MIMO Propagation Channel Modeling

Ernst Bonek Institute of Telecommunications

Technische Universität Wien Wien, Austria

ernst.bonek@tuwien.ac.at

Abstract—Propagation channel models serve as input to MIMO system-level simulations. Present-day standard models capture the spatial domain by directions of arrival and departure of individual multipath components, and by the angular spread of clusters of such components. Fully polarimetric, sophisticated measurement campaigns have provided model parameters for a large number of wireless communication scenarios, including the statistics of these parameters. Challenges remain in the areas of interference modeling, of body-worn and sensor networks, of highly non-stationary channels, of diffuse scattering, and of over-the-air testing of MIMO terminals.

Index Terms—MIMO; propagation; wireless channel; modeling; double-directional; multilink; multiplexing; vehicle-to-vehicle communications; spatial correlation; OTA testing

I. MIMO BASICS This presentation deals with modeling electromagnetic

wave propagation when multiple antennas are present at both ends of the radio link. It is the addition of the spatial domain to previous models of communication systems that makes the difference. Multiple-Input Multiple-Output (MIMO) has been a scientifically fruitful area because the diverse fields of electromagnetic wave propagation and signal processing met. Their combination exploits the spatial dimension of the radio propagation channel [1,2,3].

The seemingly paradoxical, arbitrary multiplication of Shannon's capacity has fuelled the theoretical interest in MIMO, and its potential to reach Gbits/s in wireless data transmission [4] has driven the practical interest. After some hesitation, 3GPP has fully endorsed MIMO as the enabling technology for high spectral efficiency in specifying the 4G mobile broadband standard Long Term Evolution (LTE) [5,6].!In Single-User MIMO (SU-MIMO), more than one LTE transport blocks, comprising a number of OFDM sub-carriers in a time-slot, are dedicated to a single multi-antenna terminal. Multi-User MIMO (MU-MIMO) or virtual MIMO connects several single-antenna terminals with the multi-antenna base station in shared resource blocks.

In principle, MIMO offers three different benefits, namely spatial diversity, beamforming gain, and spatial multiplexing. To dismiss cure-all expectations from MIMO, let us acknowledge that beamforming, diversity, and multiplexing are rivaling techniques that require trade-offs in implementation.

Radio signals from different directions in space give rise to spatial diversity. Radio engineers have applied diversity to

increase the transmission reliability of the fading radio link ever since. For a spatially white MIMO channel, i.e. completely uncorrelated antenna signals, the diversity order may reach a maximum of MT·MR, the number of sub-channels between MT antenna elements at the transmitter (Tx) and MR antenna elements at the receiver (Rx).

Beamforming is also very familiar from antenna array technology. When the Tx and Rx antenna patterns are directed towards each other in Line-Of-Sight (LOS) conditions, the Rx and Tx gains add up, leading to an upper limit of MT·MR for the beamforming gain of a MIMO system. Proper choice of antenna weights also allows for placing nulls into specific directions, in order to suppress interference, at the cost of loosing some gain.

The stunning novelty of MIMO, however, was that channels could support parallel data streams by transmitting and receiving on orthogonal spatial channels (spatial multiplexing). The number of usefully multiplexed streams depends on the rank of the instantaneous channel matrix, which, in turn, depends on the spatial properties of the radio environment. The spatial multiplexing gain may reach min (MT, MR) in a sufficiently rich scattering environment. Spatial correlation – manifest as channel matrix rank reduction - will reduce this gain, as it will reduce diversity order.!!

This presentation will highlight the role of the propagation channel, which sets the ultimate limits of what MIMO systems may actually achieve. But I cannot leave out Signal Processing (SP) completely: SP has played a key role in sophisticated measurements of MIMO channels; some propagation modeling philosophies make sense only in conjunction with the MIMO processing that will actually be applied.

The complexity of MIMO propagation modeling has been irritating systems engineers, but the discussions in the past two decades about what to model in MIMO and what not have converged.

II. KINDS OF MODELS

A. Modeling at which level? At which level should the model operate: propagation,

channel, link, or system? Electromagnetic wave propagation provides the basis for propagation or ‘physical models’. A physical model characterizes the environment on the basis of propagation effects, such as reflection, diffraction, and scattering. Internationally agreed-on environments, i.e.

reference scenarios, simplify comparison of models and their performance.

Specifying a system bandwidth and the antenna arrays at both link ends, by fixing the number of antenna elements, their geometrical configuration, their angular patterns, and their polarizations, turns the propagation model into a MIMO channel model. The output of a channel model is a set of impulse responses of all sub-channels between Tx and Rx antenna elements, at a specific point in space and time, i.e. an instantaneous channel transfer matrix, H. In a measurement, a channel sounder would provide this data, also. (Accounting for mobility of a terminal completes a channel model on link level.)

