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Aspects of Favorable Propagation in Massive MIMO
Hien Quoc Ngo, Erik G. Larsson, Thomas L. Marzetta
TL;DR
The paper addresses how to assess favorable propagation when channel norms differ and how favorable practical channel environments can be. It analyzes capacity, introduces a distance-from-favorable-propagation measure and an urns-and-balls model, and finds nearly favorable behavior for i.i.d. Rayleigh and UR-LoS channels, with scope boundaries for the proxies and UR-LoS assumptions.
Problem
Existing condition-number assessments are inadequate when terminal channel norms differ, while favorable propagation across practical channel environments remains insufficiently studied.
Method
The paper relates favorable propagation to sum-capacity, proposes a capacity-gap measure, studies i.i.d. Rayleigh and UR-LoS channels, and models UR-LoS using urns and balls.
Results
Both i.i.d. Rayleigh fading and UR-LoS offer asymptotically favorable propagation, while UR-LoS has mostly concentrated large singular values and a few very small ones.
Takeaways & Limitations
UR-LoS can be made approximately favorable by dropping a few terminals, underscoring the importance of user selection in massive MIMO.
Takeaways & Limitations
The condition number has sound operational meaning only for equal channel norms and disregards singular values beyond the minimum and maximum.
Abstract
from arXiv · showhide
Favorable propagation, defined as mutual orthogonality among the vector-valued channels to the terminals, is one of the key properties of the radio channel that is exploited in Massive MIMO. However, there has been little work that studies this topic in detail. In this paper, we first show that favorable propagation offers the most desirable scenario in terms of maximizing the sum-capacity. One useful proxy for whether propagation is favorable or not is the channel condition number. However, this proxy is not good for the case where the norms of the channel vectors may not be equal. For this case, to evaluate how favorable the propagation offered by the channel is, we propose a ``distance from favorable propagation'' measure, which is the gap between the sum-capacity and the maximum capacity obtained under favorable propagation. Secondly, we examine how favorable the channels can be for two extreme scenarios: i.i.d. Rayleigh fading and uniform random line-of-sight (UR-LoS). Both environments offer (nearly) favorable propagation. Furthermore, to analyze the UR-LoS model, we propose an urns-and-balls model. This model is simple and explains the singular value spread characteristic of the UR-LoS model well.
1. INTRODUCTION
Massive MIMO relies on nearly orthogonal terminal channel vectors, enabling simple linear processing and motivating detailed study of favorable propagation across channel environments.
- Massive MIMO uses hundreds of base-station antennas to serve several tens of terminals simultaneously.The systems can provide high throughput, communication reliability, and power efficiency with linear processing.
- Favorable propagation means terminal channel vectors are nearly orthogonal, allowing linear processing to achieve optimal performance.Matched filtering can cancel uplink noise and interference, while downlink beamforming can transmit multiple streams without mutual interference.
- The channel condition number is a useful favorable-propagation proxy only when channel norms are equal.Different terminal locations can produce unequal channel norms, making the condition number unreliable in practice.
- The paper studies favorable propagation by proving its sum-capacity optimality, evaluating Rayleigh and UR-LoS channels, and modeling UR-LoS with urns and balls.The analysis focuses on the uplink of a single-cell system.
2. SINGLE-CELL SYSTEM MODEL
The single-cell uplink model has K independently transmitting single-antenna terminals and an M-antenna base station, with channel vectors incorporating large- and small-scale fading.
- K single-antenna terminals independently and simultaneously transmit symbols over the same time-frequency resource to an M-antenna base station.The model considers the uplink of a single-cell system.
- The received signal is represented using the transmitted-symbol vector, channel matrix, channel vectors, and additive noise vector.The channel matrix is formed by the terminal channel vectors, while the noise elements are modeled as i.i.d. CN(0, 1) random variables.
- Each channel vector combines large-scale fading, which depends on the terminal but not the antenna, with small-scale fading.The large-scale coefficient is denoted β_k, while the small-scale fading varies across channel elements.
3. PRELIMINARIES OF FAVORABLE PROPAGATION
Favorable propagation means pairwise orthogonal channel vectors and represents the capacity-optimal channel structure under suitable constraints. The section also contrasts the condition number with distance from favorable propagation as evaluation measures, especially when channel norms differ.
- Definition: Favorable propagation is defined by pairwise orthogonality among the channel vectors, with asymptotic favorable propagation describing the corresponding large-M behavior.Exact orthogonality is generally unattainable in practice, so approximate and asymptotic forms are also considered.
