Lu TIAN,Hangcheng HAN,Wenhui CAO,Xiangyuan BU,Shuai WANG

School of Information Science and Electronics,Beijing Institute of Technology,Beijing 100081,China

Adaptive suppression of passive intermodulation in digital satellite transceivers

Lu TIAN,Hangcheng HAN*,Wenhui CAO,Xiangyuan BU,Shuai WANG

School of Information Science and Electronics,Beijing Institute of Technology,Beijing 100081,China

Available online 20 April 2017

*Corresponding author.

E-mail address:hanhangcheng@bit.edu.cn(H.HAN).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2017.03.010

1000-9361?2017 Production and hosting by Elsevier Ltd.on behalf of Chinese Society of Aeronautics and Astronautics.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

For the performance issues of satellite transceivers suffering passive intermodulation interference,a novel and effective digital suppression algorithm is presented in this paper.In contrast to analog approaches,digital passive intermodulation(PIM)suppression approaches can be easily reconfigured and therefore are highly attractive for future satellite communication systems.A simplified model of nonlinear distortion from passive microwave devices is established in consideration of the memory effect.The multiple high-order PIM products falling into the receiving band can be described as a bilinear predictor function.A suppression algorithm based on a bilinear polynomial decorrelated adaptive filter is proposed for baseband digital signal processing.In consideration of the time-varying characteristics of passive intermodulation,this algorithm can achieve the rapidness of online interference estimation and low complexity with less consumption of resources.Numerical simulation results show that the algorithm can effectively compensate the passive intermodulation interference,and achieve a high signal-to-interference ratio gain.

?2017 Production and hosting by Elsevier Ltd.on behalf of Chinese Society of Aeronautics and Astronautics.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Adaptive digital filters;

Intermodulation model;

Interference suppression;

Passive devices;

Satellite communication systems

1.Introduction

As the information propagation rate rises,both the high power of transmitters and the high sensitivity of receivers are needed in satellite communication systems.The threat of passive intermodulation(PIM)interference has been exposed in those systems with transmit-receive shared antennas.1,2PIM is a phenomenon that additional signals at new frequencies(not just harmonic frequencies)are generated when signals containing two or more different frequencies are processed at passive devices with nonlinearities.3The linearization of most passive system components can be easily changed under high-power conditions.4The power of PIM products is probably over the thermal noise power as well as the duplexer isolation limitation in high-power systems.More severely,PIM can be received by sensitive receivers as a type of interference when it falls into the receiving band.5PIM has become a major obstacle for satellite communication,especially for systems in frequency division duplexing(FDD)transmission schemes.6

There has been a lot of research done on PIM in the past half century,focusing on the phenomena,distortion mechanism,and measurement techniques.7,8Nevertheless,research on PIM suppression is not sufficient,especially for those using digital processing algorithms.Specif i c methods focusing on metal-to-metal interface planes have been introduced to minimize the generation of PIM.9,10Low-PIM blankets and special manufacturing process have also been used for the reduction of PIM,11,12but this method is obviously limited because components cannot maintain linearization with aging and corrosion.13,14Band planning and frequency hopping are good choices to reduce PIM distortion for narrowband communication systems.15However,this simple technique does not adequately address PIM interference in some communication systems that adopt very wide bandwidth,such as long term evolution(LTE)systems.An effective method of reducing PIM has been proposed utilizing the addition of a nonlinear interposer network.16Nevertheless,it is not practical because additional hardware is prohibitive in current communication systems.

Since the transmitting signal is known,the intermodulated part leaking into the receiving signal path can be estimated and compensated due to its high correlation.This approach has been utilized for cancelation of second-order active intermodulation distortion.17,18Meanwhile,the third-order intermodulation product is concentrated to cancel.19However,appropriate band planning can easily avoid the production of these low-order PIM products.For cancelation of higherorder ones,a new method is illustrated in a patent.20The coefficients of the interference model are dynamically estimated from test signals and measured interference signals.Then,the PIM interference can be estimated and canceled due to these coefficients.It is not practical in current communication systems for interrupting real-time communication and reserving time slots for coefficients estimating.Another PIM cancelation scheme is based on the adaptive filter algorithm.21Nevertheless,complex analog circuitry is required to generate a reference signal for the adaptive algorithm.

