I. INTRODUCTION
STAP for GPS Receiver
Synchronization
SEUNG-JUN KIM
RONALD A. ILTIS, Senior Member, IEEE
A space-time adaptive processing (STAP) algorithm for delay
tracking and acquisition of the GPS signature sequence with
interference rejection capability is developed. The interference
can consist of both broadband and narrowband jammers, and is
mitigated in two steps. The narrowband jammers are modeled as
vector autoregressive (VAR) processes and rejected by temporal
whitening. The spatial nulling is implicitly achieved by estimating
a sample covariance matrix and feeding its inverse into the
extended Kalman filter (EKF). The EKF estimates of the code
delay and the fading channel are used for a t-test for acquisition
detection. Computer simulations demonstrate robust performance
of the algorithm in severe jamming, and also show that the
algorithm outperforms the conventional delay-locked loop (DLL).
Manuscript received November 20, 2002; revised August 7, 2003;
released for publication September 30, 2003.
IEEE Log No. T-AES/40/1/826460.
Refereeing of this contribution was handled by L. M. Kaplan.
This work was supported in part by ONR Contract
N00014-01-G-0460 and the University of Washington, and by a
grant from the International Foundation for Telemetering.
Authors’ address: Dept. of Electrical and Computer Engineering,
University of California, Santa Barbara, CA 93106, E-mail:
(iltis@ece.ucsb.edu).
c 2004 IEEE
0018-9251/04/$17.00 °
132
The Global Positioning System (GPS) is a
satellite-based network that provides precision 3-D
position and velocity estimates by tracking the
time-of-arrival of spread spectrum signals. However,
due to its extremely weak received signal power,
the GPS receiver design must take into account the
effect of RF interferences from either intentional or
unintentional sources [1]. Military GPS receivers must
be able to operate reliably in the presence of malicious
multiple wideband/narrowband jamming signals [2].
It is well known that these interferers can
severely impair synchronization performance of
the receiver [3]. To partially alleviate the effects
of the interference, one can narrow the bandwidth
of the tracking loop, at the expense of increased
pull-in time [1]. However, sophisticated signal
processing algorithms may be necessary to deal
with challenging multiple jammer scenarios [1, 4, 5].
While narrowband interference in GPS systems
can be effectively mitigated using temporal or
frequency-domain filtering [5, 6] wideband
interference must be addressed by taking advantage
of the spatial dimension using adaptive antenna array
techniques.
Previous work on interference nulling based
on STAP in GPS receivers includes [2] and [7—9].
Assuming that the direction of arrival of the GPS
signal can be obtained from an INS (inertial
navigation system) [1], beam and null steering
techniques have been developed using criteria
such as maximum signal-to-interference ratio [7],
minimum mean square error [7], and minimum
output power [8]. However, specific results on
synchronization performance and the integrated design
of code synchronization algorithms with space-time
processing have not been extensively addressed in the
literature.
A tracking and acquisition algorithm for the
GPS C/A code delay using space-time processing is
developed here. Instead of preprocessing the samples
of the received signal with a space-time processor and
feeding the output to a conventional synchronization
algorithm (e.g. a tracking loop), we consider the use
of the extended Kalman filter (EKF) to obtain an
estimate of the code delay directly from the antenna
output vectors. The algorithm can also provide
estimates of the flat fading channel, narrowband
jammer parameters, and in some cases, even the
direction of arrival of the GPS signal.
The narrowband jammers are modeled as
vector autoregressive (VAR) processes and filtered
temporally. The autoregressive (AR) parametric
model of interference has been used in radar signal
processing [10—12]. Here, it is shown that exact
temporal whitening can actually be achieved under
certain conditions. For practical implementation, it is
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 1
JANUARY 2004
also shown that the VAR coefficient matrices need
only be scaled identity matrices, thus greatly reducing
computational complexity.
The temporally whitened signal vector has only
spatial correlation and an exact analogy to the
code-division multiple-access (CDMA) multiuser
detection problem can be established [13]. To spatially
whiten the wideband jammers and the residual
narrowband jammer power, we use a moving-average
estimate of the residual correlation matrix and feed its
inverse into the EKF.
The t-test [14] is employed in the acquisition
algorithm to address the problem of initial coarse
timing acquisition in the presence of unknown
jamming. The EKF estimates of the time delay,
channel fading, and possibly the GPS signal steering
vector are utilized to form the acquisition decision
statistic.
The paper is organized as follows. In Section II,
the signal model for the GPS receiver is described. In
Section III, the temporal whitening of the narrowband
jamming signals is developed. In Section IV, we apply
the EKF to the state-space formulation. Section V
develops the space-time acquisition algorithm.
