Signal Acquisition and Tracking ofChirp-Style GPS Jammers

Ryan H. Mitch, Mark L. Psiaki, Steven P. Powell and Brady W. O’Hanlon,Cornell University, Ithaca, NY

BIOGRAPHIES

Ryan H. Mitch is a Ph.D. candidate in the SibleySchool of Mechanical and Aerospace Engineering atCornell University. He received his B.S. in MechanicalEngineering from the University of Pittsburgh and hisM.S. in Mechanical Engineering from Cornell Univer-sity. His current research interests are in the areas ofGNSS technologies and integrity, nonlinear estimationand filtering, and signal processing.

Mark L. Psiaki is a Professor in the Sibley School ofMechanical and Aerospace Engineering. He received aB.A. in Physics and M.A. and Ph.D. degrees in Me-chanical and Aerospace Engineering from PrincetonUniversity. His research interests are in the areas ofGNSS technology, applications, and integrity, space-craft attitude and orbit determination, and generalestimation, filtering, and detection.

Steven P. Powell is a Senior Engineer with the GPSand Ionospheric Studies Research Group in the De-partment of Electrical and Computer Engineering atCornell University. He has M.S. and B.S. degrees inElectrical Engineering from Cornell University. He hasbeen involved with the design, fabrication, testing, andlaunch activities of many scientific experiments thathave flown on high altitude balloons, sounding rock-ets, and small satellites. He has designed ground-basedand space-based custom GPS receiving systems pri-marily for scientific applications.

Brady W. O’Hanlon is a Ph.D. candidate in the Schoolof Electrical and Computer Engineering at CornellUniversity. He received both his M.S. and B.S. in Elec-trical and Computer Engineering from Cornell Univer-sity. His interests are in the areas of GNSS technologyand applications, GNSS security, and GNSS as a toolfor space weather research.

ABSTRACT

This paper investigates one method of acquiring andone method of tracking various chirp-style GPS jam-mers. A signal model that uses polynomial descrip-tions of frequency versus time is developed. The de-veloped model can track ideal linear chirps as well asmore complicated, and even some non-repeating, poly-nomial frequency patterns. The developed model canalso be used for jammer classification. Two slightlydifferent measurement models that both make use ofFast Fourier Transforms (FFTs) are also developed.An ad-hoc chirp-style signal acquisition method thatuses FFTs of a moderately strong jamming signal isalso developed. The jamming signal model, observa-tion model, and acquisition procedure are combinedto create a chirp-style signal tracking Kalman filter.The developed Kalman filter is verified by applicationon three sets of laboratory data and on two sets offield data. Tracking of a chirp-style jammer is demon-strated for a distance of approximately 1.8 kilometersbetween the receiving station and the jammer. Thejammer signal models and the Kalman filter are alsoexpanded to the scenario where multiple jamming sig-nals are present, and the modified Kalman filter is eval-uated on one set of laboratory data.

INTRODUCTION

The Global Positioning System’s location and time-synchronization capabilities are used in many areasof civilian life. Some common civilian uses includenavigation through GPS-enabled smart phones ordash-mounted navigation units, and geotagging pho-tographs. Common commercial uses include trackingtrucking and shipping [1], aircraft and maritime navi-gation [2], and high precision timing applications [3, 4].

Copyright © 2013 by Ryan H. Mitch, Mark L. Psiaki,

Brady W. O’Hanlon, and Steven P. Powell. All rights reserved.

Preprint from ION GNSS+ 2013

Government agencies, such as the police or FBI, canuse GPS for tracking suspected criminals [5].

Unfortunately, the interests of some individuals can beserved by interfering with GPS. A simple example isthat of a thief who steals a vehicle that is GPS en-abled and wishes to interfere with GPS so that thevehicle cannot be recovered before dismantling. An-other example is that of an employee in a commercialtrucking corporation who wishes to interfere with theGPS tracking device on his company truck so that hemay run personal errands while being paid to makedeliveries. A less malignant use of GPS interferencewould be that of someone attempting to enforce anenvelope of privacy around their personal vehicle [6].

In the above examples the GPS interference could beprovided by a civil GPS jammer, also known as a Per-sonal Privacy Device (PPD). This has led to severalincidents, of which the so called “Newark Incident” isthe most commonly recognized. In the Newark Inci-dent a truck driver with a GPS jammer in his vehicledrove by Newark airport and periodically interferedwith the airport’s GPS equipment [2]. The truck driveronly wanted to enforce an envelope of privacy aroundhis vehicle and was not intending to interfere with theairport equipment. There was also a less benign in-cident in Great Britain where a group of car thievesused GPS jammers to try to disrupt the geolocationand recovery efforts of the relevant authorities [1].

The above incidents have motivated a number of re-searchers to investigate PPDs [7, 8, 9, 10, 11, 12, 13]in general and their geolocation [14, 15, 16, 17, 18]in specific. This paper furthers the work of [15] andprovides a set of algorithms to acquire and track vari-ous chirp-style GPS jammers. Jammer signal trackinghas applications in jammer geolocation [15], as wouldbe useful for law-enforcement actions. Although, notinvestigated in detail, this work may also have appli-cations in jammer classification and interference miti-gation.

The remainder of the paper is divided into eight sec-tions. The first section presents background informa-tion on the civilian GPS chirp-style jammers. Thesecond section develops a model and state parameter-ization for the PPDs’ chirp-style signals. The thirdsection briefly discusses jammer classification usingthe developed models. The fourth section discussesand selects an observation model for use in jammersignal tracking. The fifth section outlines an ad-hocstrategy for acquiring a chirp-style jammer that hasa moderately strong carrier-to-noise ratio. The sixthsection combines the state parameterization, observa-tion model, and acquisition procedure and applies the

resulting algorithm to data collected from several in-dividual jammers. The seventh section extends thejammer modeling to scenarios where multiple jammersare present and it also considers the new complica-tions that arise when multiple jamming signals mustbe tracked. Additionally, the section applies the multi-jammer tracking algorithm to multi-jammer labora-tory data. The final section summarizes the paper’sdevelopments and draws appropriate conclusions.

