In this paper, we experimentally validate the functionality of a developed algorithm for spectrum cartography using adaptive Gaussian radial basis functions (RBF). The RBF are strategically centered around representative centroid locations in a machine learning context. We assume no prior knowledge about neither the power spectral densities (PSD) of the transmitters nor their locations. Instead, the received signal power at each location is estimated as a linear combination of different RBFs. The weights of the RBFs, their Gaussian decaying parameters and locations are jointly optimized using expectation maximization with a least squares loss function and a quadratic regularizer. The performance of adaptive RBFs based spectrum cartography is shown through measurements using a universal software radio peripheral, a customized node and LabView framework. The obtained results verify the ability of adaptive RBF to construct spectrum maps with an acceptable performance measured by normalized mean square error (NMSE).acceptedVersionnivå

Beferull-Lozano, Baltasar

Cenkeramaddi, Linga Reddy

Hamid, Mohamed

Idsøe, Henning

Jordbru, Thomas

Agder University Research Archive

Spectrum Cartography using Adaptive Radial BasisFunctions: Experimental ValidationHenning Idsøe, Mohamed Hamid, Thomas Jordbru, Linga Reddy Cenkeramaddi and Baltasar Beferull-LozanoIntelligent Signal Processing and Wireless Networks (WISENET) Lab.Department of Information and Communication TechnologyUniversity of Agder, 4879, Grimstad, NorwayAbstract—In this paper, we experimentally validate the func-tionality of a developed algorithm for spectrum cartographyusing adaptive Gaussian radial basis functions (RBF). TheRBF are strategically centered around representative centroidlocations in a machine learning context. We assume no priorknowledge about neither the power spectral densities (PSD) ofthe transmitters nor their locations. Instead, the received signalpower at each location is estimated as a linear combinationof different RBFs. The weights of the RBFs, their Gaussiandecaying parameters and locations are jointly optimized usingexpectation maximization with a least squares loss function anda quadratic regularizer. The performance of adaptive RBFsbased spectrum cartography is shown through measurementsusing a universal software radio peripheral, a customized nodeand LabView framework. The obtained results verify the abilityof adaptive RBF to construct spectrum maps with an accept-able performance measured by normalized mean square error(NMSE).Keywords— Spectrum cartography, power spectrum maps,Adaptive radial basis functions, Experimental validation.I. INTRODUCTIONMany wireless communications related applications includ-ing network planning, frequency reuse, coverage prediction,interference management, opportunistic spectrum access andcognitive radios require spatial radio spectrum awareness[1]–[4]. This spectrum awareness is achieved by buildingspatial, frequency and time dependent radio environment maps(REMs) under a framework called spectrum cartography.Throughout the rest of this paper, REMs and spectrum cartog-raphy will be used interchangeably to refer to received signalpower maps. Spectrum cartography is a regression problemwhich is performed by collecting geo-localized measurementsfollowed by spatial regression [5].There are several techniques proposed for spectrum car-tography including Kriging interpolation [5], [6], dictionarylearning [7], sparsity aware regression [8], basis expansion [9],matrix completion [10] and reproducing kernel Hilbert space(RKHS) regression [11]–[13]. These techniques are brieflysurveyed in [14]. To the best of the authors’ knowledge, eventhough there exist diverse schemes for spectrum cartography,each of these schemes however has at least one of thefollowing three limitations [14].1) Some information regarding the transmitters’ parametersand locations are needed.2) Spatially dense measurements are