Favorable Propagation with User Cluster Sharing

Favorable propagation (FP) is a key concept underpinning the potential of massive MIMO systems [1]. As the number of base station (BS) antennas grows without bound, FP describes the resulting mutual orthogonality of user channel vectors. It is of intrinsic interest as a fundamental measure of interference [1] and is essential for the effective performance of matched filtering (MF) [1]. While alternative, interference-cancelling techniques such as zero-forcing (ZF) and minimum mean squared error (MMSE) combining do not require FP to combat interference, their performance is still aided by FP. This is shown in [2] where the performance of ZF and MMSE is analytically linked to a measure of FP.

FP is also an intrinsically valuable property for the practical realisation of massive MIMO systems. Algorithms are being designed based on the assumption that FP holds [3] which has been shown to simplify various aspects of radio resource management [4]. We note that MF is still the most preferred multiuser processing choice from an implementation viewpoint at the BS, since it only consumes a small fraction of the total baseband field programmable gate array (FPGA) resource usage, needing no real-time optimization routines. This is in stark contrast to the more sophisticated matrix inversion-based processing techniques, which require memory-consuming high-level synthesis to optimize and manage the real-time dataflow, in order to meet the overall post-scheduler processing latency requirements of 5G New Radio (5G-NR) systems.

There exists a great amount of analysis on FP for multi-user MIMO (MU-MIMO) systems with classical statistical channel models, ranging from simple i.i.d. Rayleigh fading [1], to more complex heterogeneous correlated Ricean models [5]. While providing valuable insight, their conclusions are by their nature linked to the statistical model parameters and not to the underlying physical and electromagnetic properties of the system and environment. Furthermore, propagation measurements suggest that millimeter-wave (mmWave) channels are better described by spatial or ray-based models [6, 7]. While explicitly incorporating the effects of different environmental properties and antenna topologies, the resulting complex structure of these models leads to often intractable analysis. Hence, a large portion of literature which examines FP with these physically-motivated models does so through simulation or ray-tracing [8, 9]. For example, FP is examined via ray-tracing of a dense urban location for a uniform linear array (ULA), uniform circular array (UCA), and uniform rectangular array (URA) in [8], and for a ULA and UCA in [9]. The latter additionally simulates the WINNER II channel model. The majority of works which provide a mathematical analysis rely on various assumptions around the angular distributions [10, 7, 11, 12]. The authors of [10] examine FP conditions using a ray based model with uniform angles for a ULA, horizontal URA (HURA), and UCA, while [11] and [12] do so for a ULA only. The analysis in[7] uses Gaussian and Laplacian distributed sub-rays, but still imposes the assumption of uniform cluster angles. In recent literature, there are only a handful of examples which provide generic closed-form analysis of MU-MIMO systems for ray-based models with arbitrary angular distributions [13, 2, 14].

To the best of the authors’ knowledge, all of the analytical work to date applies exclusively to ray-based models in the absence of user cluster sharing, making the channels of different users statistically independent. However, models such as the COST 2100 model [15] can generate a significant probability that a scattering cluster will be visible to multiple users. This prevalence of common scatterers is supported by measurement [16]. Although the significant impact of common scatterers on inter-user channel correlation (and thus FP) has been shown via simulation [17], all analytical progress neglects this and is instead facilitated by the assumption that the user channels are independent. In anticipation of the potential effects of common clusters, we examine the FP condition for a MU-MIMO system with ray-based channels featuring user cluster sharing and a range of antenna topologies. More specifically, our contributions are as follows.

• •

  While the existence of FP has been proven for a uniform linear array (ULA) and a horizontal uniform rectangular array (HURA) without cluster sharing in [18], we prove that FP holds with cluster sharing for ULA and HURA configurations. We also conjecture that FP holds for a UCA with and without user cluster sharing and provide a mathematical basis for the conjecture which may have applications to more general topologies (see Sec. III).

