ONJAMMIXG OF GPS S I G N U S

by

Beni Priel Ben-Zion Bobrovsky

I Introduction

The mean time to loselock (MTLL) of tracking loop has recentely attained much attention. Analytic expressions for the leading terms of the MTLL of first and second PN code tracking loops were obtained in [ 1]-[31. In most works it is assumed that the system is excited by white noise. However it is well known that generally the noise observed from a GPS receiver is correlated. Therefore an analysis of coloured noise is practidly veq important for GPS receivers . In [ 41 analysis of coloured noise in in a specific dynamical. system is presented. . We present solution for the MTLL of a firs order PN coherent tracking loop in the case where the system is escited by coloured noise with small intensity

We show that for a fised jammer poxer there is an optimal jammer bandwidth, and therefore a wideband jammer is superior to a singletone jammer with the same power. T i e results are used for practical jammer des@.. The jammer power is determined to amin desired IvlTLL for a @\en receiver bandwidth and velocity.

Beni Priel: Taniam I .AI P.0 75 Y a W Israel. Ben Zion BobrovsLy : Departement of Electrical Engineering ,Tel A\iv Uniyersiq- . Ramat Aviv,Tel Aviv ,Israel.

The loop equations are presented in sec. II. The exit problem and the domain of attraction are discussed in section Ill . The MTLL for reciever that is excited by coloured noise is given in section IV, and in section V the MTLL is used as an index of performance for practical jammer design.

I1 Loop Equations

The received signal is modeled in Fig 1 as

where P is the evarage signal poiver , n(t) is the additive coloured noise term , c(t) is the spreading code nith code rate R, , and T is the instantaneous propagation delay of the channel. The delay T vary in time due to the ve:ociF v

assuming the receiver \,elocit). is v the OEX:

is a, =- where c is the velocie of

light .

U

=,c

The coloured noise is given b).

N, . i\ = -%n(t) + 3. - VI ( 3 ) d , nhere iv is an additive standard white noise, and N, is the power specnral

5.2.3 -a-

I

0 5

- U 0 m

-0.5

- 1

density and h i s the 3 db bandwidth of n(t) .

It is well known [j] that the coherent delay lcck loop can be modeled by the "baseband equivalent" of Fig 2 .. The timing error e satisfies the loop equation

e = a , -S(e)k-n(t)k (4)

where the pieavise linear discriminator S(e) curve is given in Fig. 3.

I

gr PS-CODE & ^. t:g 1. Coherent PN-code trackins loop.

Fig. 2. Second-order baseband equivalent md:l.

-2 -1.5 -1 -0.5 0 0.5 1 1 5 2 C

Fig. 3 . Thc d~scriminntor characrcristic (S curve) .

M e r time scaling [3] t* = a t where a=k loop equtions are given by

e(t')=-+ -S(e)-n(t') a

where

(jc,d

(In the sequel the abreviation of the scaled time is omitted.).

lT7 The Exit Problem

From a mathematical point of view (5 a c ) present a small stochastic pertrubation of the dynamic system with p =O. The unpertrubated system has a stable equilibrium at

e, =a12 na = Q (6a)

and nonstable ( saddle) paint at

e, =1.5-a nb = Q for a>O (6b)

The stable equilibrium is an attractor to all trajectories starting at some neighbourhood to i t This neighbourhood is called the domain of attraction and its boundary is denoted by dD . From an engineering point of .view when the state is inside D the loop is locked. All trajectories beginning inside D v-dl coriyerge to the attracrtor fer the deterministic On the other hand m-en thc slighlsst psrtrubation is sure to cause crossing a t ri-e boundary aD ( named separatris) in sonic finite time and thus resulting In loss of lock The separatris from the domain of attraction through separatns is demonstrated by simulation.

is shown in Fig 4 , the exit

Folloning [ 1 ] the time to reach tile bounday is given aproslmately by

7 = C(2vlk,a)exp(~). Thexaluc

the minimal value of thc quasi potential 41 on the boundary D . @ gives the lonest potential barrier between the boundary and the attractor. It had been shown in [ 1 ] tlmt

& IS P

i 5.2.3 -2-

the trajectories will hit the boundary mainly at the point were $ is minimum . This agrees with the intuition that it is more likely to exit

( 2 k + / , ) P3 = k (9 a 4

from the lowest potential barrier. It will be shown that C 1 for Mk > I , therefore our main concern is to find

Nest we obtain the quasi potential function for I e I > 0.5

Denote the ausilliary function A

$=min{$ E D ) (7)

W(e,n) satisfies the Fokker Plank equation

Fig, 4. Exit from separatrix y{ W(e,n)=O , ( e , n ) E D ) (1 1 a-b)

1 W( e,, n 0 1 = 1 I

3 . . > 4 * . &.~.utl cI arJaaion

2 -

1 -

-1 -

Less of !c& lomain

e

In the nex- section we find asymptotic solution for the MTLL for p e l .

If the system equations ( 5 a-c ) were linear then W(e,n) would be proportional to the joint Gaussian probability function with g(e,n) some constant. Inserting the solution of the ( 10 ) into the Fokker Plank equation and assuming p< < 1 we select the leading terms (namely the terms multiplied by l /p ) and obtain the “Eikonal” equation

ff( e, n, 9 rl 1 = (a -S(e) b-n ) ( a$ / de) - Mk n a* / an t X’ /kz (a$ / an)’ = o (12)

where

(13 ab) IV Mean Time to Lose Lock. (IMTLL)

We first obtain the quasi potential for I e I c 0.5. Denote by rg the function Q in the reqion

This equation is solved by the method of characteristics [G 1.

