Use of Wavelet Transforms for InterferenceMitigation

Fabio DovisAssistant Professor

Department of ElectronicsPolitecnico di Torino

Turin, Italy +39-011-5644175Email: fabio.dovis@polito.it

Luciano MusumeciPh.D. Student

Department of ElectronicsPolitecnico di Torino

Turin, Italy +39-011-2276447Email: luciano.musumeci@polito.it

Abstract— The research work described in this paper is focusedon the topic of the Radio Frequency Interference (RFI) mitigationtechniques in the field of the Global Navigation Satellite Systems(GNSS). The paper focuses on an innovative interference mitiga-tion algorithm based on the Wavelet Transform. The first part ofthis paper provides a description of the Wavelet decompositionmethod and its use in the GNSS field. All the steps of the detectionand mitigation algorithms exploiting this mathematical tool aredescribed as well as its application on narrow-band interferencemitigation. The proposed method exploits the compact time-frequency behavior of Wavelet functions; they allow for agood indentification of the interfering source and a syntheticreconstruction of the signal that can then be used for mitigationpurposes. Simulation results showing the benefits of this Waveletbased mitigation technique on the acquisition and tracking stagein L1 GPS band, are presented.

I. INTRODUCTION

In the coming years, new radio navigation signals for theGPS system will be broadcast, and with the introductionof a new European GNSS system, more attention has beendevoted to the design of signal structure which is expectedto increase accuracy, availability, integrity and continuity ofservice, especially in the field of the Safety of Life applications(e.g. the accuracy needed during the landing of an aircraft).As it has been already presented in several papers, theinterference environment for civil aviation is threatening;several Aeronautical Radio Navigation systems such as Dis-tance Measuring Equipment (DME) or Tactical Air Naviga-tion (TACAN), based on pulsed signals broadcasting can beconsidered as a real in-band interference source for the 𝐿5GPS band and 𝐸5𝑎/𝐸5𝑏 Galileo bands [1]. The ARNS bandis not the only one where the interference issue is present;other kinds of interference such as Continuous Wave (CW)signals originated from the Digital Video Broadcast Terrestrial(DVB-T) service can be considered as a threat for the othersRadio Navigation Satellite Service (RNSS) signals [2]. Inliterature, several interference mitigation strategies have beenproposed and investigated, and each of them differs accordingto the domain (frequency, time or spatial) in which theyoperate. In the latest years, the research on this topic hasbeen focused on those mitigation algorithms which work ina hybrid domain, since the interference is mainly located in

a limited region of the time-frequency space. In addition tothe classical time-frequency representation of the signal, othertransformations may be beneficial for a better distinction ofthe useful component from disturbances.As an example, in the framework of the navigation satellitesystems, the Wavelet Transform has been already extensivelyused; in fact, in literature, it is possible to find several researchworks in which the Wavelet Transform has been adoptedmainly to cope with the multipath issue. In particular, in[3] a new trend extraction technique for multipath mitigationin carrier phase measurements domain using wavelet multi-resolution analysis, is presented. More detailed descriptionsof this algorithm can be found in [4] [5] [6] [7].Multipath mitigation is not the only one framework whichWavelet Transform has been used. For instance, in [8], asingularity detection technique for GPS cycle slips based onthe wavelet decomposition is described. In [9] an EmpiricalMode Decomposition (EMD) exploiting the Wavelets’ prop-erties is described as a method to reduce the carrier phasemeasurements error. Finally in [10], methodology based onWavelet Transform to evaluate the terrain and extract featuresalong the vehicle path is presented. Of particular interestare those features which can be hazardous to a followingvehicles path. In the framework of the interference mitigationtechnique, [11] gives a description of a Wavelet based methodable to identify in the time-frequency plane the presence ofpulsed interference signal.As far as narrow-band interference mitigation is concerned,it is not possible to define a tailored mitigation technique;in fact, the performance for a certain time-frequency miti-gation algorithm strongly depends on the characteristics ofthe interfering signal that despite of the narrow-band spectralfeature may behave in very different ways. As well known,an interfering Continuous Wave appears in the spectrum ofthe received signal as a spike; in this case the best option formitigation is to operate an interference excision using a notchfilter. If the interfering signal has a narrow-band spectrum,notch-filtering could not remove all the interference frequencycomponents. In [12] several time-frequency mitigation algo-rithm are presented.In this paper the attention is mainly focused on a Wavelet

978-1-61284-4577-0188-7/11/$26.00 c©2011 IEEE116

based mitigation approach for narrow-band interference; thework that will be presented in this paper aims to verify thebenefits of using Wavelet Transform for narrow-band inter-ference mitigation. Software implementation of this algorithmand performance validation are presented.

