier in this section so the notation for subband index is omitted. Then the received pilot signals of all active UEs are

where $\mathbf{W}\in\mathbb{C}^{\tau\times M}$ are the noise matrix with $\sim i.i.d.\ \mathcal{CN}(0,1)$ entries.

For Gold sequence, we only consider 3GPP channel model for small-scale fading. According to Fig. 1, the correlations among antennas are weak, thus the LMMSE channel estimation approach given in Section III-D1 can be adopted for Gold sequences. Specifically, when using 3GPP channel models, matrix $\mathbf{C}$ in (7) should be changed to $\mathbf{C}=\text{diag}\{[\text{const}_{1},\text{const}_{2},...,\text{const}_{K}]\}$.

We adopt optimal MMSE MIMO receiver in our design and denote by $\mathbf{v}_{k}^{\text{MMSE}}$ the receiver vector for the $k$th UE. Due to the ultra-low latency requirement of URLLC, instantaneous SINR is adopted for performance evaluation and the expression for the $k$th UE is given by:

where $\eta_{k}$ is the power control coefficient for the $k$th UE with the constraint $0\leq\eta_{k}\leq 1$, $\rho_{u}$ is the normalized SNR of each uplink data symbol. Using Rayleigh-Ritz Theorem, $\mathbf{v}_{k}^{\text{MMSE}}$ for $k\in\mathcal{B}$ can be obtained as

We use effective per-UE throughput as the performance evaluation metric. If the $k$th UE belongs to both sets $\mathcal{A}$ and $\mathcal{B}$, then its effective throughput is expressed as

where $10^{-1/10}$ accounts for $1$dB SINR degradation due to decoding error³³3For a channel with a given SINR, we can transmit at any rate $R=\log_{2}(1+\text{SINR})-\epsilon$ with $\epsilon$ an arbitrary small positive number, and achieve an arbitrary small probability of decoding error $P_{de}$ if a good, for example random, infinitely long error correcting code with optimal decoding is used. In reality a finite length code will be applied and we cannot use optimal decoding (complexity is too high). Thus, we assume that we will manage to achieve a target $P_{de}$ (e.g., $10^{-5}$) with $R=\log_{2}(1+\text{SINR}\times 10^{-1/10})$.. If the $k$th UE belongs to set $\mathcal{A}$ and not to set $\mathcal{B}$, this UE is mis-detected and its throughput is set as 0.

In our simulations the cell radius is set as $150$m, the number of URLLC UEs (i.e., $K$) is set as 50, and at a given moment, the average number of active UEs (i.e., $K_{a}$) is set as 10. To enable comparisons with the 3GPP compliant pilot setting presented later, the length $\tau_{c}$ of the coherence interval is set as 168 OFDM symbols, which equals the number of resource elements (RE) in a PRB. Note that in practical scenarios, $\tau_{c}$ depends on the mobility of environments and can be larger or less than 168. In addition, full power transmission is considered in this section. The per-UE throughput performance with orthogonal pilots under UMa NLoS scenario is shown in Fig. 2.

As mentioned in Section I the URLLC requirements are 250 bytes/1ms ($2\text{Mbps}$) with 99.999% reliability. Since 99.999% reliability corresponds to the probability $10^{-5}$ in a CDF curve, we say that the URLLC requirements are satisfied if the per-UE throughput is larger than 2 Mbps at probability $10^{-5}$. As observed from Fig. 2, with orthogonal pilots URLLC requirements are satisfied in both i.i.d. and 3GPP small-scale fading and the probabilities of mis-detection are 0 since no UE has 0 throughput. Furthermore, the performance using NP detector is almost equal to the performance with perfect detection regime, in which we assume that the set of active UEs is known to BS and therefore $\mathcal{B}=\mathcal{A}$.

Fig. 3 shows the per-UE throughput performance of the scenario of using Gold sequences under UMa NLoS channels. The pilot length is set as 24 which is the maximum pilot length supported by 3GPP [3] in a PRB, and for comparison the throughput performance of the scenario with orthogonal pilots is also included. It is observed from Fig. 3 that even with perfect detection, the challenging URLLC requirements is not satisfied using Gold sequences. On the other hand, the performance obtained by ML detection is very close to the performance with perfect detection, but MMV and NNLS detection lead to relatively high probability of misdetection ($\text{P}_{\text{MD}}$) and their performance is farther from the URLLC requirements than the ML detection.

