Assessment of Various Window Functions in Spectral Identification of Passive Intermodulation

electronics

Article Assessment of Various Window Functions in Spectral Identiﬁcation of Passive Intermodulation

Khaled Gharaibeh

Department of Civil and Architectural Engineering, Applied Science University-Bahrain, Al Ekir 3201, Bahrain; [email protected]

Abstract: Passive Intermodulation (PIM) distortion is a major problem in wireless communications which limits cell coverage and data rates. Passive nonlinearities result in weak intermodulation (IMD) products that are very difﬁcult to diagnose, troubleshoot and model. To predict PIM, behavioral models are used where spectral components of PIM are estimated from the power spectral density at the output of the model. The primary goal of this paper is to study the effect of window functions on the capability of the power spectral density computed from the periodogram of signal realizations to predict low power PIM components in a wideband multichannel communication system. Different window functions are analyzed and it is shown that windows with high side-lobe level fail to predict PIM components which are close to the main channels due to their high spectral leakage; while window functions with low roll-off rate of the side lobes fail to predict low power higher order PIM components especially when the frequency separation between the main carriers is high. These results are supported by simulations of the power spectral density computed using signal realizations of an LTE carrier aggregated system.

Keywords: passive intermodulation; piecewise linear model; LTE; behavioral models; threshold decomposition Citation: Gharaibeh, K. Assessment of Various Window Functions in Spectral Identification of Passive Intermodulation. Electronics 2021, 10, 1. Introduction 1034. https://doi.org/10.3390/ Passive intermodulation (PIM) distortion has been a significant problem in modern electronics10091034 mobile communication systems as it limits system performance and capacity. Unlike active nonlinear distortion which is usually manifested as intermodulation (IMD) products with Academic Editor: Yasir Al-Yasir significant power levels, passive nonlinearities result in very weak IMD products which make it very difficult to measure and model [1]. In carrier aggregated systems such as 4G Received: 21 March 2021 systems, PIM is manifested as IMD components which result in interference with the main Accepted: 18 April 2021 Published: 27 April 2021 channels in the receive band. A simple illustration of PIM in wireless systems is shown in Figure1a where the input f Publisher’s Note: MDPI stays neutral to the passive nonlinearity consists of two band pass signals centered at frequencies 1 with regard to jurisdictional claims in and f2. The power spectral density (PSD) of the output of the nonlinearity is shown in published maps and institutional affil- Figure1b which consists of the spectra at the fundamental frequencies in addition to the iations. IMD components at frequencies 2 f1 − f2 and 2 f2 − f1 which may lie within the receive band 2. In this scenario, the IMD components produced by the passive nonlinearity can result in significant interference in the receive band which limits system performance and capacity. To predict PIM in in such scenarios, behavioral models that relate the output PIM to Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. the input signal level and frequency are used [2–10]. Behavioral models are developed This article is an open access article using measured characteristics of the nonlinear device and then they can be used to predict distributed under the terms and PIM from the computation of the power spectrum at the output of the model. conditions of the Creative Commons Estimating PIM from the power spectral density at the output of the behavioral model Attribution (CC BY) license (https:// using direct application of the Fast Fourier Transform (FFT) to a signal realization can be creativecommons.org/licenses/by/ a simple task when the power level of the IMD components is high as the case of active 4.0/). nonlinearities (above −60 dBc). However, with passive nonlinearity, the IMD components

Electronics 2021, 10, 1034. https://doi.org/10.3390/electronics10091034 https://www.mdpi.com/journal/electronics Electronics 2021, 10, 1034 2 of 12

can have power levels of less than −140 dBc which makes it difﬁcult to predict PIM [7,8]. The reason for this is that the ﬁnite length of signal realizations results in high levels of spectral leakage in the computed power spectrum which tends to obscure the IMD Electronics 2021, 10, x FOR PEER REVIEWcomponents as spectral leakage level are usually much higher than the intermodulation2 of 12

power level.

(a)

2f1 − f2 f1 f2 2f2 − f1

(b)

FigureFigure 1. 1.Spectra Spectra in in a widebanda wideband receiver: receiver: (a ()a Input) Input spectrum spectrum of of two two RF RF signals signals to to a a passive passive nonlin- non- earitylinearity and (andb) the (b Output) the Output spectrum; spectrum; dashed dashed line indicates line indicates PSD computed PSD computed using FFTusing with FFT rectangular with rectangular window indicating spectral leakage. window indicating spectral leakage.

