Design of Non-orthogonal and Noncoherent Massive MIMO for Scalable URLLC Beyond 5G

We consider a massive multiple-input multiple-output (MIMO) system consisting of $K$ single-antenna users transmitting simultaneously to a base station (BS) with $M$ ($M\gg K$) receiving antennas on the same time-frequency grid. By using a discrete-time complex baseband-equivalent model, the received signal at the antenna array of BS in the $t$-th time slot¹¹1Each time slot refers to one symbol duration throughout this paper., defined as $\mathbf{y}_{t}=[y_{1,t},\ldots,y_{M,t}]^{T}$, can be expressed by

where $\mathbf{x}_{t}=[x_{1,t},\ldots,x_{K,t}]^{T}$ represents the transmitted signals from all $K$ users, ${\boldsymbol{\xi}}_{t}$ is an additive circularly-symmetric complex Gaussian (CSCG) noise vector with covariance $\sigma^{2}\mathbf{I}_{M}$. We let $\mathbf{H}=\mathbf{G}\mathbf{D}^{1/2}$ denote the $M\times K$ complex channel matrix between the receiver antenna array and all users, where $\mathbf{G}$ characterizes the small-scale fading caused by local scattering while $\mathbf{D}={\rm diag}\{\beta_{1},\cdots,\beta_{K}\}$ with $\beta_{k}>0$ capturing the propagation loss due to distance and shadowing. All the entries of $\mathbf{G}$ are assumed to be i.i.d. complex Gaussian distributed with zero mean and unit variance. The channel coefficients are assumed to suffer from block fading which are quasi-static in the current block and change to other independent values in the next block with a channel coherence time $T_{c}\geq K$. We consider a space-time block modulation (STBM) [31] scheme over $T$ time slots and the received signal vectors can be stacked together into a matrix form given by

where $\mathbf{Y}_{T}=[\mathbf{y}_{1},\ldots,\mathbf{y}_{T}]$, $\mathbf{X}_{T}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{T}]$ and ${\mathbf{\Xi}}_{T}=[\boldsymbol{\xi}_{1},\cdots,\boldsymbol{\xi}_{T}]$.

In this work, we apply a noncoherent ML detector which is optimal for uniformly distributed discrete input signals in terms of error probability. We note that (1) can be reformulated as $\mathbf{Y}_{T}^{H}=\mathbf{X}_{T}^{H}\mathbf{D}^{1/2}\mathbf{G}^{H}+{\boldsymbol{\Xi}}_{T}^{H}$. With the help of [33], the vectorized form of the received signal can then be written as

As all the entries of $\mathbf{G}$ and $\boldsymbol{\Xi}$ are i.i.d. CSCG, we immediately have $\mathbb{E}[\mathbf{y}]=\mathbf{0}$, and the covariance matrix of $\mathbf{y}$ can be calculated as $\mathbf{R}_{\mathbf{y}|\mathbf{X}_{T}}=\mathbb{E}[\mathbf{y}\mathbf{y}^{H}]=\mathbf{I}_{M}\otimes(\mathbf{X}_{T}^{H}\mathbf{D}\mathbf{X}_{T}+\sigma^{2}\mathbf{I}_{T})$. The conditional distribution of the received signal $\mathbf{y}$ at BS for any transmitted signal matrix $\mathbf{X}_{T}$ can then be given by $p({\mathbf{y}|\mathbf{X}_{T}})=\frac{1}{\pi^{KM}\det({\mathbf{R}_{\mathbf{y}|\mathbf{X}_{T}}})}\exp(-\mathbf{y}^{H}\mathbf{R}_{\mathbf{y}|\mathbf{X}_{T}}^{-1}\mathbf{y})$, where $\mathbf{R}_{\mathbf{y}|\mathbf{X}_{T}}=\mathbf{I}\otimes(\mathbf{X}_{T}^{H}\mathbf{D}\mathbf{X}_{T}+\sigma^{2}\mathbf{I}_{T})$. The noncoherent ML detector can estimate the transmitted information carrying matrix from the received signal vector $\mathbf{y}$ by resolving the following optimization problem:

From (2), we can observe that the detector relies on the sufficient statistics of the transmitted signal matrix $\mathbf{R}_{\mathbf{y}|\mathbf{X}_{T}}=\mathbf{I}\otimes(\mathbf{X}_{T}^{H}\mathbf{D}\mathbf{X}_{T}+\sigma^{2}\mathbf{I}_{T})$. The detailed discussion regarding the signal design is given in the following subsection.

In this subsection, we first identify what conditions the transmitted signal matrix must satisfy to ensure the unique identification of the transmitted signal matrix $\mathbf{X}_{T}$. We can observe from (2) that, to achieve reliable communication between all users and the BS in the considered nn-mMIMO system, the BS must be able to uniquely determine each transmitted signal matrix ${\mathbf{X}}_{T}$ once $\mathbf{R}=\mathbf{X}_{T}^{H}\mathbf{D}\mathbf{X}_{T}$ has been identified, which can be formally stated as follows:

The proof is provided in Appendix-A.

Inspired by Proposition 1, to facilitate our system design, we introduce the concept of uniquely-factorable multiuser space-time modulation (UF-MUSTM), the formal definition of which is given as follows:

Definition 1 motivates us to design a UF-MUSTM codebook for the considered nn-mMIMO system. Therefore, our primary task in the rest of this paper is to develop a new framework for a systematic design of such UF-MUSTM ${\mathcal{S}}^{K\times T}$.

Before proceeding on, it is worth clarifying here that the UF-MUSTM code design is fundamentally different from existing noncoherent space-time code/modulation designs. Specifically,

• •

  For the considered UF-MUSTM based nn-mMIMO system, the signals transmitted from different users cannot fully collaborate, and hence the widely used unitary space-time code design is intractable for the considered system. This is fundamentally different from most of conventional space-time code designs for point-to-point MIMO system, where unitary space-time code is feasible since all the transmitting antennas are connected to the same transmitter [14, 31]. Note that the error performance analysis of non-unitary codeword of MUSTM is very challenging as shown in [30].

• •

  Our design is asymptotically optimal when the number of BS antennas goes to infinity while keeping the transmitted power fixed. This is in contrast to most previous space-time coding designs which considered the asymptotic regime with the signal-to-noise ratio (SNR) going to infinity [30, 32, 31].

In practice, the computational complexity of the optimal noncoherent ML detector described in (2) could be prohibitively high. Furthermore, the error performance analysis results available for the block transmission with general block size and ML receiver are too complicated to reveal insightful results for the input codeword design and the corresponding power allocation [30]. To resolve these problems as well as to reduce the receiver complexity, our main idea is to input a small block size into the ML receiver. If only one time slot is involved in the ML detector given in (2), i.e., when $T=1$, the correlation matrix $\mathbf{R}=\mathbf{X}_{T}^{H}\mathbf{D}\mathbf{X}_{T}$ degenerates into a real scalar $\mathbf{x}_{1}^{H}\mathbf{D}\mathbf{x}_{1}=\sum_{k=1}^{K}\beta_{k}|x_{k,1}|^{2}$, where the phase information of the transmitted symbols is lost and information bits from all users can only be modulated on the amplitudes of the transmitted symbols. Such a design typically has a low spectral efficiency [21, 26, 22]. To improve the spectrum efficiency by allowing constellation with phase information being transmitted by all users, we need to feed the signals received in at least two time slots into the ML decoder [30, 34, 31].

