w
kq
are all known (after adaptation), this filter
function is readily constructed. If the GPS satellite
emits a signal with a Fourier transform S(f), then, in
the absence of signal multipath, the signal output after
passing through the adaptive filter is H(f)S(f), which
canbewritteninthetimedomainas
r(t)=
Z
1
¡1
df H (f)S(f) exp(i2¼f t) (18)
where, for notational convenience we have suppressed
the angular dependence of H. The GPS receiver
estimates time delay by using [15, 16] the cross
correlation
3
of this received signal with the known
signal
r
0
(t)=
Z
1
¡1
df
0
S(f
0
)exp(i2¼f
0
t): (19)
If the signal is a stationary random process, then
hS(f)S
¤
(f
0
)i = P(f)±(f ¡f
0
) (20)
where P(f) is the power spectrum of the signal. If
(20) is used, along with (18) and (19), we see that the
cross correlation is
C(¿)=hr(t)r
¤
0
(t + ¿)i
=
Z
1
¡1
df P (f)H(f)exp(¡i2¼f ¿): (21)
Typically, P(f) is a positive, symmetric function of
f, so that in the absence of H(f) the correlation peak
occurs at ¿ =0.However,H(f) can introduce both
a broadening of the correlation peak and a shift of
that peak from the correct value at ¿ = 0. This shift
(or bias) is introduced by the phase Á(f)ofH(f).
In addition, the presence of H(f)couldleadtoa
phase shift at the value of ¿ where the correlation
peak occurs, and this could affect the high-precision,
differential GPS systems that use carrier-phase
tracking. Thus, in a later section it is necessary to
carefully examine the effect of the adaptive antijam
filter H(f) on the cross correlation function. Of
course, if necessary, both the bias error and any phase
shifts, but not the broadening effect, can be corrected
by following the adaptive FIR filter with the filter
4
G(f)=H
¤
(f). Then, the cross correlation becomes
C
0
(¿)=
Z
1
¡1
df P (f)jH(f)j
2
exp(¡i2¼f ¿): (22)
Because P(f)jH(f)j
2
is real, the peak of jC(¿)j now
lies at the correct location (¿ = 0) and, clearly, the
phase of C
0
(0) is zero, as desired. However, the
3
This is referred to in GPS literature as the “code tracking loop.”
4
It should be recognized that G(f)=H
¤
(f) is not the only option
for compensation. In fact the filter G(f)=H
¤
(f)=jH(f)j
2
is
preferable because the cross correlation C
0
(¿) after compensation
is then undistorted. However, whereas H(f) is a simple FIR filter,
G(f) is not.
correlation peak can potentially be broadened. Thus,
there will be no bias in the pseudorange
5
estimate,
but the standard deviation of the error in the estimated
platform location may be increased. In a later section
we sho w that this compensation filter G(f)israrely
needed. Unfortunately the compensation filter H
¤
(f)
does not cure the range estimation problems caused
by signal multipath. Signal multipath causes [17] the
correct GPS signal transform S(f) to be replaced by
S(f)(1 + ¢(f)), where ¢ is of the form
¢(f)=
R
m
X
r=1
a
r
exp(¡i2¼f ¿
r
)(23)
and R
m
is the number of multipath sources, and a
r
, ¿
r
are the strength and delay of each multipath scatterer,
respectively. Note that because the GPS satellite is
at a different location than the interferers, the signal
multipath delays are different than the interferer
multipath delays. In the presence of signal multipath,
(22) is replaced by
C
0
(¿)=
Z
1
¡1
df P (f)jH(f)j
2
[1 + ¢(f)]exp(¡i2¼f ¿):
(24)
Unless ¢(f) is quite small it is evident from (24) that
the signal multipath can introduce both a bias and a
broadening of the correlation peak.
V. SUBOPTIMUM SPACE-FREQUENCY PROCESSING
We now wish to explore a suboptimum approach
[18] that may possibly reduce the computational
complexity without greatly sacrificing performance.
One method that readily suggests itself is to process
in the frequency rather than the time domain. That
is, perhaps we can split the operating band B into
M subbands of bandwidth B=M , and then in each
subband calculate weights for each antenna that
cancels the interference by placing spatial nulls on
interferers within that bin while preserving the signal.
If the discrete Fourier transform (implemented as
a fast Fourier transform (FFT)) really did behave
as a “brick wall” filter bank that fully isolated each
frequency bin from all others, we expect that this
procedure would rival the full time-domain processing
discussed earlier. However, because of leakage
between bins, even with well-designed windows,
it is necessary to consider what is happening in
some adjacent bins when attempting to minimize
interference in a particular frequency bin. We now
discuss a method for accounting for this spillover
among bins.
5
Because the GPS satellite and receiver clocks are not identical, a
GPS system does not measure true range, but rather pseudo-range.
If the same clock were used then the differential delay would yield
true range.
552 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 2 APRIL 2000