Multi-scale Spectrum Sensing in 5G Cognitive Networks

A multi-scale approach to spectrum sensing is proposed to overcome the huge energy cost of acquiring full network state information over 5G cognitive networks. Secondary users (SUs) estimate the local spectrum occupancies and aggregate them hierarchically to produce multi-scale estimates. Thus, SUs obtain fine-grained estimates of spectrum occupancies of nearby cells, more relevant to resource allocation tasks, and coarse-grained estimates of those of distant cells. The proposed architecture accounts for local estimation errors, delays in the exchange of state information, as well as irregular interference patterns arising in future fifth-generation (5G) dense cellular systems with irregular cell deployments. An algorithm based on agglomerative clustering is proposed to design an energy-efficient aggregation hierarchy matched to the irregular interference structure, resilient to aggregation delays and local estimation errors. Measuring performance in terms of the trade-off between SU network throughput, interference to PUs and energy efficiency, numerical experiments demonstrate a 10 regular tree construction, for a reference value of interference to PUs, and up to 1/4th of the energy cost needed to acquire full network state information.



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I Introduction

The recent proliferation of mobile devices has been exponential in number as well as heterogeneity [4], demanding new tools for the design of agile wireless networks [5]. Fifth-generation (5G) cellular systems will meet this challenge in part by deploying dense, heterogeneous networks, which must flexibly adapt to time-varying network conditions. Cognitive radios [6] have the potential to improve spectral efficiency by enabling secondary users (SUs) to exploit resource gaps left by legacy primary users (PUs) [7]. However, estimating these resource gaps in real-time becomes increasingly challenging with the increasing network densification, due to the signaling overhead required to learn the network state [8]. Furthermore, network densification results in irregular network topologies. These features demand effective interference management to fully leverage spatio-temporal spectrum access opportunities.

To meet this challenge, we develop and analyze spectrum utilization and interference management techniques for dense cognitive radios with irregular interference patterns. We consider a multi-cell network with a set of PUs and a dense set of opportunistic SUs, which seek access to locally unoccupied spectrum. The SUs must estimate the channel occupancy of the PUs across the network based on local measurements. In principle, these measurements can be collected at a fusion center [9, 10, 11], but centralized estimation may incur unacceptable delays and overhead [12, 8]. To reduce this cost and provide a form of coordination, neighboring cells may inform each other of spectrum they are occupying [13]; however, this scheme cannot manage interference beyond the cell neighborhood, which may be significant in dense topologies.

Fig. 1: System model (see notation in Sec. III).

We address this challenge by designing a cost-effective multi-scale solution to detect and leverage spatio-temporal spectrum access opportunities across the network, by exploiting the structure and irregularities of interference. To do so, note that the interference caused by a given SU depends on its position in the network, as depicted in Fig. 1: PUs closer to this SU will experience stronger interference than PUs farther away. Therefore, such SU should estimate more accurately the state of nearby PUs, in order to perform more informed local control decisions to access the spectrum or remain idle. In contrast, the state of PUs farther away, which experience less interference from such SU, is less relevant to these control decisions, hence coarser spectrum estimates may suffice. With this in mind, the goal of our formulation is the design of a cost-effective spectrum sensing architecture to aid local network control, which enables each SU to estimate the spectrum occupancy at different spatial scales (hence the name "multi-scale"), so as to possess an accurate and fine-grained estimate of the occupancy of PUs in the vicinity, and coarser estimates of the occupancy states of PUs farther away. To achieve this goal, we use a hierarchical estimation approach resilient to delays and errors in the information exchange and estimation processes, inspired by [14] in the context of averaging consensus [15]: local measurements are fused hierarchically up a tree, which provides aggregate spectrum occupancy information for clusters of cells at larger and larger scales. Thus, SUs acquire precise information on the spectrum occupancies of nearby cells – these cells are more susceptible to interference caused by nearby SUs – and coarse, aggregate information on the occupancies of faraway cells. By generating spectrum occupancy estimates at multiple spatial scales (i.e., multi-scale), this scheme permits an efficient trade-off of estimation quality, cost of aggregation, estimation delay, and provides a cost-effective means to acquire information most relevant to network control. We derive the ideal estimator of the global spectrum occupancy from the multi-scale measurements, and we design the SU traffic in each cell in a decentralized fashion so as to maximize a trade-off among SU cell throughput, interference caused to PUs, and mutual SU interference.

