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Weighted Centroid Algorithm for Estimating Primary User Location: Theoretical Analysis and Distributed Implementation
Jun Wang, Paulo Urriza, Yuxing Han, Danijela Čabrić
TL;DR
The paper addresses how to analyze and deploy WCL for non-cooperative primary-transmitter localization in cognitive radio networks. It develops a parameterized error-distribution framework and a distributed cluster-based implementation, finding comparable accuracy to centralized WCL with greatly reduced total power consumption.
Problem
Primary-transmitter localization needs theoretically characterized WCL error distributions and practical implementations beyond centralized data aggregation.
Method
The paper derives a WCL error framework parameterized by node density, placement, shadowing, correlation distance, and sensor-positioning errors, then proposes distributed cluster-based WCL.
Results
The distributed method achieves comparable accuracy with its centralized counterpart and greatly reduces total power consumption.
Takeaways & Limitations
The analysis provides deployment guidelines on node number and placement for achieving WCL performance in energy-efficient cognitive-radio localization.
Abstract
from arXiv · showhide
Information about primary transmitter location is crucial in enabling several key capabilities in cognitive radio networks, including improved spatio-temporal sensing, intelligent location-aware routing, as well as aiding spectrum policy enforcement. Compared to other proposed non-interactive localization algorithms, the weighted centroid localization (WCL) scheme uses only the received signal strength information, which makes it simple to implement and robust to variations in the propagation environment. In this paper we present the first theoretical framework for WCL performance analysis in terms of its localization error distribution parameterized by node density, node placement, shadowing variance, correlation distance and inaccuracy of sensor node positioning. Using this analysis, we quantify the robustness of WCL to various physical conditions and provide design guidelines, such as node placement and spacing, for the practical deployment of WCL. We also propose a power-efficient method for implementing WCL through a distributed cluster-based algorithm, that achieves comparable accuracy with its centralized counterpart.
I. INTRODUCTION
The paper motivates WCL for non-cooperative primary-user localization and develops theoretical and distributed approaches to make it practical in cognitive radio networks.
- Primary-user location information supports spatio-temporal sensing, location-aware routing, and spectrum policy enforcement in cognitive radio networks.
- Because primary users do not cooperate, conventional time-based localization techniques cannot be used directly.
- WCL estimates primary-user location from RSS-weighted secondary-user positions, while retaining low computational complexity and no need for primary-user cooperation.
- Earlier WCL studies relied on simulations or field experiments and did not analyze the localization-error distribution under shadowing.
- The paper presents a theoretical WCL error analysis, quantifies robustness to channel and placement conditions, and proposes a distributed cluster-based implementation.
- The distributed implementation addresses centralized WCL communication costs, including overhead and transmit-power demands that limit scalability.
II. SYSTEM MODEL
The system model represents sensors around a primary transmitter, models imperfect sensor positions and correlated shadowing, and defines centralized WCL from received powers.
- The model places N sensor nodes within a circle around a primary transmitter and uses nodes above a received-power sensitivity threshold.
- The analysis assumes the primary transmitter is surrounded by enough sensors to reduce edge effects in position estimation.
- Sensor self-localization is imperfect, with independent position errors characterized by standard deviation σ_l.
- Received power combines path loss and shadowing, with shadowing correlation governed by correlation distance X_c.
- Centralized WCL sends RSS measurements from participating sensors to a fusion center, which forms the primary-transmitter estimate using non-negative RSS-based weights.
III. THEORETICAL ANALYSIS OF WCL ERROR DISTRIBUTION
The theoretical analysis derives WCL localization-error distributions by reducing the estimator to one-dimensional random variables and extending the results to two dimensions.
- The analysis derives analytical expressions for one-dimensional and two-dimensional WCL localization-error distributions.
- In the independent-shadowing case, the one-dimensional estimate is treated as a ratio of two correlated Gaussian variables.
- A Gaussian approximation is verified numerically against closed-form mean and variance calculations for different node numbers.
- For N ≥30, the approximation error for the means is essentially zero and less than 3% for the variances.
- The two-dimensional error analysis accounts for correlation between the x- and y-axis estimation errors.
2) Correlated Shadowing:
The correlated-shadowing analysis models WCL’s one-dimensional estimates using correlated Gaussian variables and derives localization-error statistics for fixed or random node placements. A Gaussian approximation is validated for N ≥20, after which the two-dimensional error distribution is obtained through decorrelation and numerical integration.
