Multi-Link Propagation Modeling for Beyond Next Generation Wireless

Multi-Link Propagation Modeling for Beyond Next Generation Wireless (Invited Paper) Claude Oestges ICTEAM, Universit´e catholique de Louvain Louvain-la-Neuve, Belgium Email: [email protected]

Abstract—In this review paper, a number of recent contributions in the field of multi-link propagation are presented and analyzed, with a particular focus on beyond next generation wireless communication networks. Advances in channel sounding, characterization and modeling are described in three areas: (i) regarding cooperative channel fading statistics, the impact of node mobility is highlighted; (ii) metrics for characterizing the separation between multiple single- and/or multi-antenna channels are detailed; (iii) new geometry-based approaches with multi-link capabilities, including WINNER and COST models, are presented and compared. Finally, a couple of key features not sufficiently included in current models are discussed.

I. I NTRODUCTION Multi-link propagation has become a central paradigm for beyond newt generation wireless systems. By definition, multilink channels involve communications between more than two nodes that are spatially distributed over large areas. This formalism extends the concept of MIMO transmissions, where each node itself is based on a compact antenna array. Applications of multi-link transmissions include cooperative and relaying schemes, as well as multi-user and/or multibase station signal processing, and distributed networking. As an example, the architecture developed within the FP7 BuNGee project (Beyond Next Generation Mobile Broadband) aims to improve the overall infrastructure capacity density of mobile networks to an ambitious goal of 1Gbps/km2 in economical cellular deployment [1]. It relies on an aggressive frequency reuse made possible by combining a multi-beam Hub Base Station (HBS) connected to the operator back-haul with distributed multi-antenna Access Base Stations (ABSs) connected to the HBS by a self-backhaul link on one side and to mobile terminals (MTs) on the other side. In this example, multi-link propagation issues arise when considering not only each HBS-ABS-MT link (which can be seen as a regenerative relay transmission), but also the interference between various HBS-to-ABS links (from one HBS to different ABSs sharing the same frequency). The above example clearly illustrates two different situations where so-called multi-link channels are encountered in practice. This work was partially supported by the European Commission in the framework of the FP7 STREPS BuNGee (contract no. 248267), by the Vienna Science and Technology Fund in the FTW project PUCCO and by the Belgian Fonds de la Recherche Scientifique FNRS (FRS-FNRS). It was also carried out in cooperation within the European COST 2100 and IC1004 Actions.

In the past, simple channel models for cooperative/multilink signal processing techniques were developed ∙ taking possibly path-loss into account (making the SNR on each link deterministically related to the inter-node distance), ∙ often without considering shadowing and/or shadowing correlation, ∙ for Rayleigh fading channels, ∙ for single-antenna nodes or, in the case of multi-antenna nodes, for independent and identically distributed (i.i.d.) channels on different links. However in real-world, shadowing is always present, randomizing the relationship between path loss and distance. Furthermore, it may correlated between different links. Also, both the transmitter and receiver can be mobile in a peer-to-peer cooperative scenario, so that the Rayleigh fading assumption might not hold true anymore. Finally, each node might be equipped with multiple antennas: these multi-antenna channel channels can be correlated at the link level (invalidating the socalled i.i.d. assumption on the link channel matrix description), but more importantly, the degree of correlation between multiple links cannot be described by shadowing correlation only, but must involve the different link subspaces. These issues open up a number of modeling problems barely addressed in the past literature. The challenges faced by propagation researchers are therefore to model ∙ shadowing and shadowing correlation properly, ∙ fading statistics for peer-to-peer and multi-hop channels, ∙ MIMO channel separation for multi-antenna networks. In wireless propagation modeling, different approaches can naturally be used. On the one hand, empirical and stochastic models rely on measurements: their derivation consists mostly in curve-fitting and is thus very straightforward, at the price of a lack of generality. On the other hand, deterministic approaches, such as ray-tracing, are theoretically rigorous but suffer from a high complexity (so that any practical implementation requires simplifying assumptions compromising the accuracy) and must be computed for every new environment. Subsequently, geometry-based models (such as those developed by COST and WINNER projects) appear as an interesting alternative balancing generalization and complexity. The paper is organized as follows. In Section II, we present