A set of MIMO channel matrices provides the analytical framework for designing transmit and receive techniques for a MIMO link, e.g. space-time codes. Hence the name ‘analytical model’.!!

In a network, where typically multiple MIMO links are active, interference becomes a key issue. To model interference properly, it is advisable to step back to physical modeling and to work one’s way up again via link models to a system model on network level. This level will not be discussed here.

So what is the main difference between physical and analytical modeling? Physical modeling is preferred if antenna configurations are not yet fixed, whereas analytical modeling takes these as given. Therefore, if we want to optimize MIMO antenna arrays, we will choose physical modeling, even if that requires more parameters than a nonphysical approach. On the other hand, with given antenna configurations and system bandwidth, sets of MIMO transfer matrices provide a powerful mathematical basis for information theory and SP. We can easily convert a physical model into an analytical one, but ‘getting physical’ from analytical is difficult. Sections V, VI will discuss a foremost example of each modeling approach.

B. Physical Models [7] Deterministic models, e.g. derived from ray-tracing (RT),

are easy to reproduce. However, being site-specific they may not be really representative for the environment to be modeled, and their parameters cannot be changed easily. Sophisticated ray-tracers render path loss well, but not so well delay spread and angular spreads [3]. The required environmental databases characterizing the propagation environment, which are expensive to produce, have been the prohibitive argument against RT use in standard MIMO models so far. The original idea to add diffuse scattering to such databases [8], however, has recently rendered good results for angular spreads and cross-polarization [9].

Stochastic models describe the radio channel by probability density functions (pdfs) of model parameters. These pdfs should be derived from extensive measurements in clearly defined, distinct environments. Simulation runs with them are fast because of tapped-delay-line realizations of legacy Single-Input Single-Output (SISO) models, a fact that makes such stochastic models popular. Stochastic models are difficult to parameterize over large areas.

Geometry-based stochastic channel models (GSCM) combine the better of the two worlds by starting with geometrical input about the environment and then superimposing statistical information. Such information can be entered at various levels, e.g. at the level of individual propagation paths (‘MultiPath Components, MPCs) or at the level of clusters of MPCs. Such models excel in simulating both interference and the temporal evolution of the channel when terminals or scatterers move. They are well suited for simulating relaying situations and interference. Another major advantage of GSCMs is that they are inherently system-independent.

The standards’ pets are Clustered Delay Line (CDL) models, because they fit so well to the world of pre-MIMO propagation modeling. This requirement for backward compatibility has, in fact, hampered truly measurement-based propagation modeling. Models adopted by 3GPP, i.e. Spatial Channel Model - Extended (SCME) [10], and IEEE 802.11n [11] prescribe temporal and spatial characteristic of the MIMO radio channel by aggregate parameters (see Section IV).!!

Analytical models specify MIMO channel matrices and their correlation properties. Some extract these correlation properties from elaborate MIMO measurement campaigns, like the so-called ‘Kronecker’ [12] or Weichselberger models [13]. Other models, such as the 'i.i.d. model' simply assume random Gaussian entries in the H matrix - very popular with theoreticians.

III. THE ROLE OF MEASUREMENTS AND PARAMETER ESTIMATION

Propagation measurement must precede modeling. Schemes and models that violate this principle have plagued MIMO development, resulting in too optimistic expectations of MIMO benefits.

Reference [14] summarized the essentials of double-directional channel sounding with multi-element near-omni-directional arrays [15]. Besides path loss, DoDs, DoAs, and delay, polarization and Doppler of MPCs has been measured. Rx and Tx must operate in synchronism, either by cables, rubidium clocks, or satellite signals. Otherwise the all-important phase relation between Rx and Tx side signals gets lost. To increase the accuracy of the channel transfer matrix’s spatial and temporal structure, high-resolution parameter estimation techniques (MUSIC, ESPRIT, SAGE) have been pushed to their limits.

Without further refinement, however, these techniques miss a further important part in the impulse response, the so-called diffuse or dense multipath [16].

Clustering of MPCs helps to reduce the number of MIMO model parameters. For a long time clusters had been identified by 'visual inspection'. Such identification is, of course, subject to individual idiosyncrasy, although different human brains seemed to work amazingly similar. Reference [17] introduced a practical automatic clustering framework to MIMO modeling. The method is particularly useful to process the huge amount of data that measurement campaigns tend to produce.

Does the Wide-Sense Stationary Uncorrelated Scatterers assumption hold? I think: only in few cases. WSSUS channels have been taken more or less for granted in wireless communications since the pioneering work of Bello, despite the repeated warning of experimenters, such as “dominance of nonstationary behavior emphasizes the need to test whether measured data series satisfy the WSS requirement before estimating channel parameters“ [18]. Correlated scatterers do occur in rooms, corridors, tunnels, and vehicle-to-vehicle channels, violating the US assumption.

