Journal of Global Positioning Systems (2005)
Vol. 4, No. 1-2: 258-267
Null-steering LMS Dual-Polarised Adaptive Antenna Arrays for GPS
W C Cheuk, M Trinkle & D A Gray
CRC for Sensor Signal and Inf ormation Processing, Dept of Electrical and Electronic Engineering, University of Adelaide, Adelaide SA
e-mail: [e-mail] Tel: + 61 83036425; Fax: [phone]
Received: 27 November 2004 / Accepted: 12 July 2005
Abstract. The implementation of a null-steering antenna
array using dual polarised patch antennas is considered.
Several optimality criterion for adjusting the array
weights are discussed. The most effective criteria
minimises the output power of the array subject to
maintaining a right hand circular polarisation (RHCP)
response on the reference antenna. An unconstrained
form of this criteria is derived, in which the reference
channel is the RHCP output of the reference antenna and
the LHCP output of the reference antenna is included as
one of the auxiliary channels. An FPGA implementation
of the LMS algorithm is then described. To prevent
weight vector drift a variant of the circular leakage LMS
algorithm was used. The implementation details of a
simplified circular leakage algorithm more suited to an
FPGA implementation are presented. This simplified
leakage algorithm was shown to have a similar steady
state weight vector as the full algorithm.
Key words: GPS, Polarisation, Interference Mitigation,
Adaptive filters.
1 Introduction
A GPS receiver is relatively susceptible to interference
and a number of antenna and signal processing
techniques have been investigated to overcome this
deficiency. These include :
• Fixed antenna enhancements: Manz (2000),
Kunysz (2000).
• Single Channel Adaptive Filters: Dimos et al
(1995), Trinkle and Gray (2001).
• Adaptive Beamformers: Trinkle and Gray
(2001), Zoltowski and Gecan (1995), Jian et al
(1998), Fante and Vaccaro (2000).
• Polarisation Cancellers: Brassch et al (1998),
Trinkle and Cheuk (2003).
• Adaptive Beamformers with Polarisation
Diversity: Nagai et al (2002), Fante and Vaccaro
(2002), Compton (1981).
• Modifying the tracking loop of the GPS
receivers: Manz et al(2000),Legrand et al
(2000).
• Integrating GPS and INS sensors: Soloviev and
van Grass (2004).
In this paper combined spatial and polarisation null
steering are considered.
A single GPS antenna with an adaptive polarisation
response can be used to reject interferences with
polarisations other than the GPS signal. The GPS signal
uses RHCP (right hand circular polarisation), i.e., it
contains both horizontal and vertical polarizations 90
degrees out of phase; thus it is possible to remove any
linearly polarised signal with only a 3 dB loss in GPS
signal power see e.g. Brassch et al (1998). The key idea
of polarisation cancellers is thus to reject th e interference
by adaptively mismatching the polarisation response of
the antenna to the polarisation of the interfering signal.
Adaptive polarisation antennas can be implemented by
adaptively combining the two outputs of a dual polarised
patch antenna see e.g. Brassch et al (1998), Compton
(1981). Clearly this technique becomes ineffective if the
interference has the same polarisation as the GPS signal
(i.e., RHCP ) .
Single antenna polarisation cancellers operate purely in
the polarisation domain, while adaptive beamformers
operate only in the spatial (angle domain). It is possible to
combine these two domains in a single algorithm by
applying beamforming techniques to an antenna array
with dual polarised antenna elements. Using a such an
array, will approximately double the degrees of freedom
available for interference cancellation compared with
using a conventional antenna array see e.g. Jain (1998),
Trinkle and Cheuk (2003), Nagai et al (2002), and it has