To tailor the aggregation tree to the interference pattern of the network, we design an agglomerative clustering algorithm [16, Ch. 14]. We measure the end-to-end performance in terms of the trade-off among SU cell throughput, interference to PUs, and the cost efficiency of aggregation. We show numerically that our design achieves a small degradation in SU cell throughput (up to 15% under a reference interference-to-noise ratio of 0dB experienced at PUs) compared to a scheme with full network state information, while incurring only one-third of the cost in the aggregation of spectrum estimates across the network. We show that the proposed interference-matched tree design based on agglomerative clustering significantly outperforms a random tree design, thus demonstrating that it provides more relevant information for network control. Finally, we compare our proposed design with the state-of-the-art consensus-based algorithm [17], originally designed for single-cell systems without temporal dynamics in the PU spectrum occupancy, and we demonstrate the superiority of our scheme thanks to its ability to leverage the spatial and temporal dynamics of interference in the network and to provide more meaningful information for network control.

Related work: Consensus-based schemes for spectrum estimation have been proposed in [17, 18, 8, 19]: [8] proposes a mechanism to select only the SUs with the best detection performance to reduce the overhead of spectrum sensing; while [19] focuses on the design of diffusion methods. Cooperative schemes with data fusion have been proposed in [9, 10, 11]: [9] investigates the optimal voting rule and optimal detection threshold; [10] proposes a robust scheme to filter out abnormal measurements, such as malicious or unreliable sensors; [11] analyses and compares hard and soft combining schemes in heterogeneous networks. However, all these works focus on a scenario with a single PU pair (one cell) and no temporal dynamics in the PU spectrum occupancy state. Instead, we investigate spectrum sensing in multi-cell networks with multiple PU pairs and with temporal dynamics in the PU occupancy state, giving rise to both spatial and temporal spectrum access opportunities. Similar opportunities have been explored in [20], but in the context of a single PU, and without consideration of SU scheduling decisions. In contrast, in our paper we investigate the impact of spectrum sensing on scheduling decisions of SUs.

Another important difference with respect to [17, 18, 8, 19, 9] (with the exception of [20]) is that we model temporal dynamics in the occupancy states of each PU, as a result of PUs joining and leaving the network at random times; in time-varying settings, the performance of spectrum estimation may be severely affected by delays in the propagation of estimates across the network, so that spectrum estimates may become outdated. We develop a hierarchical estimation approach that compensates for these propagation delays. A setting with temporal dynamics has been proposed in [21, 20] for a single-cell system, but without consideration of delays.

Finally, [22] capitalizes on sparsity due to the narrow-band frequency use, and to sparsely located active radios, and develops estimators to enable identification of the (un)used frequency bands at arbitrary locations; differently from this work, we develop techniques to track the activity of PUs, and use this information to schedule transmissions of SUs, hence we investigate the interplay between estimation and scheduling tasks, and the role of network state information.

We summarize the contributions of this paper as follows:

  1. We propose a hierarchical framework to aggregate network state information (NSI) over a multi-cell wireless network, with a generic interference pattern among cells, which enables spectrum estimation at multiple spatial scales, most informative to network control. We study its performance in terms of the trade-off between the SU cell throughput and the interference caused to the PUs. We design the optimal SU traffic in each cell in a decentralized fashion, so as to manage the interference caused to other PUs and SUs.

  2. We show that the belief of the spectrum occupancy vector is statistically independent across subsets of cells at different spatial scales, and uniform within each subset (Theorem

    1), up to a correction factor that accounts for mismatches in the aggregation delays. This result 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.

set of cells, with occupancy state of cell at time ,
steady-state distribution

memory of the Markov chain

INR generated by tx in cell to rx in , cf. (II) SU traffic in cell at time ,
# of SUs in cell at time Binomial with

trials and probability

level- cluster heads level- cluster heads associated to
cells associated to h-distance between cells and , cf. Def. 1
cells at h-distance from cell , cf. Def. 2
delay from cell to level- cluster head delay between
and its upper level- cluster head
local estimate at cell delay mismatched aggregate estimate at
h-distance from cell , cf. (25)
SU cell throughput lower bound, cf. (II) estimated SU interference at cell , cf. (8)
INR caused by SUs in cell , cf. (11)-(12) estimated PU interference at cell , cf. (9)
utility function, cf. (II) local belief in cell
TABLE I: Table of Notation

Ii System Model

Network Model: We consider the network depicted in Fig. 1, composed of a multi-cell network of PUs with cells operating in downlink, indexed by , and an unlicensed network of SUs. The receivers are located in the same cell as their transmitters, so that they receive from the closest access point. Transmissions are slotted and occur over frames. Let be the frame index, and be the PU spectrum occupancy of cell during frame , with if occupied and otherwise. We suppose that are independent and identically distributed (i.i.d.) across cells and evolve according to a two-state Markov chain, as a result of PUs joining and leaving the network at random times. We define the transition probabilities as

where is the memory of the Markov chain, which dictates the rate of convergence to its steady-state distribution. Hence, at steady-state. We denote the state of the network at time as .
We assume that PUs and SUs coexist in the same spectrum band. Let be the number of SUs in cell at time , which may vary over time as a result of SUs joining and leaving the network. We collect in the vector .

