This work presents insights on the application of the Bayesian information criterion (BIC) to fix the optimum number of coefficients in the Volterra series applied to the modeling and linearization of power amplifiers (PAs). The BIC is transformed from a rule to be applied after selection techniques to a stopping criterion, which enables the halting of the algorithm when a condition is reached. This study reveals that the BIC is equivalent to allow a certain identification normalized mean square error (NMSE) decrease after the inclusion of a model component. Experimental results of the digital predistortion of a class J PA are provided, demonstrating the proposal applicability in the attaining of the optimum number of coefficients. A comparison is made between the results obtained when the stopping rule is applied to the hill climbing (HC) and the doubly orthogonal matching pursuit (DOMP) algorithms

Becerra González, Juan Antonio

Crespo Cadenas, Carlos

Madero Ayora, María José

Noguer, Rafael G.

English

idUS. Depósito de Investigación Universidad de Sevilla

IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. XX, NO. X, JUNE 2020 1On the Optimum Number of Coefficients of SparseDigital Predistorters: a Bayesian ApproachJ. A. Becerra, Senior Member, IEEE, M. J. Madero-Ayora, Senior Member, IEEE,R. G. Noguer, and C. Crespo-Cadenas, Senior Member, IEEEAbstract—This work presents insights on the application of theBayesian information criterion (BIC) to fix the optimum numberof coefficients in Volterra series applied to the modeling andlinearization of power amplifiers. The BIC is transformed froma rule to be applied after selection techniques to a stoppingcriterion, that enables the halting of the algorithm when acondition is reached. This study reveals that the BIC is equivalentto allow a certain identification normalized mean square error(NMSE) decrease after the inclusion of a model component.Experimental results of the digital predistortion of a class J poweramplifier are provided, demonstrating the proposal applicabilityin the attaining of the optimum number of coefficients. Acomparison is made between the results obtained when thestopping rule is applied to the hill climbing (HC) and the doublyorthogonal matching pursuit (DOMP) algorithms.Index Terms—Bayesian information criterion, Digital pre-distortion, Doubly orthogonal matching pursuit, Hill climbingalgorithm, Order reduction, Power amplifier.I. INTRODUCTIONIn a practical situation of model fitting, the model order isgenerally unknown. Many order selection methods have beenproposed, each one of them following a different approach.The Akaike Information Criterion (AIC) [1] estimates theinformation lost by a model, providing a means for modelselection through the trade-off between the goodness of the fitand the penalty because of the number of estimated param-eters. Another popular method is the Bayesian informationcriterion (BIC) [2] that selects the best model by maximizingthe value of the likelihood function. Despite the extensiveliterature on model selection techniques, the most commonorder selection criteria remain the AIC and the BIC [3].Regarding the compressed-sensing recovery of the coeffi-cients of the model, there are a variety of algorithms that havebeen used in practical applications [4]. There exist multiplescenarios in which the dependence on just a certain number ofregressors is desired. This tuning can be achieved by applyingmodel selection techniques, which aim at choosing one modelfrom a set of candidate models.Manuscript received March 17, 2020; revised June 12, 2020, August 14,2020 and September 2, 2020. This work was supported by the Spanish Na-tional Board of Scientific and Technological Research under Grant TEC2017-82807-P, and by the European Regional Development Fund (ERDF) of theEuropean Commission.The authors are with the Departamento de Teorı́a de la Señal y Comu-nicaciones, Escuela Técnica Superior de Ingenierı́a, Universidad de Sevilla,Sevilla, 41092, Spain (e-mail: jabecerra@us.es; mjmadero@us.es; rafgar-nog@alum.us.es; ccrespo@us.es).Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.In the context of behavioral modeling and digital predis-tortion (DPD) of power amplifiers (PAs), the use of Volterraseries as the underlying model poses a serious drawbackbecause of their rapid grow in number of coefficients withthe nonlinear order and memory depth. Coefficient selectiontechniques enable their use by reducing their computationalcomplexity. Recently, the BIC has been applied to fix theoptimum number of coefficient [5]–[9] and there exist a vastliterature on order reduction techniques that can benefit fromits features, such as the hill climbing (HC) [10] algorithm andthe doubly orthogonal matching pursuit (DOMP) [11].In this letter, we focus on the application of the BIC toiterative selection techniques by transforming the BIC costfunction to a stopping criterion in order to provide an optimalupper bound for the number of coefficients from informationtheory perspective. Considering that the BIC was previouslyused after the DOMP run, here we enable the algorithm haltin the corresponding iteration, which produces remarkablesavings in execution time. Also, we give formalism to thead-hoc rule that was applied to the HC, providing a uniquemodel-selection scheme. The remaining of this communicationcontinues with the notation and prerequisites in Section II. Thetransformation of the BIC into a stopping rule is provided inSection III. Section IV deals with experimental design andresults and Section V concludes this work.II. NOTATION AND PREREQUISITESThe recovery of unknown model parameters h ∈ CN after aspatial transformation is usually modeled by the measurementequationy = Xh + e, (1)where y ∈ CM is the output of the system, X ∈ CM×N is themeasurement matrix that holds one of the N model regressorsin each of its columns, e is the measurement error and M isthe number of sample points. A common approach to obtain anestimation ĥ is the minimization of the quadratic error ‖e‖22,that leads to the normal equation and to the least-squares (LS)solutionĥ =(XHX)−1XHy, (2)where XH stands for the Hermitian transpose of matrix X.The estimation of the output signal is attained with ŷ = Xĥ.Although this well-established procedure provides a uniquesolution to the regression, it also finds a dependence of thesystem output with the whole set of model regressors. Theapplication of compressed-sensing techniques to this schemeallows a sparse recovery of the solution.IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. XX, NO. X, JUNE 2020 2Selection techniques aim at obtaining an ordered supportset S which contains the indices of the measurement matrixcolumns sorted by importance. In each iteration the algorithmadds one component —or a set of them, depending on thetechnique— to the support set, an estimation of h is performedand the residual error r is calculated. This loop is repeateduntil a stopping criterion is achieved. Order selection supposethat there exists a set of candidate models, amongst which theminimum of a cost function will define the optimal model.Since sorting algorithms select one or more components periteration, that is, they are incremental, it is reasonable totransform the criterion that has to be evaluated after the loopof the algorithm to a stopping rule that breaks the executiononce the condition is fulfilled.III. THE BIC AS A STOPPING RULEThe BIC is defined as the sum of a term that depends onthe error and a penalty that is related to the number of modelcomponents n [5]BIC = 2M ln σ̂2e + 2n ln 2M, (3)where σ̂2e is the error variance and M is the number of samplesused for the model identification. The BIC in (3) acts as atrade-off between the error and the number of components. Inan iterative sorting algorithm, the modeling error is commonlyknown as the residual error r, thereforeσ̂2e = Var[e] = Var[y − ŷ] = Var[r] = E[|r|2]− |E[r]|2, (4)where Var[·] represents the variance of its argument, E[·] isthe expected value and | · | stands for its module. Since it is acommon practice to center the variables, without any loss ofgenerality E[r] = 0 can be assumed. Therefore,E[|r|2] = 1MM∑i=0|ri|2 =1M‖r‖22, (5)where ‖·‖2 stands for the `2 norm of its argument.Combining the above results with (3) and