On the Robustness of Look-Up Table Digital Predistortion in the Presence of Loop Delay Error

On the Robustness of Look-Up Table Digital Predistortion in the Presence of Loop Delay Error You-Jiang Liu,Bin Lu,Tao Cao,Bang-Hua Zhou,Jie Zhou,and Yi-Nong LiuAbstract—Look-up table(LUT)predistortion is the most promising technique to linearize RF power amplifiers(PAs). Before it can be employed successfully,loop delay errors must be estimated and compensated accurately.In this paper,the effect of loop delay errors on LUT predistortion is analyzed.It is shown that LUTsfluctuations happen to both amplitude and phase tables.Then,a novel loop delay estimation algorithm,with a high degree of robustness and accuracy,is presented.To further improve the robustness of LUT predistortion to loop delay errors, the smoothingfilter(SMF)method to polish LUTsfluctuations is proposed here.LUT predistortion can suffer bigger loop delay errors because of the proposed SMF method,but retains good lin-earization performance close to the optimal one.The performance of the proposed loop delay estimation algorithm is investigated by comparative study with a state-of-the-art algorithm.Its ro-bustness is also demonstrated under a noisy feedback path by simulations.The effectiveness of the SMF method is assessed by both simulations and parison among different predistortion systems is also presented.The results clearly show that the proposed techniques are useful for LUT predistortion. Index Terms—Digital predistortion(DPD),linearization tech-niques,look-up table(LUT),loop delay,power amplifiers(PAs).I.I NTRODUCTIONT O make efficient use of the sparse but valuable spectrum in modern communication systems,new signal modula-tion formats such as quadrature amplitude modulation(QAM), wideband code division multiple access(WCDMA)and orthog-onal frequency division multiplexing(OFDM)have been de-veloped during the past decade.In order to get linear amplifi-cation for these signals(with non-constant envelope and high peak-to-average ratio(PAR)),nonlinear distortion of power am-plifiers(PAs)has been a critical problem.Hence,linearization technique for PAs is a very important research pared with alternative linearization techniques,digital predistortion (DPD)[1]–[6]is the most promising one,which exhibits anManuscript received July22,2011;revised September29,2011,November 24,2011;accepted November27,2011.Date of publication February07,2012; date of current version September25,2012.This paper is recommended by Associate Editor A.Neviani.Y.J.Liu is with the Department of Engineering Physics,Tsinghua Uni-versity,Beijing100084,China and is also with the Institute of Electronic Engineering,China Academy of Engineering Physics,Mianyang621900, China(e-mail:liuyj04@).B.Lu,T.Cao,B.H.Zhou,and J.Zhou are with the Institute of Electronic En-gineering,China Academy of Engineering Physics,Mianyang621900,China. Y.N.Liu is with the Department of Engineering Physics,Tsinghua Univer-sity,Beijing100084,China.Color versions of one or more of thefigures in this paper are available online at .Digital Object Identifier10.1109/TCSI.2012.2185332Fig.1.Basic architecture of digital predistortion(DPD).inverse nonlinearity of amplifier to compensate for its nonlin-earity.Fig.1shows the basic architecture of DPD,the predis-torter block can be some common structures,such as look-up ta-bles(LUTs)[1]–[3],polynomial[4],[5],neural network[6],andso on.Among them the LUT predistortion is the most cost-effec-tive one because of its simplicity of hardware implementation[1]–[3].The adaptation block usually adopts algorithm like theleast mean square(LMS),or the recursive least square(RLS),togenerate the predistorter’s coefficients.It should be noted that,loop delay errors between the original input and the delayedfeedback output should be compensated beforehand by the loopdelay estimation block.The comparator in the adaptation block compares the originalinput with the delayed feedback output,detecting the pseudoerror,which lead to an incorrect update of the predistorter.Bynoticing that the detrimental effects of loop delay errors on DPDhave been studied[7]–[9],this paper instead presents a new ef-fect on LUT predistortion.It is shown that,LUTsfluctuationshappen to both amplitude and phase tables because of loop delayerrors.Thefluctuations lead to AM-AM(amplitude modulationto amplitude modulation)fluctuations and AM-PM(amplitudemodulation to phase modulation)fluctuations of the overall pre-distortion system,ultimately degrading the linearization perfor-mance.The loop delay estimation algorithm proposed in[10]has the problem of low accuracy and iterative stability.It isquite sensitive to the iterative control step parameter.Particu-larly,long iterations are needed if a long loop delay is encoun-tered.The work in[11]increases the computational complexityby using fast Fourier transform(FFT).The algorithm in[12]requires a high oversampling ratio of64to achieve