IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 12, DECEMBER 2013

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Distributed Spectrum Estimation for Small Cell Networks Based on Sparse Diffusion Adaptation Paolo Di Lorenzo, Member, IEEE, Sergio Barbarossa, Fellow, IEEE, and Ali H. Sayed, Fellow, IEEE

Abstract—The goal of this letter is to propose an adaptive and distributed approach to cooperative sensing for wireless small cell networks. The method uses a basis expansion model of the power spectral density (PSD) to be estimated, and exploits spectral sparsity to improve estimation accuracy and adaptation capabilities. An estimator of the model coefficients is developed based on sparse diffusion strategies, which are able to exploit and track sparsity while at the same time processing data in real-time and in a fully decentralized manner. Simulation results illustrate the advantages of the proposed sparsity-aware strategies for cooperative spectrum sensing applications. Index Terms—Distributed spectrum estimation, small-cell networks, sparse diffusion adaptation.

I. INTRODUCTION

T

HE deployment of small cell networks has the potential to increase the spectral efficiency of modern cellular systems due to the strong spatial reuse of radio resources [1][2]. To make this possibility economically feasible, it is necessary to deploy low-cost, low-complexity and low-power base stations, also known as small cell eNode B (SC-eNB) in the LTE (Long Term Evolution) terminology, that are able to cover cells with a radius in the order of tens of meters. Small cell networks are fully compliant with the cellular standard but, unlike macro networks, SC-eNBs are mainly installed by subscribers and they are deployed and maintained without global planning. Hence, a potential massive deployment of SC-eNBs might induce an intolerable interference toward macrocell users. For this reason, SC-eNBs are required to sense the spectrum before allocating radio resources, in order to avoid excessive interference toward macrocell users. A feature of the observed signal that can be advantageously exploited to improve the estimation accuracy is the sparsity of the parameter vectors to be estimated. The sparsity of the Macro Base Station’s (MBS’s) transmission is due to the fact

Manuscript received September 02, 2013; revised October 13, 2013; accepted October 17, 2013. Date of publication October 25, 2013; date of current version November 01, 2013. This work was supported by TROPIC Project under Grant 318784 . The work of A. H. Sayed was supported in part by NSF Grant CCF1011918 . The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Vincenzo Matta. P. Di Lorenzo and S. Barbarossa are with the DIET Department, Sapienza University of Rome, 00184 Rome, Italy. A. H. Sayed is with the Electrical Engineering Department, University of California, Los Angeles, CA 90095 USA (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2013.2287373

that, in every time slot, the MBS typically uses only a small subset of the available channels. Furthermore, an important feature of the MBS’s activity is its intrinsic time-variability, thus motivating the need for adaptive methods to track time-varying phenomena. In recent studies, motivated by the LASSO (Least Absolute Shrinkage and Selection Operator) technique [3] and by connections with compressive sensing [4], [5], several algorithms for sparse adaptive filtering have been proposed based on Least-Mean Squares [6][7], Recursive Least Squares (RLS) [8][9], and projection-based methods [10]. Nevertheless, local sensing can be impaired by shadowing effects that can deteriorate individual non-cooperative estimation of the power spectrum. For this reason, cooperative sensing techniques have been studied quite extensively in recent years, see, e.g., [11]–[16]. A distributed sensing algorithm incorporating the LASSO constraint was proposed in [13], where the authors used the alternating-direction method of multipliers (ADMM) [17] to enable a parallel solution amenable for decentralized networks. Diffusion adaptation strategies have also been used to solve distributed spectrum estimation problems in [14][15], but without incorporating a sparsity constraint. Our purpose in this work is to use both adaptive and distributed solutions that are able to exploit and track sparsity while at the same time processing data in real-time and in a fully decentralized manner. Investigations on sparsity-aware, adaptive, and distributed solutions appear in [16], [18], [19], [20], and [21]. The goal of this letter is to exploit recent results on sparse diffusion adaptation from [18], [19] and apply them to the distributed spectrum sensing problem in small-cell networks. First a basis expansion model is introduced to model the MBS’s transmitted PSD. Then, the distributed estimation problem is formulated as the minimization of the network mean-square error regularized by a sparsity-inducing function. Since the MBS’s transmission can be sparse in the frequency domain, sparse diffusion adaptation strategies from [18][19] lead to several advantages, as illustrated by the simulation results. II. SPARSE DISTRIBUTED SPECTRUM ESTIMATION SC-eNBs communicating with Let us consider a set of each other through a connected, undirected graph . The network is deployed inside a macrocell area served by a single symMBS. The transmitted data are parsed into blocks of bols and guard intervals are introduced to avoid inter-block interference. The channel is constant within each block, but it denote the signal remay vary from block to block. Let by SC-eNB , given by the convoceived at discrete-time and a linear, poslution of the MBS’s transmitted signal sibly time-varying, fading tapped-delay-line channel with im-