and compare three main sounding techniques specifically developed for multi-link channels, together with a selection of recent experimental campaigns. In Section III, we deal with cooperative channels, with a focus on the impact of node mobility upon the fading statistics. Section IV is dedicated to multi-link channel separation metrics, for both single- and multi-antenna links. Finally, newly developed geometry-based approaches able to model multi-link properties are detailed in Section V. The following notations are used throughout the paper: ℰ {⋅} designates the expectation, Tr(⋅) is the trace, det [⋅] is the determinant, ∥⋅∥𝐹 is the Frobenius norm, and J0 and K0 (⋅) are the Bessel/modified Bessel functions of the first/second kind at the zero-th order.

however is to operate two compatible channel sounders and to maintain the synchronization of multiple clocks. B. Selected Measurement Campaigns Various properties of distributed channels have been measured for stationary or mobile nodes in specific scenarios. ∙

∙

II. M ULTI -L INK C HANNEL S OUNDING A. Multi-Link Channel Sounding Techniques While channel sounders have been used for many years, the experimental characterization of multi-link propagation requires innovative solutions to extend the inherently singlelink capability of classical channel sounders. Three main techniques have been developed over recent years that allow to measure multiple links, i.e. links between more than two nodes. 1) Single-sounder sequential measurements: The first method is based on the sequential use of a single sounder (SISO or MIMO): that implies that each link is successively measured in a classical way [2], [3]. Naturally, the drawbacks of such method are evident: not only the environment may change between runs [4], but also the sounder calibration might vary, depending on the time lag between each run, as clocks tend to drift over time. Hence, even if the environment is kept constant, phase-synchronized multi-user measurements are impossible. However, in controlled environments, this represents a cheap and possibly accurate technique. 2) Single-sounder multi-node measurements: The second method still relies on a single MIMO sounder, connecting the equipment ports to distributed nodes by means of long RF cables [4], [5]. While this guarantees the phase synchronization (up to the sounder phase stability), the measurement range is limited to indoor and outdoor-to-indoor scenarios owing to the length of the cables and the available SNR, even with low-loss cables (typically, 0.2dB/m at 2.5 GHz). Furthermore, in mobile scenarios, cable movements may introduce phase jitter. An alternative is to use optical fibers, which allow for longer distances. Note that this solution has two weak points: it is expensive, and its calibration procedure is heavy (as the calibration must be carried out for every cable separately, because of RF/optical transducers). 3) Multi-sounder (transceiver) measurements: Whenever possible, using multiple MIMO channel sounders is the best solution [6], as this is the closest to a real-world scenario. This solution is more detailed in the companion paper by Haneda [7]. Typically, such measurement architecture involves a single transmitter with multiple receivers. The main challenge

∙

∙

∙

∙

∙

In [8], various transmit diversity schemes were considered based on measuring distributed channels in an indoor environment, and mobile multi-link measurements were presented in [9] for indoor MIMO channels with two base stations and two users. In both cases, the multi-link issue is not addressed. In [10], various properties of indoor peer-to-peer channels have been analyzed for static nodes, and a model of body shadowing is derived, while the fading statistics are estimated by a Gamma distribution. In [6], indoor multi-link MIMO channel measurements were performed using two wideband MIMO channel sounders at 5.3 GHz with 120 MHz bandwidth. The sounding equipment consisted in a single mobile terminal and two access points synchronizing the MS. The MS and BSs equip spherical or cylindrical dual-polarized antenna arrays, leading to measurements of a 30 × 32 MIMO channel matrix for each link. The main output of this work is the identification of so-called common scatterers [11], [12], whose concept was applied in the COST 2100 channel model [7], [13] (see Section V-A). In [4], [5], [14], [15], outdoor-to-indoor and indoor multiuser MIMO measurements were conducted in two different office environments (a typical cubicle-style office room and different rooms aligned along a corridor and separated by brick or plasterboard walls), where a number of indoor single- and multi-antenna terminals were kept static or moving. Both campaigns used long RF cables to connect the various nodes to the sounder. The measured results were used to characterize the peer-to-peer channels (see Section III-C) as well as the distance between multiple MIMO links (see Section IV) Outdoor narrowband multi-user MIMO measurements were conducted in [16] using a cellular testbed. While this experiment targeted the testing of various multiple access techniques, multi-link statistics could also be derived. Multi-BS outdoor measurements were carried out in [17], [18], using coherent sounding techniques. In particular, the correlation of large-scale parameters (shadowing, angle-spreads) between different links (i.e. from one user to multiple base stations or one base station to multiple users) was investigated: in [18], it is shown that angular spreads can be correlated in a multi-base station scenario if the base station separation is small (though the correlation is significantly smaller than the shadowing correlation), but that for widely separated base stations, large-scale parameters are essentially uncorrelated. Similar results were observed in [3], where a sequential sounding technique was used to compare the achievable