desired ac-curacy.The technique proposed in[13]gets only a coarse es-timation of the loop delay by using a complicated fourth-ordercross-correlation.In[14],it requires both a large number of mul-tiplications for cross-correlation and time-consuming iterationsfor fractional delay calculation.Another more,the method in[15]involves a complicated iterative process for fractional delayestimation,and the estimation accuracy is also limited.In thispaper,we propose a novel loop delay estimation algorithm with 1549-8328/$31.00©2012IEEEhigh accuracy.For computational complexity reduction,this es-timator is performed in two steps:integer delay estimation and fractional delay estimation,like[14]and[15].And the novel loop delay estimation algorithm can fulfill the1/64criterion in [10]properly.In practice,the real loop delay in predistortion system is always unknown.Though loop delay can be estimated,a small residual loop delay error may always exist.Furthermore,PAs temperature drift or unexpected circuit levelfluctuations would also cause shift of the instantaneous loop delay.This makes the previously estimated loop delay different from the real value again,consequently causing LUTsfluctuations.Even if the small loop delay error is less than1/64of a symbol duration [10],LUT predistortion may exhibit limited ability in reducing spectral regrowth at the PA output.To improve the robustness of the LUT predistortion to loop delay errors,we also introduce a smoothingfilter(SMF).This paper is organized as follows.Section II analyzes a new effect of loop delay errors on LUT predistortion in details.In Section III,the novel loop delay estimation algorithm and the smoothingfilter(SMF)method are proposed.Section IV gives the simulations and experimental results.Conclusions are drawn in Section V.Throughout this paper,afinite impulse response (FIR)filter is used in the address calculation block of LUTs to compensate for the memory effect by providing weighted sum of previous inputs as an index to the LUTs[16].II.E FFECT OF L OOP D ELAY E RRORS ON LUT P REDISTORTION Using discrete time notations for all the signals in Fig.1, and are the original input of predistorter and the delayed output,and is the output of PA without loop delay errors.When normalized,there is, where is the loop delay.Thus,one has relation between and in polar form(1)(2) where and,with the corresponding subscripts,repre-sent the amplitude and phase of signals.are amplitude errors and are phase errors.Fig.depicts the ampli-tude errors and phase errors versus,from which the dis-persions of the errors are clearly shown.In thisfigure,the loop delay error is set to be and(is the sam-pling interval of the baseband WCDMA signal,with10sam-pling ratio).We can see that the larger the loop delay error is, the severer the dispersions would be.Least mean square(LMS)method is widely used as the adap-tation algorithm for LUTs updating,namely(3)(4) where is the iteration index,is the amplitude table and is the phase table,is the desired linear gain of the am-fisystem,and are small positive step size constants,and is the square magnitude of the LUTs’input.In the following analysis,is omitted to simplify the notation.Fig.2.(a)Amplitude errors and(b)phase errors between the PA output and the delayed feedback output with and loop delay errors for 10WCDMA signal.When loop delay errors are present,the amplitude table and phase table values of the-th iteration(and),would de-viate from their ideal values,and.Denoting the deviations as and,one has and. By ignoring the shift in the operation point of PA[17],the am-plitude errors at the PA output is(5) where and are the amplitudes of the PA output with and without loop delay errors,is PA’s complex gain cor-responding to the operation point of the predistorted signal and is the amplitude of.Similarly,the phase at the PA output is(6) where and are the phases of the PA output with and without loop delay errors.Following(1),(3),(5)and(7) Similarly from(2),(4),(6)and(8) Following(7)and(8),errors of amplitude table and phase table after the-th iteration can be written as the following two recursion relations(9)(10) Because of and,wefinally have(11)(12) The results show that the LUTsfluctuation in ampli-tude table is the function of,and that in phase table is the function of.Notice thatand are expressed as weighted sum of and,respectively.And the weights are determined by and.Recalling Fig.2,different combinations of and may result in over-or lack-updating to different LUT-entries.That is to say,at some entries,there are expanding effects(or).While at some other entries,there are compressing effects(or).The entries converge to their optimal values only when or.The expanding and compressing effects will make the LUT-entry values to deviate from their optimal ones.Predistorted signal by such kind of LUTs predistorter will also lead to AM-AMfluctuations and AM-PMfluctuations of the overall system.From the point