• •

  In Sec. IV, we derive analytical expressions for the mean squared distance (MSD) of the FP metric from its large system limit for a finite-antenna system equipped with a ULA, HURA, and UCA with cluster sharing. We use this as a measure of the “distance from FP” for a finite-antenna system.

• •

  In Sec. V we examine the ergodic rate of convergence to FP for a ULA with and without cluster sharing.

• •

  We observe that user cluster sharing has a detrimental impact on FP behavior, while topologies with larger azimuth footprints promote FP behavior. We discuss how the superior spatial resolution of such topologies provides resilience to the correlated channel conditions caused by cluster sharing, reducing the MSD from FP.

We consider uplink (UL) transmission in a single-cell MU-MIMO system where a base station (BS) is equipped with $M$ antennas and serves several single-antenna users (UEs). We adopt a clustered ray-based channel model. Within each drop, the angular parameters for the UEs are defined by the following process. $C_{T}$ scattering clusters are each assigned a random central azimuth angle of arrival (AAoA), $\phi_{c}$, and central elevation angle of arrival (EAoA), $\theta_{c}$, where $c=1,2,\dots C_{T}$. Each user $l$ is randomly allocated (with equal probability) a set of $C$ visible clusters, $\mathcal{C}^{(l)}$, where $C\leq C_{T}$ and $\mathcal{C}^{(l)}\subset\{1\dots C_{T}\}$. Each cluster $c$ scatters a user’s signal into $S$ sub-rays with random, instantaneous angular offsets $\Delta_{c,s}^{(l)}$ and $\delta_{c,s}^{(l)}$ in azimuth and elevation, respectively (exact distributions are discussed later in the text). Hence the AAoA of ray $s$ from user $l$ through cluster $c$ is $\phi_{c,s}^{(l)}=\phi_{c}+\Delta_{c,s}^{(l)}$ and similarly the EAoA is $\theta_{c,s}^{(l)}=\theta_{c}+\delta_{c,s}^{(l)}$. While one or more central angles may be shared by multiple UEs for whom a common cluster is visible, the subray offsets are assumed to be i.i.d. as different UE locations will lead to different reflection points along a common scatterer. The resulting $M\times 1$ channel for user $l$ is given by

$\displaystyle\mathbf{h}_{l}=\sum_{c\in\mathcal{C}^{(l)}}\sum_{s=1}^{S}\gamma_{c,s}^{(l)}\mathbf{a}\left(\phi_{c,s}^{(l)},\theta_{c,s}^{(l)}\right),$

(1)

where $\gamma_{c,s}^{(l)}=\sqrt{\beta_{c,s}^{(l)}}e^{j\Theta_{c,s}^{(l)}}$ is the ray coefficient, $\Theta_{c,s}^{(l)}\sim\mathcal{U}(0,2\pi)$ are i.i.d. phase shifts modelling fast fading, and the ray powers, $\beta_{c,s}^{(l)}$, satisfy $\sum_{c\in\mathcal{C}^{(l)}}\sum_{s=1}^{S}\beta_{c,s}^{(l)}=\beta^{(l)}$. We define $\beta^{(l)}$ as the overall link gain between user $l$ and the BS, which is divided amongst the visible clusters such that the cluster powers, $\beta_{c}^{(l)}$, add up to $\beta^{(l)}$. Each subray $s$ has ray power $\beta_{c,s}^{(l)}=\beta_{c}^{(l)}/S$. Note that the large-scale fading simply scales the user channel vectors. It has no impact on the existence of FP, and as shown in Sec. IV-A, it scales the MSD from FP. We include $\beta^{(l)}$ in the analysis for completeness but set $\beta^{(l)}=1\hskip 2.84544pt\forall\hskip 2.84544ptl$ for all numerical results. This normalises the channel vectors.