A fully-digital adaptive suppression algorithm of high-order PIM will be presented in this paper,without changing the communicationprotocolorrequiringextrahardware.Theproposed method is online and adaptive,and it is an important supplement to conventional PIM suppression methods for digital satellite transceivers.The received signal with PIM interfered iscompensatedbyalinearlyreconstructedPIMinterferencesignal.The contributions of the current paper may be summarized in the following four aspects.Firstly,more precisely over the previous works on PIM suppression,the memory effect of PIM is taken into consideration.Secondly,the number of PIM products for evaluation declines,especially when dealing with high-order PIM products,since only those PIM products falling into the receiving band are taken into consideration.What is more,the PIM products are decorrelated from each other to achieve a faster convergence rate and a stronger stationarity.Thirdly,by utilizing the correlation between the PIM signal and downlink signals,a bilinear solution to this nonlinear problem achieves a low computational complexity.Last but not least,in consideration of the time-varying characteristics of PIM,an adaptive algorithm is adopted to suppress PIM interference in real time,achieving a 20 dB-signal to-interference ratio(SIR)gain and a 1 dB signal-to-noise(SNR)loss in a 10-2bit error rate(BER)for results.

The rest of this paper is organized as follows.Section 2 describes the system model and behavioral model of PIM.Section 3 introduces the bilinear reconstruction of PIM and the details of the adaptive suppression algorithm.Section 4 presents the numerical simulation results of suppression performance,followed by a brief conclusion in Section 5.

Notations:Throughout the discussion,N+,RM×N,and CM×Nstand for positive integers,(M×N)-dimensional real and complex valued matrices,respectively;*,E（·）, （·）*, （·）T,and （·）Hare operators for convolution,mathematic expectation,conjugate,transpose,and conjugate transpose,respectively;besides,given bold lowercase letters likea∈ RM×1or CM×1,a［i］represents itsith element,while for bold capital letters likeA∈ RM×Nor CM×N,A［i，j］andA［:，j］represent its（i，j）th element and itsjth column,respectively.

2.System model

The problem that passive intermodulation falls into the receiving band occurs in FDD transmission schemes in both satellites and ground stations.This research focuses on the PIM present in satellites,as depicted in Fig.1.The digital downlink signals in baseband are converted into analog signals by the digital-to-analog converter(DAC).The downlink signals in baseband are mixed with the local oscillator(LO)signal for up conversion,which is expressed as LODL.The downlink signals are transmitted after amplified by the power amplifier(PA).The PIM source is an antenna port of the duplexer in this example.The PIM source signal passes through lownoise amplifier(LNA),down converter(mixed with LOUL)and channel select filter(CSF),finally getting into the analog-to-digital converter(ADC)to be processed in the digital receiver mixed with the uplink signal.

Ignoring other interferences for simplicity,we can idealize a final baseband system model,which consists of the uplink signal,PIM interference,and noise.The equivalent received signal in baseband is

wheresUL（t）,p（t）,andns（t）stand for the equivalent uplink signal,PIM interference signal,and noise in baseband,respectively,which means the opposite signals after passing through the down converter and the channel selective filter.

High-power downlink signals produce PIM signals when passing through passive microwave devices,such as cables,antennas,and duplexers.The PIM phenomenon of multicarrier systems is obvious,so in this study the IF downlink signal is assumed to consist of multiple channels with different carrier frequencies.The modulated IF downlink signal composed ofKchannels is

whereBk（t）,fk,and φkexpress the baseband signal,subcarrier frequency,and initial phase of channelk.