Simulation results are given in Section VI, and a short
summary follows in Section VII.
II. SIGNAL MODEL
The GPS C/A code signal can be modeled as [6]
p
s(t) = Ref 2Ps d(t)PN(t) exp(j2¼fc t)g
(1)
where Ps is the transmitted signal power, d(t) 2
f¡1, 1g is the binary phase-shift keyed (BPSK)
data modulation at 50 bit/s, and fc is the L1 carrier
frequency of 1575.42 MHz [15]. The pseudorandom
noise (PRN) waveform PN(t) is defined by
PN(t) =
1 LX
ca ¡1
X
m=¡1 k=0
ck PTc (t ¡ kTc ¡ mTca )
(2)
where Lca = 1023 is the length of the C/A code
in chips, Tca = 10¡3 is the period of the C/A PRN
sequence given in seconds, Tc = Tca =Lca is the
chip duration, and ck 2 f¡1, 1g is the C/A code
sequence [1]. The pulse function PTc (t) satisfies PTc (t)
= 1, t 2 [0, Tc ), and zero otherwise.
The GPS signal is assumed to be transmitted
through a flat fading channel and corrupted by K
jammers. After the signal is downconverted and
lowpass filtered, we can represent the sampled
received signal vector from the antenna array by
r(n) = a0 (n)as (n)d(nTs ¡ ¿ (n))PNlp (nTs ¡ ¿ (n))
+
K
X
Here, the Nyquist sampling interval Ts is equal to
Tc =4 corresponding to the approximate bandwidth
of the C/A code signal of 2=Tc . ¿ (n) is the time
delay, jk (n) is the kth jamming signal with power
Pjk , n(n) is additive white complex Gaussian noise
with covariance ¾n2 I, and a0 (n) is the fading channel
coefficient with average power P. The time-varying
fading channel coefficients are modeled as Rician with
Rice factor K0 and Doppler spread fd Hz. The steering
vector for the desired GPS signal is as (n) and the
steering vector for the kth jammer is ajk (n) . Note
also that PN(t) in (1) is replaced by PNlp (t), which
is the ideal lowpass filtered version of the PRN
waveform, with the cutoff frequency 2=Tc , defined
by [16]
" µ
¶
1 LX
ca ¡1
X
PNlp (t) =
ck
m=¡1 k=0
(3)
k=1
KIM & ILTIS: STAP FOR GPS RECEIVER SYNCHRONIZATION
4¼
(t ¡ kTc ¡ mTca )
Tc
¡ Si
µ
4¼
(t ¡ (k + 1)Tc ¡ mTca )
Tc
¶#
(4)
where
¢
Si(x) =
Z
x
0
sin y
dy:
y
(5)
The signal-to-noise ratio (SNR) in dB after coherent
reception is defined by
P
SNR = 10 log10 2
(6)
¾n =2
and the jammer-to-signal power ratio J=S for the kth
jammer is given by
Pj
J=S = 10 log10 k :
(7)
P
III. TEMPORAL WHITENING
When the jammers are narrowband, the jamming
vector can be whitened using a VAR whitening filter
given by [10]
H(z
¡1
) = I¡
L
X
Hl z ¡l :
(8)
l=1
The following theorem shows that we can find the
VAR coefficients matrices Hl that achieve exact
whitening when the individual jammers are AR
processes of order less than or equal to L and the
number of jammers K is less than or equal to the
number of antenna array elements M.
THEOREM 1 Assume that each jammer can be
described by an AR(L) process given by
jk (n) =
ajk (n)jk (n) + n(n):
1
Si
¼
L
X
l=1
µk,l jk (n ¡ l) + ek (n),
k = 1, 2, : : : , K
(9)
133
where µk,l is the associated AR coefficient and ek (n) is a
white noise process. Then the jammer vector defined by
j(n) =
K
X
ajk jk (n)
(10)
with each other. If the sum of the jamming signals
defined by
K
X
jk (n)
(19)
j(n) =
k=1
k=1
can be modeled by an AR(L) process, i.e.,
with the M-by-1 steering vector for the kth jammer
ajk can be exactly whitened by the filter given by
(8), provided that M ¸ K, and the ajk are linearly
independent.
j(n) =
l=1
PROOF From (9) and (10), j(n) can be rewritten as
j(n) = A
L
X
l=1
diag(µ1,l , : : : , µK,l )j̃(n ¡ l) + Aẽ(n)
(11)
where
A =[aj1 (n) aj2 (n) ¢ ¢ ¢ ajK (n)]
¢
j̃(n) =[j1 (n) j2 (n) ¢ ¢ ¢ jK (n)]T
¢
ẽ(n) =[e1 (n) e2 (n) ¢ ¢ ¢ eK (n)]T :
(20)
where e(n) is a white random process with variance ¾e2 ,
then each (scalar) element of the jammer vector j(n)
can be exactly whitened by the whitening filter given by
(8), where the VAR matrices Hl are given by Ál I, with I
denoting the identity matrix.