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Fig. 1. Structure of macro-block consisting of

blocks.

pulse response , with denoting the length of the delay line, and observed in the presence of additive white noise . For estimation purposes, let us consider a macroblock composed of consecutive blocks, each containing samples. The macro-block is indexed by , the blocks by , and the single samples by , so that denotes the data received on the -th block, inside the -th macro-block, as illustrated in Fig. 1. To respect the block fading assumption, is chosen smaller than the channel coherence time, so that the impulse response can be considered invariant within each macro-block. In the following, it is assumed that the channels are known for every node at every macro-block time . This assumption is reasonable in the context of small cell networks, where MBSs and SC-eNBs are part of the same licensed system. Indeed, since the MBS commonly uses pilot signals to ensure synchronization with the receiver, assuming that the pilot tones transmitted by the MBS are known, every SC-eNB can estimate the channel gains from a number of measurements. The training is repeated at regular intervals since the channel state can vary (albeit slowly) over time. Let denote the PSD of the signal transmitted by the MBS. The PSD can be represented as a linear combination of some preset basis functions, say, as: (1) where is the vector of basis functions evaluated at normalized frequency , is a vector of weighting coefficients representing the power transmitted by the MBS over each basis, and is the number of basis functions. For sufficiently large, the basis expansion in (1) can well approximate the transmitted spectrum. Several choices for the set of basis are possible like, e.g., rectangular functions, raised cosines, or Gaussian bells [13], [22]. The propagation medium introduces a frequency-selective channel gain between MBS and the SC-eNBs. Letting be the channel transfer function between the MBS and the -th SC-eNB in the -th macro block, the PSD received by node can be expressed as

(2) where , and is the receiver noise power at the -th SC-eNB. Expression (2) models the power received by node in terms of the unknown vector

; this vector represents the expansion of the received power in the basis defined by the vectors . The sensing strategy aims at obtaining by relying on estimates for . First, fast Fourier transforms (FFTs) of the signals , within the blocks, namely , , are computed. Then, the periodogram estimate is achieved by averaging the FFTs inside the -th macroblock, thus evaluating . Since the average periodogram estimate attains the real spectrum in expectation [23], at every time index , it can be assumed that every node has access to a noisy version of the true PSD described by (2). More specifically, we assume that every node has access to samples of the approximate PSD computed at frequencies within a certain range . We denote these samples by the discrete model: (3) The term denotes observation noise on the frequency and is assumed to have zero mean and variance . The receiver noise power can be estimated with high accuracy in a preliminary step, using, e.g., an energy estimator over an idle band, and then subtracted from (3). Thus, collecting measurements over contiguous frequencies, we obtain: (4) , with , and where is a zero mean random vector with covariance matrix denoted by . Given the measurements , the SC-eNBs can cooperate to estimate the vector in an adaptive and sparsity-aware manner in order to mitigate local shadowing effects. They can do so by minimizing the following global cost function though a distributed approach [19]: (5) where denotes the expectation operator, and is a realvalued regularization function weighted by the parameter , which is used to enforce sparsity of the solution. Sparse diffusion adaptation schemes have been developed for such purpose in [18][19]. For the problem in (5), we employ a vector version of the Adapt-then-Combine (ATC) sparse diffusion algorithm proposed in [18][19], which reads as:

(adaptation step) (diffusion step)

(6)

denotes the sub-gradient of the sparsity-inducing where function , and the weighting coefficients are real, non-negative and satisfy: , , if , , , with , and . The symbol denotes the set of neighbors of node . The first step in (6) is an adaptation step, where the coefficients determine which nodes should share their measurements

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with node , and the term takes the role of a zero attractor tuned by the regularization parameter . The second step is a diffusion step where the intermediate estimates , from the neighbors , are combined through the coefficients . Several sparsity inducing functions have been proposed in the literature [3]–[6], [18]–[21]. In particular, the weighted -norm regularization construction leads to the following sub-gradient column vector to update the algorithm in (6): (7)

Fig. 2. SC-eNB’s Network.

This choice leads to what we refer to as the reweighted zero-attracting (RZA) diffusion algorithm, see, e.g., [5], [6], [18], [19], [20], [21]. Ideally, sparsity is better represented by the norm as the regularization factor; this norm denotes the number of non-zero entries in a vector. Considering that norm minimization is a Non-Polynomial (NP) hard problem, the norm is generally approximated by a continuous function. A popular approximation [7], [21] is , which leads to the following sub-gradient column vector: (8) This choice leads to what we refer to as the -sparse diffusion algorithm. In the following section, numerical results illustrate the advantages of using the sparse diffusion algorithm (6), with sparsity-inducing subgradients (7), (8), to solve the spectrum estimation problem in (5). III. NUMERICAL RESULTS In these simulations, we consider a network composed of SC-eNB’s estimating the spectrum emitted by one MBS, as illustrated in Fig. 2. The channels between the SC-eNBs and MBS are modeled as static -tap FIR filters, with , for all . Each tap is modeled as a zeromean complex Gaussian random variable with variance . The SC-eNBs scan frequencies over the frequency axis, which is normalized between 0 and 1, and use non-overlapping rectangular basis functions to model the expansion of the transmitted spectrum. The basis functions have amplitude equal to one. To estimate the spectrum profile, we employ the sparse ATC diffusion algorithm in (6) where the step-sizes are set equal to , for all . Furthermore, we consider the combination matrix , whereas is chosen such that for all , . In Fig. 3 we illustrate the result of the distributed spectrum estimation carried out by different algorithms, namely, the global sparsity agnostic Least Squares (LS) solution (i.e, the solution of (5) with ), the sparsity agnostic ATC diffusion adaptation in (6) (with ), and the -sparse diffusion algorithm in (6) (with and ). We assume that the MBS transmits over 6 basis functions (i.e., 12 channels), thus leading to a sparsity ratio equal to 6/50. The power transmitted over each basis function is set equal to 1. The true transmitted spectrum is also reported in Fig. 3. The noise in (3) is assumed to be Gaussian with a variance dependent on the interference level on the frequency , according to

Fig. 3. Example of distributed spectrum estimation.

, where the receiver noise variance is set equal to , for all . As we can notice from Fig. 3, both LS solution and ATC diffusion are able to identify the channels where the MBS is transmitting. Nevertheless, they both produce several spurious terms, which degrade the estimation performance. From Fig. 3, it is also clear how the -sparse diffusion algorithm is able to strongly reduce the effect of the spurious terms, thus fitting much better the transmitted spectrum. To assess the performance of the proposed technique, in Fig. 4 we report the behavior of the steady-state network Mean Square Deviation (MSD), defined as , obtained at convergence by different algorithms, versus the signal to noise ratio (SNR). In particular, we consider the results achieved by the sparsity agnostic ATC diffusion adaptation in (6) (with ), and the -sparse diffusion algorithm in (6), considering different values of the regularization parameter . The steady-state values are obtained by averaging over 200 independent experiments and over 100 time samples after convergence. The simulation considers all the parameters defined in the previous scenario. The SNR is modified by acting on the noise variance , while keeping fixed channels and transmitted power. As expected, from Fig. 4, the estimation performance gets worse as the SNR decreases. Since the vector to be estimated is sufficiently sparse (i.e., 6/50), the -sparse diffusion algorithm leads to better performance in almost every SNR regime. What is interesting to notice from Fig. 4 is the effect of the regularization parameter . Indeed, increasing the value of the parameter , the zero-attracting capability of the