- Gaussian approximation: The numerator is approximated as a Gaussian random variable using the central limit theorem, with the approximation validated for N ≥20.This provides the basis for the subsequent one-dimensional localization-error analysis.
- Correlated one-dimensional estimates: The analysis treats the location estimate as a ratio of correlated Gaussian variables and derives the required statistical properties.The numerator and denominator depend on shared shadowing variables, so their covariance must be calculated explicitly.
- Two-dimensional error distribution: The two error dimensions are dependent Gaussian variables with different variances, so decorrelation transforms them into independent Gaussian variables before evaluating the error norm.An orthogonal eigenvalue decomposition diagonalizes the covariance matrix.
- Two-dimensional error distribution: For fixed sensor-node positions, the analysis derives the localization-error probability density and computes its mean and variance by numerical integration.The two-dimensional error norm is formed from the transformed one-dimensional error variables.
- Random node placement: For randomly placed nodes, averaging over node coordinates extends the theoretical performance calculation beyond a particular fixed placement.The framework considers both i.i.d. and correlated shadowing cases.
IV. DISTRIBUTED IMPLEMENTATION OF WCL
The distributed WCL implementation addresses the communication and power costs of centralized WCL by organizing nodes into proximity-based clusters. It selects a high-average-RSS cluster and then performs WCL through cluster-level coordination.
- Motivation: Centralized WCL requires many nodes for high accuracy, creating substantial message and transmit-power overhead.The distributed implementation is introduced to address this cost in cognitive-radio networks.
- Two-phase design: The proposed algorithm splits distributed WCL into two phases: selecting the cluster with the highest average RSS, then performing the WCL calculation.The selected cluster acts as coordinator for the second phase.
- Cluster organization: Nodes are organized into proximity-based clusters, whose heads exchange RSS information and decide which clusters detect the primary user.Active clusters can be further reduced using a threshold based on known transmit power and desired node count.
- Cluster organization: Cluster geometry is preferably regular because WCL performs best with a radially symmetric node distribution, although arbitrary proximity-based clustering is allowed.The primary user is assumed to be randomly located across the deployment area.
- Cluster organization: A cluster head maintains neighboring-cluster and member-location information, enabling cluster-level coordination for distributed WCL.Each cluster selects one node as its head.
2) Distributed WCL (DWCL):
DWCL selects a head cluster using local RSS comparisons, then estimates the transmitter location from nodes near the selected cluster’s strongest node. Its design limits communication while addressing border bias and mobility-related power costs.
- Head-cluster selection: Neighboring clusters compare average RSS values to select the cluster likely closest to the primary user.The negative RSS-gradient direction points toward the location of maximum RSS.
- Communication reduction: Transmitting only locations and RSS measurements from nodes inside the selected circle reduces exchanged information, while cluster-level weighted centroids can distribute the final calculation.The approach is motivated by WCL’s averaging structure.
- Head-cluster selection: The proposed direction metric approximates the primary user’s direction without requiring multiple antennas at cluster heads.It subtracts the WCL result for cluster nodes from their geometric centroid to approximate the within-cluster RSS gradient.
- Final WCL calculation: After selecting the head cluster, DWCL computes WCL using nodes within radius R* of the strongest node.R* is the minimum of the distance to the nearest map edge and the cluster radius.
- Final WCL calculation: The adaptive radius reduces border bias by preserving a radially symmetric node distribution when the primary user is near the network edge.Additional nodes from adjacent clusters can improve localization accuracy.
- Mobility-aware operation: A mobility-aware restriction to the previous head cluster and adjacent clusters can limit the most power-consuming head-selection phase.This assumes the primary user remains in or moves to a neighboring cluster between successive estimates.
B. Analysis of Communication Overhead
The communication-overhead analysis compares centralized and distributed WCL in transmit power, message count, and computational operations. The distributed design replaces network-wide reporting with cluster-level exchanges and localized calculations.
- Analysis framework: The analysis evaluates total transmit power and computational complexity for centralized WCL and the proposed distributed WCL.It models single-link transmit power and counts basic operations using fixed costs for arithmetic operations.
- Centralized WCL: Centralized WCL’s total messages consist of node reports sent to the fusion center, while its operation count is O(25N).The power calculation uses statistical averages over node distances and shadowing.
- Distributed WCL: For distributed WCL, Algorithm 1 requires O(ML + 2L + 2ηKL) messages, with M nodes per cluster and L = N/M clusters.The expression accounts for cluster-level communication and the fraction η of clusters involved in the head-selection step.