FigureTo predict1b shows PIM an in illustration in such scenarios, of this phenomenon behavioral models where spectral that relate leakage the output that results PIM to fromthe theinput Fast signal Fourier level Transform and frequency (FFT) makesare used it impossible[2–10]. Behavioral to predict models IMD componentsare developed fromusing weak measured passive characteristics nonlinearities. of In the principle, nonlinear the device frequency and then of intermodulation they can be used spectra to pre- anddict their PIM power from the level computation depend on of the the frequency power spectrum separation at the between output theof the main model. channels and theEstimating input power PIM and, from thus, the it power is important spectral to usedensity an appropriate at the output spectral of the estimation behavioral algorithmmodel using so that direct these application IMD components of the Fast can Fourier be predicted. Transform (FFT) to a signal realization canSpectral be a simple leakage task canwhen be the minimized power level through of the windowing IMD components where signalis high samples as the case are of weightedactive nonlinearities by a given time (above window −60 which dBc). enhances However, the with accuracy passive of spectral nonlinearity, estimation the [ 11IMD]. Thecomponents type of the can window have haspower a significant levels of less impact than on −140 spectral dBc which leakage makes where it low-peakdifficult to side pre- lobedict of PIM the [7,8]. frequency The reason response for ofthis the is windowthat the finite function length indicates of signal small realizations spectral leakage.results in Severalhigh levels window of spectral functions leakage have in been the usedcomputed in the power literature spectrum for suppression which tends of to spectral obscure leakagethe IMD [12 –components14]. However, as the spectral choice leakage of the window level are function usually is much a compromise higher than between the inter- the amountmodulation of spectral power leakage level. and the frequency resolution and, hence, the computational complexityFigure of 1b computing shows an the illustration windowed of FFTthis [phenomenon3]. where spectral leakage that re- sultsThis from problem the Fast has Fourier been studied Transform in the (FFT) context makes of prediction it impossible and reduction to predict of IMD harmonics compo- innents power from systems weak where passive it was nonlinearities. shown that In the principle, direct application the frequency of the of FFT intermodulation has inherent performancespectra and limitations, their power such level as spectral depend leakage on the and frequency the picket separation fence effect between [11]. However, the main limedchannels research and hasthe beeninput done power on and, the effect thus, ofit selectionis important of window to use an functions appropriate on spectral spectral identificationestimation algorithm of PIM. This so that paper these aims IMD to components provide an assessmentcan be predicted. of the performance of variousSpectral window leakage functions can onbe inminimized the ability through of the power windowing spectral where density signal to predict samples the are IMDweighted components by a given in a wideband time window system. which The significanceenhances the of thisaccuracy research of spectral in that the estimation results presented[11]. The cantype be of used the window in the design has a ofsignificant simulation impact models on spectral and measurements leakage where setups low-peak for accurateside lobe prediction of the frequency or measurement response of PIM of the in wireless window systems. function indicates small spectral leakage.This paper Several provides window an analysisfunctions of have passive been nonlinearity used in the in aliterature carrier aggregated for suppression system of usingspectral a polynomial leakage [12–14]. based behavioralHowever, modelthe choice and of develops the window the output function of the is nonlinearitya compromise forbetween two carrier the aggregatedamount of spectral signals whichleakage represents and the thefrequency practical resolution scenario byand, which hence, PIM the computational complexity of computing the windowed FFT [3]. This problem has been studied in the context of prediction and reduction of harmonics in power systems where it was shown that the direct application of the FFT has inherent performance limitations, such as spectral leakage and the picket fence effect [11]. However, limed research has been done on the effect of selection of window functions on spectral identification of PIM. This paper aims to provide an assessment of the perfor-

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is manifested in wireless system. Next, the paper provides a theoretical study of power spectral density of the output of the passive nonlinearity where the characteristics of various window functions which are used to overcome spectral leakage are explored. The last section provides simulation results which quantify the ability of power spectrum computed using various window functions to predict passive intermodulation levels.

2. Modeling of Passive Intermodulation Perhaps, the most common approach to modeling nonlinearity is the Volterra like approaches [15]. One of the most common simpliﬁcations of Volterra models is the memoryless polynomial model which has extensively been used to model active nonlinearities with no memory effects. The response of an RF nonlinearity to multiple signals in a carrier aggregation system such as LTE system was studied in [16], where the response of a memoryless polynomial model to multiple communication signals at multiple carrier frequencies of the form. M 1 x(t) = x (t)ej2π fmt (1) ∑m=−K 2 m

where xm(t) is the complex envelope of the input signals centered at frequencies fm and K indicates the number of signals. The analysis in [17] was based on multinomial analysis of the polynomial model which can be described as:

( ) = N ( ) y t ∑n=1 yn t (2) n where yn(t) = anx (t), an represents the n-th order coefﬁcient of the memoryless nonlinearity and N is the nonlinear order. Note that only odd order polynomial terms are considered in this model as the passive nonlinearity is considered to have an odd symmetry in its I/V characteristic [17]. This means that the model can only model IMD components that lie within the proximity of the carrier in the frequency spectrum, which is a reasonable assumption as low frequency IMD components that result from even IMD products can usually be removed by ﬁltering. To analyze the response of the nonlinearity to multiple signals, the output of the nonlinearity can be found by inserting (1) into (2) where the n-th order nonlinearity can be written as: ! a n a n n y (t) = n x (t)ej2π fmt = n x ej2π f t + x∗ e−j2π f t (3) n n m ∑ n ∏ mi mi 2 2 n−M . . . ., nM n−M f−M+...+nM fM= f i=1

The case of two signals at frequencies f 1 and f 2 is unique as it makes the analysis more tractable and; at the same time, this enables nonlinear distortion to be quantiﬁed and predicted at IMD frequencies in a wideband carrier aggregated wireless system. For a two- signal input, the passive nonlinearity results in IMD components at the IMD frequencies. In this scenario, the input to the passive nonlinearity is represented as

x(t) = A1(t) cos(2π f1t + φ1(t)) + A2(t) cos(2π f2t + φ2(t)) (4)

