Multi-scale spectrum sensing is proposed to overcome the cost of full network state information on the spectrum occupancy of primary users (PUs) in dense multi-cell cognitive networks. Secondary users (SUs) estimate the local spectrum occupancies and aggregate them hierarchically to estimate spectrum occupancy at multiple spatial scales. Thus, SUs obtain fine-grained estimates of spectrum occupancies of nearby cells, more relevant to scheduling tasks, and coarse-grained estimates of those of distant cells. An agglomerative clustering algorithm is proposed to design a cost-effective aggregation tree, matched to the structure of interference, robust to local estimation errors and delays. Given these multi-scale estimates, the SU traffic is adapted in a decentralized fashion in each cell, to optimize the trade-off among SU cell throughput, interference caused to PUs, and mutual SU interference. Numerical evaluations demonstrate a small degradation in SU cell throughput (up to 15% for a 0dB interference-to-noise ratio experienced at PUs) compared to a scheme with full network state information, using only one-third of the cost incurred in the exchange of spectrum estimates. The proposed interference-matched design is shown to significantly outperform a random tree design, by providing more relevant information for network control, and a state-of-the-art consensus-based algorithm, which does not leverage the spatio-temporal structure of interference across the network. December 7, 2018 DRAFT arXiv:1802.08378v3 [cs.IT] 6 Dec 2018 greatly facilitates the estimation of the interference caused to PUs (Lemma 3). 3) We address the design of the hierarchical aggregation tree under a constraint on the aggregation cost based on agglomerative clustering [16, Ch. 14] (Algorithm 1).Our analysis demonstrates that multi-scale spectrum estimation using hierarchical aggregation matched to the structure of interference is a much more cost-effective solution than fine-grained network state estimation, and provides more valuable information for network control. Additionally, it demonstrates the importance of leveraging the spatial and temporal dynamics of interference arising in dense multi-cell systems, made possible by our multi-scale strategy; in contrast, consensus-based strategies, which average out the spectrum estimate over multiple cells and over time, are unable to achieve this goal and perform poorly in dense multi-cell systems.This paper is organized as follows. In Sec. II, we present the system model. In Sec. III, we present the proposed local and multi-scale estimation algorithms, whose performance is analyzed in Sec. IV. In Sec. V, we address the tree design. In Sec. VI, we present numerical results and, in Sec. VII, we conclude this paper. The main proofs are provided in the Appendix. Table I provides the main parameters and metrics.
II. SYSTEM MODELNetwork Model: We consider the network depicted in Fig. 1, composed of a multi-cell network of PUs with N C cells operating in downlink, indexed by C≡{1, 2, . . . , N C }, and...