Robust Beamforming for GNSS Synthetic Antenna Arrays

ION GNSS 2009, Session A5, Savannah, GA, 22-25 September 2009 1/15

Robust Beamforming for GNSS Synthetic

Antenna Arrays

Tao Lin†, Ali Broumandan†, John Nielsen‡, Cillian O’Driscoll† and Gerard Lachapelle†

Position Location And Navigation Group

† Department of Geomatics Engineering

‡Department of Electrical and Computer Engineering

Schulich School of Engineering

University of Calgary

BIOGRAPHY

Tao Lin is a Ph.D. student in the PLAN Group of the

Department of Geomatics Engineering at the University

of Calgary. He received his BSc. from the same

department in May 2008. His research interests include

the fields of GNSS software receiver design, digital signal

processing, satellite-based navigation, inertial navigation

and ground-based wireless location.

Ali Broumandan is a Ph.D. candidate in the PLAN

Group of the University of Calgary. He holds a BSc.

degree from the Department of Electrical Engineering, K.

N. Toosi University of technology (2003) and a MSc.

degree from the Department of Electrical and Computer

Engineering, University of Tehran (2006). His current

research focuses on cellular network based positioning,

array processing, detection and estimation theory.

Dr. John Nielsen is an Associate Professor in the

Department of Electrical and Computer Engineering. Two

main areas of his research are Ultra-Wideband technology

that is applicable for high rate data communications and

short-range imaging radar. The other area is mobile

positioning based on TOA/AOA using CDMA and GPS

signals.

Dr. Cillian O’Driscoll received his Ph.D. in 2007 from

the Department of Electrical and Electronic Engineering,

University College Cork. His research interests are in the

area of software receiver for GNSS, particularly in

relation to weak signal acquisition and ultra-tight

GPS/INS integration. He is currently with the Position,

Location And Navigation (PLAN) group at the

Department of Geomatics Engineering in the University

of Calgary.

Dr. Gérard Lachapelle is a Professor of Geomatics

Engineering at the University of Calgary where he is

responsible for teaching and research related to location,

positioning, and navigation. He has been involved with

GPS developments and applications since 1980. He has

held a Canada Research Chair/iCORE Chair in wireless

location since 2001.

ABSTRACT

The impetus for this paper and the underlying research is

to investigate and develop robust beamforming algorithms

with synthetic antenna arrays for GNSS signals. The

concepts of synthetic array based beamforming and

Angle-Of-Arrival (AOA) estimation for Global

Navigation Systems (GNSS) signals are described in this

paper. The core idea is the utilization of GNSS antenna

motion to synthesize a virtual antenna array for GNSS

spatial processing (beamforming and AOA estimation).

Since only a single antenna is used, the amount of data for

processing is the same as other conventional temporal or

frequency processing techniques. However, the

visualization of received signals in the spatial domain

provides a great potential for signal separation, such as

interference and multipath mitigation. Several classical

beamforming techniques previously developed for a

physical real array are applied to a synthetic array based

system. Their performance is compared theoretically in

terms of the antenna beamforming beam-pattern. Angle of

arrival estimation for GNSS signals is successfully

demonstrated with both simulated datasets from an

advanced hardware simulator and field measurements.

The potential synthetic array based spatial processing for

the case of unknown antenna trajectory is introduced at

the end of the paper.

INTRODUCTION

The performance of Global Navigation Satellite Systems

(GNSS) strongly depends on signal reception quality

which is degraded by the presence of in-band interferer

signals and multipath. Sophisticated beamforming


algorithms, based on multiple antennas, can

simultaneously track the desired satellite signal via

several relevant multipath directions as well as tracking

the in-band interferer signals. Subsequent processing

would attempt to null out interference/multipath thereby

improving parameter estimation for the desired satellite

signal.

Conventional beamforming techniques are based on

multi-element antenna arrays which are physically bulky

and hardware intensive. This renders them unsuitable for

integration into a small handheld portable GNSS receiver.