to a 𝜒2 distribution (with two degrees of freedom), ( ) 1 𝑠 exp − . (2) 𝑝𝜒2 (𝑠) = 2𝜎1 𝜎2 2𝜎1 𝜎2

1 h and h (Rayleigh) 1

2

0.9

h1.h2, uncorrelated (double−Rayleigh)

0.8

h1.h2, correlated (χ )

2

Probability density

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5 Fading amplitude

2

2.5

3

Fig. 1. Global channel fading distribution for relay channels (ℎ1 and ℎ2 are Rayleigh distributed with 2𝜎12 = 2𝜎22 = 1

sum-rate capacity using cooperative multi-cell transmissions, and in [19] where the different base stations were connected and synchronized via fiber cables. III. C OOPERATIVE C HANNEL S TATISTICS A. Multi-Link Aspects in Cooperation As mentioned above, one example of multi-link propagation modeling is the cooperative channel, encountered for instance in the various implementations of relaying techniques [20]. In this Section, we detail two types of relay channels: amplify & forward (AF) and regenerative (also known as decode and forward). In the former, the relay is transparent, and is only used as an amplifier for the second link, while in the latter, the relay decodes the message sent by the transmitter before sending it to the final receiver. B. Amplify & Forward Relay Channel In that scenario, the global channel is simply the multiplication of Tx-Relay and Relay-Rx channels. Neglecting shadowing at this stage, and denoting by ℎ1 and ℎ2 the fading channels on the Tx-to-relay and relay-to Rx links, this implies that the global channel reads as ℎ𝐴𝐹 = 𝐴ℎ1 ℎ2 , where 𝐴 is the amplification factor at the relay (which can be arbitrarily set to 1). Usually, ℎ1 and ℎ2 are taken as Rayleigh distributed with energy 2𝜎12 and 2𝜎22 , so that the global channel statistics depend on the correlation between ℎ1 and ℎ2 , as illustrated in Figure 1: ∙ if the individual channels are independent, which is usually the case as far as fading is concerned, the amplitude 𝑠 = ∣ℎ𝐴𝐹 ∣ is double-Rayleigh distributed ( ) 𝑠 𝑠 𝑝𝐷𝑅 (𝑠) = 2 2 K0 − , (1) 𝜎1 𝜎2 𝜎1 𝜎2 ∙

if the individual channels are correlated, the distribution is more complex, and tends, for fully correlated channels,

Note however that the Rayleigh assumption used above on the individual links implies that the relay is mobile (and Tx/Rx are fixed) or that the relay is fixed and both Tx and Rx are mobile. However, if the relay, Tx and/or Rx are all moving, this assumption is questionable. In [5], [14], [15], [21], it is shown that for peer-to-peer channels in various office environments, the links (some of them involving two mobile nodes) were for non-negligible part of the time following a distribution worse than Rayleigh (up to double-Rayleigh in some cases). It would be fallacious however to conclude that double-mobile links yield double-Rayleigh, and vice-versa. This also depends on whether single or double bounce scattering is encountered. To obtain a double-Rayleigh fading on a single link, the following conditions must be jointly met: ∙ ∙

all multipaths must undergo a double (or more) scattering process, the Tx-Rx distance must be large (compared to the distance from Tx/Rx and the local Tx/Rx scatterers).