Assumption 1.

are i.i.d. across cells, stationary and independent of , that is

where is a time interval and is a delay. Additionally, (dense network).∎

Assumption 1 guarantees that spectrum estimates are “statistically symmetric” [23], i.e., they exhibit the same statistical properties at different cells and delay scales. An example which obeys Assumption 1 is when is a Markov chain taking values from , i.i.d. across cells. The SUs opportunistically access the spectrum to maximize their own cell throughput, while at the same time limiting the interference caused to other SUs and to the PUs. Their access decision is governed by the local SU access traffic for SUs in cell . We assume an uncoordinated SU access strategy so that, given , all the SUs in cell access the channel with probability , independently of each other.111We assume that is known in cell , and a local control channel is available to regulate the local SU traffic . Therefore, represents the expected number of SU transmissions in cell . We let .
Transmissions of SUs and PUs generate interference to each other. We denote the interference to noise ratio (INR) generated by the activity of a transmitter in cell to a receiver in as , collected into the symmetric (due to channel reciprocity) matrix . Typically,


(see, e.g., [24]), where is the transmission power, common to all PUs and SUs, is the noise power spectral density and is the signal bandwidth; is the large-scale pathloss at a reference distance , based on Friis’ free space pathloss formula, and is the distance dependent component, with and the distance and pathloss exponent between cells and . We assume that the intended receiver of each PU or SU transmission is located within the cell radius, so that is the SNR to the intended receiver in cell . In practice, the large-scale pathloss exhibits variations as transmitter or receiver are moved within the cell coverage. Thus, can be interpreted as an average of these pathloss variations, or a low resolution approximation of the large-scale pathloss map. This is a good approximation due to the small cell sizes arising in dense cell deployments, as considered in this paper. In Sec. VI (Fig. 5), we will demonstrate its robustness in a more realistic setting.
Network Performance Metrics: We label each SU as , denoting the th SU in cell . Let be the indicator of whether SU transmits based on the probabilistic access decision outlined above; this is stacked in the vector . If the reference SU transmits, the signal received by the corresponding SU receiver is


where we have defined the interference signal


is the fading channel between SU and the reference SU , with the unit energy transmitted signal; is the fading channel between PU (transmitting in downlink) and SU , with the unit energy transmitted signal; is the large-scale pathloss between cells and , see (II); is circular Gaussian noise; we assume Rayleigh fading, so that . The transmission is successful if and only if the SINR exceeds a threshold ; we then obtain the success probability of SU , conditional on and ,


Noting that

is circular Gaussian with zero mean and variance


where is the number of SUs that attempt spectrum access in cell , we obtain

Then, the throughput in cell , conditional on the SU traffic and PU network state , is obtained by noting that each of the SUs succeed with probability ; hence, taking the expectation with respect to the number of SUs performing spectrum access,

(binomial random variable with probability

and trials), we obtain


where the second equality is obtained by the change of variable , with . The computation of the SU cell throughput using this formula has high complexity, due to the outer expectation. Therefore, we resort to a lower bound. Noting that the argument of the expectation is a convex function of and , Jensen’s inequality yields

Cell selects based on partial NSI, denoted by the local belief that . Taking the expectation over conditional on and using Jensen’s inequality, we obtain


where we have defined


The terms and represent, respectively, an estimate of the interference strength caused by SUs and PUs operating in the rest of the network to the reference SU in cell . Additionally, due to channel reciprocity and the resulting symmetry on , represents an estimate of the interference strength caused by the reference SU to the rest of the PU network.

Herein, we use in (II) to characterize the performance of the SUs. Since this is a lower bound to the actual SU cell throughput, using as a metric provides performance guarantees. Note that the performance depends upon the network-wide SU activity via ; in turn, each is decided based on the local belief , which may be unknown to the SUs in cell (which operate under a different belief ). Therefore, maximization of can be characterized as a decentralized decision problem, which does not admit polynomial time algorithms [25]. To achieve low computational complexity, we relax the decentralized decision process by assuming that is known to cell in slot . This assumption is based on the following practical arguments: due to the Markov chain dynamics of , varies slowly over time, hence can be estimated by averaging the SU traffic over time; additionally, the spatial variations of are averaged out in the spatial domain since is a weighted sum of across cells, yielding slow variations on due to mean-field effects. In Sec. IV, we will present an approach to estimate and based on hierarchical information exchange over the SU network.