(12)
e(n) =
(13)
(14)
This can be rewritten as
L
X
l=1
L
X
l=0
Á̃l j(n ¡ l)
(21)
where Á̃0 = 1 and Á̃l = ¡Ál , l = 1, : : : , L. Now compute
¢
the correlation matrix Re (n, m) = Efe(n)e(m)H g.
Re (n, m)
H
¡1
H
diag(µ1,l , : : : , µK,l )(A A) A Aj̃(n ¡ l) + Aẽ(n):
=E
(15)
Since Aj̃(n) = j(n), it follows that
j(n) =
L
X
l=1
Hl j(n ¡ l) + Aẽ(n)
(16)
where Aẽ(n) is temporally white and Hl is identified
as
(17)
Hl = A diag(µ1,l , : : : , µK,l )(AH A)¡1 AH :
The VAR coefficients Hl can be estimated by the
least square error criterion as in [10]
°
°2
n °
L
°
X
X
°
°
H1 , H2 , : : : , HL = arg min
Hl j(i ¡ l)°
°j(i) ¡
°
°
H1 ,H2 ,:::,HL
i=L
l=1
(18)
and the corresponding RLS algorithm can also be
derived. j(i) can be approximated by subtracting
from the received signal vector r(n) the estimated
desired signal component using the tracking algorithm
described in Section IV. However, in practice,
estimation of LM 2 parameters is cumbersome,
leading to much distortion in the desired signal and
failure in the subsequent signal tracking. In fact, the
combination of the RLS-based whitening filter and the
EKF cannot claim any optimality. A practically viable
approach is to assume that the “sum” of the jamming
signals is AR, as delineated in the following theorem.
THEOREM 2 Assume that the individual jammers jk (n)
are zero-mean wide-sense stationary and uncorrelated
134
Ál j(n ¡ l) + e(n)
PROOF The suggested whitening filter output e(n) is
¢
j(n) = A
L
X
=
8Ã
K
< X
:
K
X
k=1
k=1
aj aH
j
k
k
aj
k
L
X
l=0
Á̃l jk (n ¡ l)
L X
L
X
l=0 l0 =0
!Ã K
X
k 0 =1
aj
k0
L
X
l0 =0
Á̃l0 jk0 (m ¡ l0 )
!H 9
=
Á̃l Á̃¤l0 Rj (n ¡ m + l0 ¡ l):
;
(22)
k
Therefore, the diagonal elements of Re (n, m) equal
¾e2 ±n,m since from (20)
Efe(n)e(m)¤ g =
K X
L X
L
X
k=1 l=0
l0 =0
Á̃l Á̃¤l0 Rjk (n ¡ m + l0 ¡ l) = ¾e2 ±n,m
(23)
and the diagonal elements of
ajk aH
jk
are always unity.
The vector e(n) is not temporally white as the
off-diagonal elements of Re (n, m) are generally
non-zero for m 6= n. However, by approximating e(n)
as white, a more stable algorithm results, since only L
parameters have to be estimated, as opposed to LM 2
in (18).
It should be noted that after applying Theorem 1,
an exact analogy to the CDMA problem can be
established if we think of the residual whitening filter
output as multiuser interference with ajk being the
interfering user’s code sequence [13]. Hence, we can
apply a variety of multiuser detection-type methods
that have already been developed [13, 17, 18].
The jammer AR coefficients fÁl g can be
effectively estimated jointly with other parameters
including the time delay using Kalman filter-type
algorithms. Slightly modifying the assumption (20)
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 1
JANUARY 2004
and modeling the sum of the jamming signal and
additive white Gaussian noise as an AR(L) process,
we have
L
X
j(n) + n(n) =
Ál [j(n ¡ l) + n(n ¡ l)] + e(n)
l=1
(24)
where we assume e(n) to be temporally white.