of view of time domain,the PA output is disturbed by high frequency jitters due to LUTsfluctuations.From the point of view of frequency domain,the AM-AMfluctuations and AM-PMfluctuations may result in spectrum regrowth at the PA output,and the out-of-band spectrum regrowth shows a“flat”effect.Furthermore,because and functionally determine the weights in(11)and(12),the LUTsfluctuations become controllable by selecting proper values of and.III.P ROPOSED L OOP D ELAY E STIMATION A LGORITHM ANDS MOOTHING F ILTER M ETHODA.Loop Delay Estimation AlgorithmLike[14]and[15],the loop delay estimation algorithm in this work is performed in two steps:integer delay estimation and fractional delay estimation.But our estimator covers two differ-ences compared to that in[14]and[15].First,we reduced the computational complexity in the integer delay estimation step by using the‘absolute difference criterion’instead of the corre-lation-based method.Second,in the fractional delay estimation step,we developed a novel fractional delay estimator instead of involving a complicated iterative process.Unlike the iterative method in[14]and[15],where longer fractional delay requires longer iterations and vice versa,the novel fractional delay es-timator here is capable of estimating either long or short frac-tional delay with an equivalent computational complexity.In the following paragraphs,the loop delay is quantified by base-band sampling interval.As mentioned in[15],although the delayed feedback output is severely distorted by PA’s nonlinear distortion,the PA’s curve is almost monotonic for the original input below saturation level.This fact guarantees that a larger feedback signal is always related to a larger input amplitude and vice versa.Fortunately,most of the original input signalsare located far away from PA’s saturation region,as one canfind from the probability density function of the modulation signals (QAM,WCDMA,OFDM,etc).Thus,the sum of the absolute amplitude-difference between and,can be used to determine integer delay(i.e.,‘absolute difference criterion’). By storing samples of the two vectors andwith a length of,the‘absolute difference criterion’im-plemented as an average of the sum(13) where and are amplitudes of and,respectively.are the integer in-dexes and is an empirical constant dependent on the real loop delay.In general,the real loop delay is at the level of several microseconds[10],and can be decided by.For a10sampled WCDMA with3.84MHz chip rate,a value of several hundreds is good enough for.The index corresponding to the minimum value of indicates the integer shift between two data vectors.Practically,the in-teger delay is(14) With integer delay being eliminated from(14),recalling the assumption of(see Section II),the delayed feedback output elimination is(15) where,is a residual delay value,which is a small fraction of the baseband sampling interval due to misalignment betweenthe sampling time of and within one sampling in-terval.This residual located in the range of, requires more accurate estimation.It can be estimated by the following process,which includes the fractional interpolation operation and the‘relative difference criterion’.Atfirst,the fractional interpolation operation is applied to the input vector via linear interpolation[18]by a set of interpolation factors,where any single value corresponds to unique interpolated sequence .The interpolation factor is defined as(16) where are a set of equally spaced values in the range ofwith step size of1,and is the frac-tional time step factor.For each value,interpo-lation for gives the sequence,where ele-ment is(17) Second,from(15)and(17),the two vectors and ,with a length of,are used to perform the‘rel-ative difference criterion’,to give(18) where is the starting index of the vectors,and is the upper limit of.For each,by minimizing the set of,the estimatedtherefore(19) Computing the average of the sequence to give the final estimation of the fractional delay as(20) Until now,from(14)and(20),the estimated loop delay is.Specifically,in the fractional delay estima-tion process,utilization of the average operation in(20)can guarantee a high degree of estimation accuracy even under a noisy feedback path.Considering the parameter setting of the pair,typically,a selection of(100,100)can achieve better estimation accuracy than that of(10000,1)(refer to Section IV.D,Fig.8).However,the former one(100,100) needs far lower computational complexity and less memory access.B.Smoothing Filter MethodThe fractional delay estimation is a critical step in the pre-distortion function synthesis process.However,there would be more or less residual estimated error due to system’s non-ideal conditions(e.g.,PAs temperature drift,unexpected circuit level fluctuations,feedback noise,etc).In order to eliminate LUTs fluctuations caused by such residual estimated error and en-hance system’s robustness to loop delay errors,we introduce the