We adopt the ML approach in Algorithm 1 for active UE detection but incorporate the covariance matrix among different subcarriers in a PRB. The procedures are the same as in Algorithm 1 except that step 9 is changed to: $\text{Update}\ \bm{\Sigma}\leftarrow\bm{\Sigma}+\sum_{s=2i+1}\mathbf{C}_{k}^{s}\bm{\phi}_{k}(\bm{\tilde{\phi}}_{k}^{s})^{H}$ for $i=0,1,...,5$. Here $\mathbf{C}_{k}^{s}\triangleq\text{blkdiag}([c_{k}^{s,1}\mathbf{I}_{4},c_{k}^{s,3}\mathbf{I}_{4},...,c_{k}^{s,11}\mathbf{I}_{4}])\in\mathbb{C}^{24\times 24}$ where $c_{k}^{s,s^{\prime}}$ denotes the covariance between the channel coefficients in the $s$th and the $s^{\prime}$th subcarriers in one PRB for the $k$th UE with $s,s^{\prime}=1,3,...,11$. $\tilde{\bm{\phi}}_{k}^{s}\in\mathbb{C}^{24\times 1}$ is a column vector with zero elements except the $2(s+1)-3$th to the $2(s+1)$th elements being $\bm{\phi}_{k}(2(s+1)-3:2(s+1))$.

According to the form given by (14), standard LMMSE estimation can be applied to estimate $\tilde{\mathbf{g}}_{[m]}$ and the corresponding estimator is given as

where $\tilde{\mathbf{C}}_{\mathcal{B}}=[(\tilde{\mathbf{C}}_{\mathcal{B}}^{1})^{T},(\tilde{\mathbf{C}}_{\mathcal{B}}^{3})^{T},...(\tilde{\mathbf{C}}_{\mathcal{B}}^{s})^{T},...,(\tilde{\mathbf{C}}_{\mathcal{B}}^{11})^{T}]^{T}\in\mathbb{C}^{6|\mathcal{B}|\times 6|\mathcal{B}|}$, $\tilde{\mathbf{C}}_{\mathcal{B}}^{s}=[\mathbf{C}_{\mathcal{B}}^{s,1},\mathbf{C}_{\mathcal{B}}^{s,3},...,\mathbf{C}_{\mathcal{B}}^{s,s^{\prime}},...,\mathbf{C}_{\mathcal{B}}^{s,11}]\in\mathbb{C}^{|\mathcal{B}|\times 6|\mathcal{B}|}$, and $\mathbf{C}_{\mathcal{B}}^{s,s^{\prime}}=\text{diag}([c_{\mathcal{B}(1)}^{s,s^{\prime}},c_{\mathcal{B}(2)}^{s,s^{\prime}},...,c_{\mathcal{B}(|\mathcal{B}|)}^{s,s^{\prime}}])$. Here $\mathcal{B}(b)$ denotes the $b$th component of the predictive UE set $\mathcal{B}$ and $b=1,2,...,|\mathcal{B}|$.

In this section, we adopt UMi NLoS channel model with cell radius being $100$m and the remaining system settings are the same as in Table I. We also drop $5$% UEs with the smallest values of large-scale fading of all $\tilde{K}$ UEs. Define by $|\mathcal{R}|$ the remaining UE set after dropping poor UEs, an open-loop power control is applied which is given as

Fig. 5 shows the per-UE throughput performance with Gold sequence ($\tau=24$) under 3GPP compliant and coherence interval based pilot settings. It is observed that the performance with ML detection is very close to that with perfect detection but does not meet the URLLC reliability requirements, which indicates that these requirements can be achieved only for cells of a smaller radius, or number of service antennas $M$ should be increased. On the other hand, the performance under the coherence interval based pilot setting is significantly higher than the URLLC requirements.