When only some multipaths are doubly-scattered, or when the Tx-Rx distance is small to average, the distribution will be a mix between Rayleigh and double-Rayleigh fading. With this observation on mind, a versatile probability density function was introduced in [21], assuming that the scalar channel could be expressed as ℎ = 𝑤0 𝑒𝑗𝜃 + 𝑤1 𝐺1 + 𝑤2 𝐺2 𝐺3 , where 𝐺1 , 𝐺2 , 𝐺3 are i.i.d. complex normal random variables with zero mean and unit variance, and 𝜃 is a constant phase angle in [0, 2𝜋]. The three terms can be interpreted as a line-ofsight component, a single-bounce Rayleigh-fading component and a double-bounce double-Rayleigh-fading component, respectively. The weighting factors 𝑤0 , 𝑤1 , 𝑤2 > 0 determine the relative powers of the three components. The probability density function of 𝑠 = ∣ℎ∣ is then given by the so-called second-order scattering fading (SOSF) distribution, ∫ ∞ 2 2 4J0 (𝑠𝜔) J0 (𝑤0 𝜔) 𝜔 𝑒−𝑤1 𝜔 /4 𝑑𝜔. (3) 𝑝SOSF (𝑠) = 𝑠 4 + 𝑤22 𝜔 2 0 Note that ℰ{𝑠2 } = 1 is achieved when 𝑤02 + 𝑤12 + 𝑤22 = 1 so that the distribution can be specified by two parameters [21], 𝑤22 , + 𝑤12 + 𝑤22 𝑤02 𝛽= 2 , 𝑤0 + 𝑤12 + 𝑤22

𝛼=

𝑤02

(4) (5)

where (𝛼, 𝛽) are constrained to the triangle 𝛼 ≥ 0, 𝛽 ≥ 0, 𝛼+ 𝛽 ≤ 1. The SOSF distribution naturally encompasses Ricean fading (𝛼 = 0, with 𝛽/(1 − 𝛽) being the K-factor), Rayleigh fading (𝛼 = 𝛽 = 0), as well as Double-Rayleigh fading (𝛼 = 1, 𝛽 = 0). In such cases, the global AF channel naturally follows more complex distributions than (1) or (2). If shadowing is accounted for, the global channel can be decomposed as a fading part modeled as above, and a shad-

owing part resulting from the multiplication of both shadowing levels. The global shadowing (in dB) is therefore expressed as the sum of two (possibly correlated) Gaussian variables. This calls for a good characterization of the shadowing correlation, as dealt with in Section IV-A. C. Regenerative Relay Channel In this case, the relay decodes and re-encodes the information sent by the transmitter. The global Tx-to-Rx channel is therefore made of two successive links, with decodeencode process in- between. Since both transmissions are decoupled, each segment can be well characterized in terms of performance, and classical fading models, such as Rayleigh or Rice, might remain valid. While the small-scale fading can be assumed as independent on both links, two other aspects need again to be considered: ∙ the shadowing correlation highly impacts the correlation between the performance on both links, hence the global performance (see Section IV-A), ∙ the fact that both link ends might be mobile similarly questions the validity of the traditional Rice/Rayleigh fading assumptions. IV. M ULTI -L INK C HANNEL S EPARATION A. Shadowing Correlation Models The first metric characterizing the dissimilarity (or separation or distance) between two links is the shadowing correlation. Defined by the correlation coefficient 𝜌𝑠𝑐 , this is a complex mechanism, which occurs when two links share some dominant propagation paths. The goal of shadowing correlation models is to propose simple methods to evaluate the shadowing correlation between two links, usually considering cellular communications, i.e. one base station and multiple users, or one user communicating towards two base stations. 1) Measurement-based analytical models: Analytical models of 𝜌𝑠𝑐 can be classified as distance-only, angleonly, distance-angle-separable and finally, distance-angle. In distance-only models, the shadowing correlation is modeled as a function of the distance between the two users. These models actually better fit the shadowing auto-correlation than the cross-correlation between users. They are generally fitted by 𝜈 (6) 𝜌𝑠𝑐 = 𝑒−(Δ𝑅/Δ𝑅0 ) where Δ𝑅 is the distance between the users, Δ𝑅0 corresponds to the decorrelation distance, and 𝜈 is a tunable parameter. In the original model of Gudmundson [22], this exponent is set to unity. Another type of models [23] considers that shadowing correlation is related to the azimuth difference ΔΘ between the links, and express 𝜌𝑠𝑐 as 𝜌𝑠𝑐 = 𝑒−𝛼ΔΘ