I e I <0.5 . Then.the pdfof the’dynamicai system ( j a c ) is Guaussian and by applyins

The equations of characteristics are

the ray method (see [2]) the pdf is

approximated by esp( --) . The solution

for the potential function is therefore given by

P=={P, ( e - e , )’ + 2p,(e-e , >( n - n,,

e = a H /aq , = a , /k -S(e) -n(t) P P

1 n = a H / a q 3 = - m + 2 * h 2 / k 2 q ,

where 2 k + l

2 , p 2 = 2 k -

( 2 k +A)’ P1 = 2 ].’ (14 a-e)

The initial conditions are

(15 a d )

For every uo weget aclifferent characteristic which may cross the bounclaq aD in the neighbourhood of the saddle point , the quasi potential is minimum at the saddle, therefore we look for the initial no such that the characteristics hits the saddle. This method of “shooting“ characteristics to get the the minimal I$ is very usefd for general S(e). For the specific shape of the S curve an analytic soIution can be derived .

nQ =

(16)

The solution to the quasi potential hiction for I e I > 0.5 is given by

@ = fie,n,I+JT (17)

where W is given by (11 e).

where @,eono) is g i x n by ( S) , and q2(Q) by ( l j c ).

The pre eqonent C depends on ?Jk ana theoffset a and is found bysimul as explained in the sequel, however it follows from [A] that for large values of X C(2I k,a) - 1 therefore in this case the eqonential term in (18 ) IS dominant and we obtain

Fig 5 shows the quasi potential as a function of the scaled frequebcy Mk and the scaledoffset a.

The esTression for MTLL is c o d m e d by simulations. We find by simulations the first time that the trajectory hits the separatix ( see Fig 4.) , and the estimated quasi potential & R I k, a ) is found as the slope of the curve

b h(t) = b(C) +- (21) P

The results are demonstrated in Fig 6 . For example for a=O,3 and Uk=0.6 ir is found that @ = 0.52 ,and the true d u e is i ) = 0.55 . Similiar simulations with different values of a and ’Wk confirm that the estimated and expiic

arethesame. The Fre found by simulalions from ( 21) u;d is based on the value of 7 thst is found ‘c] simulations The dependency of C on >- :i

shown in fig 7 . The results confin x:c\ we11 with those of [1 J for >>> 1,

V Practical Jammer Design

We propose to use the LLTLL 6b -- R cn;cr,o:: for practical jammer desigri. The firs: approach is to fix the jammer poner . The optimal value of the MTLL depends on C? jammer bandlvidth A I L (see fig S). IVe notice that a wideband jammer achieves a lower MILL then a singletone jammer vith the same power, therefore it is superior to

5.2.3 4-

Fig. 6. Ln(t) for a=0.3

singletone jammer whose power is the same.

Another design approach is to fix the jammer bandwidth and to determine the jammer power to achieve the MTLL criterion. For example, choosing Lkc-10 the jammer power to achieve a value of MTLL is determined from (20)

where

w is the jammer power and is given by

1 Iro

Fig. 7. C(>/k,0.3)

1001 1

In Fig. 9 the MTLL is shown for a=0.3 and W l O as a function of the jammer power w The MTLL strongly depends on w , and decreases rapidly as w increases.

We notice that for a fised noise to signal ratio p , the MTLL is almost the same for values of Mk>l . The reduction of the MTLL is attained by increasing jammer power. Therefore a good jammer design approach is to limit the jammer bandwidth to the range of the the reciever bandwidth , the high frequency range that is much greater then the reciever bandwidth has not any practical efffect on the jammer performance. We notice that we can approach the white noise jammer performance by bandlimited jammer.

A

Fig. 5. 6

.=o. 1

n

Fig. 8. Log(MTL1) w=l a=0.3

2-5 5

0 1 2 3 4 A k

Fig. 9. Log(MTLL) a=0.3 h/k=lO

- 0 0.5 1

N k watt

VI Discussion

Explicit solution to the MILL is derived. and the analytic espresion of the MTLL is \{ell confirmed by simulation results. The principle of the jammer design is to choose jammer power such that desired MTLL is achieved. The new criterion is useful1 to choose between two kinds of jammers with the same power . One is wideband and the second a singletone jammer . The criterion commonly used for jammer design is the jitter error. The jitter is the Same for a singletone and a wideband jammer nith tht same power. It is well known that there are several anti jamming methods such as limiters or adaptive filters that enable to reject a single tone jammer. We show in sec. V that the better jammer is the wideband, as it is more effective to achiwe receiver loss of lock then a singletone jammer. .It is also shown that good practical jammer design is to choose a limited jammer bandwidth, the jammer banwidth should be of the same order of magnitude as the receiver bandwidth.

References

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2.) 2. Schuss "Theorq. and application of stochastic daerential equatims".X\;sn I-ici;: Wiley, 1980

3) A.L Welti , B.Z Bobrox s b I' X k n iir- to lose lock for coherent second oidei TT code traciung loop - the singular pertrubation approach" EEE J. on Selzzs: Areas in Communication Vol8 no 5 June 1990

4 ) ?VI. Klosek D y a s . B. J.IM2tko~x 5%; . Z.Schuss " Unifomi asymptouc exTansiccr in dynamical qsterns driyen by coloured noise", Phisical review vol 35 no. 5 September 195s.

6.) Chester "Techniques in partial differential equations 'I, New York. McGrav Hill ,1971.

,

5 . ) R.E Zienier R.L..Peterson "Digital comunication and spread spectrum systems'' New York Macniilan ,19S5.