As an initial attempt, in this paper we focus on a slot-by-slot ML detection over the first and $t$-th time slots, which is similar to the differential modulation with the hard-decision based noncoherent multiuser detection. More specifically, we let the transmitted signal matrix be $\mathbf{X}_{T}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{T}]$. For detection purpose, we now stack the transmitted signal of the first and the $t$-th time slot as $\mathbf{X}_{t}=[\mathbf{x}_{1},\mathbf{x}_{t}]$, and then make the decision on $\mathbf{Y}_{t}=[\mathbf{y}_{1},\mathbf{y}_{t}]$ by using (2). For simplicity, we consider the transmitted signal from the first and second time slots, i.e., $\mathbf{X}_{2}=[\mathbf{x}_{1},\mathbf{x}_{2}]$, hereafter, and the case of $\mathbf{X}_{t}$ follows similarly. We denote $\mathbf{R}_{\mathbf{y}|\mathbf{X}_{2}}=\mathbf{I}\otimes\mathbf{R}_{2}$, in which

By (3), we have

As a consequence, the ML receiver can be reformulated as follows

where $\mathbf{y}_{1}$ and $\mathbf{y}_{2}$ are the received signal vectors in the first and second time slots, respectively. It can be observed that the diagonal entries in (3) are $\mathbf{x}_{1}^{H}\mathbf{D}\mathbf{x}_{1}=\sum_{k=1}^{K}\beta_{k}|x_{k,1}|^{2}$ and $\mathbf{x}_{2}^{H}\mathbf{D}\mathbf{x}_{2}=\sum_{k=1}^{K}\beta_{k}|x_{k,2}|^{2}$, in which the phase information is lost, while the off-diagonal term is $\mathbf{x}_{1}^{H}\mathbf{D}\mathbf{x}_{2}=\sum_{k=1}^{K}\beta_{k}x_{k,1}^{*}x_{k,2}=\sum_{k=1}^{K}\beta_{k}|x_{k,1}||x_{k,2}|\exp\big{(}j\arg(x_{k,2})-j\arg(x_{k,1})\big{)}$, indicating that we can transmit a known reference signal vector $\mathbf{x}_{1}$ in the first time slot and then transmit the information bearing signal vector $\mathbf{x}_{2}$ to imitate a “differential-like” transmission [35]. The exact PEP is extremely hard to evaluate for the matrix $\mathbf{X}_{2}$ given above. Moreover, the exact expression for the PEP does not seem to be tractable for further optimization. Inspired by the Chernoff-Stein Lemma, when the number of receiver antennas $M$ goes to infinity, the PEP will goes to zero exponentially where the exponent is determined by the KL divergence [31]. Hence, in this paper we propose to use the KL divergence between the conditional distributions of the received signals for different inputs as the design criterion thanks to its mathematical tractability.

We now derive the KL divergence between the received signals induced by the transmitted signals matrices $\mathbf{X}_{2}=[\mathbf{x}_{1},\mathbf{x}_{2}]$ and $\widetilde{\mathbf{X}}_{2}=[\tilde{\mathbf{x}}_{1},\tilde{\mathbf{x}}_{2}]$, which is also the expectation of the likelihood function between two received signal vectors. Essentially, the likelihood function between the received signal vectors corresponding to the two transmitted signals converge in probability to the KL-divergence as the number of receiver antennas increases [31]. More specifically, the KL-divergence between the received signals corresponding to the transmitted matrix $\mathbf{X}_{2}$ and $\widetilde{\mathbf{X}}_{2}$ can be calculated as

in which

We can observe from the above expression that $\mathcal{D}_{\rm KL}(\mathbf{X}_{2}||\widetilde{\mathbf{X}}_{2})$ is actually the KL-divergence when there is only one receiving antenna. Due to the assumption of the independence of channel coefficients, and the KL divergence with $M$ antennas $\mathcal{D}_{\rm KL}^{(M)}(\mathbf{X}_{2}||\widetilde{\mathbf{X}}_{2})$ is $M$ times of $\mathcal{D}_{\rm KL}(\mathbf{X}_{2}||\widetilde{\mathbf{X}}_{2})$.