(7)

𝜌𝑠𝑐 = 𝐴 cos ΔΘ + 𝐵

(8)

or

or also via piecewise-linear functions fitted to measured data [24]. In (7) and (8), 𝐴, 𝐵 and 𝛼 are tunable parameters. Naturally, hybrid models can be created by multiplying expressions provided by distance-only and angle-only models. Finally, more elaborate models have been developed, where the distance and azimuth dependences cannot be separated. As an example, the model introduced in [25] not only depends upon Δ𝑅 and ΔΘ, but also on the individual ranges 𝑅1 and 𝑅2, ⎧ √ 𝑅1 ⎨ , ΔΘ < ΔΘ0 √ ( 𝑅 2 )𝛾 𝜌𝑠𝑐 = (9) 𝑅1 ΔΘ0 ⎩ , ΔΘ ≥ ΔΘ 0 𝑅2 ΔΘ Δ𝑅0 , Δ𝑅0 being the decorrewhere ΔΘ0 = 2 arcsin 2 min{𝑅 1 ,𝑅2 } lation distance introduced in (6). For interested readers, a more detailed analysis of shadowing correlation models is provided in [24]. 2) Random field-based model: A simple model for both outdoor and indoor fixed wireless multi-link channels was proposed in [26], based on wide sense stationary Gaussian random fields. The formulation allows to calculate the shadowing correlation between any two link pairs. However, the model always produces a positive shadowing correlation, whereas previous measurements [14], [27] also reveal negative correlations, and it has been shown in [28] that some cooperative protocols are extremely sensitive to the sign of the shadowing correlation. 3) Measurement-based stochastic models: It is sometimes not possible to relate the shadowing correlation with geometry considerations, especially in indoor scenarios. In [27], the interlink shadowing correlation was found to vary between -0.5 and 0.5 in office scenarios. In [5], [14], [15], peer-topeer channels were investigated in various office environments, consisting in a large indoor area containing individual cubiclestyle offices and in an office building with rooms distributed along a corridor. As an example, it was found that in the cubicle-style office, the shadowing correlation could be very high (positively or negatively) and was related to the node mobility: the average correlation between two links is found to be more positive when both links share a common moving node.

B. MIMO Multi-Link Separation So far, we have considered the separation between to SingleInput Single-Output (SISO) links, which is mainly characterized by the shadowing correlation (as small-scale parameters, such as fading, are usually decorrelated). Considering Multiple-Input Multiple-Output (MIMO) links, the angular properties should be considered. If, for two links sharing a common Tx, the Tx angular spectrum is similar, it is intuitive to understand that using space division access at the Tx is not possible, as both links are likely to be seen through the same subspace by the Tx array. The subspace separation appears therefore as a critical parameters for the performance of multilink signal processing techniques. However, the separation between two MIMO links can be expressed through many distance metrics [29], [30], the most

important of them being detailed below. 1) Channel Matrix Collinearity: The similarity between two (complex-valued) matrices H0 and H1 of identical can be measured by the collinearity [4]:  ( )   Tr H0 H𝐻 1  𝑐(H0 , H1 ) = (10) ∥H0 ∥𝐹 ∥H1 ∥𝐹 The channel matrix collinearity actually compares the subspaces of both matrices, and ranges between zero (for orthogonal matrices) and one. A channel matrix distance may naturally be expressed as 1 − 𝑐(H0 , H1 ). 2) Correlation Matrix Distances: Various metrics have been introduced to measure the distance between the correlation matrices of two links. The first one is introduced by [31] and is defined as { } ( ) Tr R0 R1 𝑑𝑐𝑜𝑟𝑟 R0 , R1 = 1 − , (11) ∥R0 ∥𝐹 ∥R1 ∥𝐹 A second metric is the geodesic distance, measured on the convex cone formed by the space of Hermitian and positive definite matrices (such as correlation matrices), ) ( [ ( ) ]  1/2 ∑  log 𝜆 −1  , (12) 𝑑𝑔𝑒𝑜𝑑 R0 , R1 = 𝑘

