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Correlated Link Shadow Fading in Multi-hop Wireless Networks

Piyush Agrawal, Neal Patwari

arXiv:0804.2708v2stat.AP

TL;DR

Multi-hop physical-layer models often assume independent link shadowing despite geographically proximate links experiencing correlated environmental fading. The paper measures link pairs across deployed networks, develops a correlated shadowing model, and analyzes multi-hop path connectivity. Measurements find significant correlations up to 0.33, while correlated-shadowing analyses show substantially higher path-failure probabilities than independent-shadowing models, including 120% and 200% increases in three- and four-node examples.

  • Problem

    Existing multi-hop channel models treat path losses as independent, although geographically proximate links can share environmental shadowing effects.

  • Method

    The paper measures full-link path loss across an ensemble of deployed networks, models shadowing through an underlying spatial loss field, and analyzes multi-hop paths under correlated and independent shadowing.

  • Results

    Significant non-zero correlations occur for 15 of 28 link geometries, with a maximum coefficient of 0.33; correlated shadowing raises path-failure probability by 120% in a three-node case and 200% in a four-node case.

  • Takeaways & Limitations

    Independent-shadowing models can substantially overestimate multi-hop path connectivity, especially as paths become longer and networks are designed for higher reliability.

Abstract

from arXiv · show

Accurate representation of the physical layer is required for analysis and simulation of multi-hop networking in sensor, ad hoc, and mesh networks. This paper investigates, models, and analyzes the correlations that exist in shadow fading between links in multi-hop networks. Radio links that are geographically proximate often experience similar environmental shadowing effects and thus have correlated fading. We describe a measurement procedure and campaign to measure a large number of multi-hop networks in an ensemble of environments. The measurements show statistically significant correlations among shadowing experienced on different links in the network, with correlation coefficients up to 0.33. We propose a statistical model for the shadowing correlation between link pairs which shows strong agreement with the measurements, and we compare the new model with an existing shadowing correlation model of Gudmundson (1991). Finally, we analyze multi-hop paths in three and four node networks using both correlated and independent shadowing models and show that independent shadowing models can underestimate the probability of route failure by a factor of two or greater.

I. INTRODUCTION

The paper targets inaccurate multi-hop radio-channel models by modeling correlated shadow fading between geographically proximate links. It separates fading into shadowing and other losses and contrasts realistic correlations with independent-shadowing assumptions.

  • Motivation: Current physical-layer models inadequately represent multi-hop radio channels, creating a disconnect between simulation, analysis, and real-world deployment.The paper seeks improved statistical models to reduce this difference.
  • Contribution: The paper presents a statistical joint path-loss model for static nodes that relates correlated link shadowing to network connectivity, interference reliability, and energy consumption.Joint path losses and transmit powers determine these communication properties during power control.
  • Contribution: Shadowing losses are hypothesized to correlate across geographically proximate links because shared environmental obstructions affect their propagation paths.The model separates total fading into shadowing loss X_i,j and non-shadowing losses Y_i,j.
  • Related models: The circular coverage and i.i.d. log-normal models represent opposite extremes and fail to capture spatially correlated, irregular coverage behavior.The i.i.d. model has no spatial memory, so nearly overlapping links can be modeled as independent.
  • Motivation: An obstacle that strongly attenuates one link direction is likely to impose additional path loss on other receivers behind the same obstacle.This contrasts with the i.i.d. model, which assumes shadowing across such links is independent and can exaggerate connectivity.

C. Correlation Limits Link Diversity

The paper studies how channel correlation limits multi-hop diversity, addressing a gap in research on correlated shadowing across links in sensor, mesh, and ad hoc networks. It uses full-link measurements from frequency-hopping sensor deployments to estimate correlations across repeated geometries.

  • Motivation: Multi-hop networking provides network-layer diversity through multiple paths, but channel correlations can limit the reliability gains of diversity.Prior work has shown analogous limits for time, space, frequency, and multipath diversity.
  • Approach: The study experimentally quantifies link-shadowing correlations using full-link measurements from an ensemble of deployed multi-hop networks.It proposes a joint path-loss model and evaluates the effect of correlation on source-to-destination path statistics.
  • Related work: Prior studies of multi-hop link correlations were limited by measuring only a single network, whereas this study examines repeated link geometries across multiple networks.The campaign includes link pairs both with and without a common node.
  • Measurement system: The NCMS measures received power for every deployed link across a range of frequencies and records the data for later analysis.The system uses mica2 motes operating in the 902–928 MHz band with programmable transmit power and RSS reporting.
  • Measurement system: Nodes hop across 14 center frequencies, with three-second intervals between frequency hops.The measurement protocol also synchronizes frequency-hopping nodes and uses TDMA broadcasts to avoid interference.