Hence such devices are typically limited to a single small

antenna. However, the potential spatial processing gains

associated with the antenna array can be obtained if the

antenna is physically moved as the signal is being

captured by the receiver, which is equivalent to realizing a

spatially distributed synthetic antenna array. In addition,

unlike a physical real array, a synthetic antenna array is

not affected by inter-channel phases, gains and mutual

coupling between antenna elements. Therefore, it does not

require calibration, which is a serious problem in multi-

antenna array processing.

The concept of a synthetic array based on a single moving

antenna has been applied in radar for several decades, and

has been introduced to GNSS community in the past two

years. In (Broumandan et al 2007), a synthetic array based

AOA estimation algorithm to estimate the direction of the

incoming interference signals was proposed. (Pany et al

2008 a) and (Pany et al 2008 b) have demonstrated the

potential application of a synthetic antenna array with

circular antenna motion for synthesizing a uniform

circular array in multipath mitigation. In (Soloviev et al

2009), a fast Fourier transform (FFT) - based method was

applied for beamforming with a synthetic antenna array.

As mentioned in (Pany et al 2008 a) and (Soloviev et al

2009), one of the major advantages of synthetic array is

that the effective aperture is motion dependent. A very

large array aperture can be synthesized, which will be

beneficial for multipath mitigation and GNSS-based

synthetic aperture radar (SAR) imaging. However, the

effective aperture for our targeted application –

interference and multipath mitigation for a handheld

device – could be relatively small, depending on user

motion. Therefore, some adaptive beamforming

algorithms are required for synthetic arrays to provide

additional spatial zeros in the direction of undesired

signals. Adaptive beamforming algorithms, including

AOA estimation based algorithms such as Minimum

Variance Distortionless Response (MVDR), Minimum

Power Distortionless Response (MPDR) and reference

signal based algorithms such as Least Mean Squares

(LMS) have been studied for a multi-element antenna in

the wireless communication and radar communities for

many years and have demonstrated the capability for

interference suppression.

Needless to say, for reference signal based algorithms,

reference signals are required. In wireless communication,

a training sequence is used as a reference signal for

adaptive beamforming. For GNSS, the navigation data

bits from the data channel could be used as the reference

signals. However, in this case, the conflict between the

LMS beamforming algorithm and the Phase Lock Loop

(PLL) needs to be resolve (De Lorenzo 2007). This

conflict arises because both LMS and the PLL will try to

track the phase component of the signal. To optimally

apply LMS for GNSS adaptive beamforming, LMS and

PLL should be redesigned and combined together.

In this paper, the focus will be on the AOA estimation

based adaptive beamforming algorithms. Essentially the

robustness will degrade in the presence of AOA estimate

mismatch of the desired signals and the undesired signals

(i.e. interference and multipath). Several robust

beamforming algorithms such as Linear Constraint

Minimum Variance (LCMV) beamformer and Linear

Constraint Minimum Power (LCMP) beamformer have

been developed for improving robustness in the presence

of AOA mismatch (Van Trees, 2002). For AOA

estimation, three algorithms (Beam Scan, Minimum

Variance Distortionless Response (MVDR) and Multiple

Signal Classification (MUSIC)) previously developed for

physical real array are implemented and tested with the

simulated datasets from an advanced hardware simulator

and the collected field measurements.

SYNTHETIC ARRAY CONCEPTS

The core of synthetic array processing is the

transformation from the temporal samples received by a

moving antenna to spatial samples. When an antenna is

moving, the receiver receives signals with different

phases at different times. If the propagation channel is

stationary, this transformation is trivial. Figure 1

demonstrates the concept of a synthetic array. Assuming

there are M real GNSS antennas located uniformly on a

circle, signal samples are collected about every tsynthetic.

seconds. The batch of samples collected by each antenna

can be divided into M equal parts. If the channel is

stationary during the data collection period, the signal

blocks used for beamforming or AOA estimation can be

asynchronous in time. A synthetic array uses this property

to create a virtual antenna array. Instead of collecting M×

tsynthetic second signals, a single rotating antenna collects

signals represented by shaded cells in Figure 1.