We define the average INR experienced by the PUs as a result of the activity of the SUs as


where is the average number of active PUs at steady-state. In fact, the expected number of SUs transmitting in cell is , so that is the overall interference caused by SUs in cell to the PU in cell . is then obtained by averaging this effect over the network. Herein, we isolate the contribution due to the SUs in cell on (10), yielding


so that . By computing the expectation with respect to the local belief and using the symmetry of , we then obtain


Since the goal of SUs is to maximize their own cell throughput, while minimizing their interference to the PUs, we define the local utility as a payoff minus cost function,


where is a cost parameter which balances the two competing goals. Given , the goal of the SUs in cell is to design so as to maximize . Since this is a concave function of (as can be seen by inspection), we obtain the optimal SU traffic


where denotes the projection operation onto the interval . It can be shown by inspection that both and are non-increasing functions of , so that, as the PU activity increases ( increases), the SU activity and the local utility both decrease; when is above a certain threshold, then and ; indeed, in this case the PU network experiences high activity, hence SUs remain idle to avoid interfering. Additionally, is a convex function of . Then, by Jensen’s inequality,


where is the Kronecker delta function centered at , reflecting the special case when is known, so that represents the utility achieved when , known. Consequently, the expected network utility is maximized when is known (full NSI). Thus, the SUs should, possibly, obtain full NSI in order to achieve the best performance. To approach this goal, the SUs in cell should obtain in a timely fashion. To this end, the SUs in cell should report the local and current spectrum state to the SUs in cell via information exchange, potentially over multiple hops. Since this needs to be done over the entire network (i.e., for every pair ), the associated overhead may be impractical in dense multi-cell network deployments. Additionally, these spectrum estimates may be noisy and delayed, hence they may become outdated and not informative for network control. In order to reduce the overhead of full-NSI, we now develop a scheme to estimate spectrum occupancy based on delayed, noisy, and aggregate (vs timely, noise-free and fine-grained) spectrum measurements over the network.

Iii Local and Multi-scale Estimation Algorithms

In this section, we propose a method to estimate and at cell based on hierarchical information exchange. To this end, SUs exchange estimates of the local PU spectrum occupancy , denoted as , as well as the local SU traffic decision variable . For conciseness, we will focus on the estimation of in this section; however, the same technique can be applied straightforwardly to the estimation of as well. In fact, and have the same structure – they both are a weighted sum of the respective local variables and , with weights , see (8)-(9), hence they can be similarly estimated.

Iii-A Aggregation tree

To reduce the cost of acquisition of NSI, we propose a multi-scale approach to spectrum sensing. To this end, we partition the cell grid into sets , and define a tree on each , designed in Sec. V. Since each edge in the tree incurs delay, disconnected trees are equivalent to a single tree where the edges connecting each of the subtrees to the root have infinite delay (and thus, provide outdated, non-informative NSI). Hence, without loss of generality, we assume where, possibly, some edges incur infinite delay.

Level- contains the leaves, represented by the cells . To each cell, we associate the singleton set .222Note that represents cell , containing SUs (Assumption 1). At level-, let be a partition of into non-empty subsets, each associated to a cluster head . The set of level- cluster heads is denoted as . Hence, is the set of cells associated to the level-1 cluster head , see Fig. 1.

Recursively, at level-, let be the set of level- cluster heads, with . If , then we have defined a tree with depth . Otherwise, we define a partition of into non-empty subsets , each associated to a level- cluster head, collected in the set . Let be the set of cells associated to level- cluster head . This is obtained recursively as


We are now ready to state some important definitions.

Definition 1.

We define the hierarchical distance (h-distance) between cells as

In other words, is the smallest level of the cluster containing both and . It follows that the h-distance between cell and itself is , and it is symmetric ().

Definition 2.

Let be the set of cells at h-distance from cell : , and, for all , , (then, is the level- cluster head of cell )

In fact, contains all cells at h-distance (from cell ) less than (or equal to) . Thus, we obtain by removing from all cells at h-distance less than (or equal to) , (note that this is a subset of , since ). For example, with reference to Fig. 1, (cell is at h-distance from itself), (cells , and are at h-distance from cell ), (cells , , and are at h-distance from cell ).

Iii-B Local Estimation

The first portion of the frame is used by SUs for spectrum sensing, the remaining portion for data communication. Thus, spectrum sensing does not suffer from SU interference.

Remark 1.