IV. TRACKING ALGORITHM
To develop an EKF-based tracking algorithm, a
dynamic model for the parameters to be estimated has
to be assumed. Since the time delay ¿ (n) is generally
a non-zero-mean random process, it is reasonable to
model it as a sum of a zero-mean Markov process and
a constant process, both of which are estimated by the
EKF [19]. That is,
¿ (n) = ¿m (n) + ¿c (n)
(25)
¿m (n) = f¿ ¿m (n ¡ 1) + v¿ (n ¡ 1)
(26)
where
¿c (n) = ¿c (n ¡ 1)
and v¿ (n) is a zero-mean white Gaussian random
process with variance ¾¿2 . The time-variation of the
jitter process ¿m (n) is set by f¿ , and ¾¿2 accounts for
its variance. These parameters depend on various
factors such as the quality of the oscillator used and
the relative motion of the receiver and the space
vehicle (SV), and thus are determined empirically.
The constant process ¿c (n) can alternatively be
viewed as an extremely narrowband AR process
that accommodates the long-term motion of the
platform. Note that this model is different from
the simple first-order AR model widely used for
the delay process in the literature in that the sum
of a constant and an AR process is in fact an
autoregressive moving average (ARMA) process
[16, 20].
A simple AR model is adopted for the Rician
fading channel. First, a lump parameter a(n) is
defined to be the product of the fading channel a0 (n)
and the data modulation d(nTs ). Then a(n) is modeled
as
a(n) = fa a(n ¡ 1) + va (n ¡ 1):
(27)
To take the BPSK data modulation into account,
a modification is made for the model at the bit
boundary, following [21]. At the bit boundary,
8
with probability
>
< fa a(n ¡ 1) + va (n ¡ 1)
a(n) = fa a(n ¡ 1) + [va (n ¡ 1) ¡ 2fa a(n ¡ 1)]
>
:
with probability
Therefore, the variance of va (n) is modeled by
½ 2
¾¯ a + 2fa2 ja(n ¡ 1)j2
at the bit boundary
2
¾a (n) =
2
¾¯ a
otherwise
(29)
where fa and the nominal variance ¾¯ a2 of the
zero-mean white Gaussian process va (n) are
determined so that the power spectrum of the fading
process closely resembles that of a realistic fading
channel with the given Rice factor and Doppler
spread. Justification for using an AR fading channel
model can be found in [22]. It is also straightforward
to extend the model to a higher order AR model
[23].
The AR parameters for the narrowband jammers
are modeled as zero-mean first order AR processes.
Ál (n) = fÁ Ál (n ¡ 1) + vÁl (n ¡ 1),
l = 1, : : : , L
(30)
the variance of the white Gaussian
where fÁ and
process vÁl (n), are determined by the dynamics of the
parameters.
One can rely on external information such as
almanac and/or an INS to get an estimate of the
steering vector for the desired signal as (n) [24], or
alternatively, use the proposed algorithm to directly
estimate it. In the latter case, the model
¾Á2 ,
ais (n) = fas ais (n ¡ 1) + vas (n ¡ 1),
i = 2, : : : , M
(31)
can be used, where
is the ith component of the
vector as (n) and vas (n) has the variance ¾a2s . The first
element of as (n) is not estimated as it is assumed
to correspond to the reference antenna element and
thus a1s (n) is fixed at 1. Since jais (n)j = 1 for all i, this
nonlinear constraint is applied at every Kalman update
step to enhance the estimation performance [25].
These models can be combined in the plant
equation
ais (n)
x(n) = F(n ¡ 1)x(n ¡ 1) + v(n ¡ 1)
(32)
where the state vector x(n) 2 C3+L+M¡1 is given by
x(n) = [¿m (n), ¿c (n), a(n), Á1 (n), Á2 (n), : : : ,
T
ÁL (n), a2s (n), a3s (n), : : : , aM
s (n)]
(33)
the transition matrix F(n) is given by
F(n) = diagff¿ , 1, fa , fÁ , : : : , fÁ , fas , : : : , fas g
| {z } | {z }
(34)
M¡1
L
and the zero-mean white Gaussian noise vector v(n)
has the covariance matrix given by
1
2
:
1
2
(28)
KIM & ILTIS: STAP FOR GPS RECEIVER SYNCHRONIZATION
Q(n) = diagf¾¿2 , 0, ¾a2 (n), ¾Á2 , : : : , ¾Á2 , ¾a2s , : : : , ¾a2s g:
| {z } | {z }
L
M¡1
(35)
135
The measurement model can be identified from (3)
and (24) as r(n) 2 CM
r(n) = h(x(n), rL ) + e(n)
(36)
= a(n)as (n)PNlp (nTs ¡ ¿ (n))
+
L
X
l=1
Ál (n)[r(n ¡ l) ¡ a(n)as (n)PNlp ((n ¡ l)Ts ¡ ¿ (n))]
(37)
+ e(n)
where rL is defined by fr(n ¡ 1), r(n ¡ 2), : : : , r(n ¡ L)g
and e(n) is assumed to be zero-mean white Gaussian
with covariance Re (n). a(n) and as (n) are assumed to
be constant over LTs s.