II. WAVELET TRANSFORM THEORY

This mitigation technique in the time-scale domain is basedon the decomposition of a signal into its constituent compo-nents through the use of a local basis functions. These areobtained from a mother wavelet through scaled and shiftedversion [2]. Hence, the original signal can be represented bylinear combinations of a basis functions (wavelets) localizedin time and frequency domains. The basic idea is to providedetection and mitigation in the wavelet domain defining athreshold in each scale of the Wavelet decomposition, isolatingthe scale affected by the interference. In this way, bothdetection and mitigation can be applied only in a limitednumber of blocks. In the following subsection, a theoreticalpresentation on the Wavelet transform will be given.

A. Continuous Wavelet Transform

The continuous wavelet transformation for a time-dependingfunction 𝑓(𝑡) is defined as:

𝛾(𝑠, 𝜏) =

∫ ∞

−∞𝑓(𝑡)𝜓∗

𝑠,𝜏 (𝑡)𝑑𝑡 (1)

where ∗ denotes the complex conjugation and 𝜓∗𝑠,𝜏 (𝑡) is called

the wavelet with its variables scale 𝑠 and translation 𝜏 . Thewavelet functions are deduced from one single wavelet, theso-called mother wavelet 𝜓(𝑡), by scaling and translationaccording to:

𝜓𝑠,𝜏 (𝑡) =1√𝑠𝜓

(𝑡− 𝜏

𝑠

)(2)

where 𝑠 represents the scaling and 𝜏 the translation.The characteristics of the wavelet mother functions are de-scribed in [13].

B. Discrete Wavelet Transformation

Working with sampled data requires the definition of a dis-crete Wavelet Transformation. The discrete Wavelet functionis given by:

𝜓𝑗,𝑘(𝑡) =1√𝑗𝑠0

𝜓

(𝑡− 𝑘𝜏0𝑗𝑠0

𝑗𝑠0

)(3)

with integers 𝑗 and 𝑘, discrete scaling steps 𝑠0 and timedilations 𝜏0 but continuous with respect to the time axis 𝑡.Also with discrete Wavelets an infinite number of scalings andtranslations are still needed to fully represent the original inputfunction. From the Wavelet characteristics defined in [13], itcan be deduced that Wavelet functions offer a band-pass likespectrum. As well known, a compression in time correspondsto a stretching in the frequency domain together with a shiftingup to higher frequencies. This can be used to construct a seriesof Wavelets that cover the whole frequency range of our input

signal. Practically, it is impossible to cover all spectrum of aninput function down to zero.

A solution is shown in Figure 1, where in order to cover allthe frequency range down to zero, a scaling function is used,and it can be interpreted as a low-pass filter . This scaling

Fig. 1. Introduction of Scaling Function

function can be interpreted in terms of its Wavelets as:𝜙(𝑡) =

∑𝑗,𝑘 𝛾(𝑗, 𝑘)𝜓𝑗,𝑘(𝑡).

Graphically the use of scaling functions can be represented asshown in Figure 1.In the same way, we can also add the Wavelet next to the scalefunction to it, obtaining a scaling function with a spectrumtwice as much as before. Mathematically it can be expressedas shown in [13] as:𝜙(2𝑗𝑡) =

∑𝑘 ℎ𝑗+1(𝑘)𝜙(2

𝑗+1𝑡− 𝑘).The above equation means that one scaling function can beexpressed as a function of translated scaling at the next smallerscale. The Wavelet that was replaced by the scaling functioncan be expressed as:𝜓(2𝑗𝑡) =

∑𝑘 𝑔𝑗+1(𝑘)𝜙(2

𝑗+1𝑡− 𝑘).Finally the function to be transformed into its representationwith Wavelets can be approximated as:𝑓(𝑡) =

∑𝑘 𝜆𝑗−1(𝑘)𝜙(2

𝑗−1𝑡−𝑘)+∑𝑘 𝛾𝑗−1(𝑘)𝜓(2

𝑗−1𝑡−𝑘).Assuming to have orthonormal scaling functions 𝜓𝑗,𝑘(𝑡)and orthonormal Wavelet functions 𝜓𝑗,𝑘(𝑡), the coefficients𝜆𝑗−1(𝑘) and 𝛾𝑗−1(𝑘) can be calculated by a scalar productbetween the scaling Wavelet function and the initial function𝑓(𝑡).Other mathematical details on the Wavelet theory can be foundin [14].