GPS JAMMER BACKGROUND INFORMA-TION

Civilian GPS jammers/PPDs can be found in a vari-ety of form factors, but are on average approximatelythe size of a hand-held cellular telephone [8]. Threedifferent civilian GPS jammers are shown in Fig. 1.

Figure 1 Three different form factors of civilian GPSjammers/PPDs.

The algorithms used in the processing of signals fromGPS jammers can benefit from an understanding ofthe RF output of those same jammers. The typicaloutput of a civil GPS jammer is shown in Fig. 2. Thehorizontal axis is time and the top plot’s vertical axis isfrequency. Each vertical slice of the top plot in the fig-ure is a Fast Fourier Transform (FFT) of the RF sam-pled signal, centered at the GPS L1 frequency. Thez, or color axis, is power, with red denoting a largevalue and blue denoting a small value. The bottomplot’s vertical axis is power. The figure shows a classicexample of a chirp signal, or a tone whose frequency re-peatedly ramps linearly upwards and then resets backto the starting frequency.

The plot shown in Fig. 2 is a common example of aGPS jammer’s output spectrum, but other minor vari-ations exist and will be addressed later. Further infor-

2

Figure 2 A common GPS jammer spectrum. The topplot displays vertical slices of 64-point Hamming-win-dowed FFTs, and the bottom plot is of power.

mation can be found in the survey of civilian GPSjammers in Ref. [8].

JAMMER MODELING AND PARAMETER-IZATION

There are multiple parameterizations that could be de-veloped for the chirp-style jammer signal introduced inthe previous section. This paper considers a param-eterization that is similar to that used in Ref. [15],but with several modifications. The parameterizationstarts by assuming a linear chirp, or a first-order rateof change of the frequency versus time, as shown inFig. 3.

Figure 3 Time-history of the signal frequency of a lin-ear first-order chirp, with appropriate states labeled.

The following state vector is a moderately low-order

parameterization of a chirp-style jammer:

x =

θfαu

αd

Atu

td

T

(1)

where the entries of the state vector are as follows: θ isthe phase in units of cycles, f is the frequency in unitsof Hertz, αu is the upward frequency rate of changein units of Hertz per second, αd is the downward fre-quency rate of change in units of Hertz per second, A isthe amplitude in units of Volts, tu is the ramp up timestart for the current ramp and is in units of seconds,td is the ramp down time start for the current rampand is in units of seconds, T is the chirp period and isin units of seconds. The ramp times could have sepa-rate periods, such as Tu and T d, but experimentationwith separate periods did not dramatically change theresults presented later in this paper.

The above parameterization assumes that the jammerchirp has a linear first-order polynomial rate of changeof frequency versus time. For the majority of the chirp-style jammers the above parameterization is a suffi-ciently accurate model for most signal processing algo-rithms. However, there are several jammers that havesignificantly different behavior. The following two fig-ures show the spectra of two jammers that are notideally approximated by linear chirps. Fig. 4 appearsto have a non-first-order polynomial ramp down in fre-quency, whereas Fig. 5 has a non-first-order ramp upin frequency.

Figure 4 A jammer power spectra, with a non–first-order polynomial ramp down in frequency versustime.

The model state can be modified in several ways toaccount for higher-order variations in the frequency

3

Figure 5 A jammer power spectra, with a non–first-order polynomial ramp up in frequency versustime.

ramps. The modification method selected in this paperis to add new states that allow the frequency ramps tofollow higher-order polynomials, as is shown in Fig. 6.The polynomial expansion is in some ways analogousto a Taylor series expansion of a nonlinear term.

Figure 6 Possible time-history of the signal frequencyfor a polynomial-type chirp, with appropriate stateslabeled.

The general form of the new state vector is as follows:

x =

θf cu1...

cuMu

cd1

...cdMd

Atu

td

T

(2)

All of the cuj and cdj states are coefficients of the Taylor-series-type polynomial expansion of the frequency be-havior, including the first order terms cu1 and cd1 thathave replaced αu and αd. The states of the form cujare the coefficients of a ramp-up frequency polynomial,and the states of the form cdj are the coefficients ofa ramp-down frequency polynomial. The frequencypolynomial is defined as follows:

f (x, t) =

f +

Mu∑j=1

cuj (t− tu)j, tu ≤ td

f +

Md∑j=1

cdj(t− td

)j, td < tu

(3)

and it can be integrated to determine the correspond-ing phase:

θ (x, t) =θ + f (t− tu) +

Mu∑j=1

cuj(t−tu)(j+1)

(j+1) , tu ≤ td

θ + f(t− td

)+

Md∑j=1

cdj(t−td)

(j+1)

(j+1) , tu > td

(4)

where the top case in Eqs. 3 and 4 is for times whenthe frequency is ramping upwards and the bottom caseis for times when the frequency is ramping downwards.

The order of the polynomial, and the associated statescu1–cuMu

and cd1–cdMd, can be considered a tuning pa-

rameter. More states will allow for a model that canmore closely replicate the behavior of the true jam-mer. The practical upper bound on the number ofcoefficients used in the model is primarily set by thesampling rate and numerical conditioning limitations.If the sampling rate is very low, then the number ofaccumulations that can be computed for each chirp is

4

reduced and it will be difficult to estimate the higherorder terms accurately. Numerical conditioning issuesmay arise due to the dramatically different state mag-nitudes in the above parameterization, and this mayrequire that the states be modified to improve the nu-merical stability of the algorithm. It should also benoted that using more polynomial coefficients thannecessary is a waste of computational effort. There-fore, the order of the polynomial should be limited toa reasonable number.

A 5th order polynomial is considered for both the fre-quency ramp-up and ramp-down in the remainder ofthis paper’s modeling work. Additionally, the poly-nomial coefficients have been modified from the def-inition in Eqs. 2–4. The new polynomial coefficientshave been multiplied by the sample time Ts, raised tothe corresponding power, as shown below for the rampup:

cu1,new = cu1 (Ts)1

...

cuMu,new = cuMu(Ts)

Mu (5)

and for the ramp down:

cd1,new = cd1(Ts)1

...

cdMd,new= cdMd

(Ts)Md (6)

The new states will have magnitudes that are closerto unity, and therefore will be less likely to cause nu-merical instabilities in the developed algorithms. Forexample, a reasonable numerical value for cu1 might be1012, and a system with a sampling rate of 10 MHzwould cause cu1,new to have a numerical value closer to105. It would also be possible to use a fixed numberthat is unrelated to sample rate.