smoothingfilter(SMF)method as a low passfilter for LUTs. The SMF method is applied to LUTs after each iteration (every time all the LUT-entries are updated),somehow like that in our previous work in[19].Compared with[19],there are two improvements to the SMF method this time:1)A simple digital differentiator is adopted beforehand tofind the indexes of the peak and quasi-peak values of both amplitude table and phase table;2)New interpolation functions are involved,which uses four neighbors of the interpolated point instead of only its two neighbors[19].Taking amplitude table as an example, the SMF method includes three steps as follows(similar SMF method can be applied to phase table as well).First,the digital differentiator is realized as[20](21) where is the entry-value in the-th LUT-entry.The peak and quasi-peak values appear for,where is a proper threshold(,in our experiments,we set in the range of(0,0.1]).Denote the indexes as, which correspond to the peak and quasi-peak values. Second,to eliminate bigfluctuations in LUTs,interpolation operations are applied to,namely(22) wherewhere is a constant(,typically,).(22) can be performed iteratively.The interpolation is actually a weighted average of the four neighbors of,with the nearest two weighing more.This kind of interpolation can reduce thefluctuations amongand.Notice that the interpolation doesn’t affect LUTs’predistortion performance significantly.However,LUTsfluctuations can be reduced greatly.Third,based on the interpolated results of,the residual LUTsfluctuations can be eliminated by a polynomial fitting for the sequence of.In most cases,afifth order polynomialfitting is good enough,resulting in(23) where are the LUT-entry indices,is the size of the LUT,and arefitted coefficients,which can be calculated by using least squares method.Gener-ally speaking,the polynomial order of (23)should be equal to or larger than that of PA’s highest nonlinear order.Compared with the old SMF method in [19],on one hand,adding the digital differentiator allows improving the accuracy of the SMF method,which can properly find the entries that need to be dealt with in the second step.Because only the LUT-entries found by the digital differentiator are involved for interpolation,hence the computational complexity of (22)is reduced.On theother hand,(22)uses the four neighbors offor inter-polation instead of only two,which enhances system’s tolerance to the interpolation operations without loss much information of LUTs’inverse nonlinearity for PA.IV .S IMULATIONS AND E XPERIMENTAL R ESULTSFor validation purpose,simulations and experiments are de-scribed in this section.The size of the LUT is 64both for ampli-tude table and phase table.In the simulations,a memory poly-nomial model as(24)is used as the model of PA.Where,and are input and output of the model,and the model coef ficients areIn the experiments,an S-band class-AB ampli fier based on GaN HEMT is used,which has 10-W saturated output power and 15dB gain.The excitation signal is a 2.14GHz single-carrier 10WCDMA with 5MHz bandwidth and 7.8dB PAR.A.LUTs FluctuationsIn order to evaluate LUTs fluctuations,we invoke the fol-lowing equations to express the fluctuations.The fluctuation in amplitude table is(25)And the fluctuation in phase table is(26)where,and are entry-values in the -th LUT-entry of amplitude table and phase table (with loopdelay error),andand are the corre-spondingdesired entry-values (without loop delay error).is the size of the LUT.For different loop delay errors,Fig.3shows the simulation results of LUTs fluctuations versus loop delay error,where weFig.3.Simulated results of LUTs fluctuations versus loop delay error.Fig.4.Simulated LUTs fluctuations versus step size constants .set and .It is shown that longer loop delay error causes severer LUTs fluctuations.When loop delay erroris quite large,which means thatand in (11)and (12)are big,the fluctuations would increase sharply.Fur-thermore,LUTs fluctuations are strongly related to the step size constants and .Fig.4gives simulated LUTs fluctuationsversus step size constants,with aloop delay error,where we set.It proves that LUTs fluctuations become severer for larger step size.In practical applications,smaller and would make the LUT predistorter more robust to loop delay er-rors.However,small and would cost a long convergence process.In Fig.5,simulation results present the output spectra of PA with LUT predistortion,under different loop delay errors.The “flat”effect of spectral regrowth at the out-of-band emerges very clearly.B.Performance of Integer Delay EstimationThe ‘absolute difference criterion’gives a good integer delay estimation even if the PA behaves nonlinearity.Fig.6shows thesimulated results ofin (13)versus integer time index with ,where the actual loop delay is set separately to be and .The results illustrate that theestimated integer delays areand for these three cases.Following that,the residual fractional delay