The complex envelop of the input signal can be written as

φ1(t) −2π f0t φ2(t) 2π f0t xe(t) = A1(t)e e + A2(t)e e (5)

f2− f1 where f0 = 2 . As shown in [17], by inserting (5) into (3), the output of the nonlinearity in (2) consists of the fundamental signals at frequencies − f0 and f0 in addition to the IMD components at Electronics 2021, 10, 1034 4 of 12

the IMD frequencies as shown in Table1 below. The output of the polynomial model at the fundamental frequencies can be as [17]:

n−1 N 2 n−1 −l n+1 −l z (t) = b ul u 2 u 2 ul ef0 ∑ ∑ n,l e−2 e−1 e1 e2 (6) n=1 l=0

where n−1 n bn,l = an2 n−1 n+1 l, 2 − l, 2 + l, l

Table 1. IMD frequencies for a two-tone input.

Upper Intermod Upper Intermod Lower Intermod Lower Intermod N (Band Pass) (Envelop) (Band Pass) (Envelop)

3 2 f2 − f1 3 f0 2 f1 − f2 −3 f0

5 3 f2 − 2 f1 5 f0 3 f1 − 2 f2 −5 f0

7 4 f2 − 3ω1 7 f0 4 f1 − 3 f2 −7 f0

9 5 f2 − 4 f1 9 f0 5 f1 − 4 f2 −9 f0

11 6 f2 − 5 f1 11 f0 6 f2 − 5 f1 −11 f0

This expression represents the output of nonlinearity at the ﬁrst carrier and captures nonlinear distortion that results from the interaction of the two signals by nonlinearity. In a similar manner, the third order IMD components at the lower IMD frequency 3f 0 can be found as [17] n−2 N 2 n−3 −l n+1 −l z (t) = b ul+1u 2 u 2 ul e3 f0 ∑ ∑ n,l e−2 e−1 e1 e2 (7) n=1 l=0

where ! a n = n bn,l n−1 n−3 n+1 2 2 − l, l + 1, l, 2 − l and higher order IMD components can be derived in a similar way. The above analysis enables PIM in carrier aggregation systems to be predicted. In the above formulation, PIM is manifested as gain compression, spectral regrowth and cross modulation distortions at the carrier frequency plus a nonlinear distortion component with a wider bandwidth at the intermodulation frequency. Figure2 shows the output spectra of a passive nonlinearity to the sum of two OFDM signals each of which has a 20 MHz bandwidth with a frequency separation of 120 MHz between them. The ﬁgure shows the IMD components which appear at 360, 600, 840, 1080 MHz respectively. Note that the widened spectrum at the fundamental frequency −f 0 represents the spectral regrowth and cross modulation while the spectral components at the IMD frequencies which may interfere with other channels in a carrier aggregation system or with the receive bands. ElectronicsElectronics2021 2021, ,10 10,, 1034 x FOR PEER REVIEW 55 ofof 1212

FigureFigure 2.2. PSDPSD ofof thethe outputoutput of of nonlinearity nonlinearity for for an an input input that that consists consists of of the the sum sum of twoof two OFDM OFDM signals withsignals frequency with frequency separation separation of 100 MHz. of 100 MHz.

3.3. EstimationEstimation ofof PowerPower SpectralSpectral DensityDensity ofof thethe OutputOutput ofof aa PassivePassive NonlinearityNonlinearity TheThe powerpower spectralspectral densitydensity ofof aa randomrandom signalsignal isis deﬁneddefined asas thethe statisticalstatistical averageaverage ofof thethe signalsignal periodogramperiodogram ofof signalsignal realizationsrealizations ofof inﬁniteinfinite lengthslengths [[11]:11]:

𝑆(𝜔) =𝐸{𝑙𝑖𝑚𝑝,(𝜔)} Sxx(ω) = E{ lim→px,M(ω)} (8)(8) M→∞ where where M−1 2 1 2 1 −jωn px,M(ω) = M |DTFT x(n)| | = M ∑ x(n)e 1 1 𝑝 (𝜔) = |𝐷𝑇𝐹𝑇(𝑥(𝑛)| | = n=0 𝑥(𝑛)𝑒 (9) , 𝑀 M−1−|l| 𝑀 1 ⇔ M ∑ x(n)x(n + |l|) n=0 (9) || is the periodogram of the output signal1 which is basically the squared magnitude of the ⇔ 𝑥(𝑛)𝑥(𝑛 + |𝑙|) DTFT of the signal realization of a finite𝑀 length M. Therefore, the periodogram represents an unbiased estimate of the power spectral density (PSD) which is a random variable with zero mean, but has a non-zero variance as it represents a single realization of the signal [11]. Note that the periodogram of finite length signal does not represent the PSD of the signal as theis the PSD periodogram in (8) assumes of athe signal output of infinite signal length;which is however, basically the the accuracy squared of magnitude the periodogram of the inDTFT representing of the signal the reali PSDzation increases of a asfinite the length ofM. the Therefore, signal realization the periodogram used is represents increased. Thean unbiased finite length estimate of the of signal the realizationpower spectral results density in spectral (PSD) leakage which where is a random extra frequency variable componentswith zero mean, are generated but has a in non-zero the vicinity variance of the as main it represents frequency a component single realization [13]. Spectral of the leakagesignal [11]. results Note in thethat inability the periodogram of the periodogram of finite length to represent signal other does frequencynot represent components the PSD ofof thethe signalsignal suchas the as PSD harmonics in (8) assumes and IMD a signal components. of infinite length; however, the accuracy of the periodogramTo overcome in the representing problem of the spectral PSD increa leakage,ses as the the input length signal of the is signal multiplied realization by a windowused is increased. function where The anyfinite finite length length of signalthe signal is viewed realization as having results an infinitein spectral length leakage when viewedwhere extra from afrequency window ofcomponents a finite length are and,generate hence,d in the the periodogram vicinity of canthe bettermain frequency represent thecomponent actual PSD [13]. from Spectral a finite leakage length results realization in th ofe inability the signal of as the will periodogram be seen in the to followingrepresent section.other frequency The periodogram components of the of windowedthe signal such signal as isharmonics called the and modified IMD components. periodogram and is expressedTo overcome as [11]: the problem of spectral leakage, the input signal is multiplied by a

window function where any finite length signal is viewed as having 2an infinite length 1 1 M−1 when viewed from a (window) = of a finite( length( )|2 =and, hence, the( ) periodogram−jωn can better pxw,M ω DTFT xw n ∑ xw n e (10) represent the actual PSD fromM a finite length realizationM n of=0 the signal as will be seen in the following section. The periodogram of the windowed signal is called the modified peri- whereodogramxw( andn) = isx expressed(n)w(n) is as the [11]: windowed signal and w(t) is the window function and M is the signal size. The windowed periodogram can enhance the accuracy of the spectral analysis of the PIM components and produce better estimates of nonlinear distortion. 1 1 Note that the𝑝 periodogram(𝜔) = is|𝐷𝑇𝐹𝑇 an estimate(𝑥 (𝑛)| of| = the PSD of 𝑥 the(𝑛) signal.𝑒 The accuracy(10) of , 𝑀 𝑀 the estimate depends on the bias of the estimate while its precision depends on the mean squared error. The bias of the estimate results in spectral leakage while the variance of the

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where 𝑥(𝑛) =𝑥(𝑛)𝑤(𝑛) is the windowed signal and 𝑤(𝑡) is the window function and M is the signal size. The windowed periodogram can enhance the accuracy of the spectral analysis of the PIM components and produce better estimates of nonlinear distortion. Note that the periodogram is an estimate of the PSD of the signal. The accuracy of the estimate depends on the bias of the estimate while its precision depends on the mean squared error. The bias of the estimate results in spectral leakage while the variance of the estimate results in inconsistency of the estimate. The periodogram of a finite length Electronicssignal2021, 10, 1034is always a biased estimate of the PSD but this the bias can be controlled by win-6 of 12 dowing. A larger window size (Ts) which means higher resolution (1/Ts) results in smoother transitions which means less bias of the estimate and hence, less spectral leakage. Therefore,estimate the choice results inof inconsistencythe window of thefunction estimate. depends The periodogram on a compromise of a finite length be- signal tween the amountis alwaysof spectral a biased leakage estimate and of the the PSD frequency but this the biasresolution can be controlled and, hence, by windowing. the A larger window size (Ts) which means higher resolution (1/Ts) results in smoother computational complexitytransitions whichof computing means less biasthe ofwindowed the estimate andFFT hence, [13]. less In spectral general, leakage. window Therefore, functions with narrowthe choice main of lobe the window have a function smaller depends frequency on a resolution compromise while between those the amount with of wide main lobe resultspectral in leakagehigh frequency and the frequency resolution. resolution On the and, other hence, hand, the computational the low level complexity of side-lobes and theirof computinghigh roll-off the windowedrates result FFT in [13 lower]. In general, spectral window leakage functions than with those narrow with main high side-lobe levelslobe or have low a roll-off smaller frequencyrates [13]. resolution while those with wide main lobe result in high frequency resolution. On the other hand, the low level of side-lobes and their high roll-off On the other rateshand, result the inconsistency lower spectral of leakage the estimator than those depends with high on side-lobe the variance levels or lowof the roll-off estimate which decreasesrates [13]. as the number of segments of the signal is increased. Therefore, for a fixed T seconds Onof thedata, other there hand, is the a consistencytrade-off between of the estimator the length depends of on the the segments variance of the which controls theestimate bias and which number decreases of assegmen the numberts, which of segments controls of the the signal variance is increased. of the Therefore, es- for a fixed T seconds of data, there is a trade-off between the length of the segments which timate. In principle,controls the variance the bias andthe numberperiodogra of segments,m does which not tend controls to zero the variance as the ofdata the estimate.length In L tends to infinity andprinciple, hence, the it variance is not thea consis periodogramtent estimator does not tendof the to zeroPSD; as however, the data length it canL tends be to a useful estimate ofinfinity the PSD and when hence, itthe is notdata a consistentrecord long. estimator of the PSD; however, it can be a useful Figure 3 showsestimate the spectrum of the PSD whenof various the data types record long.of windows which shows that the Figure3 shows the spectrum of various types of windows which shows that the rectangular window has the lowest main lobe width while the Blackman Harris window rectangular window has the lowest main lobe width while the Blackman Harris window has the widest mainhas lobe. the widest Table main 2 shows lobe. Tablethe 2va showsrious the properties various properties of these of windows these windows [11,14]. [11,14 ].