III. WAVELET BASED MITIGATION ALGORITHMS FOR

NARROW-BAND INTERFERENCE

In this section, a description of the steps performed forthe narrow-band interference detection and mitigation is pre-sented. The interference scenario has been generated by usinga fully software signal generator, named N-FUEL, which isable to provide samples at intermediate frequency (IF), andallows the user to define signal characteristics as frequencydoppler, code delays, 𝐶/𝑁0 and interference parameters. Moredetails about N-FUEL software signal generator can be foundin [15] [16].

A. Discrete Wavelet Transformation

The first step of this mitigation algorithm is the Waveletdecomposition of the incoming GNSS signal. This decom-position has been achieved through the use of orthogonal

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local basis function derived from the Meyer mother Wavelet.We should mention that the Wavelet decomposition has beenperformed in order to isolate the interference frequency com-ponents. Figure 2 shows, as an example, the interfered L1 GPSsignal in the frequency domain.

0 1 2 3 4 5 6 7 8 9−160

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−150

−145

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−130

−125

−120

−115

−110

Frequency [MHz]

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dB

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Fig. 2. Power Density Spectrum of the L1 GPS Interfered Signal

B. Interference Detection

Figure 3 shows the Power Spectral Density (PSD) of theWavelet coefficient obtained at the output of the third stageof the Wavelet filters bank, iterated on the Low frequencycomponents.Once the received signal is decomposed, a detection strategy isneeded to identify the presence of the inteference in each scale.A possible detection approach is based on the determinationof a threshold according to a required false-alarm probabilityor missed detection probability. In this way, exploiting the

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−110

Frequency [MHz]

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dB

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LHH signalLHL signalLLH signalLLL signal

Fig. 3. Wavelet Decomposition

compact time-frequency behavior of the Wavelet functions,interference isolation can be achieved.

C. Interference Excision

The last stage of this Wavelet based algorithm is the inter-ference excision. Once the interference frequency componentsare identified and isolated, a synthetic reconstruction of theinterference signal in the time domain is provided, performingan inverse Wavelet Transformation. In this way, it is possibleto subtract this reconstructed interference signal from theoriginal interfered signal, obtaining a new signal without theinterference frequency component, as it can be seen in Figure4 .

0 1 2 3 4 5 6 7 8 9−160

−155

−150

−145

−140

−135

−130

−125

−120

−115

−110

Frequency [MHz]

Po

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sity [

dB

/Hz]

Interfered signalReconstructed signal

Fig. 4. Comparison Between the Received Interfered Signal and the Signalat the output of the Excisor

IV. BENEFITS OF WAVELET MITIGATION TECHNIQUE

In this section the performance of the above describedWavelet detection and mitigation algorithm are shown, in orderto show the benefits obtained in acquisition and tracking stage.The validation of the technique has been performed using theN-GENE software receiver [17].

Desired signal GPS L1 C/A code (PRN 1)Nominal 𝐶/𝑁0 45 dB-HzCoherent integration time 1 msLoop bandwidths DLL: 2 Hz

PLL: 20 HzRF front-end bandwidth 4.092 MHzInterference bandwidth 300 kHzInterference power −130 dBW

−145 dBWInterference carrier 0 Hzoffset 300 kHz

600 kHz

TABLE I

SIMULATED SCENARIO PARAMETERS

A. Acquisition Stage

Through a software implementation of the acquisition al-gorithm within a GNSS receiver, it has been possible toanalyze the advantages brougth by the Wavelet detection and

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mitigation algorithm. The considered acquisition metric isdefined as:

𝛼𝑚𝑒𝑎𝑛 = 10 log10 ⋅(𝑅𝑝

𝑀𝑐

)2

(4)

where 𝛼𝑚𝑒𝑎𝑛 represents the ratio in dB between the highestcorrelation peak 𝑅𝑝 and 𝑀𝑐, which is the mean value of thecorrelation floor on the search space. In Figure 5 the searchspace obtained during the acquisition of the interfered signalis shown. Here the interference power is equal to −130 dBWwith a bandwidth of 300 kHz and its carrier frequency is equalto L1. As it can be seen, in this case the search space isreally noisy, and the receiver is not able to acquire the receivedsignal.