required which iscostly in terms of energy and communication bandwidth.3) Basis functions are chosen statically with no adaptationbased on the measurements.In [14], the aforementioned limitations are overcome bydeveloping a spectrum cartography algorithm that adapts theparameters of the basis functions based on a relatively lowerdensity of measurements with no prior information about thetransmitters. This goal is achieved by using RBF that arecentered at strategic informative centroids locations insteadof using kernels directly centered on sensors locations.In view of the theoretical foundations of adaptive RBFbased spectrum cartography provided in [14], this paperpresents an experimental validation of the algorithm in twoaspects. At first, the functionality of the algorithm is shownin real radio propagation environment. Secondly, in contrast tothe simulations findings, measurements’ performance is evalu-ated. Moreover, to the best of our knowledge, no experimentalstudies have been conducted regarding spectrum cartographyprior to our campaign being reported in this paper. Therefore,the measurements setup and methodology is a stand alonecontribution of this paper which can be used for implementingother spectrum cartography algorithms for comparison andbench marking purposes.The remaining of this paper is structured as follows. SectionII shows the system model. Section III presents the maincontribution of [14] which is adaptive RBF with representativecentroids based cartography. Measurements setup, results andfindings are shown in Section IV. Finally, Section V concludesthe paper.Notation - Upper case bold letters are used to denotematrices such as C ∈ RM×N while the element correspondingto the ith row, jth column of C is denoted as [C]ij . Columnvectors are denoted by lower case bold letters as c with cibeing the ith entry of vector c. For scalars, non-bold lettersare used. CT and C−1 are the transpose and inverse of matrixC, respectively. The Identity matrix of size L is denoted byIL. 1 denotes an all one vector.II. SYSTEM MODELThe task of spectrum cartography is to construct a completeREM having a finite number of sensors. Consider an areaA ⊆ R2 that contains a set of N sensors that are awareof their locations. We define X to be the set of the senorslocations where X ⊂ A = {x1,x2, ...,xn}, 1 ≤ n ≤ N . Eachsensor located at xi is providing a fusion center (FC) with ameasurement of the received signal power yi(t) on its locationat time t. Therefore, we define the measurements set at timet as Y(t) ∈ R = {y1(t), y2(t), · · · , yn(t))}, 1 ≤ n ≤ N . The© 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. - DOI: 10.1109/SPAWC.2017.8227752 Fig. 1: Graphical illustration of adaptive RBF algorithm based spectrum cartography. The left sub-figure shows an arbitrarythree initial centroids locations and their Gaussian decaying parameters as shown by the red curves while the green dotsrepresent arbitrary sensors locations. The right sub-figure depicts the adapted RBF centroids and their Gaussian decayingparameters after convergence. All the curves and data in the figure are for illustrative purposes.learning set that contains both the measurements and the sen-sors is denoted as Z(t) ∈ R3 = {z1(t), z2(t), · · · , zN (t)|zi =(xi, yi(t))}. Throughout the rest of this paper we considerinstantaneous spectrum cartography with no learning alongthe time dimension, accordingly we omit the time dependencyhereafter. The estimated cartography is denoted from here onas hˆ(x) : R3 → R while the actual cartography is denotedas h(x). Cartography estimation is performed by the FC thatreceives all measurements.The RBF based learning assumes that each point on thelearning set, (xn, yn), affects the target function on anylocation x as a function of the Euclidean distance betweenx and xn. Considering REM as a function to be learned withthe standard form of RBF, the learning REM is given by:hˆ(x) = w0 +N−1∑i=1wiexp(−γn ‖x− xi‖2) (1)where