(a) (b)

Figure 3. Cont.

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(e) (f) FigureFigure 3. Spectrum 3. Spectrum of various of various window window functions: functions: (a) Rectangular (a) Rectangular window; (b) Hanning window; window; (b) Hanning (c) Blackman window; window; (c) (d) Blackman-HarrisBlackman window; window; (d) (Blackman-Harrise) Hamming window window; and (f) Kaiser (e) Hamming window. window and (f) Kaiser window.

Table 2. Table 2. Properties of variousProperties window of various functions. window functions. Width of the Main Side Lobe Max Side Lobe Window w(n) WidthLobe of (3-dB the BW) Main SideRoll-Off Lobe Rate Level (dB) (π samples ) Max Side (dB/Octave) Lobe (3-dBcycle BW) Roll-Off Window 𝒘(𝒏) ( Lobe Level 1 0 ≤ n ≤ N 𝐬𝐚𝐦𝐩𝐥𝐞𝐬0.011719 −13.3 Rate−6 Rectangular 𝝅 (dB) 0 Otherwise 𝐜𝐲𝐜𝐥𝐞 (dB/Octave) Hanning 10≤𝑛≤𝑁1 1 − cos 2πn , 0 ≤ n ≤ N 0.019531 −31.6 −18 Rectangular 2 N 0.011719 −13.3 −6 0 Otherwise Hamming 2πn 0.019531 −43.7 −6 1 2𝜋𝑛 0.54 + 0.46 cos N , 0 ≤ n ≤ N Hanning 1 − cos , 0≤𝑛≤𝑁 0.019531 −31.6 −18 2 𝑁 0.42 − 0.5 cos 2πn + 0.08 4πn , Blackman 2𝜋𝑛 N N 0.023438 −58.2 −18 Hamming 0.54 + 0.46 cos ,0≤𝑛≤𝑁0 ≤ n ≤ N 0.019531 −43.7 −6 𝑁 − 2πn 2𝜋𝑛 0.358754𝜋𝑛0.48829 cos N Blackman–Harris 0.027344 −92.2 −6 Blackman 0.42 − 0.5 cos + + 0.08 4πn,0≤𝑛− ≤𝑁 6πn 0.023438 −58.2 −18 𝑁 0.14128 cos 𝑁N 0.01168 cos N q 2𝜋𝑛2 I β 1−(1− 2n ) 0.35875 − 0.488290 cos N ( ) 𝑁 , 0 ≤ n ≤ N Blackman–Harris I0 β 0.027344 −92.2 −6 Kaiser 4𝜋𝑛(β is a parameter that determines6𝜋𝑛 the 0.011719 −14.7 −6 +0.14128 cos − 0.01168 cos 𝑁tradeoff between main lobe𝑁 width and side lobe levels) 2 2 1 −1 − 0 , 0As≤ discussed ≤ in Section1, spectral leakage causes low power spectral components Kaiser 0( ) to be obscured especially when their0.011719 power level is low.− In14.7 the problem of−6 estimating (𝛽 is a parameter that determinesPIM from the the tradeoff PSD at the output of a nonlinear model, this is manifested as inability of the estimated PSD to show the PIM components at the IMD frequencies when the input power between main lobe width andis lowside as lobe PIM componentslevels) usually have power levels of 100 dB less than the power of the main channel. In the following section, simulations of the PSD of the output a passive As discussed nonlinearityin Section usingI, spectral various leakag windowe causes functions low are power presented. spectral components to be obscured especiallySimulation when their Results power and Discussion:level is low. In the problem of estimating PIM from the PSD at the outputThe theoretical of a nonlinear concepts model, presented this in theis manifested paper are veriﬁed as inability by simulations of the es- results. The behavioral model used in the simulations is based on a hyper tangent function which timated PSD to showhas beenthe PIM shown components in [18,19] to followat the measured IMD frequencies nonlinear I/Vwhen characteristics the input ofpower a passive is low as PIM componentsnonlinearity usually over an have input power range levels of moreof 100 than dB 30 less dB. than The hyper the power tangent of function the is main channel. In giventhe following by 9 section, simulations of the PSD of the output a passive nonlinearity using various window functionsI(t) are= g 0presented.[1 + k1tanh( k2V(t))] (11) Simulation Results and Discussion: The theoretical concepts presented in the paper are verified by simulations results. The behavioral model used in the simulations is based on a hyper tangent function which has been shown in [18,19] to follow measured nonlinear I/V characteristics of a passive nonlinearity over an input power range of more than 30 dB. The hyper tangent function is given by 9