R0 R1

𝑘

] [ where 𝜆R−1 R1 is the 𝑘 th eigenvalue of R−1 0 R1 . Alter0 𝑘 natively, one may also define a distance between correlation matrices as ( )    (13) 𝑑𝐹 R0 , R1 = I𝑛 − R−1 0 R1 𝐹 , where the dimension of the correlation matrices is 𝑛 × 𝑛. 3) Ratio of Condition Numbers: This metric is given by ( ) 𝜎max (H0 ) / 𝜎max (H1 ) 𝜒(H0 , H1 ) = 10 log10 , (14) 𝜎min (H0 ) 𝜎min (H1 ) where 𝜎max (H) denotes the largest singular value of H. With this metric, the channel similarity is indicated by values close to 0 dB. Note that the condition number ratio and the matrix collinearity provide a different notion of the (dis)similarity of the spatial structure: while the condition number ratio illustrates whether some channels are more directive, the collinearity measure is sensitive to the alignment or nonalignment of the preferred directions. 4) Mutual Information under Interference: Another measure tightly related to the capacity of multi-user communications is the average spectral efficiency under interference [32]. This measure is defined as ⎧ [ ( )−1 ]⎫ 𝑛𝑖 ⎬ ⎨ ∑ 𝑛 𝑡 I𝑛𝑟 + H𝑘 H𝐻 , ℰ log2 det I𝑛𝑟 + H0 H𝐻 0 𝑘 ⎭ ⎩ 𝜌 𝑘=1

(15) where H0 is the MIMO channel matrix for the considered link and H𝑘 (𝑘 = 1, . . . , 𝑛𝑖 ) are the interfering channel matrices of the 𝑛I interfering links. A convenient property of this measure is that its value depends on both the singular values and the

eigenstructure of the channels. Further elaborating on this concept, the authors in [30] replace the channel matrices in (15) by the correlation matrices R𝑟,0 and R𝑟,I at the receiver (i.e. the base station in an uplink scenario), } { = UΛ0 U𝐻 , R𝑟,0 = ℰ H0 H𝐻 (16) {𝑛 0 } 𝑖 ∑ R𝑟,I = ℰ H𝑘 H𝐻 (17) = VΛI V𝐻 . 𝑘 𝑘=1

Hence, a spectral efficiency metric is defined as [ ( )−1 ] 𝑛𝑡 𝐽 = log2 det I𝑛𝑟 + R𝑟,0 R𝑟,I + I𝑛𝑟 . 𝜌

(18)

For fixed eigenvalues of R𝑟,0 and R𝑟,I , the value of 𝐽 depends on the degree of alignment between the subspaces characterized by the respective eigenvectors U and V. The worstcase interference (𝐽 = 𝐽min occurs when the eigenspaces of signal and interference are identical (the strongest interference mode affects the strongest signal mode), whereas the largest mutual information metric is achieved (𝐽 = 𝐽max ) when the strongest eigenmode of the interference aligns with the weakest eigenmode of the signal and vice versa. To emphasize the role of the subspace alignment, a scaled metric (bounded between 0 and 1) is eventually defined as 𝐽 − 𝐽min . 𝐽˜ = 𝐽max − 𝐽min

(19)

V. G EOMETRY-BASED S TOCHASTIC C HANNEL M ODELS WITH M ULTI -L INK C APABILITIES A. COST 2100 Multi-Link MIMO Channel Model The COST 2100 channel model [13] is the latest of the COST channel model family, building in particular on COST 259 and 273 model formalisms. The former was originally established for the simulation of systems with multiple antenna elements at either the base station (BS) or the mobile terminal (MT) [33], [34]. The model describes the joint impacts of small-scale as well as large-scale effects, and covers macro-, micro- and picocellular scenarios. It consists of three different layers. ∙

∙

At the top layer, there is a distinction between different radio environments, which represent groups of environments that have similar propagation characteristics (e.g. Typical Urban). The goal of the second layer is to model non stationary large-scale effects, i.e., the variations of channel characteristics as the mobile terminal moves over large distances (typically, 100 wavelengths or more). These effects include the appearance/disappearance of remote scattering clusters, shadowing, changes of direction of departure/arrival (DoA/DoD), or in the delay-spread. The large-scale effects are described by their probability density functions, whose parameters differ for the different radio environments.

∙

Single cluster

The bottom layer deals with small-scale fading, caused by multipath interference. Statistics of the small-scale fading are determined by the large-scale effects.