The nonlinearity effect of this cascade system constituted by a series of passive microwave components can be mathematically abstracted as Volterra series.Its advantage is the ability to capture memory effects compared with the Taylor series.Volterra series has been used by numerous groups to describe the behavioral models of nonlinear systems.22

Referring to Frechet’s approximation theorem,the order of Volterra series should be sufficiently high with a number of variables,which is an essential problem in applications.Due to the poor practicality of Volterra series for high-order intermodulation production,we adopt Hammerstein model,which is a simplified model of Volterra series remaining memory effects.The RF downlink signalsRF（t）is

wherefDLdenotes the frequency of the downlink local oscillator for the transmitter,and φDLrepresents the initial phase of the downlink local oscillating signal.We can get the PIM signal from Hammerstein model as

wherehPIM（t）is used to represent the memory effect of PIM,and the coefficients αnis assumed to decrease rapidly in magnitude,because only mild nonlinearities are expected in PIM.23The down converted PIM signal is

where,likewise,fULdenotes the frequency of the uplink local oscillator for the receiver,and φULrepresents the initial phase of the uplink local oscillating signal.

Meanwhile,as a bandpass filter,the channel selective filter will filter out most interference signals.We can find out the PIM products falling into the receiving band according to the basic information of downlink signals.The equivalent baseband PIM signal is

wherehCSF（t）is the impulse response of the channel selective filter.Combined with the above equations,

wheresP（t）will be simplified in the following section.

3.Adaptive suppression algorithm

In order to utilize the correlation between the PIM signal and downlink signals in time domain to suppress PIM interference,the PIM signal should be reconstructed.In addition,for the sake of adopting a linear suppression algorithm with low complexity,the reconstruction should be in a linear way.Therefore,we can get the PIM suppressed signal

3.1.Reconstruction of PIM

It is the PIM products falling into the receiving band that interfere the performance of receivers,so only these will be taken into consideration.The amount of intermodulation products in multi-carrier systems is formidably large.24However,a simple and effective way to find the lowest order of PIM falling into the receiving band is to look for the lowest order of PIM from a two-tone signal.The frequencies of the two-tone signal are the lowest and highest frequencies of the downlink signals.Meanwhile,the lowest order of first-zone

(The first zone of a nonlinearity output refers to the output in the vicinity of the input carrier frequencies,as distinguished from those around the harmonics of the output carriers.)PIM products from a two-tone signal can be easily calculated as a Diophantine problem.3The first-zone PIM product with the lowest order is of particular importance because it contains the most power if the carriers have similar amplitudes.23Naturally,this lowest-order PIM product will be the focus to deal with.In this situation,the constructed baseband PIM signal is

where φ is the initial phase andNis the lowest order.Nis definitely an odd number,under the condition offirst zone.Additionally,the harmonic signals from the local oscillator are considered to be filtered.We assume thatfDLequalsfUL,and

Finally,we can get a simplified constructed baseband PIM signal as

A more practical approximation can be

whereb1= αN+2/αNandbM= αN+2M/αN,withMnew coefficients to estimate.Instead of infinite higher-order terms,onlyMones are taken into consideration,owing to the weak nonlinearity of passive devices.In addition,a proper value ofMwill be discussed in the simulation section.The above equation can be rewritten as

whereb0=1.WhenM=0,we get the simplest approximation,which is exactly the former one.Assuming thatxm（i）is the digital discrete expressions of（t）,and（t）is replaced by a finite-impulse-response(FIR)filter withLtaps,which can be regarded as a PIM selective filter.wkis the tap coefficients,wherek=0，1，...，L-1.Thus the digital discrete constructed baseband PIM signal is

where there are two groups of coefficients to estimate.（i）is a bilinear predictor function.The matrix formulation of the above equation is

wherew= ［w0，w1，...，wL-1］Tare the coefficients of the PIM selective filter,b= ［b0，b1，...，bM］Tare the weights of PIM products,and

3.2.Bilinear polynomial decorrelated normalized least mean square algorithm

In order to solve the bilinear regression problem,the coordinate descent algorithm is a good choice,which iteratively updateswwithbfixed and thenbwithwfixed.