A common pitfall in measuring MIMO is to throw together measurement data from different environments into one, single ensemble and then estimate parameters. This will inevitably result in an average model (central-limit theorem!) of a virtual environment that is never encountered in practice.

IV. PARAMETERS OF THE MIMO RADIO CHANNEL The angularly resolved impulse response hij(τ,t,φ) is what

distinguishes MIMO from conventional channel modeling. Here, hij is the impulse response between Tx antenna j and Rx antenna i, as a function of delay τ, time t, and (symbolic) angle φ. The double-directional view of the propagation environment provides the necessary angular information at both Tx and Rx [19]. Under far-field conditions we may model MPCs as local plane waves and trace their propagation from Tx to Rx via their wave vectors. Thus, each identified MPC has a Direction-Of-Arrival (DOA) at Rx and, surprising in case of omni antennas, a Direction-Of-Departure (DOD) at Tx as well, Fig 1.

Figure 1. The double-directional, angularly resolved impulse response characterizes the MIMO propagation environment.

Aggregate parameters. Already SISO modeling derived the following aggregate parameters from the impulse response. The squared magnitude of hij yields an instantaneous Angular Delay Power Spectrum (ADPS) and, after time averaging, an average ADPS. This we may integrate over angle (or delay) to obtain a Power Delay Profile (PDP) (or an Angular Power Spectrum (APS)). The second central moments of these latter quantities finally yield the well-known rms Delay Spread (rms DS) and the rms Angular Spread (rms AS).

Now enter MIMO: the rms AS alone will not suffice to model different environments correctly. Channels with identical rms AS may have widely differing MIMO responses because of different intra-cluster angular spreads of each cluster of MPCs [20] (Fig. 2). In 2x2 MIMO angular resolution is so poor that intra-cluster angular spread differences are barely noticeable. This will radically change for 4x4 and 8x8 MIMO.

Figure 2. Intra-cluster angular spread is different when viewed from Rx or Tx

Polarization. Another important aggregate parameter is cross-polarization discrimination (XPD). So actually we have to model four angularly resolved impulse responses for each sub-channel, not one. Two different origins of cross- polarization have to be considered: the incomplete XPD of the antennas at Tx and Rx, and de-polarization that electro-magnetic waves undergo when they propagate through the environment. Poor XPD will diminish the particular advantages that polarization offers to MIMO [21].

Correlation. The familiar steering vector of phased array technology uniquely relates the phases at each element in free space. However, the stochastic multipath of mobile wireless communications has changed the picture completely. MIMO link antennas experience stochastic phase changes between elements, which are best described by correlation matrices. I want to stress that in MIMO not the correlation of antenna signals matters, but the correlation of the i x j sub-channels between antennas [1].

When the channel is Rayleigh or Rice fading, the so-called full correlation matrix RH completely describes MIMO correlation, between all sub-channels [12]. There are, however, two problems: (i) the MT·MR x MT·MR elements of RH lack physical interpretation, except the diagonal elements, which describe power transfer; (ii) the full correlation matrix becomes very large as soon as MT, MR exceed 2. A meaningful approximation is the so-called Kronecker model, which factorizes RH into the two independent correlation matrices at Rx and Tx, RRx and RTx. Such a description, however, neglects correlation across the link, which does matter once the angular resolution of the arrays becomes better than that of just two omni-antenna elements. Another, more realistic approximation of RH will be discussed in Section VI.

V. WINNER II AND BEYOND The WINNER II model [22] excels among the GSCMs

because it has been chosen by ITU-R for evaluation of radio access technologies for IMT-Advanced (‘4G’)[23]. It relies on the double-directional view, which separates the antenna arrays from the propagation environment, and allows free choice of antenna configurations.

Each path is characterized by a sum of N individual MPCs (=‘rays’) with delay, DOD, DOA, gain, and cross-polarization. During model synthesis, MPCs are generated not individually but in M clusters. The WINNER model introduces randomness by random draws, from specified pdfs, of rms DS, rms AS (departure and arrival), shadow fading variance, and K-factor. These so-called large-scale parameters are generated as

correlated variables. One such drawn realization is called a ‘drop’. Within a drop, the temporal change of the channel can be simulated by terminal movement over a short distance. Adding up all N·M MCPs in all four possible polarization combinations results automatically in small-scale fading and spatial correlation (Fig. 3).

Figure 3. Structure of WINNER II model [22]. The S (= MT in usual notation) antennas of Array 1 transmit via N paths to U (=MR) receive antenna elements of Array 2. Each ‘path’ (≈ cluster) combines M ‘sub-paths’ or ‘rays’ (= MPCs). The location vectors rtx,j and rrx,i, the orientation of each element and its pattern can be chosen freely for versatile, polarimetric simulations.