The state dynamics are defined in a chirp-by-chirpmanner, with the frequency and phase defined at theramp times tu and td. The phase and frequency canbe evaluated at any time, but are only propagated for-ward and updated when the time crosses the ramp-down time, td, or ramp-up time, tu, by evaluatingEqs. 3 and 4.

The resulting state dynamics equation is as follows:

xk+1 = Φk (tk+1, tk;xk)xk + Γk (tk+1, tk;xk) vk (7)

where xk+1 is the state at time tk+1, xk is the state attime tk, Φk is the state transition matrix from time tkto tk+1, Γk is the process noise influence matrix, andvk is the zero-mean, unity-covariance, white, Gaussian

process noise vector. The process noise is expected toenter only at the beginning of each ramp interval, tu ortd. It should be noted that the state transition matrixand the process noise influence matrix are both statedependent, and are defined as follows:

Φ (tk+1, tk;xk) =

Φ (tk+1, tk)up , if A

Φ (tk+1, tk)down , if B

I, otherwise

(8)

Γ (tk+1, tk;xk) =

Γ (tk+1, tk)up , if A

Γ (tk+1, tk)down , if B

0, otherwise

(9)

where the conditions A and B are defined as follows:

Condition A: t = td (10)

Condition B: t = tu (11)

The state transition matrices for this system are theidentity matrices, except when Eqs. 10 and 11 are sat-isfied, i.e. when the system time equals td or tu. Whenthe system time equals td the phase and frequencyare propagated as defined by the up-ramp polynomialfrom time tu up to time td. Similarly, when the sys-tem time equals tu the phase and frequency are prop-agated as defined by the down-ramp polynomial fromtime td up to time tu. The time difference terms inEqs. 3 and 4 can be rewritten as rows of Φ (tk+1, tk)up

or Φ (tk+1, tk)down because the states, the coefficientsof the polynomials, enter linearly into those equations.The amplitude, A, is defined as a constant and thetime states are defined by the following equations:

tu =

{tu + T, if t = td

tu, otherwise(12)

td =

{td + T, if t = tu

td, otherwise(13)

where the time states tu and td are incremented bytheir period T when the system time equals td or tu,respectively. The rows corresponding to tu or td inΦ (tk+1, tk)up or Φ (tk+1, tk)down will have 1 entries forthat state, and if the conditions of Eqs. 10 or 11 aresatisfied, it will also have a 1 entry for the period in-crement for the corresponding state.

Propagation of the state from a time before tu or td

to a time after tu or td will require multiple matrices.For example, if the state were propagated from time tkto tu and then from tu to tk+1 the resulting equationswould be:

Φ (tk+1, tk, xk) = Φ(tk+1, t

u+

)Φ(tu+, t

u−)

Φ(tu−, tk

)= I Φ (tk+1, tk)down I

= Φ (tk+1, tk)down (14)

5

where the time tu− designates a time infinitesimally be-fore tu and time tu+ designates a time infinitesimally af-ter tu. If multiple ramp changes were included in thepropagation interval then matrix multiplication seriesof Eq. 14 would contain more terms.

The default process noise influence matrices for thissystem are zero matrices, except when the conditionsin Eqs. 10 or 11 are met, i.e. when the system timeequals td or tu. All of the states, except for θ, areconsidered to have some process noise enter at the be-ginning of each ramp. The introduction of processnoise creates non-zero entries in the Γ matrices, whichare assumed to be diagonal. If the non-zero diagonalentries are each set to 1, then the process noise co-variance matrix Q becomes the square of the standarddeviation of the process noise expected for the entireramp. Q is formally defined as:

Q = E[(vk − E[vk]) (vk − E[vk])∗] = E[vkv

∗k] (15)

where E [] is the expected value of the quantity in thebrackets. A reasonable assumption for the Q matrixis that all of the noise inputs are uncoupled, i.e. Q isa diagonal matrix.

The amount of process noise that is assumed to bepresent in each signal state is a tuning parameter thatcan be modified to improve the performance of the fil-ter. The signal tracking results of the single GPS jam-mers were not extremely sensitive to the assumed levelof process noise. However, the multi-jammer scenarioswere decidedly more sensitive to the process noise tun-ing. It should be noted that each jammer can have adifferent level of process noise, and a thorough surveycould be conducted to determine a statistically sig-nificant estimate of the process noises that could beexpected from any GPS jammer seen in the field. Asurvey of that many jammers is beyond the scope ofthis current work.

The resulting output of the above jammer model is:

y′i = A′i ∗ cos(2π ∗ φ(x, ti)) (16)

where φ(x, ti) is the evaluation of the relevant phasepolynomial, and A′i is the broadcast signal amplitude,at time ti. The signal as seen at a receiver station withmixing frequency fmix, and with the high-frequencymixing term removed, is as follows:

yi =Ai2∗ cos (2π ∗ (φ (x, ti)− fmixti))

=Ai2∗ cos (Θ (x, ti, fmix)) (17)

where the amplitude in Eq. 17 is the actual trackedamplitude state. Eq. 17 is the final form of the signalmodel that is used in the remainder of this paper.

JAMMER CLASSIFICATION

In Ref. [8] the authors found that jammers with thesame physical appearances produced significantly dif-ferent frequency versus time behavior. The unique fre-quency versus time behavior can potentially be usedto identify individual jammers. Jammer identificationcan be useful for determining how many different jam-mers are seen at one station. It can also be useful ifa person is being prosecuted for operating a PPD andthe authorities wish to know where else that personhas broken the law by jamming GPS. Jammer identi-fication only requires that at least one station receivethe jamming signal, as opposed to most automatedgeolocation methods which require several stations.