2) Receiver Base:

The campaign measures path loss across repeated network realizations while varying environmental obstructions. The analysis estimates single-link model parameters and treats frequency-averaged fading as approximately Gaussian in dB.

  • Measurement campaign: The campaign uses one classroom environment and randomly varies object locations instead of deploying identical geometries across many buildings.This approach addresses the practical difficulty and expense of accessing multiple buildings and reproducing identical layouts.
  • Measurement campaign: Fifteen network realizations are measured by repeatedly changing the locations of randomly placed obstructions around a 4x4 node grid.The grid uses 16 mica2 nodes with 4 ft (1.22 m) spacing, and each realization runs for 10 minutes.
  • Data analysis: The analysis first estimates path-loss parameters and then analyzes shadowing correlations between link pairs.The deployment index m ranges over M = 15 experiments.
  • Fading model: Frequency averaging motivates representing the fading term as Gaussian in dB despite underlying frequency-selective mechanisms.The paper assumes shadow fading is constant across the measured frequency band and separates it from non-shadow fading.
  • Single-link analysis: A linear regression of frequency-averaged received powers against known distances estimates the reference-power term and path-loss exponent for each experiment.The regression also determines the variance of the fading term, while equal transmit powers and battery voltages avoid requiring exact transmit-power knowledge.

B. Analysis of Link Correlations

The analysis computes correlations for repeated link geometries across deployments and tests whether measured correlations differ from zero. This supports identifying which geometric relationships produce statistically significant correlated fading.

  • Link geometry: Link geometry is defined by the relative coordinates of the endpoints of two links.In a grid, the same geometry can recur many times, allowing repeated measurements.
  • Correlation computation: For each repeated geometry, vectors Z_a and Z_b collect all measured total-fading values for the two links across repetitions and experiments.The vectors are LM×1 and contain the samples used to compute the correlation coefficient.
  • Correlation computation: The correlation coefficient between two links is computed by treating the corresponding collected vectors as samples of their total fading.This aggregates measurements for link pairs sharing a particular geometry.
  • Statistical testing: The study computes correlations for varied link-pair geometries and applies a hypothesis test for statistical significance.The reported probability of measuring the observed correlation under H0 is given in Table I.

C. Discussion

Measurements found statistically significant, non-zero correlations for many link-pair geometries, especially when links are geographically proximate. The measured correlations were substantial enough to provide information about fading on one link from proximate links.

  • 15 of 28 studied link geometries showed statistically significant non-zero correlation, consistently involving geographically proximate links.The links’ transmitter-to-receiver lines partially or nearly overlapped.
  • The highest measured correlation coefficient was 0.33, while six geometries had ρ > 0.20 and eleven had ρ > 0.10.
  • Although proximate-link fading does not determine another link’s loss, it can provide substantial information about that loss.

VI. JOINT PATH LOSS MODEL

The joint path loss model represents link shadowing as an integral over a spatial loss field whose correlation structure produces correlated losses on geographically related links. Its assumptions and numerical evaluation are designed to match observed shadowing behavior.

  • The model treats shadowing on network links as arising from an underlying spatial loss field p(x), with higher loss where a link crosses high-loss areas.
  • The spatial loss field is modeled as an isotropic wide-sense stationary Gaussian random field with zero mean and exponentially decaying spatial correlation.
  • The covariance function is justified by its basis in a Poisson spatial random process, which is proposed as a model for randomly arranged attenuating obstructions.
  • A. Single-Link Properties: The model reproduces approximately constant dB shadowing variance with path length and Gaussian shadow fading for single links.
  • B. Joint Link Properties: For two links, the model computes covariance and correlation from their shadowing variables, with numerical integration used to evaluate the correlation coefficient.

B. Total Fading Model

The total fading model combines correlated shadowing loss with independent non-shadowing loss. Under the paper’s spacing assumption, total-link correlation is a scaled version of shadow-fading correlation and its parameters are fitted to measurements.