Extensive measurements provided reliable data for parameterization of 18 ‘typical’ scenarios, such as urban, indoor-outdoor, moving networks, etc. A particular strength of the WINNER model philosophy is its suitability for interference simulations of MU-MIMO on network level.

The problems of existing GSCMs [10,22] to model the temporal evolution of MIMO transmission throughput is tackled by the COST 2100 Multilink MIMO Channel Model through time-dependent ‘visibility regions’ of clusters [3]. This fully polarimetric model characterizes the channel by geometric positioning of individual clusters. Such a cluster-oriented structure is more flexible when characterizing multi-link scenarios. So-called twin clusters are placed in such a way as to satisfy angular spread requirements at both BS and terminal sides, and delay [24] As another highlight, this model also incorporates the diffuse part of impulse responses.

VI. THE WEICHSELBERGER MODEL [13] This is an interesting example of ‘Analytical Models’.

Starting from an eigenmode analysis of the full correlation matrix RH, this model derives directly from measurements and gives refreshingly intuitive insights about which MIMO benefit can be reaped in which propagation environment. Furthermore, it has been independently validated for frequencies between 0.3 and 5.8 GHz [25], rendering ergodic capacity and diversity better than other correlation-based analytical models.

First, we have to distinguish between eigenbases and eigenvalues [13]. The eigenbases reflect the radio environment, i.e. number, positions, and strengths of the scatterers. They are independent of the excitation of transmit antenna elements. The eigenvalues, on the other hand, do depend on the transmit weights, and show how the scatterers are illuminated by the radio waves propagating from the transmitter. Spatial correlation of the transmit weights determines how much power is radiated into which directions and polarizations.

With two simplifying assumptions, Weichselberger turns this rather formal approach into a useful MIMO model. Key element is the so-called ‘coupling matrix’, Ω. It describes which Tx eigenmodes couple into which Rx eigenmodes. Thus, the structure of Ω tells us how many parallel data streams can be multiplexed, which degree of diversity is present at either link side, and whether beamforming gain can be achieved. Think of the eigenmodes as discrete direction. This is not generally true, but aids intuition. Figure 4 shows three examples of coupling matrices (square root of magnitude, linear scale), taken in a MIMO setup with 15 Tx antennas at cluttered street level and 8 Rx antennas high above tree tops. (http://measurements.ftw.at/MIMO.html and W. Weichsel-berger).

Figure 4. (a) Tx eigenmode 1 couples only to Rx eigenmode 1: good for beamforming at both link ends. (b) Twelve Tx eigenmodes couple to one Rx

eigenmode: good for Tx diversity and Rx beamforming. (c) Nine Tx eigenmodes couple strongly to five Rx eigenmodes. Space-time coding will

exploit multiplexing or diversity. Such a channel is not Kronecker separable!

Reference [26] ponders how to make use of this modeling approach in system level simulators.

VII. CHALLENGES TO MEET.

A. Body-Area Networks [27,28] In personal-area networks the convenient separation of

antennas and propagation environment breaks down. By necessity, antennas are in close proximity of body parts, which now become part of the antenna system. Similar difficulties are encountered in sensor networks, when sensors are located close to ground, vegetation, or other objects.

B. Diffuse multipath Caused by numerous multiple reflections or diffraction, modeling without this component underestimates delay spread by some 25%, while affecting capacity only little [29]. However, it will matter more and more as the number of antennas at base station and terminal will increase to 8x8 (MIMO channels with many strong eigenvalues).

C. Vehicle-to-vehicle MIMO Vehicle-to-vehicle communications provide a particularly

challenging propagation environment, violating not only WSS, but also US (tunnel walls, road signs). Also, scenarios to be modeled differ considerably from the usual ones [30]. Current standards do not require MIMO operation, so little work has been done so far [31], but MIMO’s potential for diversity could reduce latency, a critical parameter for safety.

D. Over-the-Air (OTA) Testing OTA testing of MIMO terminals for standard conformity is

intimately related to MIMO channel modeling. Three different testing methods have been proposed. Which will eventually make it to standardization is a question of politics [32]. Technologically, the main issue is how to synthesize a spatial correlation environment that captures both antenna and propagation characteristics [33]. Consideration of close-by body parts constitutes a further tremendous complication in standardized testing.

E. Interference Interference modeling will become absolutely necessary for

LTE-Advanced to fulfill the IMT-A specifications at cell edges. On the propagation level, GSCMs with well-positioned scatterers are poised to do the job. Analytically [34,35], we may expect many interesting schemes for interference management.

ACKNOWLEDGMENT I gratefully acknowledge the numerous contributions of

COST 231/259/273/2100 groups to the art of MIMO channel modeling. Over many years, A1 Telekom Austria has generously supported my PhD students whose work is mentioned in this paper, and so have Alcatel, Stuttgart, Nokia Research Center, Helsinki, and Elektrobit, Oulu, Finland.

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