Cheuk: Null-steering LMS Dual-Polarised Adaptive Antenna Arrays for GPS 259
recently been shown in Jian et al (1998) and Fante and
Vaccaro (2002) that GPS polarimetric antenna arrays can
cancel more interferences than a standard antenna array
with the same number of antenna elements. Thus
polarisation diversity can be used to significantly reduce
the size of GPS antenna arrays, which is important in
many applications.
Field Programmable Gate Array, FPGA, technology has
lately become an attractive alternative for the
implementation of a wide range of DSP applications
because of its flexibility and speed. An FPGA allows a
large number of multipliers and accumulators to be
configured and inter-connected in such a way as to suit a
particular algorithm. This fine-grain parallelism allows
most adaptive beamforming techniques to be
implemented much more efficiently than on a standard
DSP. Algorithms implemented on the largest of the
current generation FPGAs can achieve several hundred
Giga Operati o ns per Seconds (GOPs)
The FPGA implementation of an adaptively null-steering
dual-polarised antenna array is proposed. The novelty of
this approach lies in
• The choice of the reference and auxiliary
channels in the power minimisation algorithm.
• The use of a modified version of the complex
LMS algorithm to minimise computations.
• The use of Hilbert transforms to generate the
analytic signal.
• The use of the circular leakage LMS algorithms
rather than the standard leaky LMS to prevent
weight vector drift.
• A simplification of the circular leakage LMS
algorithm more suited to FPGA implementation.
2 Digital Null-Steering Structure
System Architecture
A polarimetric adaptive antenna array is placed in front of
the GPS receiver. Each element of the array is a patch
antenna with two outputs corresponding to the horizontal
and vertical polarizations of the received signal. See
Figure 1 for a four channel digital null-steering system
using two dual-polarized antenna elements.
Figure 1. Basic structure of digital null-steerer using a polarimetric antenna array
y : Beamformer output
w : Beamformer weight
vector
e : Estimation error
u : Antenna output.
HT : Hilbert Transform

260 Journal of Global Positioning Systems
In this example the horizontal component of the first
antenna is used as a reference channel as originally
proposed in Fante and Vaccaro (2002). The subscript
H denotes the horizontal polarization, V denotes the
vertical polarization, I denotes the real part of the
complex signal, and Q denotes the imaginary part of
the complex signal.
The output of each polarization channel is converted
into a digital signal using an analog to digital (A/D)
converter. Since the L1 band GPS signal is transmitted
at 1.575GHz and most A/D converters do not have an
input bandwidth large enough to sample the GPS
signal, a down-converter is used to convert the
incoming GPS signal to a lower intermediate
frequency. The digital signal from the A/D converter is
then fed into an FPGA, see Figure 1, which
implements further filtering and the complete adaptive
antenna array processing. The single channel output, y,
of the adaptive antenna array is then converted back to
the analogue domain by a D/A converter and after
further analogue filtering and mixing the signal can be
played into a standard GPS receiver.
Adaptive Al g orithm
Optimum null-steering algorithms are usually applied
to antenna arrays by simply minimising the output
power of a weighted sum of receiver outputs subject to
one of the multiplier weights (the reference channel)
being fixed at unity, see e.g. Trinkle and Gray (2001).
This leads to an optimum weight vector given by
cRc
cR
wH1
1
−
−
=
where
T
c]0,,0,0,1[ …=
and
R
is the covariance matrix of the receiver outputs
defined by
{
}
)()( kukuER H =.
The vector )(ku contains the antenna outputs at time
s
kt where s t is the sampling interval. With reference
to Figure 1, it is assumed that each channel is Hilbert
transformed allowing the use of the analytic (complex)
signal representation.
For polarimetric antenna arrays, better performance
may be achieved by choosing as the reference channel
the right hand circularly polarized component of the
reference antenna, thus avoiding a 3dB loss in SNR of
desired GPS signals in the absence of interferences see
e.g. Trinkle and Cheuk (2003). To handle the dual
channels of each polarised antenna the vector of
complex receiver outputs is written as:
(
)
T
NVNHVHVH kukukukukukuku ][],[],...[],[],[],[)( 2211
=
(2.1)
where
][][][kjukuku nHQnHInH +
=
and
][][][ kjukuku nVQnVInV +
=
.
For right hand circular polarisation the horizontal and
vertical components of the electric field are –90
degrees out of phase with each other. Thus, for an ideal
narrowband RHCP signal the complex vector of
outputs from the first reference channel, is given by
⎥
⎦
⎤
⎢
⎣
⎡
−
=
⎥
⎦
⎤
⎢
⎣
⎡
=j kts
ku
ku
ku s
V
H1
][
][
][
)(
1
1
1
Combining, with complex weights, H w1 and V w1, the
outputs of the horizontal and vertical channels, such
that the response due to a circularly polarised signal of
unit amplitude is fixed at unity implies that the
complex weights are constrained to satisfy
1
1
*
1
*=− VH jww .
Generalising for the full array, the constraint becomes
1=cwH
where
T
jc ]0,,0,,1[ …−= .
From Trinkle and Cheuk (2003), the optimum weight
vector becomes
cRc
cR
wH1
1
−
−
=. (2.2)
The advantage of this cons traint, is that it allows, for an
N channel receiver, N-1 nulls to be formed as opposed
to the previous approaches which restricted the number
of nulls to be N-2 see e.g. Trinkle and Cheuk (2003).
In order impose this constraint with an unconstrained
LMS adaptive algorithm, an orthogonal projection
matrix is used. This transformation takes the
horizontally and the vertically polarized complex
signals from the dual polarized antenna and converts
them to a RHCP signal and another signal that is
orthogonal to the RHCP signal (LHCP) as illu strated in
Figure 2 below.