One possible identification method would be to storea short amount of time of high-sample-rate data frommany jammers in an RF signature library and thencompute cross correlations between the received dataand the library to determine if the signals match. Anew method using the modeling in this paper wouldalso start with a short recording of data from each jam-mer in the library. The jammer state would then beestimated for many chirps. The resulting time-seriesof state estimates would then comprise many samplesfrom a probability density function (pdf) of the jam-mer’s state perturbed by process noise. A similar pro-cedure would then be applied to the new received sig-nal and the resulting two sets of pdfs would be com-pared to each other. Relevant similarity metrics andhypothesis tests could be developed to determine ifthe new signal is the same as one of the signals in thelibrary. This method of jammer identification mightrequire a smaller amount of storage space when com-pared to one which used a sampled RF data library.This is a potentially deep research area and will requirefurther investigation to explore.

JAMMER OBSERVABLES

The standard pair of observables used to track anRF signal are the In-Phase and Quadrature accumu-lations. The accumulation formulas at time tk are:

Ik =

N∑i=1

yicos(ΘNCO (ti)

)(18)

Qk =

N∑i=1

yisin(ΘNCO (ti)

)(19)

where N is the number of samples used in the accu-mulation, yi is the sampled RF data, and ΘNCO (ti) is

6

the numerically controlled oscillator (NCO) phase attime ti.

The NCO phase is arbitrarily defined by the user. Thejammer phase time history can be determined from theprevious modeling section and is as follows for a singlepolynomial ramp:

Θ (x, ti, fmix) =

2π

[θNk + fNk ∆tA +

NA∑j=1

cAj∆tj+1

A

(j + 1) (Ts)j

− fmixti](20)

where the subscript “A” denotes ramp A, ∆tA is de-fined as ti− tA, and tA is either tu or td, depending onif the ramp is up or down, respectively. If the accu-mulation time interval spans part of two ramps, rampA and then ramp B, the phase is as follows:

Θ (x, ti, fmix) =

2π

[θNk + fNk ∆tA +

(NA∑j=1

cAj∆tj+1

BA

(Ts)j

[∆tBA(j + 1)

+ ∆tB

])

+

(NB∑j=1

cBj∆tj+1

B

(j + 1) (Ts)j

)− fmixti

](21)

where ∆tBA is defined as tB − tA, and tB is either tu

or td and tA is the other, i.e. if the first ramp is an upramp then ramp A starts at time tu and tB = td. Ifmore than two ramps are spanned by an accumulationinterval then the received phase equation simply gainsadditional terms.

If the signal contains no noise, then the NCO phasethat produces the largest accumulation is the samephase as the incoming signal that one wishes to track.In a traditional signal tracking architecture the ac-cumulations would then constitute the measurementsthat would be sent to the Kalman filter. The corre-sponding measurement models would then be definedby substituting the signal model and the signal modelestimates into the yi term in the accumulation modelsof Eqs. 18 and 19. The measurement models can besimplified further by converting the discrete sums tocontinuous integrals and attempting to solve the re-sulting integrals in closed form. The integral of thehigher order phase terms is at simplest, in the case ofa perfectly linear ramp, a Fresnel integral. The higherorder polynomials will further complicate the integra-tion. The resulting measurement model is complicatedand requires significant computational overhead. Ad-ditionally, a single pair of long time-span accumula-tions is likely to prove insufficient for filter convergenceand signal tracking. This is because the accumulationswill result in appreciable power only if the NCO phase

time-histories are close to that of the real signal beingtracked.

The slow evaluation speed and a strong preference formultiple accumulations suggest that a different mea-surement model that does not have the same draw-backs should be considered. It should be noted that,unlike GPS, the GPS jammers broadcast very power-ful signals, and their signal can be easily seen abovethe noise floor in a given area around each jammer. Afaster measurement that computes accumulations andspans the frequency range of the physical sampling sys-tem is the Fast Fourier Transform (FFT). The FFTcomputes accumulations at fixed frequencies spacedevenly between the positive and negative Nyquist fre-quency bounds. If the accumulation lengths are shortenough that they do not lose power due to the chang-ing frequency of the incoming signal then they willprovide a measurement of the jammer power in eachfrequency bin across the entire Nyquist range.

The FFT measurements are defined as follows:

z(L) = I + iQ

=

N∑n=1

w(n)y(n)e−i2π(L−1)(n−1)

N

=

N∑n=1

w(n)y(n)e−iζ(L,n,N) (22)

∀L ∈ [1, ..., N ]

where w is the time domain windowing function usedto reduce sidelobes in the FFT, y is the sampled signal,and L is the frequency bin number of the FFT. Thepresent work used the Hamming window, but otherwindowing functions would likely produce similar re-sults.

It is important to note that it would be unreasonableto expect that the phase can be precisely tracked inthis system. Precise tracking of the phase state θ re-quires at least the following two conditions be met.The first condition is that the frequency polynomialestimate be very accurate, such that there are onlyvery minor fluctuations in the frequency that cannotbe modeled by the polynomial. The second conditionis that the jamming signal always stays within the re-ceiver’s Nyquist frequency bounds.

Satisfying the above two conditions will be difficult ineven the most benign situations, where the jammingenvironment can be controlled and RF data can beacquired with a high-speed recording system. The re-sults of two such scenario are shown in Figs. 7 and 8.In Fig. 7 the order of the frequency polynomial re-quired to describe the signal changes from chirp to

7

chirp, and sometimes the frequency of the signal per-forms near-instantaneous jumps that may not be ac-curately modeled by a continuous polynomial. Evena well-designed signal tracking algorithm will there-fore be unable to accurately track this jammer’s phase,unless the instantaneous jump behavior is accuratelymodeled and reliably estimated. The frequency poly-nomial model developed in this paper does not permitinstantaneous frequency jumps, but it will allow an ap-proximation to that behavior. In Fig. 8 the frequencyof the jamming signal passes outside of the receiversystem’s Nyquist range. When the frequency passesoutside of the Nyquist range the phase can no longerbe tracked. Although the frequency and phase behav-ior are governed by the polynomial parameterizationsdeveloped in this paper, they are not guarantees of thebehavior outside of the visible frequency range. TheNyquist frequency range is dependent on the equip-ment being used, but even in the above high-rate labo-ratory system the signal is not always visible, and fieldequipment typically uses a lower digitization rate.