Figure 1 Synthetic array (Broumandan et al 2007)

In reality, the propagation channel is not stationary,

because GNSS satellites are not static, and satellite clocks

and receiver clocks have different levels of drifts.

Therefore the carrier phase of the received signal samples

collected at different time and spatial points is due to not

just the motion of the antenna but also the effect of

satellite motion, satellite and receiver clock drifts. To

successfully perform spatial processing (e.g.

beamforming and AOA estimation), the phase/Doppler

change due to satellite motion, satellite and receiver clock

drifts need to be compensated prior to using the correlator

outputs for beamforming. If the location of the antenna is

exactly or approximately known, the satellite motion and

satellite clock drift can be compensated with ephemeris

information. An alternative approach is to use the Doppler

estimates from a static reference antenna for satellite

motion and satellite clock drift compensation (potential

extension of Assisted GPS), which is similar to the

concept of single differencing between the rover and the

base in a GNSS navigation solution. As shown in (Pany et

al 2008 b) the receiver clock drift can be further removed

by differencing between two satellites from the same

antenna. If a circular motion is used for a synthetic array,

the Doppler caused by the circular motion tends to be

sinusoidal, while the net Doppler caused by the satellite

motion and clock drift is approximately linear. In other

words, the Doppler due to the antenna motion can be

approximately extracted by de-trending the Doppler.

However, the net Doppler caused by the satellite motion

and clock drift is not exactly linear. This approximation

will degrade the performance of the synthetic spatial

processing (especially the AOA estimation with MUSIC).

In addition, the navigation data also must be wiped off for

synthetic array processing. At the post-correlation level

after tracking, the navigation data can be obtained from

the sign estimate of the prompt correlator. At the pre-

correlation level, external navigation data aiding is

required. For future GNSS signals, synthetic array

processing can be performed in the pilot channel.

Figure 2 Doppler estimates from a moving antenna

and a static antenna

In order to apply the complex spatial coefficients – the

steering vector/array manifold vector – to the received

signal samples, the relative antenna motion/trajectory

needs to be known precisely (on the order of cm given

that the spacing between two virtual antenna points

should be less than half of the carrier wavelength to avoid

spatial aliasing). In (Broumandan et al 2007) and (Pany et

al 2008 b), a controllable rotation table was used to

support the synthetic array system. Broumandan et al

2007 considered the use of a MEMS Inertial

Measurement Unit (IMU) to estimate the antenna motion.

In (Soloviev et al 2009), the trajectory was estimated

based on a GPS/INS navigation solution after fixing

carrier phase ambiguities. In this research work, a

controllable rotation table was used in the field test

demonstration since the main focus of this paper is the

signal processing part of synthetic array processing. The

estimation of the trajectory will be investigated fully in

future work. A synthetic array based GNSS software

receiver with aiding from an IMU is being developed. The

system diagram is shown below:

Figure 3 Proposed system

Compared to the common vector based/ultra-tight GNSS

software receivers, the major differences are the

interference detector and the spatial multipath mitigation

unit placed before and after the baseband processor. The

relative trajectory will be provided by the integrated


navigation solution. If the spatial correlation matrix can

be estimated precisely, the need of the relative trajectory

will be eliminated as will be explained at the end of the

paper.

SIGNAL MODEL FOR SYNTHETIC ARRAYS

In this section, the signal model for synthetic arrays is

introduced. As a starting point, it is assumed that there is

only GNSS signal and white Gaussian noise in the

received samples. At the post-correlation level, assuming

the local PRN code is perfectly synchronized with the

incoming signal, the prompt correlator is given as:

( ) ( ) ( ) ( ){ }

( ) ( ) ( )

, , exp ,

ˆ, , ,coh

t t t p

t t coh t T

P t a t d t j t n

t t t T

φ

φ φ φ −

= ∆ +

∆ = − −

r r r

r r r (1)

where, tr is the position vector with respected to the

reference point of the synthetic array (e.g. the centre of

the circular array) at epoch t , a is the signal amplitude,

d is the navigation data, R is the PRN code correlation

power, n is the white Gaussian noise, φ∆ is the carrier

phase difference, cohT is the coherent integration time.