𝐼(𝑡) =𝑔 [1 + 𝑘 𝑡𝑎𝑛ℎ(𝑘𝑉(𝑡))] (11)

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where g0, k1 and k2 are the model parameters which can be adjusted to fit a wide variety of nonlinearwhere g0, kcharacteristics.1 and k2 are the model Note parametersthat these whichcharacteristics can be adjusted can model to ﬁt a the wide less variety than of 3 dB/dB nonlinear characteristics. Note that these characteristics can model the less than 3 dB/dB behavior of a passive nonlinearity by the proper selection of the model parameters. The behavior of a passive nonlinearity by the proper selection of the model parameters. The passivepassive nonlinearitynonlinearity in thein simulationsthe simulations that will that follow will was follow generated wasusing generated a hyper using tangent a hyper tangent model with parameters k−15 = 6 × 10−5, k−2 3= 3 × 10−3 and g0 = 1. Figure 4a shows the model with parameters k1 = 6 × 10 , k2 = 3 × 10 and g0 = 1. Figure4a shows the I/V I/Vcharacteristics characteristics of the of hyperbolicthe hyperbolic tangent tangent model model and Figure and Figure4b shows 4b theshows input/output the input/output powerpower characteristics of of the the model. model.

(a) (b)

FigureFigure 4. Hyper tangenttangent nonlinear nonlinear model: model: (a) ( I/Va) I/V characteristics; characteristics; and (andb) Pin/Pout (b) Pin/ characteristics.Pout characteristics.

TheThe input signalsignal to to the the model model was was generated generated in Matlab in Matlab and is and based is on based the speciﬁca- on the specifi- tions of the Orthogonal Frequency Division Multiplexed (OFDM) signals that resemble LTE cations of the Orthogonal Frequency Division Multiplexed (OFDM) signals that resemble signals as shown in Table3[ 16]. The periodogram of the output of the passive nonlinearity LTEin (10) signals was computedas shown for in anTable input 3 that[16Error! consists Reference of two simulated source not OFDM found. signals]. The with periodo- gram20 MHz of bandwidththe output each of the and passive a frequency nonlinearity separation in of (10) 100 MHz.was computed The periodogram for an wasinput that consistsestimated of using two differentsimulated window OFDM functions. signals Notewith that, 20 MHz in order bandwidth for the periodogram each and ofa thefrequency separationoutput of the of nonlinearity100 MHz. The to show periodogram the IMD components, was estimated the sampling using different frequency window needs functions.to be greaterNote that, than in the order highest for IMDthe periodogram component. For of example,the output in orderof the to nonlinearity show the 9th to show theintermodulation IMD components, component, the sampling the sampling frequency frequency needs must to be be greater greater than than 9ω0 .the highest IMD component. For example, in order to show the 9th intermodulation component, the Table 3. OFDM Signal Parameters. sampling frequency must be greater than 9𝜔. No. of Sub-Carriers 1705 Table 3. OFDM Signal Parameters. 16 QAM Modulation Constellation type: Gray No. ofPulse Sub-Carriers Shaping Rectangular 1705 pulse PAPR 10.816 dB QAM Modulation Input Power rangeConstellation 0 dBm to 35 dBm type: Gray Pulse Shaping Rectangular pulse The simulationPAPR results include computing PIM distortion from10.8 the PSDdB of the output of the nonlinearityInput Power at various range IMD frequencies and power levels. 0 dBm The to power 35 dBm of the input signals is swept in a range of 0 to 35 dBm and the periodogram was computed at each power level. The power in the IMD components is computed within a bandwidth of 20 MHz.The simulation In order to studyresults the include effect of computing the intermodulation PIM distortion frequency from on the the PSD prediction of the output ofcapability the nonlinearity of the power at various spectral IMD density frequencies with a given and windowpower levels. function The to power predict of PIM the input signalscomponents, is swept the in frequency a range separationof 0 to 35 betweendBm and the the input periodogram carriers was was swept computed and the at each powerpassive level. IMD The components power in were the computedIMD components at various is frequency computed separations within a andbandwidth using of 20 MHz.different In windoworder to functions. study the effect of the intermodulation frequency on the prediction capability of the power spectral density with a given window function to predict PIM components, the frequency separation between the input carriers was swept and the passive IMD components were computed at various frequency separations and using different window functions.

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Electronics 2021, 10, 1034 9 of 12 4. Results