Figure 7 A jammer power spectra, with jumps in fre-quency. Continuous approximations to the above fre-quency behavior will likely be inaccurate and the phaseestimate will likely be untrustworthy.

Because the phase estimated state θ is unlikely to beaccurate, the phase state will be removed for the singlejammer signal tracking case. The resulting measure-ments are further simplified by considering only theabsolute value of the accumulations, as follows:

z(L) = |I + (iQ) |

= |N∑n=1

w(n)y(n)e−iζ(L,n,N)| (23)

∀L ∈ [1, ..., N ]

Figure 8 A jammer power spectra, the frequencyspan is much greater than that covered by the high--speed data sampling equipment (62.5 MHz complexsamples). Phase estimates derived from data thatspans multiple ramps will likely be erroneous.

The measurement model is defined similarly:

h(L) = |I(x) + (iQ(x)) |

= |N∑n=1

w(n)y(n, x, fmix)e−iζ(L,n,N)| (24)

∀L ∈ [1, ..., N ]

where y is the predicted RF sample obtained by prop-agating the state to the current measurement intervaland evaluating Eq. 17 for the time at sample n. The Iand Q dependencies on the non-state terms have beenomitted for the sake of notational convenience. TheJacobian, or partial derivatives of the measurementmodel with respect to the model state, is required forKalman filter estimation and is defined as:

H(L) =∂

∂x|I(x) + (iQ(x)) |

=∂

∂x|N∑n=1

w(n)y(n, x, fmix)e−iζ(L,n,N)| (25)

∀L ∈ [1, ..., N ]

The absolute value is defined as:

|a+ (ib)| =√

(a2) + (b2) (26)

and the partial derivative as:

∂

∂x|a+ (ib)| =

a(∂a∂x

)+ b

(∂b∂x

)|a+ (ib)|

(27)

8

In the case of the measurement model from Eq. 24:

a = I (x)

≈ A

2

N∑n=1

w(n)cos (Θ(x, tn, fmix)− ζ (L, n,N))

=A

2

N∑n=1

w(n)cos (µ (L, n,N, fmix, x))

=A

2

N∑n=1

w(n)cos (µ (P , x)) (28)

b = Q (x)

≈ A

2

N∑n=1

w(n)sin (Θ(x, tn, fmix)− ζ (L, n,N))

=A

2

N∑n=1

w(n)sin (µ (L, n,N, fmix, x))

=A

2

N∑n=1

w(n)sin (µ (P , x)) (29)

where P is a vector of parameters that has been intro-duced for notational convenience and includes L, n,N , andfmix. The reason that Eq. 28 and 29 are onlyapproximations is because the high frequency term,from the conversion of the products of the trigono-metric terms to the sum of the terms with combinedarguments, is assumed to sum to zero. The resultingpartial derivatives of a and b are as follows:

∂a

∂x=

aA , for x = A

−A2

N∑n=1

w(n)∂Θ∂x sin (µ (P , x)) , otherwise

(30)

∂b

∂x=

bA , for x = A

A2

N∑n=1

w(n)∂Θ∂x cos (µ (P , x)) , otherwise

(31)

Eqs. 26–31 can be combined with the measurementmodel equations, Eq. 24 and 25, to complete the ab-solute value FFT measurement model.

Sometimes it might be preferable to use the power ofan accumulation pair instead of just the absolute value,where the absolute value is effectively the square rootof the power. In that case the measurements are de-fined using the same I and Q definitions from the FFTmodel:

z(L) = [I + (iQ)]∗

[I + (iQ)]

= I2 +Q2 (32)

∀L ∈ [1, ..., N ]

The measurement model is defined as follows:

h(L) = I(x)2 +Q(x)2 (33)

∀L ∈ [1, ..., N ]

with partial derivatives:

H(L) =∂

∂x

[I(x)2 +Q(x)2

]= 2

∂I(x)

∂xI(x) + 2

∂Q(x)

∂xQ(x) (34)

∀L ∈ [1, ..., N ]

The partial derivatives of the accumulations with re-spect to the states are given by Eqs. 30 and 31.

Every RF sample includes some amount of samplenoise, which eventually becomes accumulation noise.The noise statistics of the absolute value of the FFTsis complicated and only an approximation has beenused in this paper. The results are similar to the noisestatistics for the power measurements, which are asfollows:

E[z(L)] ≈(I2true +Q2

true

)+Nσ2 (35)

cov(z(L)) = E[(z(L)− E[z(L)]) (z(L)− E[z(L)])

∗]≈ 2Nσ2

[I2true +Q2

true

]+N2σ4 (36)

where Itrue and Qtrue are the accumulations if thesignal contained zero noise.

One component of the measurement model that stillneeds to be addressed is the RF filter characteristics ofthe physical jammers and recording devices. The RFfilters will attenuate different frequencies at differentrates. The data recording equipment RF filter shapecan be determined off-line. Off-line system identifi-cation is likely not possible for the jammers, becausethe equipment is not typically available before the firstencounter in the field. It might still be possible to per-form off-line system identification on several jammersand attempt to identify repeatable RF filter shapesamong the surveyed jammers. Therefore, if the classof the jammer is known a priori then the RF filtershape can be assumed—it may provide sufficient ac-curacy for tracking purposes.

Alternatively, the RF filter shapes can be identified on-line, after signal detection and before the initializationof the tracking algorithms. There are many ways todo this, but one simple method of system identificationwould start by computing FFTs of the incoming sig-nal at a rate that is faster than the chirp period. Thenthe maximum magnitude, or the average magnitude, of

9

each FFT bin would be computed over many chirp pe-riods. The resulting RF filter’s magnitude versus fre-quency bin output would contain the effects of both thejamming and recording equipment. That output couldthen be normalized. The result of the above procedureis an attenuation factor that can be included into eachFFT computation in the measurement model.

It should be noted that the maximum time length ofeach accumulation is a function of the frequency rateof change of the jammer being tracked. A jammer thathas a slower ramp rate will be able to use more samplesbefore the frequency mismatch between the jammer’sfrequency and the FFT bin frequency causes the I-Qvector to complete a full rotation. A jammer with afast rate of change will only be able to use a fewernumber of samples.