As mentioned previously, the navigation data and the

carrier phase difference due to satellite motion and clock

drifts need to be compensated. For post-correlation

beamforming, since the carrier phase is tracked by the

PLL in the baseband processor, back compensation of the

correlator is required, which tries to restore the full carrier

phase information back to the correlator so that all the

signal information will be contained in the compensated

correlator. Assuming the signal power does not change

rapidly during the synthetic spatial processing, the

compensated correlator is:

( ) ( )

( ) ( ) ( )

( ) ( ){ }

1

0

0 0

ˆ, ,

, , ,

, exp ,

k

k k i i

i

comp k k k ref k k

comp k comp k p

t t

t t t

P t a j t n

φ φ

φ φ φ

φ

=

=

= −

= +

∑r r

r r r

r r

(2)

where, ( ),k ktφ r is the total carrier phase due to antenna

motion, satellite motion and clock drifts at epoch k,

( )ˆ ,i itφ r is the estimated carrier phase from PLL at epoch

i, ( ),ref k ktφ r is the carrier phase due to satellite motion

and clock drifts only at epoch k, n is the noise after the

compensation, ( )0 ,comp ktφ r is the compensated carrier

phase which is only due to antenna motion.

The compensated correlators can be thought as

representing the correlators from an antenna array. These

“spatial correlators” can be rewritten in a matrix form as:

( )

( ){ }

( ){ }

0 0

0

0 1

exp ,

exp ,

comp

comp p p

comp N

j t

t a

j t

φ

φ −

= +

r

P n

r

M (3)

where, pa is the amplitude of the prompt correlator, N is

the number of synthetic antennas.

The correlators above can be rewritten as a function of the

LOS signal array manifold vector ( )skv .

( ) ( )

{ }

{ }

0

0

1

exp

exp

cos cos2

cos sin

sin

Ts

comp p s p p

Ts N

s

j

t s k s

j

θ ϕπ

θ ϕλ

θ

−

= = +

=

k r

P v n

k r

k

M

(4)

where θ is the elevation angle, ϕ is the azimuth angle,

ps is signal at the reference point of the array.

The correlation matrix of the prompt correlator is as

follows:

( ) ( ) ( ) ( )2 20 0

HH

p s s nE t t s k k σ

= = +

xR P P v v I (5)

BEAMFORMING ALGORITHMS

In this section, several classical beamforming algorithms

will be discussed. Although these algorithms were

developed for physical arrays, they can be applied to

synthetic arrays after the compensation process, which

removes phase changes due to satellite motion, satellite

clock drifts and receiver clock drifts. In general,

beamforming is achieved by multiplying the spatial

correlators (at the post correlation level) or spatial signal

samples (at the pre-correlation level) by a complex weight

vector w . The algorithms described here are all linear;

therefore they can be applied at the pre-correlation level

or the post-correlation level depending on the application.

In general, interference detection/mitigation is applied at

the pre-correlation level, while multipath mitigation is

performed at the post-correlation level.


Delay-and-sum Beamformer

The Delay-and-sum beamformer, one of the fundamental

beamformers, is the optimum beamformer in white

Gaussian noise (Van Trees, 2002). It can be considered as

the spatial matched filter that can maximize the array

signal-to-noise ratio (ASNR) in the white noise channel.

The mathematical expression for the weighting vector of

the Delay-and-sum beamformer is (Van Trees, 2002):

( )1

skN

=w v (6)

Where N is the number of virtual antennas in the array;

( )skv is the true steering vector or array manifold vector

of the LOS GNSS signal.