Electronics 2021, 10, x FOR PEER REVIEW Figure 5 shows the PSD of the output of the nonlinear model for9 ofan 12 input that con- 4.sists Results of two OFDM signals separated by 100 MHz. The PSD was estimated using a peri- odogramFigure5 withshows two the PSD window of the output functions: of the nonlinear a Hanning model window for an input and that a consists rectangular window 4.ofwhen Results two OFDM the input signals power separated is much by 100 MHz.lower The than PSD the was saturation estimated using point a periodogram of the nonlinearity (Pin = with two window functions: a Hanning window and a rectangular window when the 30 dBm).Figure 5At shows this theinput PSD power, of the output the level of the of nonlinear the IMD model components for an input is that very con- low as compared input power is much lower than the saturation point of the nonlinearity (Pin = 30 dBm). At sists of two OFDM signals separated by 100 MHz. The PSD was estimated using a peri- thisto the input main power, channel the level power. of the IMDThe componentsfigure clearly is very shows low asthat compared with a to rectangular the main window the odogramchannelperiodogram power. with two The only window ﬁgure predicts clearly functions: showsthe third a that Ha withordernning a rectangularwindowPIM component and window a rectangular thebut periodogram fails window to predict all higher whenonlyorder predictsthe IMD input thecomponents power third is order much PIMdue lower componentto than the the severe saturation but fails spectral to point predict leakage.of the all nonlinearity higher In contrast, order (Pin IMD = a periodogram 30 dBm). At this input power, the level of the IMD components is very low as compared componentsthat uses a due Hanning to the severe window spectral predict leakage. all InIM contrast,D components a periodogram which that have uses power levels as toa Hanningthe main channel window power. predict The all figure IMD componentsclearly shows which that with have a rectangular power levels window as low the as periodogram−low200 as dBm. −200 only dBm. predicts the third order PIM component but fails to predict all higher order IMD components due to the severe spectral leakage. In contrast, a periodogram that uses a Hanning window predict all IMD components which have power levels as low as −200 dBm.

FigureFigureFigure 5. 5. PSD5.PSD PSD of of the theof outputthe output output of of the the passiveof passive the passivenonlineari nonlinearity nonlinearity using using a Hanning aty Hanning using window a window Hanning and and a rectangularwindow a rectangu- and a rectangular window.larwindow. window.

To analyze the effect of the window function on the capability of the PSD to predict ToTo analyze analyze the effectthe effect of the ofwindow the window function onfuncti the capabilityon on the of capability the PSD to predictof the PSD to predict PIMPIM components, components, the the periodogram periodogram was was comput computeded using using the the different window functions presentedpresentedPIM components, in in Table Table 11 andand the withwith periodogram aa frequencyfrequency separation separationwas comput ofof 2020ed MHzMHz using betweenbetween the different thethe twotwo inputinput window functions carriers.carriers.presented The powerin Table ofof PIMPIM 1 and atat the the with IMD IMD a frequencies frequenciesfrequency was wasseparation calculated calculated from of from 20 the MHzthe PSD PSD within between within the the two input thesamecarriers. same bandwidth bandwidth The power of the of inputthe of input PIM signal signal at for the various for IMDvarious input frequencies input power power levels waslevels as shown calculated as shown in Figure in fromFigure6a–d . the PSD within 6a–d.The ﬁgures The figures show show the input the input power power levels levels at which at which the periodogram the periodogram (which (which uses a uses given a windowthe same function) bandwidth can predict of the the input PIM power signal of for all nonlinearvarious orders.input power levels as shown in Figure given6a–d. window The figures function) show can predict the input the PIM power power levels of all nonlinearat which orders. the periodogram (which uses a given window function) can predict the PIM power of all nonlinear orders.

(a) (b)

Figure 6. Cont.

(a) (b)

Electronics 2021, 10, 1034 10 of 12 ElectronicsElectronics 2021 2021, 10, 10, x, xFOR FOR PEER PEER REVIEW REVIEW 1010 of of 12 12

(c()c ) (d(d) ) FigureFigureFigure 6. 6.6. Power PowerPower in in inthe the the intermodulation intermodulation intermodulation components components components vs vs. vs.Input. Input Input power power power computed computed computed from from fromthe the periodo- periodo- the peri- gramodogramgram with with withvarious various various window window window functi functi functions:ons:ons: Solid Solid o: Solid o: Hanning Hanning o: Hanning window window window Solid Solid *: Solid*: Blackman Blackman *: Blackman window, window, window, Solid Solid ∆Solid∆: Blackman: Blackman∆: Blackman Harris Harris Harriswindow, window, window, Dashed Dashed Dashed o: o: rectangular rectangular o: rectangular window, window, window, Dashed Dashed Dashed*: *: Hamming Hamming *: Hamming window window window and and Dashed ∆: Kaiser window; (a) IMD3, (b) IMD5, (c) IMD7 and (d) IMD9. andDashed Dashed ∆: Kaiser∆: Kaiser window; window; (a) IMD3, (a) IMD3, (b) IMD5, (b) IMD5, (c) IMD7 (c) IMD7 and and(d) IMD9. (d) IMD9.

TheTheThe effect effecteffect of ofof frequency frequencyfrequency separation separationseparation between betweenbetween the thethe two twotwo carriers carrierscarriers was waswas also alsoalso studied studiedstudied where wherewhere thethethe PSD PSDPSD of of thethe the periodogramperiodogram periodogram of of of the the the output output output of of theof the the nonlinearity nonlinearity nonlinearity was was simulatedwas simulated simulated using using using different dif- dif- ferentwindowferent window window functions functions functions at different at at different different frequency frequenc frequenc separationsyy separations separations and, and, and, hence, hence, hence, different different different separation separa- separa- tionbetweention between between IMD IMD frequenciesIMD frequencies frequencies and theand and fundamental the the fundamental fundamental frequencies frequencies frequencies at a carrier at at a a carrierpower carrier ofpower power 35 dBm of of 35 as35 dBmshowndBm as as inshown shown Figure in in7 Figurea–d. Figure 7a–d. 7a–d.