Cheuk: Null-steering LMS Dual-Polarised Adaptive Antenna Arrays for GPS 261
Figure 2. Orthogonal pr ojec tion matrix.
The orthogonal projection matrix transformation is
given by
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡−
=
⎥
⎦
⎤
⎢
⎣
⎡
][
][
1
1
][
][
1
1
1
1
ku
ku
j
j
ku
ku
H
V
L
R
The outputs of the orthogonal projection matrix are
then fed to the adaptive algorithm. The RHCP channel,
][
1ku R, is the reference channel, as it has the same
polarization as the GPS signal. The orthogonal
projection of the RHCP channel, i.e., ][
1ku L is the first
auxiliary channel, which is multiplied by a weight
determined by the adaptive algorithm. The other
auxiliary channels are the vertical and horizontally
polarised outputs of the second anten na. Note that there
is no need to transform these outputs into right and left
circularly polarised components as the signals are
assumed narrow-band so the required phase shift can
be incorporated into the adaptive alg orithm.
To estimate the weights we could first estimate R from
the input data and then substitute into Eq. 2.2 to give
the desired weights. This approach was not taken as it
would require a matrix inversion which is not simple to
implement in an FPGA. To avoid direct matrix
inversion, gradient descent algorithms are used to
iteratively minimise the mean square error.
As the output of the beamformer needs to be converted
back to a real signal, a simplified version, of the
standard complex LMS algorithm was applied. The
trade-off between this algorithm and the full complex
LMS algorithm has been considered in Horowitz and
Senne (1981).
3 Implementation of the LMS Algorithm
The use of finite precision arithmetic in the LMS
algorithm can cause drift in the weight vectors,
Sethares et al (1986), particularly in the presence of
strong interferences. To overcome this a leakage factor
can be incorporated into the LMS algorithm. An
investigation of a number of leaky LMS algorithms is
carried out below.
Leaky LMS Algorithm
The leaky LMS algorithm prevents weight vector drift
in finite precision implementations by inserting a
leakage factor, L
α
, into the weight vector update loop.
This leakage factor avoids weight vector drift by
containing the energy in the impulse response of the
LMS adaptive filter see e.g. Haykin (2002). The leaky
LMS algorithm is given by :
{
}
][)()()1()1( kekukwkw L
µ
µα
+
−
=
+
(3.1)
where ][ke , the error signal, is given by
)()()()()(][][ 1kukwkukukwkdke H
R
H
′
−=
′
−=
and
()
T
NVNHVHLkukukukukuku ][],[....],........[],[],[)( 221
=
′
The difference between the leaky LMS and the
conventional LMS algorithm is the leakage factor
)1( L
µα
−
, the first term on the right hand side of the
Eq. 3.1. To stabilize the algorithm (i.e., to avoid
overflows), the leakage factor, L
α
, must satisfy the
condition
µ
α
1
0<≤L (3.2)
The leakage factor can also prevent stalling in the LMS
algorithm. Stalling occurs when the correction term
(i.e. )]()([ keku
µ
) in the weight update equation is
smaller than the least significant bit (LSB), thus the
LMS algorithm will stop adapting. Stalling can also be
prevented by dithering the error term.
R
u1
L
u1