Due to the nonlinear measurement function h(¢), a
closed-form solution for the recursive Bayesian state
estimator cannot be obtained. The most widely used
suboptimal solutions include the EKF and Gaussian
sum filter (GSF) [26]. The EKF is employed here
since the delay estimation error nominally falls in the
range of 10¡3 Tc to 10¡6 Tc for the GPS code tracking
application. Therefore the first-order linearization
approximation for the delay estimate may be well
justified. The linearization of the measurement
function about the predicted state vector estimate
x̂(n j n ¡ 1) at time n given the measurements up to
time n ¡ 1 is given by
h(x(n), rL ) ¼ h(x̂(n j n ¡ 1), rL ) + H(n)[x(n) ¡ x̂(n j n ¡ 1)]
(38)
¢
where H(n) = H(x̂(n j n ¡ 1), rL ) 2 CM£(3+L+M¡1) is the
Jacobian defined by
@h(x, rL )
H(x, rL ) =
(39)
@x
which is again given by H = [h1 h2 ¢ ¢ ¢ hM+L+2 ], where
h1 = h2 = ¡a(n)as (n)PNd (nTs ¡ ¿ (n))
+
Here, PNd (t) is the derivative of PNlp (t) and ei
is a unit column vector with one for the ith
element.
In order to formulate the EKF update equations,
the correlation matrix for the white measurement
process e(n) must be estimated. A moving-average
estimate of the residual correlation matrix given
by [27] is used as follows
L
X
l=1
Ál (n)a(n)as (n)PNd ((n ¡ l)Ts ¡ ¿ (n))
R̂(n) =
n¡1
1 X
[r(l) ¡ h(x̂(l j l ¡ 1), rL )]
NR
l=n¡NR
£ [r(l) ¡ h(x̂(l j l ¡ 1), rL )]H :
(41)
The EKF update equation can now be written
P¡1 (n j n) = P¡1 (n j n ¡ 1) + HH (n)R̂¡1 (n)H(n)
x̂(n j n) = x̂(n j n ¡ 1) + P(n j n)HH (n)R̂¡1 (n)
(42)
£ [r(n) ¡ h(x̂(n j n ¡ 1))]
x̂(n + 1 j n) = Fx̂(n j n)
P(n + 1 j n) = FP(n j n)FT + Q:
Since the state vector estimate x̂ is complex in
general, the time delay estimate is obtained by taking
the real part of the corresponding complex estimate,
e.g.,
¿ˆ (n j n) = Ref¿ˆm (n j n) + ¿ˆc (n j n)g
(43)
where ¿ˆm (n j n) and ¿ˆc (n j n) are the first and the
second elements of x̂(n j n), respectively. It is
noteworthy that R̂¡1 (n) actually nulls the jammers
spatially in (42) since, if the jammer power dominates
the background noise power, R̂¡1 (n) approaches
(1=¾n2 )(I ¡ A(AH A)¡1 AH ), which is a linear projector
to the left nullspace of A [9]. Therefore, the
broadband jammers, which cannot be whitened
temporally, are rejected spatially at this stage.
The block diagram for the overall tracking
algorithm is given in Fig. 1.
h3 = as (n)PNlp (nTs ¡ ¿ (n))
¡
L
X
l=1
V.
Ál (n)as (n)PNlp ((n ¡ l)Ts ¡ ¿ (n))
(40)
hi+3 = r(n ¡ i) ¡ a(n)as (n)PNlp ((n ¡ i)Ts ¡ ¿ (n)),
i = 1, : : : , L
"
hi+L+3 = ei+1 a(n)PNlp (nTs ¡ ¿ (n))
¡
L
X
l=1
#
Ál (n)a(n)PNlp ((n ¡ l)Ts ¡ ¿ (n)) ,
i = 1, : : : , M ¡ 1.
136
ACQUISITION ALGORITHM
In order for the EKF-based tracking algorithm
to operate properly, the initial time delay estimate
must be set to a value that is within half a chip of
the true delay. This coarse acquisition of the delay
can be accomplished by means of a binary hypothesis
test. Consider the temporally whitened sequence y(n)
defined by
y(n) = r(n) ¡
L
X
l=1
Ál r(n ¡ l)
= a(n)as (n)g(¿ (n)) + e(n)
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 1
(44)
JANUARY 2004
Fig. 1. Block diagram for STAP EKF tracking algorithm.