The estimation error can be written as

The aim is to minimize the mean square estimation error at every moment,so the cost function is

In order to reach the minimum,we try to adjust the value of every coefficients to realize▽J=0.However,it is impossible for a time-variant signal,so we can only try to letJ（i+1）＜J（i）,which originates from the method of steepest descent.The gradients toandbmare

Write Eq.(23)in a matrix form as

where ▽Jw= ［▽Jw0，▽Jw1，...，▽JwL-1］Tand ▽Jb= ［▽Jb0，▽Jb1，...，▽JbM］T.The iterative relations of the PIM selective filter coefficients and the PIM product weights are

wherew（i）= ［w0（i），w1（i），...，wL-1（i）］Taretheestimated coefficients of the PIM selective filter,andb（i）= ［b0（i），b1（i），...，bM（i）］Tare the estimated weights of the PIM products at every specific time.μwand μbare the step size parameters determining the convergence and the remaining estimation error,and we adjust their values in a proper range to ensure the convergence.25

Compared with the conventional adaptive polynomial filter method,26the computational complexity is reduced from O（L× （M+1））to O（L+M+1）.However,due to the strong correlation,the condition of the correlation matrix gets inferior,and the divergence of eigenvalue becomes large.This will reduce the convergence rate in a general gradient adaptive algorithm.Correlation among the PIM products also causes imbalance in the weight vector.The convergence speed and computational complexity about adaptive algorithms are a core issue about PIM cancelation in satellite transceivers.If the convergence speed is very slow or the computational complexity is very high,adaptive algorithms will not track the changes in the PIM interference signals in time,and the residual interference will badly affect the communication quality.Usually,the convergence speed increases at the expense of the computational complexity.To ensure the convergence speed and reduce the computational complexity,this paper will find a balance between the convergence speed and computational complexity as far as possible to arrive at the best real-time effect.

A bilinear polynomial decorrelated normalized least mean square(BPDNLMS)algorithm is proposed as a modified algorithm to improve the steady state error and convergence speed of bilinear adaptive algorithms for colored and nonstationary signals.An adaptive pre-filter is adopted jointly as a decorrelation filter to both the input and desired signals in the process of updatingb,and the algorithm is depicted in Fig.2.Parameters are initialized at first,w［0］= ［0，0，...，0］T,b［0］= ［0，0，...，0］Tanda［0］= ［0，0，...，0］T,whereais theQ-length coefficients vector of the linear pre-filter.The iterations fori=1,2,...are as follows,

where d（i）is srec（i）for brevity,and~x and~e are the pre-filtered signals.To make the adaptation stable in the mean sense,0＜μa＜1 and 0＜β＜1.To overcome the numerical difficulties when the denominators are small,δwand δbare added to the denominators respectively,where δw，δb＞ 0.BPDNLMS can move the correlation between different PIM products.This whitening usually results in reducing the eigenvalue spread and,therefore,speeds up the convergence.

The computational complexity of the bilinear NLMS algorithm without decorrelation is 3（L+M+1）,and that of BPDNLMS is 3（L+M+1）+6Q.The increased complexity improves the convergence and steady state performance.As Fig.3 shows,the BPDNLMS adaptive algorithm converges a little faster,and has a lower mean square deviation ofD=E（｜b-bopt｜2）.

4.Numerical simulation results

To evaluate the performance of the PIM suppression algorithm referred in this paper,the simulation frequencies used in this contribution are based on the FDD UTRA band I.The simulation system works in a structure which is similar to the model presented in Fig.1.Table 1 shows the simulation parameters that will be used in this section.The PIM products considered in the simulation are from the 3rd order to the 23th order.The calculation indicates that the 7th order of PIM is the most effective one falling into the receiving band.