can be esti-mated in the next step of fractional delay estimation.Fig.5.Simulated spectral regrowth of PA output after LUTpredistortion,under different loop delay errors (and ).Fig.6.Simulated results of integer delay estimation.C.Performance of Fractional Delay EstimationIn the fractional delay estimation process,three critical pa-rameters are and .For evaluation,we assumethat there is aresidual fractional delay.Fig.7shows the simulated results versus different values,where we setto be (100,100),(1000,10)and (10000,1).Fromthe perspective of the 1/64criterion [10],the loop delay esti-mated error should be less than (i.e.,)for 10WCDMA signal.Therefore,shouldn’t be too big in order to ensure the estimation ually,we setvalue in the range of (0,0.2).As one can see from Fig.7,the case of shows the worst perfor-mance,with signi ficant fluctuation on the curve,while the one ofshows the best ingthe average operation in (20)can make the fluctuation phe-nomena between loop delay estimated error and smaller.D.Robustness and Accuracy of Loop Delay Estimation AlgorithmIn order to validate that the average operation in (20)can guarantee a high estimation accuracy even under a noisy feed-back path,we consider three cases with valuesof :Fig.7.Simulated loop delay estimated error versus differentvalues.Fig.8.Statistical property of the proposed loop delay estimation algorithmunder a noisy feedback path (dB).(200,200),(100,100)and (10000,1).The cases of (200,200)and (100,100)can be regarded as that with the average oper-ation,while the last one of (10000,1)is without.In this sub-section,we set signal to noise ratio (SNR)in feedback path to be 30dB.From a statistical point of view,in each case,we per-form 3000groups of calculations for loop delay estimation (the actual loop delay is assumed to be ).After statistical anal-ysis,the normalized probability distributions of the estimated results are plotted in Fig.8,where is 2000and is 0.04.The case of (200,200)shows the best performance with the smallest variance.This shows that the average operation can make the estimation more robust to feedback noise.We have performed a comparative study with a state-of-the-art algorithm described in previous work [15].In this study,we set .Considering the cases both with and without noise in feedback path,the estimated results are plotted in Fig.9.The results indicate that the proposed loop delay estimation algorithm outperforms that in [15].One can conclude that the loop delay estimated error of the algorithm in [15]leads to a linear relation between the loop delay estimated error and the actual fractional delay.That is to say,the bigger the actual fractional delay is,the lower the estimated accu-racy would be.Instead,the algorithm proposed here behaves differently.It can keep an estimated error limited withinTABLE IP ERFORMANCE S UMMARY AND C OMPARISON A MONG D IFFERENT P REDISTORTION S YSTEMS IN T HE P RESENCE OF D ELAY MISMATCHFig.9.Loop delay estimated accuracy of the loop delay estimationalgorithms.Fig.10.Simulated results of curves of both amplitude table andphase table,before and after SMF delay error is).,compared with of the onein [15].Furthermore,even under a noisy feedback path with 30dB SNR,the proposed loop delay estimation algorithm works well.E.Performance of Smoothing FilterTo evaluate the smoothing filter (SMF)method,Fig.10presents the curves of both amplitude table and phase table before and after SMF with a loop delay error.One can see thatcurves are notably smoothed after SMF.To evaluate the effectiveness of the SMF method in reducing spectral regrowth when loop delay errors exist,we perform the LUT predistortion under various loop delay errors (without loopdelay compensation).The loop delay error residual in the loop is assumed to be in the range of .We apply the SMF method to the amplitude and phase tables every time when all the LUT-entries are updated.By using ACPR at 5MHz and 10MHz as metrics,the results of the LUT predis-tortion are shown in Fig.11for 30iterations.There are four cases with respect to the step size constants (and ):Case (a),;Case (b),;Case (c),;and Case (d),.One can see that the SMF method improves the robustness of LUT predistortion to loop delay er-rors.It is clearly shown that LUT predistortion with a selection of can suffer the widest loop delay errors range of ,while the one with can onlysuffer loop delay errors in the range of.How-ever,it is worth mentioning that as and decreases,the predis-torter’s convergence speed decreases as well.Therefore,when we select the values of and ,we should make a compro-mise between system’s tolerance to loop delay errors and pre-distorter’s convergence speed.To further validate the superiority of the SMF method,com-parison to other state-of-the-art works [21]–[23]about the sys-tems’tolerance to loop delay errors is also carried out.First,for the digital feedback predistortion (DFBPD)[21],[22],we find that loop delay errors may