The main objective of this subsection is to develop a new QAM division based MUSTM design framework for the considered nn-mMIMO system. The design is built upon the uniquely decomposable constellation group (UDCG) originally proposed in [36, 37] for the commonly used spectrally efficient QAM signaling. We now introduce the definition of UDCG as follows:

As PAM and QAM constellations are commonly used in modern digital communications, which have simple geometric structures, we now give the following construction of UDCG.

With the concept of UDCG, we are now ready to propose a QAM division based UF-MUSTM for the considered nn-mMIMO system with a noncoherent ML receiver given in (III-A). The structure of each transmitted signal matrix is given by $\mathbf{X}_{2}=[\mathbf{x}_{1},\mathbf{x}_{2}]=\mathbf{D}^{-1/2}\mathbf{\Pi}{\mathbf{S}}_{2}$, in which

In our design, the diagonal matrix $\mathbf{D}^{-1/2}$ is used to compensate for the different large scale fading among various users. The vector $\mathbf{p}=[p_{1},\ldots,p_{K}]$ is introduced to adjust the relative transmitting powers between all users, and $\mathbf{s}=[s_{1},\ldots,s_{K}]$ is the information-carrying vector. The instantaneous power constraint can be given by $\mathbb{E}\{|x_{k,t}|^{2}\}\leq P_{k}$, $k=1,\ldots,K$ and $t=1,2$. We let $s_{k}\in\mathcal{X}_{k}$, where all $\mathcal{X}_{k}$’s constitute a UDCG with sum-QAM constellation $\mathcal{Q}$ such that $\mathcal{Q}=\uplus_{k=1}^{K}\mathcal{X}_{k}$ as defined in Lemma 1. The rate allocation between the $K$ users are based on the sum-decomposition such that $\sum_{k=1}^{K}N_{k}=N$ in which $N_{k}=N_{I,k}+N_{Q,k}=\log_{2}(|\mathcal{X}_{k}|)$ denotes the bit rate of the user constellation $\mathcal{X}_{k}$. The matrix $\mathbf{\Pi}=[\mathbf{e}_{\pi{(1)}},\ldots,\mathbf{e}_{\pi(K)}]^{T}$ is a permutation matrix, where $\mathbf{e}_{k}$ denotes a standard basis column vector of length $K$ with 1 in the $k$-th position and 0 in other positions. $\pi:\{1,\ldots,K\}\to\{1,\ldots,K\}$ is a permutation over $K$ elements characterized by $\begin{pmatrix}1&2&\ldots&K\\ \pi(1)&\pi(2)&\ldots&\pi(K)\end{pmatrix}$. We also let $\pi^{-1}:\{1,\ldots,K\}\to\{1,\ldots,K\}$ be a permutation such that $\pi^{-1}(\pi(k))=k$ for $k=1,\ldots,K$. From the above definition, we immediately have $\mathbf{\Pi}^{T}\mathbf{\Pi}=\mathbf{I}_{K}$.

For the transmitted signal matrix $\mathbf{X}_{2}$, we have the following desired properties:

The proof of Proposition 2 is given in Appendix-B.

To further enhance the system reliability performance, we now optimize the user-constellation assignment $\pi$ and power allocation vector $\mathbf{p}$ for the proposed nn-mMIMO framework. For the transmitted signal matrix considered in (8), we have

We can see from (III-C) that $\mathbf{X}_{2}^{H}\mathbf{D}\mathbf{X}_{2}+\sigma^{2}\mathbf{I}_{2}$ and $\widetilde{\mathbf{X}}_{2}^{H}\mathbf{D}\widetilde{\mathbf{X}}_{2}+\sigma^{2}\mathbf{I}_{2}$ are independent of the permutation function $\pi$, but depends on the power allocation vector $\mathbf{p}=[p_{1},\ldots,p_{K}]^{T}$, and the information carrying vectors $\mathbf{s}=[s_{1},\ldots,s_{K}]^{T}$ and $\tilde{\mathbf{s}}=[\tilde{s}_{1},\ldots,\tilde{s}_{K}]^{T}$. In this case, the ML receiver given in (III-A) can be further simplified as