Multipath components with similar delays and directions are usually grouped into so-called clusters. This clustering largely reduces the number of parameters required to describe the channel description. At the beginning of a simulation, clusters of scatterers (one local cluster, around the MT, and several remote scatterer clusters) are distributed at random fixed locations in the coverage area, according to a specified probability density function. Each of the clusters has a smallscale averaged delay-angle power spectrum assumed to be separable on the cluster level, the power delay spectrum being taken as exponential and and the angular power spectrum, as Laplacian in azimuth and elevation. Note that this does not imply that the total delay-angle power spectrum is separable. The spectra are parameterized by a cluster RMS delay and angular spread, and these intra-cluster spreads are correlated random variables characterized by their joint probability density functions [35], [36]. Finally, each scatterer is characterized by a random complex scattering coefficient, usually taken as complex Gaussian distributed. As the MT moves, the delays and angles between the clusters are obtained deterministically from their position and the positions of BS and MT, while large-scale effects (including changes in the intra-cluster spreads) are obtained stochastically. The angularly resolved complex impulse responses are then calculated similarly as in ray-tracing tools, but with point-like Gaussian scatterers. Analogous to ray-tracing, the COST 259 model provides continuous directional impulse responses (within a certain range of validity). A concise description of the model can be found in [33], while references to detailed descriptions of the final version are given in [34]. The COST 273 model [37]–[39] can be seen as the doubledirectional extension of the COST 259 model. Because it is double-directional, the joint transmit-receive angular power spectrum needs to be accounted for. Hence, the generation of scattering clusters is implemented as to include different scattering mechanisms, illustrated in Figure 2: local clusters around Tx and/or Rx (with a large angle-spread), singleinteraction clusters and twin-clusters for multiple interactions. Single-interaction clusters are located at randomly chosen locations in the 2-D plane, and DoD, DoA and delay are computed by means of geometrical relationships. Multiple interactions are modeled thanks to a twin-cluster concept. Each physical cluster is divided into two clusters, one corresponding to the Tx side, the other one corresponding to the Rx side. The advantage is that the angular dispersion can be modeled independently at Tx and Rx, based on the marginal angular power spectra. Note that this does not mean that the joint angular power spectrum can be modeled as the product of the marginal spectra. Indeed, each DoD is related to one DoA, and vice-versa. However, there is no geometrical relationship between DoD, DoA and delay (unlike the case of singleinteraction clusters).

: MPC Local cluster

Twin cluster

BS MT

Fig. 2. General structure of COST 273 and 2100 models, with local, singlebounce and twin clusters

The temporal evolution of the channel is governed by the concept of so-called visibility regions: a visibility region (VR) is a circular region with fixed size in the azimuth plane, which determines the visibility of a given cluster. When the MT is moving inside the visibility region associated to a given cluster, the latter smoothly increases its visibility in the link. When the MT is located in an area where multiple visibility regions overlap, multiple clusters are simultaneously active. This process actually cause clusters to appear and disappear in a natural way as the MT moves within the cell. Finally, the COST 2100 model extends the COST 273 approach by considering ∙ a polarization model of multipath contributions, ∙ the addition of dense multipath components to the specular contributions, ∙ the extension to multi-link (multi-cell, multi-user) MIMO scenarios. Let us start by noting that the single-link COST 273 and 2100 models are multi-user by definition, as the propagation environment is characterized with respect to one BS irrespective of the MT location, so that channels between one BS and multiple MTs dropped in different locations can be simultaneously modeled. A similar principle was further applied within COST Action 2100 to describe channels in multipleBS multiple-MT scenarios, simply by adding up multiple single-link channel realizations. However, since clusters and the corresponding visibility regions are generated separately and independently for each BS, there is no guarantee that the multiple links reflect the important features of the multi-link scenarios realistically, in particular the large-scale correlations, such the shadowing correlation. One possible modeling approach is therefore to consider that clusters are simultaneously visible in different links, i.e. that some clusters are common between multiple links [12]. This solution, which is further detailed in [7] requires to characterize the cluster visibility in different links, and this, without altering the physical properties of the clusters to guarantee the compatibility with the existing COST 259/273 approach. The extension to multilink scenarios is thus achieved by considering that the visibility regions now define the cluster visibility to multiple BSs, i.e. the VR associated to a given cluster determines to which BS the cluster will be connected once a MT is located inside that