also lead to LUTs fluctuations.We can obtain the LUTs fluctuations in DFBPD expressed as (A6)(please see Appendix A for details).Notice that the result in (A6)is similar to (11)and (12)in Section II.Similarly,the SMF method can also be employed to eliminate the fluctuations in DFBPD,and the results are summarized in Table I.Second,for the polar/EER transmitter [23],delay mismatch between am-plitude and phase of PA output (AM-PM delay mismatch)may also cause similar LUTs fluctuations when LUT predistortion is employed (please see Appendix B for details).The comparisons with and without SMF are also summarized in Table I.For all cases,the results clearly show that the SMF method improves systems’tolerance to loop delay errors.We also give the complexity information of the proposed SMF method,compared to the SMF method presented in [19].For both SMF methods,we assume that the interpolation oper-ations ((22)in this work or (18)in [19])are performed for iterations,to continuously eliminate the big fluctuations.For the SMF method proposed in this work,the numbers of LUT-en-tries involved in interpolation are and ,respectively for amplitude table and phase table.The complexity information is summarized in Table II.We can see that the computational complexity of both SMF methods show linear relations with theFig.11.Simulated effectiveness of the SMF method in reducing spectral regrowth when loop delay errors exist(without loop delay compensation):(a);(b);(c);and(d).TABLE IIC OMPLEXITY I NFORMATION OF SMF M ETHODSLUT size.For further comparison,we present the results of complexity and linearization performance for a loop delay error in Table III.It is shownthat the SMF method in this work outperforms the old one in[19],with respect to both complexity and linearization performance.TABLE IIIC OMPLEXITY I NFORMATION AND L INEARIZATION P ERFORMANCE OF SMFM ETHODS FOR A L OOP D ELAY E RRORNotably,the SMF method for LUT predistortion does not greatly increase system complexity since we still use the same LUT structure and the same LMS algorithm.The main differ-ence is that we adjust the LUT-entries values by additional in-terpolation and polynomialfitting processes in(22)and(23),Fig.12.Photograph of the designed Class-AB GaN amplifier.which do not need extra measurement and can be implemented in digital domain on the original DSP/FPGA devices in the real system.However,this extra SMF process can significantly im-prove linearization performance,especially when loop delay er-rors exist.F.Experiments for10-W Class-AB GaN Amplifier Experimental validations are also carried out on the widely used VSG-VSA test-bed[24].For this purpose,we designed a 2.14GHz class-AB amplifier based on GaN HEMT.A photo-graph of the PA prototype is presented in Fig.12.Under a single tone excitation,it can be pushed to a10-W saturated output power with a drain efficiency of up to65.1%.In the experi-ments,the excitation signal is a single-carrier WCDMA with5 MHz bandwidth and7.8dB PAR.The PA is excited to operate at6dB OBO(output back-off)point,corresponding to34dBm output power,with a drain efficiency of about35.5%.LUT predistortion is employed to linearize the GaN PA.Par-ticularly,we purposely introduce a loop delay error of during the adaptation of LUTs.The step size constants are set to be.Fig.13gives the AM-AM and AM-PM plots of PA before and after linearization.We can see that the results without SMF show significantfluctuations,whereas that with SMF can obtain smooth results.The measured spectra at the output of the PA prototype before and after linearization are also presented in Fig.14.The spectrum without SMF illustrates the“flat”effect at the out-of-band.However,after employing the SMF,there is an improvement of the spectrum suppression. Overall,dBc and dBc ACPR at5MHz and10 MHz are obtained after LUT predistortion with the SMF,com-pared to dBc and dBc ACPR at5MHz and10 MHz without the SMF.V.C ONCLUSIONThis paper has presented that loop delay errors would cause LUTsfluctuations for the LUT predistortion systems.This would detrimentally lead to a“flat”effect of the out-of-band spectrum at the PA output.First,we have proposed a novel loop delay estimation algorithm,which has been validated to be su-perior to state-of-the-art estimator in[15].Second,to enhance the robustness of LUT predistortion to loop delay errors,the SMF method has also been proposed to polish LUTsfluctua-tions.We have carried out extensive computer simulations and experimental measurements to demonstrate theeffectiveness Fig.13.Measured results of(a)AM-AM and(b)AM-PM plots ofthe PA output with and without SMF for LUT predistortion,with a loop delay error.Fig.14.Measured spectra at the output of the PA prototype before and after linearization,with a loop delay error.of the proposed estimator and the SMF method.We have also performed comparative studies to other state-of-the-art works[21]–[23]about the systems’tolerance to loop delay。