Fig. 5. Interfered Search Space

Figure 6 shows the search space after the interference excisionis applied; it is possible to clearly observe a reduction ofthe noise floor, which allows the acquisition scheme to betterdetect the received signal. This analysis on the search space

Fig. 6. Search Space after Wavelet mitigation

has been also carried out considering different interference sce-narios according to the chosen interference carrier frequencyoffset with respect to the L1 GPS carrier. Table II summarizesthe experimental results. It is possible to notice that, in theworse case, where the interference carrier frequency is equalto L1 GPS carrier frequency, the mitigation algorithm providesa gain up to 7 dB in terms of 𝛼𝑚𝑒𝑎𝑛.

Carrier offset [kHz] Interfered Signal Wavelet Mitigation𝛼𝑚𝑒𝑎𝑛 [dB] 𝛼𝑚𝑒𝑎𝑛[dB]

0 6.3837 13.6992300 7.4141 14.3246600 9.6943 14.5262

TABLE II

RATIO BETWEEN THE HIGHEST PEAK AND THE MEAN VALUE OF THE

CORRELATION FLOOR

B. Tracking Stage

In order to analyze the improvements provided by theWavelet based mitigation technique in terms of DLL jitter andestimated 𝐶/𝑁0, a low power interference has been chosenin order to allow the N-GENE software receiver to track theinterfered signal. Table III summarizes the chosen interferenceparameters.

Parameters ValueInterference Power −135 dBWInterference bandwidth 300 kHzInterference carrier offset 300 kHz

TABLE III

INTERFERENCE SCENARIO

In Table IV the impact of the Wavelet mitigation algorithm isshown; an improvement on the DLL jitter and also on the mean𝐶/𝑁0 can be observed. In fact, the jitter of the DLL is reducedwith respect to that one obtained without applying the Waveletmitigation. In Figure 7 a comparison between the behavior

No Interference Interference Waveletpresent Mitigation

DLL [m] 3.59𝑒− 2 5.77𝑒− 2 3.89𝑒− 2𝐶/𝑁0 [dB-Hz] 45.29 38.95 42.98

TABLE IV

STANDARD DEVIATION OF DLL AND 𝐶/𝑁0

of the 𝐶/𝑁0 is shown. After the excision of the interferencethrough the Wavelet method, the 𝐶/𝑁0 is 4 dB higher thanthe 𝐶/𝑁0 obtained without applying any Wavelet mitigationalgorithm. In the considered case the Wavelet Transformmitigation allows to acquire a signal that would be maskedin the search space, as well as an improvement at the trackingstage.

V. LIMITATIONS OF WAVELET MITIGATION TECHNIQUE

So far, the case in which the interference frequency compo-nents fall entirely within a band of the Wavelet decompositionhas been considered. However, in real condition, the GNSSreceivers have not any information about the interferencesignal which is impacting the quality of the GNSS signal; inparticular, the power, the carrier frequency and the bandwidthof the jammer are unknown. Therefore, it is possible that theinterference frequency components, which are unknown, couldfall in different bands, as in Figure 8.

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5 10 15 20 250

5

10

15

20

25

30

35

40

45

C/N0 for PRN 1

Time [s]

C/N

0 [

dB

−H

z]

No interference presentInterference presentWavelet complete mitigation

Fig. 7. Estimated 𝐶/𝑁0 for PRN 1

0 1 2 3 4 5 6 7 8 9−190

−180

−170

−160

−150

−140

−130

−120

−110

Frequency [MHz]

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sity [

dB

/Hz]

LHHH signalLHHL signalLHLH signalLHLL signalLLHH signalLLHL signalLLLL signalLLLH signal

Fig. 8. Wavelet Decomposition

In this context, it has been investigated how the algorithmshould work. In fact, the mitigation stage could remove com-pletely all the sub-bands affected by the interference, removingalso a larger part of the useful signal, or it could remove onlyone of them. In this second case that one which contains mostof the interference frequency components should be chosen.A comparison between the options previously described, hasbeen performed for the interference scenarios, summarized inTable V.