  • Total fading loss is decomposed as Zi,j = Xi,j + Yi,j, where shadowing and non-shadowing losses are modeled as independent.
  • Non-shadowing losses {Yi,j} are treated as independent because sensors in multi-hop networks typically are spaced more than a few wavelengths apart.
  • Equation (17) gives a linear relationship between total-fading correlation ρZa,Zb and shadow-fading correlation ρXa,Xb.
  • The measured total-fading correlation ρZa,Zb is used with regression analysis to fit the shadow-fading model parameters across 28 link geometries.
  • The measurement data determine the two correlation-model parameters, (δ, σX), including σ2dB = 0.29.

D. Comparison with Gudmundson Model

The proposed shadowing-correlation model predicts measured link correlations more accurately than Gudmundson’s model, which is limited in multi-hop geometries.

  • Gudmundson’s model applies only to link pairs sharing a common node in multi-hop networks.The model was designed for cellular links involving a mobile receiver and base station.
  • Gudmundson’s model ignores the common node’s location, predicting identical correlations for geometries whose measured correlations are 0.21 and 0.05.Its large-base-station-distance assumption is not generally applicable to multi-hop networks.
  • 80.4% agreement with measurements was achieved by the proposed model, versus 64.4% for Gudmundson’s model.Both models were fitted using two parameters, making the comparison valid.
  • The path-connectivity analysis compares correlated and independent shadowing using threshold-based link connectivity assumptions.A link is connected when normalized received power βm,n exceeds zero, and interference losses are omitted.

A. A Three Node Multi-Hop Path

For a three-node network, a direct link and a two-hop relay path provide alternative routes, and correlated shadowing changes their joint failure probability.

  • Two routes connect nodes i and k: direct link (i,k), or the two-hop path through relay node j.The direct link and relay links can experience related shadowing, affecting the combined path outcome.
  • The links (i,j) and (j,k) are nearly uncorrelated, so their two-hop connectivity probability is approximately the i.i.d. value.The correlated-case failure probability is derived using this near-zero correlation.

2) Case of Correlated Shadowing:

Correlated shadowing substantially increases multi-hop path failure, especially as paths lengthen and networks are designed for higher link reliability.

  • 2) Case of Correlated Shadowing:: The four-node experiment simulates 10^5 normalized received-power samples under both correlated and i.i.d. link shadowing.Four-node path-failure analysis is simulated because the analytical expression is tedious.
  • 2) Case of Correlated Shadowing:: 120% greater path-failure probability occurred under correlated shadowing than i.i.d. shadowing for the three-node network at β̄i,j = 2.The models converge only for very unreliable links, such as β̄i,j = 0.
  • 2) Case of Correlated Shadowing:: 200% increase in path-failure probability occurred for the four-node network, compared with 120% for the three-node network.The result indicates that longer paths make correlation increasingly important.
  • 2) Case of Correlated Shadowing:: The current i.i.d. shadowing model underestimates path-failure probability by a factor of two or higher.The conclusion applies to reliable multi-hop network design, where ignoring correlations produces dramatic effects.
  • 2) Case of Correlated Shadowing:: Future work will test additional indoor and outdoor deployment ensembles and quantify effects on higher-layer protocols and interference control.These extensions define the stated scope of ongoing evaluation.

APPENDIX

The appendix derives the three-node joint connectivity probability by conditioning on the direct-link margin and approximating the relay-link conditional distribution as independent.

  • APPENDIX: βi,j, βj,k, and βi,k are modeled as jointly Gaussian normalized received powers.Their conditional distributions given βi,k = b are therefore Gaussian.
  • APPENDIX: The conditional joint density of βi,j and βj,k is approximated by the product of their conditional densities.This uses the observed very small or absent correlation between the two relay links.
  • APPENDIX: The full joint density is factored into the two conditional relay-link densities and the marginal density of βi,k.This factorization supports the subsequent probability calculation.
  • APPENDIX: The intersection event requires βi,j > 0, βj,k > 0, and βi,k > 0 simultaneously.Integrating this joint event yields P[A ∩ B].
  • APPENDIX: The squared term in the resulting expression follows because the two relevant correlation coefficients are equal for the considered geometry.Specifically, ρXj,k,Xi,k = ρXi,j,Xi,k.
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