w0 is a constant offset that represent the backgroundnoise, w1, · · · , wN−1, are weighting parameters for the con-tribution of the different RBF, γi is a Gaussian decayingparameter for the ith RBF. Applying (1) to the learning setwould result inΦw = y (2)where Φ =[1N Φ˜], Φ˜ ∈ RN×N with eachelement being Φ˜nk = exp(−γk‖xn − xk‖2) ,w =[w0, w1, · · · , wN−1]T and y = [y1, y2, · · · , yN ]TAccordingly, w is found byw = Φ−1y (3)The solution given by (3) implies finding N parameters fromN observations which can be computationally expensive forlarge data sets. Moreover, Φ is not always invertible. Toovercome these two limitations, representative centroids tolocate the RBF is proposed as explained in the next section.III. ADAPTIVE REPRESENTATIVE RBFNow instead of centering the RBF around the sensorlocations, K representative centroids are chosen for the RBF.The location of the centroids are denoted as µ1, · · · , µK .Consequently, the cartography learning model represented by(1) is modified as:hˆ(x) = w0 +K∑k=1wkexp(−γk ‖x− µk‖2) (4)which is rewritable asΘw = y (5)where Θ =[1 Θ˜], Θ˜ ∈ RN×K composed as Θ˜nk =exp(−γk‖xn − µk‖2), 1 ≤ n ≤ N, 1 ≤ k ≤ K.To estimate the cartography on A, following model param-eters need to be optimized jointly1) The RBF centroids positions, µ1, · · · , µK .2) The weights vector w.3) The Gaussian decaying parameters γ1, · · · , γK .The joint optimization of these parameters, a least squaresolution is considered asminw,γ1···γk,µ1···µkN∑n=1(yn − hˆ(xn))2+ λ ‖w‖2 (9)where λ is a positive constant that trades off the estimator biasand variance [15]. Moreover, λ > 0 guarantees a solution to(9) even if ΘΘT is a singular matrix (see (6)).To solve the optimization problem (9) for w, µ1, · · · , µK ,and γ1, · · · , γK , we propose to use expectation maximizationbased optimization as in Algorithm 1.For initializing the centroids locations µ1, · · · , µK , eitherK− means clustering considering the sensors’ locations orCartesian grid over A can be used. On the other hand, theGaussian decaying parameters γ1, · · · , γK can all be assigneda same value for initialization. Fig. 1 illustrates graphically theidea of RBF adaptation exploiting Algorithm.SwitchRx (ZC706/FMCOMMS5)PC runningLabViewTx (USRP-2953R)(a) (b)Fig. 2: (a) Measurement setup. (b) A picture of measurements location and setupData: Initialize the decaying parameters, γ1, · · · , γK andthe centroids µ1, · · · , µK ;while no convergence doFix γ1, · · · , γK , µ1, · · · , µK and solve for w asw =(ΘTΘ + λIK)−1ΘTy (6)Fix w, µ1, · · · , µK and solve for γ1, · · · , γK , usinggradient desent asγk γk − αN∑n=1(yn − hˆ(xn))· ‖xn − µk‖2 ·(wkexp(−γk ‖xn − µk‖2))(7)1 Fix w, γ1, · · · , γK and solve for µ1, · · · , µK ,using gradient descent asµk µk + 2αγkN∑n=1(yn − hˆ(xn))· (xn − µk) ·(wkexp(−γk ‖xn − µk‖2))(8)endAlgorithm 1: Adaptive RBF cartographyIV. MEASUREMENTSThis section presents the measurements setup and obtainedresults which is the contribution of this paper.A. Measurements setupThe basic task of our measurements setup is to have thecomponents of the adaptive RBF based cartography. Thesecomponents are transmitters, sensors and a fusion center. Forsimplicity we adopted a setup consists of one transmitter,one sensor that moves and sense in different locations and acomputer that controls both the transmitter and the sensor(s)and acts as a fusion center. Fig. 2(b) shows the adoptedmeasurements setup with the equipment and parameters as inTable I. Fig. 2(b) shows a picture of the measurement locationand setup-100-95-90-85-80-75Fig. 3: Obtained spectrum map when using 50 measurementsfor learning and K = 10 centroids. The white crossesrepresent the measurements locationsFollowing are important remarks regarding the measure-ments setup:1) ZC706 /FCOMM5S SDR which used as a sensor is asoftware defined radio (SDR) that is developed by theauthors of this paper and and its performance is evaluatedin [16].2) The operating frequency of 5.41 GHz