MVDR/MPDR Beamformer

The MVDR beamformer is optimum in both maximum

likelihood and maximum Array Signal to Noise Ratio

(ASNR) sense in correlated or uncorrelated Gaussian

noise channels. Here, the term “noise” is meant in a

generic sense, which includes interference, multipath and

white noise. In fact, the MVDR beamformer can be

viewed as the spatial generalized matched filter. The cost

function of this beamformer is (Van Trees, 2002):

( ) ( )*1 1H H Hn s sF k kλ λ = + − + −

w R w w v w v (7)

where w is the array weighting vector; nR is the

correlation matrix of the unwanted signals (noise,

interference and multipath); ( )skv is the true steering

vector/array manifold vector.

The weight vector of the MVDR beamformer can be

shown as (Van Trees, 2002):

( )

( ) ( )

1

1

Hs nH

mvdr Hs n s

k

k k

−

−=

v Rw

v R v (8)

In practice, it is difficult to obtain nR , since the signal

component is always embedded in the received

measurements. Therefore an alternative beamformer

named MPDR was proposed. The cost function of the

MPDR is defined as (Van Trees, 2002):

( ) ( )*1 1H H Hs sF k kλ λ = + − + −

xw R w w v w v (9)

where w is the array weighting vector; xR is the

correlation matrix of the received samples (signal, noise,

interference and multipath); ( )skv is the true steering

vector.

The weight vector of the MPDR beamformer can be

shown as (Van Trees, 2002):

( )

( ) ( )

1

1

HsH

mpdr Hs s

k

k k

−

−=

x

x

v Rw

v R v (10)

If the AOA of the Line-Of-Sight (LOS) GNSS signal is

known perfectly, the performance of MVDR and MPDR

will be the same. However, the performance of MPDR

degrades more rapidly than MVDR when the LOS AOA

mismatch appears (Van Trees, 2002). One major

difference between the Delay-and-sum beamformer and

the MVDR/MPDR beamformer is that the MVDR/MPDR

beamformer takes into account the interference and

multipath in the channel. The inverse operation of the

unwanted signal correlation matrix or received sample

correlation ( 1−nR and 1−

xR ) will tend to null out the

interference and multipath signals. In other words,

interference and multipath signals will be mitigated even

if their AOAs are unknown, given that the unwanted

signal correlation matrix nR is known precisely. In

reality, nR must also be estimated. Therefore, MVDR

and MPDR are theoretically optimum but not practically

robust. In should be noted that forward-backward

smoothing techniques should be applied for any spatial

processing which needs to manipulate the correlation

matrix (eg. inversion and eigen-decomposition), since

multipath signals are coherent with the LOS signals (Van

Trees, 2002).

LCMV/LCMP Beamformer

LCMV and LCMP are the generalized versions of MVDR

and MPDR. They allow multiple linear constraints in the

cost function. The cost functions for LCMV and LCMP

are shown below (Van Trees, 2002):

*H H H HJ λ λ = + − + −

nw R w w C g C w g (11)

*H H H HJ λ λ = + − + −

xw R w w C g C w g (12)


correlation matrix of the unwanted signals (noise,

interference and multipath); xR is the correlation matrix

of the received samples (signal, noise, interference and

multipath); C is the constraint matrix that contains all the

steering vectors used in the defined constraints. g is the


constraint vector that contains all the numerical values for

all the constraints. The weight vectors for LCMV and

LCMP can be shown as (Van Trees, 2002):

1

1 1H H H Hlcmv

−− − =

n nw g C R C C R (13)

1

1 1H H H Hlcmp

−− − =

x xw g C R C C R (14)

Instead of utilizing only the distortionless constraint in

MVDR/MPDR, various constraints can be found in the

implementation of LCMV/LCMP. Four constraints

(distortionless constraint, distortion constraint, null

constraint and derivative constraint) are typically used.