If the user desires to track a very weak signal, ora signal that has a quick time-rate-of-change of fre-quency, then a different measurement model might berequired. A candidate model that has similar char-acteristics of an FFT accumulation model is one thatstarts with FFTs, but then adds a linear frequencyrate of change term. The ramping frequency term willallow for longer accumulations, and therefore a highersignal-to-noise ratio. However, there are several down-sides to this new proposed model. One down-side isthat many ramp-rates would be needed until the ac-tual ramp rate of the jammer being tracked is knownwith enough certainty that only one ramp rate couldbe used. A second down-side is that the measurementnoise covariance matrix, R, may become more compli-cated. This type of accumulation may also face lengthlimitations due to the NCO frequency passing outsideof the Nyquist bounds of the system. Finally, the ob-servability properties of this new model have not beeninvestigated.

AD-HOC ACQUISITION OF A MODER-ATELY POWERFUL CHIRP-STYLE JAM-MER SIGNAL

There are multiple ways to initialize a state for use inthe developed chirp-style GPS jammer signal trackingfilter. The method presented in this paper assumesa moderately strong jammer signal-to-noise ratio, i.e.the signal can be at least partially seen in FFTs com-puted at the receiver. The primary obstacle to a faststate initialization procedure in this system is the highdimensionality of the proposed state. If the 2-ramppolynomial parameterization model developed in thispaper is used (without phase) then the state’s dimen-sion is fifteen. This is a state space that is likely too

large to search in any brute-force manner in a smallamount of time. Instead, the initialization procedurewill attempt to extract some state values from auto-correlations and FFTs of the sampled data and thenmake some simplifying assumptions about the jammerto reduce the search dimensions to a more tractablesize. The developed algorithm is intended to allow forthe initialization of the state of a single jammer only.

The spectra of a GPS jammer from data recorded in afield campaign at White Sands Missile Range in 2012is shown in Fig. 9. Several quantities are clearly dis-

Figure 9 A jammer power spectra spanning 9 MHzon the y axis and approximately 20 µs on the x axis.

tinguishable from Fig. 9. These quantities include theperiod of the jammer, T , a time t0 at which the signalpasses through a given frequency f0, and the linearcoefficient of the ramp-up polynomial, cu1 . The fourquantities, T , f0, t0, and cu1 are labeled in Fig. 10,and code can easily be written to automate the com-putation of those quantities. The period can also becomputed more precisely by using autocorrelations ofthe raw data.

Fig. 10 seems to imply that the other states cannotreliably be initialized by directly inspecting the FFTsof two chirps, and an expanded initialization procedureshould be developed.

The additional initialization procedure presented inthis paper will collapse the remaining (large) searchspace by assuming a 1st order linear polynomial chirp.This assumption is not strictly valid, but appears tobe a reasonable low-order approximation of the manyjammers in Ref. [8]. The benefit of the 1st order lin-ear polynomial chirp assumption is that it reduces the

10

Figure 10 A jammer power spectra spanning 9 MHzon the y axis and approximately 20 µs on the x axis.Easily extractable quantities have been labeled.

number of polynomial coefficients that must be initial-ized to two. This lower-dimensional initialization canbe considered a “rough” initialization.

It might appear that there are four states that mustbe determined to complete the rough initialization: f ,cd1, tu and td (the amplitude state can be scaled at theend of the initialization procedure). That is not thecase. Coupling between states, the frequency f0, andthe time t0, can be used to reduce the independentinitialization dimension to two. The remaining twodimensions must be explored to determine the bestrough state initialization. The first dimension is thespan of the chirp. The span can be defined in termsof the frequency or time, because the two quantitiesare coupled through the cu1 term. The frequency isselected in this work. The frequency span fspan isshown graphically for two distinct chirps in the Fig. 11.Where fspan,1 is the frequency span of the first chirpand fspan,2 is the frequency span of the second chirp.

The second dimension is the starting point of the chirp.The starting point can be defined in terms of eitherthe initial frequency f or time of ramp up tu, becausethe two quantities are coupled in a manner similar tothe coupling in fspan. The frequency f is selected inthis work, so that parallelism is preserved between thetwo search dimensions. The center frequency f0 (orfL1

), t0, and cu1 are provided earlier by direct com-putation on the FFT spectra, and they constrain thechirp to lie along a line in the frequency-time plot withslope cu1 , and they prescribe an intersection point at(t0,f0). The quantities fspan, T , cu1 , and the assump-

Figure 11 Frequency-time histories for two differentvalues of the chirp frequency span, but otherwise usingsimilar states. The chirps have been offset in time inthe interest of clarity.

tion of a 1st order linear polynomial chirp prescribethe ramp-down rate cd1. The starting frequency searchis shown graphically in Fig. 12 for three different hy-pothesis starting chirp times and frequencies, f1, f2,and f3, but identical states otherwise.

Figure 12 Frequency-time histories for three differentvalues of the chirp frequency starting value, but similarstates otherwise.

The many different state hypothesis xhthat will resultfrom searching the above two dimensions must be eval-uated to determine how closely they fit the real data.There are many ways to define an appropriate ad-hoc“goodness of fit” metric. This paper’s metric will bea cost function J

(xh, y

)that seeks to minimize the 2-

norm of the frequency difference between the real dataand the hypothesis data. The actual metric considers

11

the FFT frequency bin spacing distance between peaksof FFTs computed on real data and those computedon the hypothesis data. The formal definition of thescalar cost function is as follows:

J(xh, y

)= ||d

(xh, y

)||2 (37)

where d is a vector of FFT bin distances between themaximum powered accumulation in the FFTs of thereal data and the hypothesis data.

Fig. 13 shows a possible entry of d, where the vector dis composed of many such d values, and correspondingFFTs on sequential data. In Fig. 13 d is the single dis-tance between the peaks of the real data, at bin num-ber 95, and the hypothesis data, at bin number 175,and is approximately 80 bins. Many computations ofthe form shown in Fig. 13 are combined to cover thefull chirp period of the jammer as shown in Fig. 14. Infact, the use of several chirps seems to improve the 1st

order linear frequency polynomial fit results by aver-aging to reduce the effect of measurement noise.