Parameters ValueInterference Power −130 dBW

−135 dBWInterference bandwidth 250 kHzInterference carrier offset 200 kHz

TABLE V

INTERFERENCE SCENARIO

When the GPS L1 received signal is corrupted by aninterfering signal power equal to −135 dBW, looking at theresults shown in Table VI, the value of 𝛼𝑚𝑒𝑎𝑛 is equal to9.1235. It is clear that, the acquisition in this scenario is a

challenging operation.In Figure 9, the blue curve is related to the PSD of the re-

constructed signal where only one sub-band has been removed,while in Figure 10, the PSD reconstructed by removing boththe Wavelet sub-bands is represented.

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Interfered signalReconstructed signal

Fig. 9. Reconstructed Power Density Spectrum: Interference PartiallyRemoved

0 1 2 3 4 5 6 7 8 9−160

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Interfered signalReconstructed signal

Fig. 10. Reconstructed Power Density Spectrum: Interference All Removed

In the first case, where the interference frequency compo-nents are partially removed, there is not any gain looking atthe acquisition metric 𝛼𝑚𝑒𝑎𝑛. The second case shows thatremoving all the scales which contain frequency informationabout the interference is the best solution. In fact a gain of 3dB can be observed for the 𝛼𝑚𝑒𝑎𝑛 metric.

In Table VII, it is possible to observe that in this case,even if there is a gain in the acquisition stage, there is notany improvements in terms of DLL jitter (which seems moredegraded) and 𝐶/𝑁0.However, the acquisition phase has been significantly im-proved

Figure 11 confirms that in this scenario, the Wavelet methodbrings benefits to the acquisition stage, since the green curve

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InterferencePower[dBW]

InterferedSignal

WaveletPartialMitigation

WaveletCompleteMitigation

𝛼𝑚𝑒𝑎𝑛 [dB] 𝛼𝑚𝑒𝑎𝑛[dB] 𝛼𝑚𝑒𝑎𝑛[dB]-130 6.6144 7.5284 14.2336-135 9.1235 9.4824 12.6932

TABLE VI

RATIO BETWEEN THE HIGHEST PEAK AND THE MEAN VALUE OF THE

CORRELATION FLOOR

No Interference WaveletInterference present Complete

MitigationDLL [m] 3.59𝑒− 2 4.45𝑒− 2 5.29𝑒− 2𝐶/𝑁0 [dB-Hz]

45.29 41.3 41.3

TABLE VII

STANDARD DEVIATION OF THE DLL AND 𝐶/𝑁0

reaches a stable value faster than the red curve which rep-resents the estimated 𝐶/𝑁0 when the interference is notremoved.

4 6 8 10 12 14 160

5

10

15

20

25

30

35

40

45

C/N0 for PRN 1

Time [s]

C/N

0 [

dB

−H

z]

No interference presentInterference presentWavelet complete mitigation

Fig. 11. Estimated 𝐶/𝑁0 for PRN 1

VI. CONCLUSIONS

This paper has shown that the Wavelet based mitigationapproach can represent a powerful method for narrow-bandinterference excision. Significant improvements has been ob-tained both in acquisition and tracking stage. In the ideal case,where the interference frequency components are identifiedand isolated by only one band of the Wavelet decomposition,even for high power narrow-band interference, it is possible toachieve 7 dB of gain in terms of 𝛼𝑚𝑒𝑎𝑛. The estimated 𝐶/𝑁0

shows benefits when this mitigation algorithm is applied, andsome benefits are achieved also by assessing the variance ofthe DLL output. The work tried to assess which is the besttrade off when the interference frequency components fallsin different bands. In the case presented, the interference hasbeen equally splitted in two Wavelet bands, and the results

have shown that in this situation, removing all Wavelet bandsthat contains interference frequency information represents thebest solution for the acquisition, even if a part of the usefulsignal is removed in the frequency domain.Future works will be focused on the use of the Wavelet ap-proach, considering the best trade-off for several interferencescenario, considering the best Wavelet decomposition and therefinement of the statistical threshold for the detection.

ACKNOWLEDGMENTS

The authors would like to thank Marco Rao, MaurizioFantino and Mario Nicola, researchers at the Instituto Su-periore Mario Boella, for their technical support during thisresearch activity. The Ph.D work of Luciano Musumeci ispartially funded by the European Space Agency under the ESAPrestige Programme (ref. TEC-ETN/2011.36/JS).

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