Fig. 2. RMS jitter of EKF using single element antenna and no jamming (175 simulation runs).
where g(¿ (n)) is given by
g(¿ (n)) = PNlp (nTs ¡ ¿ (n)) ¡
L
X
l=1
Ál (n)PNlp ((n ¡ l)Ts ¡ ¿ (n)):
(45)
The whitening filter is assumed to be exact in tracking
mode, that is, the EKF estimation error of the AR
coefficients fÁl g is assumed negligible. (When the
receiver is not in tracking mode, the whitening filter
need not be exact, as explained below.) Therefore,
e(n) is approximated as a temporally white Gaussian
sequence. The two hypotheses to consider are H1 ,
where the estimates of the delay ¿ˆ (n j n), the channel
â(n j n) and the steering vector âs (n j n) are equal to
their respective true values, so that acquisition has
occurred, and H0 where they are not equal. That is,
H1 : y(n) = â(n j n)âs (n j n)g(¿ˆ (n j n)) + e(n)
(46)
H0 : y(n) = a(n)as (n)g(¿ (n)) + e(n)
KIM & ILTIS: STAP FOR GPS RECEIVER SYNCHRONIZATION
for n = 0, 1, : : : , m ¡ 1. The number of samples used for
detection is m.
However, since ¿ (n), a(n), as (n), and Ál (n) are
unknown when the receiver is not in the tracking
mode, it is not possible to test for H0 exactly.
This problem can be circumvented by noting that
g(¿ˆ (n j n)) is a direct-sequence signal, and hence
is approximately uncorrelated with g(¿ (n)) when
acquisition has not been achieved. Furthermore, the
signal component is negligible compared with the
thermal plus residual jammer noise, since the GPS
receiver operates at a very low SNR. Therefore,
approximate hypotheses can be set up as
H1 : y(n) = â(n j n)âs (n j n)g(¿ˆ (n j n)) + e1 (n)
H0 : y(n) = e0 (n)
(47)
for n = 0, 1, : : : , m ¡ 1. The covariance R0 of e0 (n) is
slightly larger than the covariance R1 of e1 (n), since
e0 (n) incorporates the nonacquired sequence.
Since the covariances R0 and R1 are not known a
priori, the t-test [14] is employed to detect acquisition
137
Fig. 3. RMS jitter for four-element antenna array with no jamming (100 simulation runs).
TABLE I
Directions of Desired Signal and Jamming Signals
and can be expressed as follows:
t(m) =
maxR1 p(ym j H1 , R1 ) H1
?t
maxR0 p(ym j H0 , R0 ) H0 0
(48)
where ym = fy(0), y(1), : : : , y(m ¡ 1)g and t0 is the
detection threshold. Thus the t-test coincides with
the generalized likelihood ratio test (GLRT) [28].
With â(n) and âs (n) fixed, the maximum likelihood
estimates (MLEs) of R1 (n) and R0 (n) are given
by
R̂1 (m) =
1
m
m¡1
X
n=0
[y(n) ¡ â(n j n)âs (n j n)g(¿ˆ (n j n))]
£ [y(n) ¡ â(n j n)âs (n j n)g(¿ˆ (n j n))]H
(49)
m¡1
R̂0 (m) =
1X
y(n)y(n)H :
m
n=0
By substituting these into (48), and under the
assumption of white Gaussian residuals ei (n), the
t-statistic reduces to
t(m) =
jR̂0 (m)j
jR̂1 (m)j
:
(50)
VI. SIMULATION RESULTS
The performance of the proposed algorithm was
evaluated by simulation. The antenna array considered
consists of a four-element square configuration with
half-wavelength spacing. Both narrowband and
wideband jamming signals are considered. The K1
narrowband jamming signals are assumed to be swept
138
Azimuth
Elevation
as
aj
aj
aj
aj
aj
85±
27±
288±
80±
216±
80±
120±
80±
240±
80±
5±
80±
1
2
3
4
5
tones defined by
p
_ nT )nT )
jk (n) = Jk exp(j2¼(¢fk + ¢f
k s
s
k = 1, : : : , K1
(51)
_ is the sweep rate. The K wideband
where 2¢f
k
2
jamming signals jk (n) for k = K1 + 1, : : : , K1 + K2
are assumed to be zero-mean white Gaussian. The
SNR (6) is assumed to be ¡12 dB and a J=S ratio
of 60 dB is used throughout the simulations. The
directions of the desired GPS signal and the jammers
used in the simulations are tabulated in Table I.
For the operation of the EKF, the AR model
parameters must be determined. fa = 0:9946 and
¾¯ a2 = 0:0034123 are used, corresponding to a fading
channel characterized by fd = 5 kHz and K0 = 15 dB.