adding an explicitdependence with the iteration k ∈ N, the BIC can be rewrittenasBIC(k) = 2M ln(1M‖r(k)‖22)+ 2n(k) ln 2M, (6)where n(k) denotes the number of components at iteration k.This cost function is composed of two terms. The first termis dependent on ‖r(k)‖22. Since this magnitude measures themodeling error —or equivalently the amount of output powerleft to be captured by the model—, as more regressors areadded to the model, the basis space in which the output isgoing to be expressed is richer and the residual decreases. Thesecond term in (6) is a penalty that increases linearly with thenumber of components and also depends on a fixed numberof samples M . Therefore, the BIC is initially decreasing, anda minimum will be found when its increment is not negative.Assuming that there is no collinearity in the measurementmatrix, i.e., all the regressors are different, the BIC is amonotonically decreasing function. As the BIC is initially103104105Identification size M-10-1-10-2-10-3-10-4NMSE/n (dB)Fig. 1. Evolution of the allowed NMSE increment per iteration and componentwith the identification size for the BIC.decreasing, a minimum will be found when it starts to increase,thus ∆BIC(k) > 0. The BIC increment in one iteration followsln(1M‖r(k)‖22)− ln(1M‖r(k−1)‖22)+ ∆n(k)ln 2MM> 0,(7)where ∆n(k) = n(k) − n(k−1) stands for the increment innumber of components at iteration k, and rearranging theterms,‖r(k)‖22‖r(k−1)‖22> (2M)−1M ∆n(k). (8)The normalized mean square error (NMSE) at iteration k isdefined as NMSE(k) = 10 log10‖r(k)‖22‖y‖22, therefore∆NMSE(k)∆n(k)> − 10Mlog10 2M. (9)This relationship is plotted in Fig. 1. The stopping rule in(9) supports the following procedure: “continue running thealgorithm until it is not able to discard a certain amount oferror per iteration and component”. This error threshold isdependent on the number of samples used for the identificationof the model, which balances the error evolution shape withthe number of model components and allows order selectionwith a reduced risk of overfitting [12].A. Generalization of the BIC Stopping Rule and Relation withOther Proposals in the LiteratureFrom (8), the definition of a general stopping rule follows‖r(k)‖22‖r(k−1)‖22> α∆n(k), (10)which is expressed as inequality considering that there doesnot exist regressor repetition. As the stopping rule is expectedto be triggered when the model shows good accuracy, we canalso state that the residual decrement will be slow, that is,‖r(k−1)‖22 ≈ ‖r(k)‖22. From (9) and (10),∆NMSE(k)∆n(k)> 10 log10 α, (11)where α < 1, and its value in decibels can be interpreted asthe NMSE tolerance per coefficient.In (6) of [10] —and, equivalently, (10) of [13]— a costfunction J was defined as the trade-off between model accu-racy and model complexity,J (k) = NMSE(k) + µn(k), (12)IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. XX, NO. X, JUNE 2020 30 50 100 150 200 250Number of coefficients-44-42-40-38-36-34-32-30NMSE (dB)HC neighborsHC optimum valuesHC convex hullDOMPDOMP+BICHC optimum values + BICFig. 2. Empirical identification NMSE of the HC and DOMP algorithmsversus the number of coefficients. The BIC stopping rule indicates the momentin which the error difference decreases below a given threshold.where µ represents the NMSE tolerance per coefficient, andthe minimum of J (k) indicates the optimum model. Consid-ering (13) of [14], which states thatnBIC = arg minn[NMSE(k) +n(k)M10 log10 2M], (13)and following the same rationale, the optimum is found when∆NMSE(k)∆n(k)> µ, (14)therefore µ = 10 log10 α and µBIC = − 10M log10 2M.IV. EXPERIMENTAL DESIGN AND RESULTSIn order to validate the proposal, a generalized memorypolynomial (GMP) model [15] was regressed for the indirect-learning DPD of a PA working on its nonlinear regime.The testbench was composed by a SMU200A vector signalgenerator (VSG) from Rohde & Schwarz, a PXA-N9030Avector signal analyzer (VSA) from Keysight Technologies anda dc power supply. The device under test (DUT) was thecascade of two Mini-Circuits TVA-4W-422A+ preamplifiersand the continuous mode class J PA [16] operating at a centerfrequency of 850 MHz. The output power was