- Capacity: Favorable propagation maximizes sum-capacity when the channel-vector norms are fixed.Hadamard inequality gives equality exactly when GHG is diagonal, which is equivalent to the favorable-propagation condition.
- Capacity: Under a constraint on ∥G∥2_F, sum-capacity is maximized when propagation is favorable and all channel vectors have equal norms.The resulting bound is tight only when all channel norms are equal.
- Measures: Directly checking favorable propagation requires testing all (K −1)K/2 channel-vector pairs, motivating simpler proxy measures.The condition number and distance from favorable propagation are proposed as computationally simpler alternatives.
- Measures: The condition number is a simple proxy for favorable propagation, but it is operationally sound only when channel norms or large-scale fading coefficients are equal.It also ignores singular values between σmin and σmax.
- Measures: When channel norms differ, distance from favorable propagation measures the relative capacity gap between the achieved capacity and the favorable-propagation upper bound.A value of ∆C=0 corresponds to favorable propagation.
4. FAVORABLE PROPAGATION: RAYLEIGH FADING AND LINE-OF-SIGHT CHANNELS
The section compares favorable propagation in i.i.d. Rayleigh fading and line-of-sight channels, including asymptotic behavior, finite-array effects, and an urns-and-balls model for UR-LoS. Both environments can approach favorable propagation, but finite UR-LoS channels may contain weak singular directions requiring terminal removal.
- 4.1. Independent Rayleigh Fading: For i.i.d. Rayleigh fading, normalized inter-terminal inner products concentrate around zero with variance proportional to 1/M.
- 4.2. Uniform Random Line-of-Sight: For fixed distinct arrival angles, UR-LoS has asymptotically favorable propagation as M grows.
- 4.2. Uniform Random Line-of-Sight: UR-LoS with independently uniform sin(θk) has the same inner-product convergence rate as i.i.d. Rayleigh fading, but slightly faster convergence for finite M.
- 4.3. Urns-and-Balls Model for UR-LoS: If two UR-LoS arrival angles differ in sin(θ) by order 1/M, favorable propagation may fail for finite arrays.
- 4.3. Urns-and-Balls Model for UR-LoS: For M = 100, K = 10 and M = 200, K = 20, UR-LoS has respectively two or three very small singular values with high probability, while the remaining values cluster near their median.
- 4.3. Urns-and-Balls Model for UR-LoS: The urns-and-balls model assigns terminals independently to M orthogonal beams and drops all but one terminal from any multiply occupied beam.
- 4.3. Urns-and-Balls Model for UR-LoS: Terminal selection offers limited improvement for Rayleigh fading but can significantly improve worst-user performance in UR-LoS by dropping selected terminals.
5. EXAMPLES AND DISCUSSIONS
For M = 100 and K = 10, both Rayleigh fading and UR-LoS capacities are usually close to the favorable-propagation upper bound. In UR-LoS, only a small number of terminals typically need removal to obtain favorable propagation.
- Both Rayleigh fading and UR-LoS capacities are very close to the favorable-propagation upper bound with high probability.The exact capacity curves nearly match the bound curves derived from the maximum sum-capacity under favorable propagation.
- For M = 100 and K = 10, dropping 2 UR-LoS terminals yields favorable propagation while preserving a small capacity gap.Despite UR-LoS having a large condition number with high probability, removing two terminals is sufficient in this case.
- The probability of dropping 3 terminals is less than 1% when M = 100 and K = 10, and dropping 4 is less than 1% when M = 200 and K = 20.These probabilities correspond to cases where three or four singular values are substantially smaller than the rest.
- Guaranteeing favorable propagation requires dropping only a small number of terminals, approximately 20%.
6. CONCLUSION
Both i.i.d. Rayleigh fading and UR-LoS offer asymptotically favorable propagation, but their singular-value structures differ. These models motivate user selection in UR-LoS and suggest approximately favorable propagation in intermediate practical environments.
- Both i.i.d. Rayleigh fading and UR-LoS with uniformly random angles-of-arrival offer asymptotically favorable propagation.
- In i.i.d. Rayleigh fading, singular values are well spread between the smallest and largest values.
- In UR-LoS, most singular values cluster near the maximum while a few are very small, so dropping a few terminals makes propagation approximately favorable.
- The Rayleigh and UR-LoS models represent rich-scattering and no-scattering extremes, respectively, while practical environments may lie between them.
- UR-LoS observations underscore the importance of user selection in massive MIMO.