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Fig. 4. MSD versus SNR, for different algorithms and values of the regularization parameter .

-sparse diffusion algorithm improves. This can be useful at low SNR where, if the vector is sufficiently sparse, the effect of a large noise can be counteracted through the use of a stronger zero-attracting capability, which forces the elements of the estimated vector, associated with the zero-components of , to become smaller in magnitude. In the low SNR regime, this explains the large gain achieved by the -sparse diffusion algorithm, with respect to the sparsity agnostic diffusion algorithm, if we use a large value of . At the same time, at large values of SNR, a large value of might induce a too strong constraint that reduces the performance of the sparse diffusion algorithm with respect to the sparsity agnostic diffusion strategy. For the same reasons, a lower value of can induce a large gain at high SNR, while it becomes less effective at low SNR. This shows the capability of the -sparse diffusion algorithm to provide large gains, with respect to sparsity agnostic diffusion, in both the low and high SNR regimes, depending on how the parameter is chosen. To assess the sensitivity of the proposed algorithms to different degrees of sparsity of the spectrum, in Fig. 5, we show the behavior of the difference (in dB) between the network MSD of -sparse diffusion and sparsity agnostic diffusion versus the number of non-zero components of . The same behavior for RZA-ATC diffusion is also shown in Fig. 5. The results are averaged over 200 independent experiments and over 100 samples after convergence. We consider the same settings of the first simulation (Fig. 3), while varying the number of active basis. The parameters of RZA-ATC diffusion are and . As we can notice from Fig. 5, both algorithms lead to a large gain when the vector is very sparse. However, while the RZA-ATC diffusion leads to a loss when the number of non-zero components is greater than a certain value (around 15, in this example), the -sparse diffusion always leads to a positive gain in all cases. Finally, to illustrate the adaptation capability of the proposed technique, in Fig. 6 we report the behavior of the power estimated over an initially busy channel (the 20-th channel in Fig. 3), which the MBS ceases to use after iteration 1000, comparing the results achieved by ATC diffusion and -sparse ATC diffusion. We consider the same settings of the first simulation

IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 12, DECEMBER 2013

Fig. 5. MSD Gain versus number of non-zero components, considering different algorithms.

Fig. 6. Example of spectrum tracking capability, considering different algorithms.

(Fig. 3), except for the MBS’s transmitted power that is set equal to 0.25. From Fig. 6, we can notice how the -sparse ATC diffusion is able to track the sparse spectrum in a much more efficient way than the sparsity-agnostic ATC diffusion, thanks to its zero-attracting feature that accelerates the convergence of the zero-components. IV. CONCLUSIONS In this letter, we proposed a distributed, sparsity-aware spectrum sensing technique for small cell networks, based on a basis expansion model of the power spectral density transmitted by the Macro Base Station. The estimator relies on sparse diffusion adaptation strategies, which are able to exploit the sparsity induced by the narrow-band nature of MBS’s transmissions with respect to the broad swaths of usable spectrum. Numerical results illustrate the potential benefits of the proposed strategies. In particular, by exploiting sparsity, the -sparse diffusion algorithm is able to improve the spectrum estimation performance, and to accelerate its adaptation capability to timevarying scenarios, with respect to the sparsity-agnostic diffusion counterpart.

DI LORENZO et al.: DISTRIBUTED SPECTRUM ESTIMATION FOR SMALL CELL NETWORKS

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