set to +31 dBmcharacterized by 3.5 dB of gain compression.The signal under test was designed according to a 5G-NRscheme with 20 MHz of bandwidth, 30 kHz carrier separationand a sampling rate of 92.16 MHz. The signal segment usedfor identification was composed of 5500 samples, that through(9) corresponds to a minimum-allowable NMSE decrement ofµ = −0.0073 dB or, equivalently, α = 0.998.The proposed BIC stopping rule was applied to two differentselection techniques, the HC and the DOMP algorithms, andtheir results were compared. According to the BIC appliedto the HC algorithm, the optimum GMP model presented aset of parameters given by Ka = 5, La = 7, Kb = 3,Lb = 2, Mb = 2, Kc = 4, Lc = 4, Mc = 7. For thesake of comparison, the DOMP was applied to the neighbourwith maximum number of coefficients in the HC iterations toobtain a set of sorted regressors by importance, to which theBIC defined an optimum set. Fig. 2 shows the identificationNMSE evolution with the number of coefficients. Note that the810 820 830 840 850 860 870 880 890Frequency (MHz)-60-50-40-30-20-100Normalized PSD (dBm/Hz) w/o DPDError w/o DPDHCError HCDOMPError DOMPFig. 3. Normalized power spectral density of the PA output without DPD andwith DPD regressed with the number of coefficients indicated by the BIC.TABLE IMEASURED PERFORMANCE IN TERMS OF NMSE, ACPR, EVM ANDNUMBER OF COEFFICIENTS FOR THE PA WITH AND WITHOUT DPDNMSE ACPR -1/+1 EVM NumberExperiment (dB) (dBc) (%) of coef.w/o DPD -19.5 -29.1 / -27.0 8.6 -HC DPD -41.4 -53.4 / -52.8 1.0 206DOMP DPD -41.5 -53.8 / -53.1 1.1 137BIC was applied to both techniques independently, attainingan optimum number of coefficients of 137 and 206 in theDOMP and the HC techniques, respectively. This differenceis justified by the shape of the error evolution, which reachesa steady state sooner in the DOMP case. Note that the DOMPcurve does not match with the HC convex hull because thefirst selects the model coefficients while the latter selectsthe optimal GMP model parameters, i.e., the HC producesa complete GMP-model regressors matrix while the DOMPproduces a subset of it, namely a sparse regressors matrix.The optimum models for both HC and DOMP were usedto generate a predistorted signal, whose linearization perfor-mance and power spectral density (PSD) are shown in Table Iand Fig. 3, respectively. The effect of both techniques isevidenced by a reduction of the in-band distortion, achievingan enhancement of over 20 dB in NMSE and an errorvector magnitude (EVM) of about 1% in the linearized signal.The adjacent channel power ratio (ACPR) improvement isillustrated by the decrease in spectral regrowth, achieving alevel of below −52 dB in all the linearized cases.V. CONCLUSIONA transformation of the BIC into a stopping rule for iterativealgorithms has been presented in this letter. This modificationallows to extract information about how the optimum modelis selected amongst the set of candidates, that follows the ruleof stop adding components to the support set once the newselection is not able to discard a certain amount of error. Theresult has been related to other forms of cost functions in theliterature, providing a unique framework. Measurement resultsshow the effect of the error evolution on the optimum numberof coefficients and its applicability to techniques such as theHC and the DOMP.IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. XX, NO. X, JUNE 2020 4REFERENCES[1] H. Akaike, Information Theory and an Extension of the MaximumLikelihood Principle. Springer New York, 1998.[2] G. Schwarz, “Estimating the dimension of a model,” Ann. Statist., vol. 6,no. 2, pp. 461–464, Mar. 1978.[3] J. Ding, V. Tarokh, and Y. Yang, “Bridging AIC and BIC: A newcriterion for autoregression,” IEEE Trans. Inf. Theory, vol. 64, no. 6,pp. 4024–4043, June 2018.[4] Y. C. 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On the Optimum Number of Coefficients of Sparse Digital Predistorters: A Bayesian Approach

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On the Optimum Number of Coefficients of Sparse Digital Predistorters: A Bayesian Approach

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