is chosen afterscanning several other frequencies and that particularfrequency is found vacant at the time and location ofthe measurements.3) Binary phase shift keying (BPSK) is chosen arbitrarilyas the main concern is to have a wireless transmissionregardless of the modulation type and transmitted infor-mation.4) A BPSK signal of a sample rate of 1.0 MS/S, wouldresult in a main lobe bandwidth of 2.0 MHz. Therefore,a reception bandwidth of 3.0 MHz assures that the wholeTABLE I: Measurements Equipment and ParametersDevice/Parameter Type/ValueTransmitter NI-USRP 2953RSensor(s) ZC706 /FCOMM5S 2953R SDRCenter frequency 5.41 GHzModulation BPSKSample rate 1.0 MS/SReception bandwidth 3.0 MHzTABLE II: Obtained NMSE for measurements and simula-tions, N = 50Number of centroids Measurements5 −7.9 dB10 −8.7 dB20 −9.7 dBenergy contained in the main lobe will be received withextra energy contained in the first side lobe.5) The switch is used to control both the transmission andsensing through the same PC.The location of the measurements is a polyhedron roomwhere a Cartesian grid of 0.5m ×0.5m is made to determinethe measurements locations set Xmes ⊂ A as the crossingsof this grid. The largest dimensions of the area within theroom that is capable of hosting the measurements are 7.5mand 3.0m. Consequently, the measurements points set, Xmes,are expressible as Xmes = U × V,U = {0, 0.5, 1, · · · 3},V ={0, 0.5, 1, · · · 7.5}. The resultant measurements set consists of112 points. However, 12 points are not feasible because theyare either corners or there exist some unmovable furnitureon them. Therefore, 100 measurements points are consideredwhich is divided into a learning set, Xler, and a verificationset, Xver each of them contains 50 points, learning set pointsare chosen uniformly randomly and the rest of measurementspoints are left for verification. Hence, Xler ∩ Xver = ∅ andXler ∪ Xver = Xmes.For the quantitative evaluation, the normalized mean squareerror (NMSE) of the estimator using the measured values isused which is calculated asNMSE = E∣∣∣h(x)− hˆ(x)∣∣∣2|h(x)|2with E [·] denoting the expected value and x ∈ XverB. Measurements resultsHereafter, the obtained results from the measurements arepresented and analyzed.Fig. 3 shows the reconstructed power spectrum map usingK = 10 centroids followed by a quantitative analysis bymeans of obtained NMSE when changing the number ofcentriods as in Table II. As expected, the larger the numberof centroids, the better the performance.V. CONCLUSIONSAn experimental validation of adaptive radial basis func-tions based spectrum cartography algorithm is carried out.The theoretical essence of the algorithm is to perform spatialinterpolation for constructing power spectrum maps usingstrategically centered adaptive Gaussian radial basis func-tions. The centroids locations optimized jointly with theGaussian decaying parameters and the linear model weights.The measurements findings validate the theory and show theinfluence of the number of centroids on the performance ofthe algorithm.ACKNOWLEDGMENTThis work was supported by the FRIPRO TOPPFORSKgrant 250910/F20 from the Research Council of Norway.REFERENCES[1] A. Galindo-Serrano, B. Sayrac, S. B. Jemaa, J. Riihija¨rvi, andP. Ma¨ho¨nen, “Harvesting MDT data: Radio environment maps forcoverage analysis in cellular networks,” in 8th Int. Conf. Cognitive RadioOriented Wireless Networks, Jul 2013, pp. 37–42.[2] S. Grimoud, S. B. Jemaa, B. Sayrac, and E. Moulines, “A REM enabledsoft frequency reuse scheme,” in IEEE Global Commun. Conf., Dec2010, pp. 819–823.[3] J. D. Naranjo, A. Ravanshid, I. Viering, R. Halfmann, and G. 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Spectrum cartography using adaptive radial basis functions: Experimental validation

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Spectrum cartography using adaptive radial basis functions: Experimental validation

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