262 Journal of Global Positioning Systems
A disadvantage of the leaky LMS algorithm is that it
requires more computational and hardware resources.
It also biases the weight vector away from the optimal
solution, which degrades the performance of the filter.
Since the average weight vector noise in the leaky
LMS will never converge to the zero, the MSE
(Minimum Square Error) is normally larger than for the
conventional LMS. Decreasing the value of L
α
will
reduce this problem, but if it is too small then stalling
will occur, as the estimation error is not noisy enough.
Circular Leakage LMS Algorithm
The circular leakage LMS described in Nascimento
and Sayed (1999) is another leakage-based algorithm
to prevent the drift problem. It has an advantage over
the leaky LMS algorithm as it does not introduce a bias
in the weight estimates. The circular leakage LMS
algorithm is a modified version of the
σ
−
switching
algorithm described in Ioannou and Tsakalis (1986),
but uses less hardware resources while maintaining the
same stabilization performance in the adaptive filter.
The weight update equation for the circular leakage
LMS can be written as
)]()([)())(1()1( kekukwkkw C
µ
µα
+−=+ (3.3)
where )(k
C
α
is a nonlinear and time-varying circular
leakage term. It is defined as follows
In the above equation the 0
α
is a pre-defined constant,
21
0CC << are pre-specified levels for the magnitudes
of the components of the estimated weight vectors and
{}
12
5.0 CCD −=. Their choice is not discussed here.
As indicted above the circular leakage term has four
different regions which determine the value of the
time-varying circular leakage term to be applied to the
weight vector recursion It is also only applied to one
channel at each iteration (i.e. in a multi-channel
antenna array each channel has a single tap weight
)(kwn, where n is the channel number). At the first
iteration, the magnitude of the weight vector
component corresponding to the first channel, i.e.,
)(
1kw is checked to determine into which of the
ranges of Eq 3.4 it falls. The circular leakage term
appropriate to that range is then applied to the weight
vector update equation. In the next iteration, the
magnitude of the weight vector component
corresponding to the second channel, i.e., )1(
2+kw is
examined to determine which circular leakage term to
apply to the update equation, etc. The procedure
continues until the weights of all channels have been
examined and then, in the next iteration, the process
returns to the first channel and cycles around again.
A typical circular leakage function plot is illustrated in
Figure 3 where the positive constants chosen are : 0
α
= 0.3, 1
C = 0.6, 2
C= 0.9. These values can be
calculated using the equations given in Nascimento and
Sayed (1999).
{}
{}
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎬
⎫
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
+<<
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
<≤+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
−
≥
=
otherwise
CCkwCif
D
Ckw
CkwCCif
D
kwC
Ckwif
k
n
n
n
n
n
C
,0
5.0)(
)(
2
)(5.0
)(
2
)(
)(
211
2
1
0
221
2
2
0
0
20
α
α
α
α
α (3.4)

Cheuk: Null-steering LMS Dual-Polarised Adaptive Antenna Arrays for GPS 263
Figure 3. Circular leakage fu n ct i o n .
When implementing the circular leakage in real time
on the FPGA, one of the main factors that needs to be
consider is latency. The standard LMS weight update
loop has a delay of one clock cycle. From Eq. 3.3 and
Eq. 3.4 the circular leakage algorithm requires six
multiplications, two subtractions and two divisions per
weight vector update. This introduces significant
latency in the weight estimation loop, requiring it to
take more than one clock cycle to complete all the
calculations. Such a delay would cause the LMS
algorithm to behave like the DLMS (Delayed Least
Mean Square algorithm), resulting in increased
convergence times and an overall loss in performance.
Modified Circular Leakage LMS Algori thm
As the adaptive null-steerer is to operate in real time,
the above circular leakage algorithm was replaced by a
simplified linear algorithm that required less arithmetic
operations. The linear equations for determining the
leakage factor C
α
are given by
Eq. 3.5 has the same four regions as Eq. 3.3, but since
the quadratic terms are replaced by linear ones the
modified function uses less multiplications and no
division and so its computational complexity is
significantly reduced. As a result, the modified
function can complete the weight update calculation in
one clock cycle, making it more suitable for a real time
implementation.
Figure 4 shows a plot of c
α
for the standard circular-
leakage (blue), and the modified circular-leakage (red).
As shown in the plot, the original function rises quickly
when the weight estimation reaches the 1
C (0.88)
bound, it then goes up almost linearly as w increases,
and then rolls off smoothly shortly before reaching the
2
C(0.93) bound.
{}
{}
⎪
⎪
⎭
⎪
⎪
⎬
⎫
⎪
⎪
⎩
⎪
⎪
⎨
⎧
+<<−
<≤++−
≥
=
otherwise
CCkwCifCw
CkwCCifw
Ckwif
n
n
n
C
,0
5.0)()(4.17
)(5.058.0)9133.0(2.5
)(
2111
221
20
α
α (3.5)
)(kwn