(a(a) ) (b(b) )

(c()c ) (d(d) ) Figure 7. Power in the intermodulation components vs. frequency separation computed from the FigureFigure 7.7. PowerPower inin thethe intermodulationintermodulation components vs vs.. frequency frequency separation separation computed computed from from the the periodogram with various window functions: Solid o: Hanning window Solid *: Blackman win- periodogramperiodogram with with various various window window functions: functions: Solid Solid o: o: Hanning Hanning window window Solid Solid *: *: Blackman Blackman window, window,dow, Solid Solid ∆ ∆: Blackman: Blackman Harris Harris window, window, Dashed Dashed o: o: re rectangularctangular window, window, Dashed Dashed *: *: Hamming Hamming Solid ∆: Blackman Harris window, Dashed o: rectangular window, Dashed *: Hamming window windowwindow and and Dashed Dashed ∆ ∆: Kaiser: Kaiser window. window. and Dashed ∆: Kaiser window.

Electronics 2021, 10, 1034 11 of 12

5. Discussion Table4 summarizes the results obtained from Figure6a–d where it is shown that the Blackman and Blackman–Harris windows have best performance in predicting PIM components in comparison to other windows at all power levels and all orders of PIM. The table also shows that at low input power levels, Blackman window surpasses other window function in predicting all PIM component, while at high input power, the Blackman and the Blackman–Harris windows show a similar result in predicting PIM component. These results are consistent with the properties of the window function in Table1 which shows that although the Blackman window has side lobe level (−58 dB) which is much higher than the Blackman Harris window, it has better performance than the Blackman Harris window for higher order IMD components. The reason for this is that although the Blackman Harris window has a better side lobe level performance (−92 dB), the Blackman window has a much higher roll-off rate of the side lobes (−18 dB/decade) especially when the frequency separation between the two input signals is small and hence, high frequency PIM components. On the other hand, at high input power level, the PIM components have a higher power level and hence, both windows have similar PIM prediction performance.

Table 4. IMD estimation using various window function.

Pin = 0 dBm Pin = 30 dBm Window IMD3 IMD5 IMD7 IMD9 IMD3 IMD5 IMD7 IMD9 Rectangular −53 −58 −60.9 −62.8 −6.8 −25.7 −31.4 −33.7 Kaiser −53.4 −58.3 −61.2 −63.1 −6.8 −25.8 −31.7 −34 Hamming −70.9 −75.9 −78.8 −80.7 −7 −27.3 −43.7 −50.9 Hanning −96.7 −174.4 −211 −218 −7 −27.3 −45.8 −64.7 Blackman −97.5 −175.3 −219 −226 −7.4 −28 −46.8 −65.6 Blackman Harris −98.1 −136.5 −139.4 −141 −7.8 −28.8 −47.9 −66.8

The results in Figure7a–d show that for higher order PIM components (5th order and higher), the Blackman and Hanning windows provide the lowest IMD power and hence, the best PIM prediction capability regardless of the frequency separation. With other window functions, the ﬁgures show that the PIM level decreases with increasing the frequency separation between the carriers as a result of spectral leakage. These results comply of the properties of the different window functions in Table1 where the Blackman and Hanning windows have the best side lobe roll of (−18 dB/decade), which means that the spectral leakage decreases rapidly with increasing frequency separation and hence, the PSD can better predict IMD components of all orders than other window functions. For the third order PIM component, Figure7a shows that Hamming and Kaiser windows provide better prediction capability at high frequency separations and, at the same time, the worst PIM prediction capability at lower frequency separations. The reason for this is the low side lobe roll-off rate of these windows which results in higher spectral leakage at low frequencies. On the other hand, the Blackman and Hanning windows offer reasonable PIM prediction at all frequency separations, which make them better candidates for spectral estimation of PIM than the Hamming and Kaiser windows.

6. Conclusions In this paper, prediction of PIM in carrier aggregated system from the output spectrum of a behavioral model of a passive nonlinearity has been studied. Since PIM components usually have a very low power levels, it has been shown that spectral leakage has a signiﬁcant effect of the prediction capability of the PSD to identify PIM components. A detailed analysis and simulations of the impact of the choice of the window functions on prediction of PIM distortion has been presented. Since the frequency and the power of IMD frequencies depend on the frequency separation between the main channels and their power levels, it has been shown that accurate prediction of PIM components from the Electronics 2021, 10, 1034 12 of 12

output PSD requires window functions that compromise the side lobe levels and the side lobe roll-off rates. The simulation results showed that, in principle, window functions that have high side lobe roll-off rates (Hanning and Blackman) offer better PIM prediction when IMD components are strong and distant from the main carrier (high frequency separation between the main carriers). On the other hand, window functions with low side lobe levels (Blackman and Blackman Harris) offer better prediction of PIM components when the IMD components are weak and close to the main carriers (low frequency separation between the main carriers).

Funding: This research was funded by the Applied Science University-Bahrain. Data Availability Statement: Not applicable. Conﬂicts of Interest: The author declares no conﬂict of interest.

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