This frame structure requires accurate synchronization among SUs, achievable using techniques developed in [26]. Loss of synchronization may cause overlap between the sensing and communication phases; herein, we assume that the duration of the sensing phase is sufficiently larger than synchronization errors, so that this overlap is negligible.

In the spectrum sensing portion of frame , SUs in cell estimate . Each of the SUs observe the local state through a binary asymmetric channel, , where is the false-alarm probability ( is detected as being occupied) and is the mis-detection probability ( is detected as being unused). In practice, each SU measures the received energy level and compares it to a threshold; the value of this threshold entails a trade-off between and . We assume that these spectrum measurements are i.i.d. across SUs (given ). In principle, may vary over cells and time, but for simplicity we treat them as constant.

Then, these measurements are fused at a local fusion center at the cell level,333The optimal design of decision threshold, local estimators, fusion rules, are outside the scope of this paper and can be found in other prior work, such as [9, 10, 11], for the case of a single-cell. and then up the hierarchy, using an out-of-band channel which does not interfere with PUs. Thus, the number of measurements that detect (possibly, with errors) the spectrum as occupied in cell , denoted as , is a sufficient statistic to estimate . Let

be the prior probability of occupancy of cell

, time , given measurements collected up to (excluded). After collecting the measurements, the cell head estimates as


where the second step follows from Bayes’ rule. Note that and . Thus, we obtain


Given , the prior probability in the next frame is obtained based on the spectrum occupancy dynamics as


Iii-C Hierarchical information exchange over the tree

In the previous section, we discussed the local estimation at the cell level. We now describe the hierarchical fusion of local estimates to collect multi-scale NSI. This fusion is patterned after hierarchical averaging [14], a technique for scalar average consensus in wireless networks.

The aggregation process running at each node is depicted in Fig. 2. The cell head, after the local spectrum sensing in frame , has a local spectrum estimate . These local estimates are fused up the hierarchy, incurring delay. Let be the delay to propagate the spectrum estimate of cell all the way up to its level- cluster head . It includes the local processing time at each intermediate level- cluster head traversed before reaching the level- cluster head, as well as the delay to traverse the links (possibly, multi-hop) connecting successive cluster heads. We assume that is an integer, multiple of the frame duration; in fact, scheduling of SUs transmissions in the data communication phase is done immediately after spectrum sensing, hence a spectrum estimate with non-integer delay can only be used for scheduling decisions with delay . In the special case when , the estimate of cell becomes immediately available to the level- cluster head; if , it becomes available for data communication in the following frame, and so on.

Fig. 2: Aggregation process referred to Fig. 1, and aggregate estimates relative to cell .

We assume that , i.e., the delay augments as the local spectrum estimates are aggregated at higher levels. More precisely, let be the delay between the level- cluster head , and its level- cluster head , with . We can thus express as


where is the level- cluster head of cell . By the end of the spectrum sensing phase, the level- cluster head receives the spectrum estimates from its cluster : is received from cell with delay . These are aggregated at the level- cluster head as


each with its own delay. This process continues up the hierarchy: the level- cluster head receives from the level- cluster heads connected to it, with delay , and aggregates them as


each with its own delay . Importantly, these delays may differ from each other, hence does not truly reflect the aggregate spectrum at a given time. For this reason we denote as the delay mismatched aggregate spectrum estimate at level- cluster head . The next lemma relates to the local estimates.

Lemma 1.

Let be a level- cluster head. Then,


See Appendix A. ∎

Despite mismatched delays, in Sec. IV we show that cell can compensate them via prediction.

Remark 2.

Note that the aggregation process runs in a decentralized fashion at each node: level- cluster head needs only information about the set of level- cluster heads connected to it, , and the delays . This information is available at each node during tree formation; delays may be estimated using time-stamps associated with the control packets. The aggregation process has low complexity: each cluster-head simply aggregates the delay mismatched aggregate spectrum estimates from the lower level cluster heads connected to it, and transmits this aggregate estimate to its higher level cluster head.

Eventually, the aggregate spectrum measurements are fused at the root (level-) as


where we used Lemma 1 and . Upon reaching level- and each of the lower levels, the aggregate spectrum estimates are propagated down to the individual cells over the tree.444We include the propagation delay from the cluster head back to the single cells in .

Therefore, at the beginning of frame , the SUs in cell receive the delay mismatched aggregate spectrum estimates from their level- cluster heads ,

where we remind that is the set of cells associated to at level , and is the delay for the estimate of to propagate to the level- cluster head . From this set of measurements, cell can compute the aggregate spectrum estimate of the cells at all h-distances from itself as


To interpret as the aggregate estimate at h-distance from cell , note that Lemma 1 yields