Inserting (III-C) into (7), and after some algebraic manipulations, we have

where

Recall that the power constraints are $\mathbb{E}\{|x_{k,t}|^{2}\}\leq P_{k}$ for $k=1,\ldots,K$ and $t=1,2$. That is, for the first and second time slots, we have $\mathbb{E}\{|x_{k,1}|^{2}\}=\frac{1}{p_{\pi(k)}\beta_{k}}\leq P_{k}$, and $\mathbb{E}\{|x_{k,2}|^{2}\}=\frac{p_{\pi(k)}E_{\pi(k)}d^{2}}{\beta_{k}}\leq P_{k}$, where $E_{k}=\frac{\mathbb{E}\{|s_{k}|^{2}\}}{d^{2}}$. The power constraints can thus be expressed as:

Our design can now be formulated into the following optimization problem.

For Problem 1, we first can attain that $f_{1}(\mathbf{p},\mathbf{s},\tilde{\mathbf{s}})\geq 0$ by applying the fundamental inequality in information theory [38, Lemma 2.29], where the equality $f_{1}(\mathbf{p},\mathbf{s},\tilde{\mathbf{s}})=0$ holds if and only if

Considering the fact that the joint minimization of $f_{1}(\mathbf{p},\mathbf{s},\tilde{\mathbf{s}})$ and $f_{2}(\mathbf{p},\mathbf{s},\tilde{\mathbf{s}})$ over $\{\mathbf{s},\tilde{\mathbf{s}}:\mathbf{s}\neq\tilde{\mathbf{s}}\}$ could be extremely tedious, we consider the minimization of $f_{2}(\mathbf{p},\mathbf{s},\tilde{\mathbf{s}})$ first, which is a lower bound of $\mathcal{D}_{\rm KL}(\mathbf{X}_{2}||\widetilde{\mathbf{X}}_{2})$ as $f_{1}(\mathbf{p},\mathbf{s},\tilde{\mathbf{s}})\geq 0$. We will verify the condition when the minimum of $f_{1}(\mathbf{p},\mathbf{s},\tilde{\mathbf{s}})$ and $f_{2}(\mathbf{p},\mathbf{s},\tilde{\mathbf{s}})$ can be achieved simultaneously. Mathematically, we temporarily focus on solving the following optimization problem:

We first consider the inner optimization problem in Problem 2. The denominator of (14a) is independent of $\mathbf{s}$ and the numerator is minimized when the sum terms $\sum_{k=1}^{K}s_{k}$ and $\sum_{k=1}^{K}\tilde{s}_{k}$ are the neighboring points on the sum-constellation, where the minimum value of $\big{|}\sum_{k=1}^{K}s_{k}-\sum_{k=1}^{K}\tilde{s}_{k}\big{|}^{2}$ is $d^{2}$. For notation simplicity, we define $\tilde{\mathbf{s}}=(\tilde{\mathbf{v}}+j\tilde{\mathbf{w}})d$, where $\tilde{\mathbf{v}}=[\tilde{v}_{1},\ldots,\tilde{v}_{K}]^{T}$ and $\tilde{\mathbf{w}}=[\tilde{w}_{1},\ldots,\tilde{w}_{K}]^{T}$. As the power constraint given in (14b) is independent of $\mathbf{v}$ and $\mathbf{w}$, Problem 2 can be split into two subproblems:

and

In the following, we only present the maximization of $f_{3}(\tilde{\mathbf{v}})$ over $\tilde{\mathbf{v}}$ in (15), and the maximization of $f_{4}(\tilde{\mathbf{w}})$ over $\tilde{\mathbf{w}}$ given in  (16) follows similarly and hence is omitted fore brevity. We now rewrite the objective function (15a) as