VR. Typically, clusters will be associated to multiple VRs, and therefore, multiple BSs. B. 3GPP/3GPP2 Spatial Channel Models The 3GPP/3GPP2 spatial channel model (SCM) [40] was established specifically for the simulation of 5-MHz bandwidth third generation networks in urban and suburban macrocells as well as in urban microcells. The SCM is not defined as a continuous model, but prescribes a specific discretized implementation (i.e., DoDs, DoAs and azimuth spreads are assigned fixed values). However, the model has been constructed as to incorporate correlations between these different large-scale parameters. Hence, unlike the COST approach, it does not allow for continuous, large-scale movements of the mobile terminal, but considers different possible segments of the mobile motion within the cell. The provided implementation is a tapdelay line model. Each tap consists of several sub-paths that share the same delay, but have different directions of arrival and departure. Several options have been defined that can be switched on to obtain a better agreement with real-world data, such as polarized antennas, far scatterer clusters, line-ofsight, and urban canyons. Interference is handled by modeling strong interferers as spatially correlated, while weak interferers are taken as spatially white. Finally, shadowing correlation is assigned fixed values of 0 when considering different MTs connected to a single BS and 0.5 when considering a single MT connected to multiple BSs. The original 3GPP/3GPP2 SCM was extended by the interim WINNER model [41], in particular to include intracluster delay-spreads, a line-of-sight (LOS) and K-factor model for all scenarios, as well as time-variant shadow fading, path angles and delays. This is implemented by defining a number of large-scale parameters: the shadowing standard deviation, the Ricean K-factor, the delay-spread and the directions spreads at departure/arrival. For a given link, the model fixes the large-scale parameters according to prescribed distributions. This implies that only short segments of successive channel matrices can be generated: these short segments correspond to one sample of large-scale parameters. Different segments (i.e. different periods of time of a given link) are related by correlating the large-scale parameters as a function of the inter-segment distance, but the clusters for each segment are generated for that segment only. This means that even though two segments can be very close, the clusters (or scatterers) for each segment are generated independently: both segments share highly correlated large-scale parameters, but see totally different clusters. For any segment, the WINNER model then generates multipaths in a similar way as the COST approach, i.e. using clusters of scatterers. C. WINNER II Multi-Link MIMO Channel Model The WINNER channel model was updated in 2007 to become the WINNER II channel model [42]. Analogous to the 3GPP Spatial Channel Model (SCM), and by contrast to the COST 2100 model, the WINNER II model is a typical multilink system-level geometry-based stochastic model, in that

Fig. 3.

WINNER II multi-link model

the propagation environment is specified for each realization of all radio links between base stations and mobile terminals. Whereas multiple links between multiple base stations, mobile terminals or relays can be simulated simultaneously, as shown in Figure 3, each simulation is run for each link separately, according to the SCM philosophy. This is actually where the major difference with the COST 2100 model lies, as the latter allows for simultaneous simulation of multiple links. The correlation between multiple links is introduced by correlating the large-scale parameters, this correlation being a simple function of distance, as already implemented for multi-segment scenarios. Whereas this is also the rule that applies to correlate shadowing across multi-user links, the shadowing correlation is set to zero when considering links from a given MT to multiple BSs (note the difference with the SCM shadowing correlation). The advantage of WINNER II model is that the largescale statistics in a specific scenario are always guaranteed by any realization. However, the independent initialization of the propagation environment in each realization prohibits to readily connect different realizations, which is important when characterizing the time variations caused by the motion of the users. Also, forcing this consistency at system level makes the model quite rigid. Typically, when new largescale parameters, such as the channel correlation in multi-link communication scenarios, are included into the analysis, the entire initialization of the propagation environment must be redefined, hindering the straightforward extension of the model. VI. C ONCLUSIONS Multi-link propagation measurements and models have been reviewed and compared, in particular with regards to cooperative channel fading statistics, multi-link SISO/MIMO distance metrics and recent standardized geometry-based channel models. More effort is however necessary to (i) validate channel models, (ii) determine the applicability of the models in different environments, and (iii) identify the most crucial

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