Figure 13 An example plot of FFT power versus FFTbin number for real data (blue) and for hypothesis data(green).

The final step in the initialization procedure is the“fine” initialization. The fine initialization procedureexpands the rough initialization state correspondingto the best 1st order linear polynomial chirp approx-imation to a full polynomial. The above rough ini-tialization procedure should produce a state estimatethat is within the pull-in range of a standard non-linear Maximum Likelihood Estimator (MLE) that al-lows full polynomial variation on the frequency ramps.The Square Root Information (SRI) formulation of theMLE cost function is shown below:

J(x, y)

=[R−Ta

(z(y)− h (x)

)]T [R−Ta

(z(y)− h (x)

)](38)

where z is the vector of FFT measurements computedfrom the data y, h is the FFT measurement model of

Figure 14 Graphical representation of a 1st order lin-ear polynomial frequency approximation to a higherorder chirp. FFTs of the real data are in blue, hypoth-esis data in green, the frequency bin distances betweenthe two data (before rounding to the nearest frequencybin) are in black, and the the FFT frequencies used tocalculated the entries of d are in red.

Eq. 24, and Ra is the Cholesky factorization, or ma-trix square root, of the measurement noise covariancematrix R. Where R is defined as:

R = RTa ∗Ra = E[(ν − E [ν]) (ν − E [ν])

T]

(39)

where ν is the measurement noise vector. The mea-surement noise statistics are similar to those in theprevious section on jammer observables.

SINGLE-JAMMER KALMAN FILTER SIG-NAL TRACKING

The state, dynamics model, measurement model, andinitialization procedure can be combined to produce aKalman filter to track the RF signal of a chirp-styleGPS jammer. The data used in this section comesfrom two different sources. The first source is dataoriginally collected for use in Ref. [8], where manyGPS jammer’s signals were recorded using complexsampling at a rate of 62.5 MHz. The second source isthe data originally collected for use in Ref. [15], wheremany GPS jammer’s signals were recorded at approx-imately 8 or 9 MHz (depending on the file) during atesting event at WSMR sponsored by the DHS.

The first set of Kalman filter tracking results considersthree different jammers in a laboratory environment.The jammers progress from a nearly perfectly linear1st order polynomial jammer behavior to the jammer

12

that was least like a 1st order jammer out of all thejammer signatures collected for Ref. [8]. Each labora-tory jammer is initialized not using the previously de-veloped acquisition procedure. The initialization wasperformed by manually inspecting an FFT surface plotof the jammer data and then guessing a linear 1st or-der polynomial state. This particular initialization wasselected so that the initial state provided to the filterhad some significant error, as was likely not to be thecase with the previously developed acquisition proce-dure. The error was introduced to demonstrate someof the filter convergence properties. A full analysis ofthe filter convergence range is beyond the scope of thispaper. This paper’s ad-hoc acquisition procedure hasbeen used on the WSMR data shown later.

The results of the most benign jammer, or that whichhas a frequency behavior that most closely matches thelinear 1st order frequency polynomial model, is shownin Fig. 15. The Kalman filter’s frequency estimate ateach sample time is plotted on top of a surface plotcomprised of FFTs of the real data. Despite the ini-tial error, the filter quickly determines the correct stateof the jammer, and, although it is not shown, the fil-ter state estimates completely settle within 10 chirpperiods.

Figure 15 Spectra of the first GPS jammer in a labo-ratory environment, with the Kalman filter frequencystate estimates overlaid in green.

The results of the second jammer are shown in Fig. 16.The second jammer does not have as close of a matchto the linear 1st order frequency polynomial behavioras the previous jammer, as can be seen in the curve ofthe frequency up-ramp. The initial error can be seenin the jump of the frequency estimate at the bottomof the first chirp. The Kalman filter determines thestate of the jammer in less than two chirp periods.

The results of the least benign jammer, or that whichhas a frequency behavior that least closely matches thelinear 1st order frequency polynomial model, is shown

Figure 16 Spectra of the second GPS jammer in alaboratory environment, with the Kalman filter fre-quency state estimates overlaid in green.

in Fig. 17. Despite the fact that the jammer has adifferent ramp behavior for each chirp the Kalman fil-ter is able to modify its state estimate every ramp toclosely follow the jammer’s actual frequency.

Figure 17 Spectra of the third GPS jammer in a lab-oratory environment, with the Kalman filter frequencystate estimates overlaid in green.

The above results will naturally degrade with in-creased noise. The motivation for testing the filterwith more noise is to understand how the jammers canbe tracked in a real-world scenario. Instead of testingwith more noise the filter will be analyzed using actualdata from a real-world scenario at WSMR.

The second set of Kalman filter tracking results con-siders one jammer at two different distances from thereceiver station. The first set of results is shown inFig. 18, when the jammer is approximately 50 metersfrom the recording station. The state is initializedusing the automated acquisition procedure mentionedpreviously and the filter is able to track the signal veryaccurately from the beginning of the data set.

In Fig. 19, the Kalman filter is applied to the samejammer as in Fig. 18, but now the jammer is approx-

13

Figure 18 Spectra of a GPS jammer at WSMR whenit is very close, with the Kalman filter frequency stateestimates overlaid in green.

imately 1.8 kilometers away from the receiver station.Despite the fact that the jamming signal is only barelyvisible above the noise the acquisition procedure is ableto initialize the state and the filter is able to track thesignal. It should be noted that the automated acquisi-tion procedure required several initialization attempts,discarding several µs of data after each failed initial-ization, to produce a good state estimate for signaltracking. Methods of determining when the signal hasfailed initialization are beyond the scope of this paper,but a common test is to consider the measurementresiduals after the initialization procedure concludes.

Figure 19 Spectra of a GPS jammer at WSMR whenit is far away, with the Kalman filter frequency stateestimates overlaid in green.