The steering vector $\mathbf{a}(\phi,\theta)$ for an arbitrary ray with angles $\phi$ and $\theta$ is governed by the antenna topology at the receiver. For a ULA located along the $x$-axis with an antenna spacing of $d_{x}$ wavelengths, the $m^{\textrm{th}}$ entry of $\mathbf{a}(\phi,\theta)$ is given by [10]

$\displaystyle\left(\mathbf{a}(\phi,\theta)\right)_{m}=e^{j2\pi(m-1)d_{x}\sin{\phi}}.$

(2)

For a HURA situated in the azimuth plane with antenna spacings $d_{x}$ and $d_{y}$,

$$\mathbf{a}(\phi,\theta)=\mathbf{a}_{x}(\phi,\theta)\otimes\mathbf{a}_{y}(\phi,\theta).$$

(3)

Let $M_{x}$ and $M_{y}$ be the number of antennas along the $x$- and $y$-axis respectively, then the entries of the $M_{x}\times 1$ vector $\mathbf{a}_{x}(\cdot)$ are defined as [10]

$\displaystyle\left(\mathbf{a}_{x}(\phi,\theta)\right)_{m}=e^{j2\pi d_{x}(m-1)\sin{\theta}\cos{\phi}}$

(4)

and those of the $M_{y}\times 1$ vector $\mathbf{a}_{y}(\cdot)$ are

$\displaystyle\left(\mathbf{a}_{y}(\phi,\theta)\right)_{m}=e^{j2\pi d_{y}(m-1)\sin{\theta}\sin{\phi}}.$

(5)

Finally, the entries of the steering vector for a UCA in the azimuth plane with antenna spacing $d_{r}$ are given as [10]

$\displaystyle\left(\mathbf{a}(\phi,\theta)\right)_{m}=e^{j\frac{\pi d_{r}}{\sin(\pi/M)}\sin{\theta}\cos{\left(\phi-\psi_{m}\right)}},$

(6)

where $\psi_{m}=2\pi m/M$. We adopt the simple cluster sharing mechanism previously described to emulate that in the COST 2100 model - the most complete sharing mechanism which has been proposed thus far. In COST 2100, each cluster is allocated one or more visibility regions (VR); a cluster is said to be visible to a user if that user falls within the cluster’s VR(s). A cluster can thus be visible to, and hence shared by, multiple UEs. Our model provides a simple mechanism to introduce this feature. Analysis of cluster sharing only depends on the probability that a randomly selected ray from one user originates from the same cluster as a randomly selected ray from another user. This sharing probability is denoted by $p_{\textrm{sh}}$ and can be simply adjusted by changing $C_{T}$ as the sharing mechanism gives $p_{\textrm{sh}}=1/C_{T}$. Although this mechanism limits the possible values of $p_{\textrm{sh}}$ in simulations, it provides a much simpler simulation method for the purpose of verifying analytical results which can then be used for any value of $p_{\textrm{sh}}$ between 0 and 1 in the analysis.

Note that this model contains several approximations. For example, the visibility of clusters might be a function of the distance between the UE and the cluster. Additionally, a per-cluster shadowing might occur or, for very small distances between two UEs, the angles of the subrays within the cluster might become correlated. While incorporating these effects would further refine the channel model, it would make analytical treatment very difficult. The model we consider is thus a compromise, but importantly one that is more accurate than all the models previously used.

In this section we examine the existence of FP for ray-based channels with inter-user cluster sharing. Sec. III-A presents generic conditions for FP to hold with these models, while Sec. III-B examines these conditions for a ULA, HURA, and UCA. Recall that FP has been proven in [18] for a ULA and HURA in the absence of cluster sharing. Here, we extend these results by considering shared clusters. We also conjecture the existence of FP for a UCA with and without cluster sharing and provide a posible methodology for a mathematical proof.