Fig.4 is a sampling waveform shot,which clearly represents the effectiveness of the proposed adaptive suppression algorithm.The sampling points of receiving signals adopting the suppression algorithm coincide with those without PIM interference.In Fig.5,the error vector magnitude(EVM)decreases simultaneously with PIM suppression.

Fig.6 indicates the effectiveness of the PIM suppression algorithm at any value ofM.AsMincreases,the uncoded BER performance is improved.The value ofMis based on system design requirements,andM=4 is a good choice in the situation of Fig.6.

The simulation results in Fig.7 clearly show the impact of PIM on system performance,where a high BER is presented with an SIR less than zero.Fig.7 also illustrates a SIR gain over20 dB utilizing the proposed adaptive suppression algorithm.

Table 1 Simulation parameters.

The SNR loss when the BER equals 10-2is a value we are concerned about,considering the channel coding gain.As shown in Fig.8,the SNR performance loss is reduced to less than 1 dB by the proposed algorithm,when the SIR is-10 dB.The adaptive PIM suppression algorithm efficiently cancels the PIM interference,leading to a lower SNR loss than that in the case without any suppression.Thus,more PIM interferences can be allowed by suppressing with the proposed algorithm,which means the margin is improved.

5.Conclusions

(1)An adaptive suppression algorithm of PIM interference,based on a bilinear approximation of a system model,has been presented.It is used to suppress the PIM interference in digital satellite transceivers at baseband.

(2)The simulation results show that the BPDNLMS algorithm significantly improves the system performances.The proposed algorithm is adaptive and online,and it achieves good convergence performance with low complexity.

(3)However,since the assumption of the simplified PIM model limits the suppression performance,new PIM models will be investigated to achieve further improvements.

Acknowledgements

This study was supported by the National Natural Science Foundation of China(Nos.U1636125,61601027).

Appendix A.Proof of higher-order PIM terms producing the lowest-order one

This appendix gives a proof,which indicates that the highorder PIM terms contain PIM products with lower orders including the lowest one.

Without loss of generality,the proof is given for a system with two carriers for convenience.Assume that the input signal is

The spectrum of the first two terms in Eq.(A2)can cover that of signalxn（t）due to Fourier transform.This is a powerful evidence that the high-order PIM terms produce lowerorder ones.The proof also works whenxn+2（t）is replaced byxn+2k（t）,wherek∈N+.WhennequalsN,the lowest order,the final proof work is done.

1.Jiang J,Li TJ,Ma XF,Wang P.A nonlinear equivalent circuit method for analysis of passive intermodulation of mesh reflectsors.Chin J Aeronaut2014;27(4):924–9.

2.Bai H,Cui WZ.Simulation of PIM generated on the mesh reflectsor.European microwave conference;2015 September 7–10;Paris,France.Piscataway(NJ):IEEE Press;2015.p.602–5.

3.Kai YE,Thomas ES.The order-and-type prediction problem arising from passive intermodulation interference in communications satellites.IEEE Commun Trans1981;29(5):549–55.

4.Henrie JJ,Christianson AJ,Chappell WJ.Linear-nonlinear interaction and passive intermodulation distortion.IEEE Trans Microwave Theory Tech2010;58(5):1230–7.

5.Zhao P,Zhang XP,Yang DC.Analysis of passive intermodulation generated by broadband signals.Electron Lett2016;52(7):564–6.

6.Singh R,Hunsaker E.PIM risk assessment and mitigation in communications satellites.AIAA international communicationssatellite systems conference and exhibit;2004 May 9–12;Monterey,California,USA.Reston:AIAA;2004.p.9–14.

7.Lui P.Passive intermodulation interference in communication systems.Electron Commun Eng J1990;2(3):109–18.

8.Vicente C,Wolk D,Hartnagel HL,Gimeno B,Boria VE,Raboso D.Experimental analysis of passive intermodulation at waveguide f l ange bolted connections.IEEE Trans Microwave Theory Tech2007;55(5):1018–28.