264 Journal of Global Positioning Systems
Figure 4. Simplified/s t andard circular leakage function with 93.0,88.0 21
=
=
CC .
The linear approximation has a slightly higher
numerical value of leakage factor (c
α
) at the same
weight value (w) compared with the standard one.
This ensures stability of the algorithm. Figure 5 shows
simulation results indicating that the modified circular
leakage algorithm remains stable and behaves almost
the same as the original one albeit at what appears to
be a higher level of misadjustment noise.
4 FPGA Implementation
This section considers the FPGA implementation of the
adaptive null-steerer. The FPGA has four main
processing modules including, digital decimation
filtering, Hilbert transform filtering and the
implementation of the LMS adaptive algorithm using
the simplified circular leakage algorithm. The signal
flow diagram is illustrated in Figure 6.
Figure 5 w for standard/simplified circular leakage 1667.0=
o
µα
, 93.0,88.0 21 =
=
CC .

Cheuk: Null-steering LMS Dual-Polarised Adaptive Antenna Arrays for GPS 265
Figure 6 FPGA signal flow diagram
In this section, the FPGA implementation of the
circular leakage algorithm is discussed.
Simplified Circular Leakage LMS Algorithm FPGA
Implementation
The Xilinx System Generator was used to implement
the code for the adaptive null-steerer. A schematic
block diagram which represents the structure of the
code is shown in Figure 7.
In Figure 7, there are 8 inputs on the left hand side
(these are from top to bottom: u1HI, u1HQ, u1VI,
u1VQ, u2HI, u2HQ, u2VI and u2VQ) and one output
on the right hand side. The top four inputs on the left
hand side are fed to the orthogonal projection block,
this block converts the horizontal polarization and
vertical polarization to a RHCP and LHCP output
signal. The RHCP channel is chosen to be the reference
channel (i.e. u_RHCP_I), which is treated as the
desired signal )(kdI. Then the next pair is the real part
and imaginary part of the signal from the orthogonal
projection of RHCP channel (i.e. u_RHCP(1+j)_I and
u_RHCP(1+j)_Q), then the next pair is u2HI and u2HQ
etc. So there are a total of one reference and three pairs
of auxiliary channels.
Each of the real / imaginary signals are fed to the LMS
based adaptive algorithm, shown in Figure 8. Then all
the outputs of each LMS block are fed into the adder
and subtracted from the desired signal to give the error
signal which is then is fed back to the LMS block to
calculate the updated weights. The LMS loop operates
continuously, at the input sample rate, and the optimum
weights are continuously estimated.

266 Journal of Global Positioning Systems
Figure 7 Four channel dig i t al n u l l -steerer - FPGA schematic.
Figure 8 LMS base algorithm block - FP GA sch ematic.

Cheuk: Null-steering LMS Dual-Polarised Adaptive Antenna Arrays for GPS 267
The circular leakage algorithm is implemented in
VHDL, as shown in Figure 7 in the block called
circular leakage. It has six inputs on the right hand side
and same number of outputs in left hand side. It
implements the if/then statement of Eq. 3.5. In each
iteration, it measures the weight estimate for one
channel, and outputs the leakage value according to the
Eq. 3.5. This value is then fed back to the leak_out
(Figure 8), and subtracted from the current weight
vector component. Then in the next iteration, the
circular leakage block will check the wof the next
channel.
Conclusions
The FPGA implementation of an adaptive LMS
algorithm suitable for dual polarised GPS antenna
arrays has been presented. The LMS algorithm
minimises the output power of the array while
maintaining unity gain for RHCP signals on the
reference antenna. To prevent weight vector drift in the
LMS algorithm a circular leakage algorithm was
implemented. A simplified version of the leakage
algorithm was derived which introduces minimal
additional bias and allows the LMS updates to occur at
the input data rate. The complete algorithm was
implemented on an FPGA using the Xilinx System
Generator.
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