In this section, we examine the MSD of $\mathbf{h}_{l}^{\textrm{H}}\mathbf{h}_{l^{\prime}}/M$ from the FP limit as a means of assessing the proximity to FP conditions for finite-antenna systems. Sec. IV-A provides a generic analysis of this metric for ray-based models with user cluster sharing, while Sec. IV-B provides analytical expressions for a ULA, HURA, and UCA. Note that a similar distance metric is used in [10], while alternative metrics can be found in, for example, [1].

We now examine the rate of decay of $\kappa^{\textrm{FP}}$ for a ULA in the large system limit using the methodology from [14], extended to accommodate user cluster sharing. From (13), this decay rate is determined by the decay rates of $K_{c}$ and $K_{s}$. For $K_{c}^{\textrm{ULA}}$ we begin by reducing (16) to a single summation of the form

	$\displaystyle K_{c}^{\textrm{ULA}}$	$\displaystyle=\frac{1}{M}\left(1+2\sum_{m=1}^{M-1}\left(1-\frac{m}{M}\right)\left|\mathbb{E}\left[e^{-j2\pi d_{x}m\sin\phi}\right]\right|^{2}\right)$
		$\displaystyle=\frac{1}{M}\left(1+2\nu\right).$		(26)

In [18], it is shown that $\nu$ is $\mathcal{O}(\log M)$ in most situations, but in the absence of end-fire radiation $\nu$ is $\mathcal{O}(1)$. Hence, $K_{c}^{\textrm{ULA}}$ decays as $\mathcal{O}\left(1/M\right)\leq\mathcal{O}\left(K_{c}^{\textrm{ULA}}\right)\leq\mathcal{O}\left(\log M/M\right)$. For $K_{s}^{\textrm{ULA}}$, we have the form

	$\displaystyle K_{s}^{\textrm{ULA}}=\frac{1}{M^{2}}\sum_{m,m^{\prime}}^{M-1}\mathbb{E}\bigg{[}$	$\displaystyle e^{-j2\pi d_{x}\left(m-m^{\prime}\right)\left(\sin\left(\phi_{c,s}^{(l)}\right)-\sin\left(\phi_{c,s^{\prime}}^{(l^{\prime})}\right)\right)}\bigg{]}$
	$\displaystyle\begin{split}=\mathbb{E}_{\phi}\bigg{[}\frac{1}{M^{2}}\sum_{m,m^{\prime}}^{M-1}\bigg{|}\mathbb{E}_{\Delta}\bigg{[}&e^{-j2\pi d_{x}\left(m-m^{\prime}\right)\left(\sin\left(\phi_{c}+\Delta_{c,s}^{(l)}\right)\right)}\bigg{]}\bigg{|}^{2}\bigg{]}\end{split}.$			(27)

The contents of the expectation $\mathbb{E}_{\phi}[\cdot]$ in (V) are identical in form to (16) with an additional angular offset $\phi_{c}$ which is constant relative to the expectation. By the same process used to analyse the decay rate of $K_{c}^{\textrm{ULA}}$, we find that $K_{s}^{\textrm{ULA}}$ also decays as $\mathcal{O}(1/M)\leq\mathcal{O}(K_{s}^{\textrm{ULA}})\leq\mathcal{O}(\log M/M)$. This implies the decay rate of $\kappa^{\textrm{FP}}$ is $\mathcal{O}(1/M)\leq\mathcal{O}(\kappa^{\textrm{FP}})\leq\mathcal{O}(\log M/M)$.

This section presents numerical results and a discussion of the trends observed in $\kappa^{\textrm{FP}}$ for different antenna topologies and varying levels of cluster sharing. Recall that higher levels of $\kappa^{\textrm{FP}}$ indicate a slower convergence to FP.

Table I gives the parameters considered. The angular spread values for Scenarios 1 and 2 were obtained from [6] and [20], respectively, and a square HURA is considered. Central cluster angles are Gaussian distributed in azimuth and Laplacian distributed in elevation while subray angles are Laplacian distributed in both cases, in accordance with [6] and [20].