9.Schennum G,Rosati G.Minimizing passive intermodulation product generation in high power satellites.Proceedings of IEEE aerospace applications conference;1996 Feb 3–10;Aspen,CO.Piscataway(NJ):IEEE Press;1996.p.155–64.

10.Jiang J,Li TJ,Liu YC.Passive inter-modulation scattering analysis of reflectsor considering contact nonlinearity.Chin J Aeronaut2013;26(2):463–9.

11.Agassi YD,Oates DE.Intermodulation distortion and surface resistance in impurity-doped YBCO and MgB 2.Phys C Supercond2014;506:119–32.

12.Shitvov AP,Olsson T,Banna BE,Zelenchuk DE,Schuchinsky AG.Effects of geometrical discontinuities on distributed passive intermodulation in printed lines.IEEE Trans Microwave Theory Tech2010;58(2):356–62.

13.Liu Y,Mao YR,Xie YJ,Tian ZH.Evaluation of passive intermodulation using full-wave frequency-domain method with nonlinear circuit model.IEEE Trans Veh Tech2016;65(7):5754–7.

14.Wilkerson JR,Lam PG,Gard KG,Steer MB.Distributed passive intermodulation distortion on transmission lines.IEEE Trans Microwave Theory Tech2011;59(5):1190–205.

15.Jung H,Tonguz OK.Random spacing channel assignment to reduce the nonlinear intermodulation distortion in cellular mobile communications.IEEE Trans Veh Tech1999;48(5):1666–75.

16.Henrie J,Christianson A,Chappell WJ.Cancellation of passive intermodulation distortion in microwave networks.38th European microwave conference;2008 Oct 27–31;Amsterdam,USA.Piscataway(NJ):IEEE Press;2008.p.1153–6.

17.Lederer C,Huemer M.Simplif i ed complex LMS algorithm for the cancellation of second-order TX intermodulation distortions in homodyne receivers.45th Asilomar conference on signals,systems and computers;2011 Nov 6–9;Pacif i c Grove(CA),USA.Piscataway(NJ):IEEE Press;2011.p.533–7.

18.Gerzaguet R,Ros L,Belvze F,Brossier JM.Performance of a digital transmitter leakage LMS-based cancellation algorithm for multi-standard radio-frequency transceivers.Dig Sign Process2016;51(C):35–46.

19.Kamizuma H,Masuda T,Onishi M.Third-order intermodulation product canceller for LTE base station receiver.41st European microwaveconference;2011 Oct10–13;Manchester,UK.Piscataway(NJ):IEEE Press;2011.p.230–3.

20.Wang M,Li S,Baldemair R,Cheng J,Ghasemzadeh F,Larsson M,et al.,inventors.Telefonaktiebolaget LM Ericsson(Publ),assignee.Dynamic cancellation of passive intermodulation interference.United States patent US 20120295558;2012 Nov 22.

21.Keehr EA,Hajimiri A.Successive regeneration and adaptive cancellation of higher order intermodulation products in RF receivers.IEEETransMicrowaveTheoryTech2011;59(5):1379–96.

22.Walker AL.Behavioral modeling and characterization of nonlinear operation in RF and microwave systems[dissertation].Raleigh:North Carolina State University;2006.

23.Eng K,Yue O.High-order intermodulation effects in digital satellite channels.IEEE Trans Aerosp Electron Syst1981;3:438–45.

24.Yeary MB.An efficient intermodulation product computing technique for broadband active transmit systems.IEEE Trans Instrum Meas2008;57(2):438–43.

25.Haykin S.Adaptive filtering theory.4th ed.New Jersey:Prentice-Hall;2002.

26.Zhang J,Zhao H.A novel adaptive bilinear filter based on pipelined architecture.Dig Sign Process2010;20(1):23–38.

2 September 2016;revised 20 November 2016;accepted 23 December 2016