The timing jitter is modeled by f¿ = 0:9999 and ¾¿2 =
10¡12 , which yield a time constant of approximately
10,000 samples and a jitter on the order of 10¡6 .
Similarly, fas = fÁ = 0:9999 and ¾a2s = ¾Á2 = 10¡3
are chosen for the simulation. Note that perfect
knowledge of these parameters is assumed by the
EKF. In practice, they must be measured or fine-tuned
for proper operation of the EKF.
Fig. 2 shows the rms error of the EKF-based
tracking algorithm when only a single antenna
element is used and no jamming signal is applied.
The initial estimation error is assumed to be uniformly
distributed over [¡ 12 Tc , 12 Tc ]. The average was obtained
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 1
JANUARY 2004
the DLL is given by a discrete-time version of the
noncoherent DLL in [29]
Ã
!
jye (n)j ¡ jyl (n)j
¿ˆ (nN) = ¿ˆ ((n ¡ 1)N) + ¹Tc
jyp (n)j
(52)
where
ye (n) =
nN
X
r(k)PNlp (kTs ¡ ¿ˆ ((n ¡ 1)N) ¡ Tc =2)
nN
X
r(k)PNlp (kTs ¡ ¿ˆ ((n ¡ 1)N))
nN
X
r(k)PNlp (kTs ¡ ¿ˆ ((n ¡ 1)N) + Tc =2)
k=(n¡1)N+1
yp (n) =
k=(n¡1)N+1
yl (n) =
k=(n¡1)N+1
Fig. 4. Steering vector estimates for desired signal from 85± .
(a) At 1023rd chip. (b) At 5155th chip. (c) At 15345th chip.
(d) At 30690th chip.
from 175 independent simulation runs. Also, the
window size NR for forming the sample covariance
matrix R̂(n) is set to 200 samples. The rms value
of the error converges to 3:5 £ 10¡3 Tc at the end of
51150 chips, or 0.05 s. This reveals that the model
for the delay process ¿ (n) given by (25) and (26)
can yield good error performance for the EKF-based
tracker even at a low SNR. This is contrary to the
reports of poor performance of the EKF-based
algorithms at low SNR environments as in [20], which
did not take into account the non-zero mean of the
underlying delay process. Also shown in Fig. 2 is
the rms error of a simple first-order delay-locked
loop (DLL) for comparison. The update equation for
(53)
and r(k) is given by (37), with r(k) reduced to a
scalar. ¹ and N are design parameters that determine
the transient time and the jitter performance. In
the simulation they are chosen to be ¹ = 0:1 and
N = 4096 samples. Recall that the transient and jitter
performance of the DLL is a trade-off. From Fig. 2, it
can be seen that the EKF-based tracker outperforms
the DLL in terms of both pull-in time and jitter
performance.
Fig. 3 depicts the rms jitter of the proposed
algorithm averaged over 100 simulation runs in the
absence of jamming. Also plotted is the performance
of the simple DLL for comparison. The DLL assumes
the steering vector as to be known a priori, and uses
aH
s r(k) in place of r(k) in (53). Again, it is clear that
the EKF-based algorithm exhibits less pull-in time
and rms jitter. Note that the EKF is also estimating
as . The steering vector estimated by the EKF is
shown in Fig. 4 at different iterations by plotting
a(µ, 27± )H âs (n), 0± · µ < 360± , where a(µ, 27± ) is the
Fig. 5. RMS jitter under three narrowband and two wideband jammers (100 simulation runs).
KIM & ILTIS: STAP FOR GPS RECEIVER SYNCHRONIZATION
139
Fig. 6. Effective frequency response of whitening filter for one tone jammer and two swept CW jammers.
Fig. 7. Null pattern formed by R̂¡1 .
steering vector for the four-element planar array at
azimuth µ and elevation 27± .
Fig. 5 shows the performance of the EKF-based
tracking algorithm under mixed jamming. There are a
total of five jammers. Three narrowband jammers have
parameters ¢f1 = ¢f2 = 0, ¢f3 = 1:023 MHz and
_ = ¢f
_ = 2 GHz. Two wideband jammers
_ = 0, ¢f
¢f
1
2
3
are white Gaussian. In the figure, it is clear that the
temporal whitening with L = 6 is very effective in
mitigating the jamming signal, compared with the
case where L = 0, i.e., where the temporal whitening
is turned off. Although not depicted in the figure, the
rms jitter for the single antenna case without temporal
whitening was observed to quickly increase within a
few iterations.