MULTI-JAMMER KALMAN FILTER SIG-NAL TRACKING

The jammer models must first be extend to handlemultiple signals before the signal tracking problem canbe addressed. The dynamics of each jammer are inde-pendent of other jammers and result in the following

uncoupled dynamics equation:

xk+1

= Φ xk

+ Γ vk

(40)

where the entries of Eq. 40 are defined as follows forthe case of two jammers:

x =

[x1

x2

]v =

[v1

v2

]Φ =

[Φ1 00 Φ2

]Γ =

[Γ1 00 Γ2

]where x1 is the state of the first jammer, x2 is thestate of the second jammer, and the other terms are de-fined in a similar manner in conjunction with Eq. 7–13.Eq. 40 can be extended to handle an arbitrary numberof jammers by appending the new jammer’s state tothe current state vector, appending the new processnoise vector to the current process noise vector, andthen modifying the matrices Φ and Γ appropriately.

The measurement model is slightly more complicated.The signal received at the recording station is a linearcombination of the two, or more, jammer signals. Thecombined signal model is as follows:

y(ti, x, fmix

)= y1 (ti, x1, fmix) + y2 (ti, x2, fmix)

=A1,i

2∗ cos (Θ (x1, ti, fmix))

+A2,i

2∗ cos (Θ (x2, ti, fmix)) (41)

where y1 and y2 are defined by Eq. 17. Eq. 41 can beextended to an arbitrary number of jammers by addingthe new signals in the same manner that y2 was addedto y1.

The resulting FFT measurement model has the samefundamental form as before:

h(L) = |N∑n=1

w(n)y(n, x, fmix)e−iζ(L,n,N)| (42)

∀L ∈ [1, ..., N ]

FFT operators have the convenient property that theyare linear and can therefore be distributed among themultiple incoming signals. The resulting measurement

14

model equation has the following form:

h(L) = |N∑n=1

w(n)y1(n, x1, fmix)e−iζ(L,n,N) + ...

N∑n=1

w(n)y2(n, x2, fmix)e−iζ(L,n,N)| (43)

∀L ∈ [1, ..., N ]

The partial derivatives behave similarly and have beenomitted for the sake of brevity.

Multiple incoming jamming signals experience an ad-ditional effect that is not normally seen in single jam-ming signals: interference. The two signals are fre-quency modulated sinewaves that will likely have verysimilar frequencies at some point during signal track-ing. When two sine waves with similar frequencies arecombined they begin to constructively and destruc-tively interfere with each other. The factor that de-termines whether the interference is constructive ordestructive is the relative phase of the two signals.Therefore, the signal phase that was neglected earlierin this paper’s filter development becomes vitally im-portant. The measurement model could be modifiedagain to allow the phase of both signals to be states,but it is difficult to track both signal’s phases accu-rately enough to predict the type and level of interfer-ence. Additionally, it is the relative phase between thejammers that determines the type and extent of theinterference. Therefore, at each time step the relativephase θrel of the jammers is added as a parameter andthe measurements are optimized over it. The optimiza-tion cost function is the same as the earlier maximumlikelihood cost function, but for a single measurementset:

J(θrel, x, y

)=[

R−Ta(z(y)− h (x, θrel)

)]T [R−Ta

(z(y)− h (x, θrel)

)](44)

Determining the optimal relative phase θrel fromEq. 44 is easily accomplished by evaluating multiplephases on the unit circle and selecting the one withthe lowest cost. It was found that a grid of 40 pointsspaced evenly on the unit circle determined a relativephase value to an appropriate resolution for trackingtwo jamming signals.

The above algorithm was tested on laboratory data.A multi-jammer data set was developed by adding to-gether the samples from two different jammer datasets. The states of each jammer were initialized sepa-rately on the individual data sets before combinationbecause an efficient multi-jammer initialization algo-rithm is beyond the scope of this current work. The

spectra of the combined jammer samples is shown inFig. 20. The Kalman filter frequency estimates of thetwo jammers are overlaid in green and red in Fig. 21.

Figure 20 Spectra of two GPS jammers in a labora-tory environment.

Figure 21 Spectra of two GPS jammers in a labo-ratory environment, with the Kalman filter frequencyestimates overlaid in red and green.

The multi-jammer signal tracking work has beenshown to work on real jamming data in a laboratorysetting. A real-world test has not yet been completed,and it is not known how many jammers can reasonablybe tracked using the developed algorithms. It shouldbe noted that the optimization in Eq. 44 increases indimensionality by one every time another jammer isadded to the incoming signal, effectively slowing downthe algorithm more than might normally be expectedfrom the simple state dimension increase. It shouldalso be noted that significant tuning of the measure-ment and process noise was required to allow the filterto track the combined two-jammer signal. The resultsof the multi-signal tracking algorithm would likely im-prove, and be more stable to noise parameters, if afilter architecture that is different from the ExtendedKalman filter was implemented. A filter architecturethat might work more reliably is the particle filter orthe Gaussian mixture filter.

15

SUMMARY AND CONCLUSIONS

This paper investigated one method of acquiring andone method of tracking various chirp-style GPS jam-mers. A signal model that uses polynomial descrip-tions of frequency versus time was developed and itsutility for jammer signal tracking and classificationwas discussed. Two slightly different observation mod-els were derived. An ad-hoc chirp-style signal acqui-sition method that used FFTs of a moderately strongjamming signal was also developed. The various algo-rithm components were combined to produce a signaltracking Kalman filter. The developed Kalman filterwas verified by application on three sets of laboratorydata and on two sets of field data. One GPS jammerwas even acquired and tracked at a distance of approx-imately 1.8 kilometers. The developed models and al-gorithms were expanded to consider multiple jammerssimultaneously, and the algorithms were tested on oneset of multi-jammer laboratory data.

ACKNOWLEDGEMENTS

The authors would like to express their thanks to theDepartment of Homeland Security for their arrange-ment and sponsorship of the jamming event at WhiteSands Missile Range and to the 746th Test Squadronfor their part in the jamming event.

The authors would also like to express their thanksto the following members of the UT/Austin Ra-dionavigation Laboratory for helping to staff the re-ceiver stations at White Sands Missile Range: ToddHumphreys, Daniel Shepard, and Reese Shetrone.

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