- Distributed WCL: Algorithm 2 requires O(2K) messages for communication among neighboring cluster heads.Its transmit-power calculation follows the corresponding second step of Algorithm 1.
- Distributed WCL: The operation counts are O(27N +44L+64KL+ηKL) for Algorithm 1 and O(34KM + 26M) for Algorithm 2.These expressions quantify the computational cost of the two distributed phases.
V. EVALUATION OF WCL PERFORMANCE
The evaluation examines WCL under shadowing, correlation, positioning error, and node-placement conditions using theoretical analysis and simulations. WCL is robust to independent shadowing and positioning error, but correlated shadowing and random placement can substantially degrade accuracy.
- Evaluation setup: Normalizing mean localization error by average node spacing D removes the accuracy improvement caused solely by geometry scaling.This metric complements prior normalization by transmission range R.
- Impact of shadowing: A 7.5 dB increase in shadowing from 2.5 dB to 10 dB increases normalized error by only 5%.The gain from increasing node count saturates at approximately 200 nodes or more.
- Impact of correlation: For highly correlated shadowing with XC > 5D, increasing node count worsens error, reaching 47% of D with 400 nodes.Correlation has a degrading effect when all nodes participate, and the worst correlation distance scales with the number of participating nodes.
- Impact of positioning error: A positioning error variance corresponding to σl = 7m increases localization error by about 1%.The analysis describes the overall impact of sensor-node positioning errors as marginal in the evaluated setting.
- Impact of node placement: Random node placement can increase error by as much as three times compared with fixed placement, while random-grid placement incurs approximately 10% performance loss.Fixed-grid placement is therefore more accurate in the evaluated scenarios.
5) Impact of Degree-of-Irregularity:
The evaluation studies coverage irregularity, participation level, and comparisons with lateration and centroid localization. WCL tolerates moderate irregularity, but correlated shadowing makes selecting participating nodes important; its relative advantage grows as shadowing increases.
- Impact of Degree-of-Irregularity: At high DOI, adding more nodes slightly increases normalized mean error because denser distributions deviate further from circular coverage.The reported increase is observed at DOI values of 30% or 40%.
- Node participation: With uncorrelated shadowing, 100% node participation is optimal, whereas with even slight correlation approximately 10% participation is preferred.Nodes are selected by received signal strength, prioritizing those with higher received power.
- Comparison with other techniques: Under shadowing below 2.5 dB, lateration is more accurate than WCL, while WCL performs better than lateration at higher shadowing.At 0 dB shadowing, the range-based technique achieves perfect localization in the reported scenario.
- Comparison with other techniques: Centroid performs slightly worse than WCL because it does not use RSS information, but their performance difference narrows at higher shadowing.The comparison uses uniformly and independently distributed nodes under uncorrelated shadowing.
C. Performance of DWCL algorithms
The distributed WCL (DWCL) achieves localization accuracy between centralized WCL (CWCL) and strongest-node (SN) methods while reducing transmit power. Its performance approaches SN under high shadowing, and distributed operation increases computational operations.
- DWCL performance lies between CWCL and SN under varying shadowing conditions.
- High shadowing degrades DWCL accuracy relative to CWCL and makes it approach SN performance because cluster-head selection depends on SN.
- 15 dBm less power per node is consumed by DWCL with 16 clusters than by CWCL.
- DWCL transmit power per node decreases as cluster count increases because average communication distance becomes smaller.
- Distributed algorithms require about three times more operations than centralized algorithms, although distributed implementation reduces total transmit power.
- The paper provides a WCL error-distribution framework parameterized by node density, placement, shadowing variance, correlation distance, and sensor-position errors.
- The analysis supplies node-number and placement guidelines, variable node participation, and a distributed cluster-based implementation without a fusion center.
- The distributed method achieves comparable accuracy with centralized WCL while greatly reducing total power consumption.
APPENDIX
The appendix derives the correlation between localization errors in the x and y dimensions. It simplifies expectations involving correlated Gaussian variables through variable transformations and QR factorization before obtaining the result.
- The appendix derives the correlation coefficient between localization errors in the x and y dimensions.
- The derivation accounts for both independent and correlated shadowing cases.
- It defines t1, t2, and t3 as linear combinations of the Gaussian vector q to simplify the expectation calculation.
- Because the expectation involves products and ratios of correlated random variables, direct calculation is not possible.
- QR factorization reduces dimensionality by transforming the variable matrix into an upper-triangular form.
- Orthogonal invariance of standard Gaussian variables preserves the relevant distribution under the transformation.
- Substituting the factorized expression into the preceding equation yields the result.