140
In Fig. 6, the effective time-varying frequency
domain response of the whitening filter is plotted
under the same jamming condition as in Fig. 5,
to illustrate its capability to reject the narrowband
jammers. The effective frequency response is
calculated by
L
X
j2¼f
H(e
Á̂l e¡j2¼fl :
) = 1¡
(54)
l=1
It can be seen that the “notches” of the whitening
filter follow the instantaneous frequencies of the
stationary and swept-tone jammers to null out the
jamming signal.
Fig. 7 shows the ability of the proposed
algorithm to null the wideband jammers by plotting
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JANUARY 2004
Fig. 8. ROC of acquisition algorithm without jamming.
Fig. 9. ROC of acquisition algorithm under two narrowband and one wideband jammers.
a(µ, 80± )H R̂¡1 as (n) for 0± · µ < 360± , where 80±
corresponds to the jammer elevation. It is observed
that deep nulls are placed in the direction of the
jammers, including those of the wideband jammers,
which have not been attenuated by temporal
whitening.
The acquisition performance of the t-test-based
detector is evaluated by simulation. Fig. 8 shows
KIM & ILTIS: STAP FOR GPS RECEIVER SYNCHRONIZATION
the receiver operating characteristic (ROC) of the
proposed detector when SNR = ¡12 dB and no
jamming is present. The probabilities are averaged
over 10,000 independent runs. It is observed that
when the number of observed samples m ¸ 1000,
reliable performance for acquisition is obtained. Fig. 9
shows the detector performance when two narrowband
and one wideband jammers are present. For the
141
Fig. 10. Mean acquisition time of proposed algorithm. (a) Without jamming. (b) With jamming.
_ = 0,
narrowband jammers, ¢f1 = ¢f2 = 0 and ¢f
1
_ = 2 GHz were used. The jammer directions
¢f
2
are 288± and 216± in azimuth for the narrowband
jammers, and 120± for the wideband jammer. The
elevation is fixed at 80± and J=S = 60 dB for all
jammers. The probabilities are averaged over 2,300
independent runs. It can be seen that m ¸ 15,000
gives acceptable detection and false alarm probabilities
under the given jamming condition.
Fig. 10 shows the worst case mean acquisition
time T̄acq for the algorithm when the simple serial
search scheme is employed with and without
jamming, computed from the simulation results
associated with Fig. 8 and Fig. 9. Therefore, the
jamming condition is the same as in Fig. 9. The worst
case refers to the case where the whole uncertainty
region of the unknown code phase is searched at
least once. T̄acq in number of dwells can be computed
by [30, 31]
T̄acq =
1
[(D ¡ 1)pF (Ns ¡ 1) + Ns ]
pD
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was set to 2,046 trial delays, corresponding to the
phase update of a half chip.
VII. CONCLUSION
[1]
[2]
[3]
[4]
A code synchronization algorithm with integrated
narrowband/wideband jamming mitigation for GPS
142
signals has been developed. The vector input from
the antenna array was first whitened temporally to
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to approximate the minimum variance estimates
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acquisition algorithms was evaluated by numerical
simulations and shown to reliably acquire and track
the desired timing in the presence of strong multiple
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143
Seung-Jun Kim received the B.S. and M.S. degrees in electrical engineering from
Seoul National University, Seoul, Korea, in 1996 and 1998, respectively.
From 1998 to 2000, he served as a Korea Overseas Volunteer at Chaing
Rai Teacher’s College, Chaing Rai, Thailand. In 2001, he was with GCT
Semiconductor, Seoul, Korea. Since 2001, he has been with the University
of California, Santa Barbara, working toward a Ph.D. degree in electrical
engineering. His current research interests lie in synchronization, channel
estimation, and adaptive antenna array beamforming.
Ronald Iltis (S’83–M’84–SM’91) received the B.A. in biophysics from
Johns Hopkins University, Baltimore, MD, in 1978, the M.Sc. in engineering
from Brown University, Providence, RI, in 1980, and the Ph.D. in electrical
engineering from the University of California, San Diego, in 1984.
Since 1984, he has been on the faculty of the Department of Electrical and
Computer Engineering at the University of California, Santa Barbara, where
he is currently an associate professor. His current research interests are in
spread-spectrum communications, blind equalization, multisensor/multitarget
tracking, and neural networks. He has also served as a consultant to government
and private industry in the areas of adaptive arrays, neural networks, and
spread-spectrum communications.
Dr. Iltis was previously an editor for the IEEE Transactions on
Communications. In 1990 he received the Fred W. Ellersick award for best paper
at the IEEE MILCOM Conference.
144
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JANUARY 2004