Multi-Beam Beamformer

The beamformers introduced above can be applied for

multi-directional beamforming, although only one LOS

steering vector was used in the equations above. For

GNSS signals, there is only one desired signal after

correlation because of the pseudo-orthogonal nature the

spreading codes. The synthetic array beamforming in this

research is mostly performed at the post-correlation level.

In that case, beamforming will be performed in each PRN

tracking channel. In other words, 12 beamformers are

required for a 12 tracking channel GNSS receiver.

Synthetic array beamforming can be performed at the pre-

correlation level as well. In that case, only one multi-

beam beamformer (beams pointing at all satellites in

view) is required. The weight vectors for the

MVDR/MPDR based the multi-beam antenna array are

given as:

( )

( ) ( )

1

1

smvdr H

s s

k

k k

−

−=

n

n

R Vw 1

V R V (15)

( )

( ) ( )

1

1

smpdr H

s s

k

k k

−

−=

x

x

R Vw 1

V R V (16)


unwanted signals (noise, interference and multipath)

correlation matrix; xR is the received samples (signal,

noise, interference and multipath) correlation matrix, 1 is

a 1N × vector of ‘1’s, N is the number of synthetic

antennas, ( )skv is the true steering vector/array manifold

vector.

To illustrate these concepts, an antenna beam pattern for a

multi-beam synthetic array is simulated. An antenna is

moving circularly with a speed of 0.5 revolutions per

second to synthesize a virtual circular array. The radius of

the circle is 1.2 m. 7 satellites are simulated with

elevation and azimuths angles shown in the following

table:

Table 1 Simulated elevation and azimuth angles

Elevation (deg) Azimuth (deg)

PRN 3 53 315

PRN 6 65 60

PRN 8 77 220

PRN 13 30 15

PRN 18 45 90

PRN 21 30 150

PRN 22 40 260

Figure 4 Multi-Beam antenna beam pattern

In this simulation, satellite motion, satellite clock drift and

receiver clock drift are assumed to be completely

compensated before beamforming with a synthetic array.

Seven peaks circled in red in the plot correspond to the

simulated PRNs in space. The power of the antenna beam

pattern is normalized by the maximum value and coded in

colors. Some secondary peaks with less power can be

found in the plot, which are due to the constructive and

destructive interference of the side-lobs of the beams

which point towards the seven satellites.

BEAMFORMING BEAM-PATTERN

To illustrate some of the beamforming algorithms and the

effect of the antenna aperture of a synthetic array, a few

simulated antenna beam patterns are given below. For

these simulations, the directions of the GNSS signals are

assumed to be known and the compensation process is

performed at the correlator outputs prior to synthetic

beamforming.

Circular Synthetic Array

In the simulation, an antenna is moving circularly with an

angular speed of 0.5 revolutions/s. Two antenna apertures

are considered, with radius of 0.2 m and 1 m. The


synthetic array beamforms at the GNSS signal direction

( 53 , 315Elevation Azimuth= =o o) at the post-correlation

level. When the radius of the trajectory is 0.2 m (a small

circular trajectory), the beam pattern from a MVDR

beamformer in a pure noise channel (no

interference/multipath) is shown in Figure 5. When an

interference signal at (elevation = 60o , azimuth = 120o )

is added into the channel, the beam pattern from a MVDR

beamformer is shown in Figure 6. In Figure 7, the beam

pattern of a LCMV beamformer is provided for the same

interference and noise channel. The null point on the

interference direction is circled in red in both figures. For

the LCMV beamformer, the direction of the interference

signal is assumed to be known or estimated at the pre-

correlation level and five null constraints (the interference

direction and the four points located on elevation and

azimuth with 5± o offset from the interference direction)

and one distortionless constraint (at the GNSS signal

direction) are applied. Similarly, the beam patterns for the

case of 1 m radii are shown from Figure 8 to Figure 10.

Figure 5 MVDR beamformer, circular trajectory

radius = 0.2 m with eLOS = 53 deg, aLOS = 315 deg


radius = 0.2 m with eLOS = 53 deg, aLOS = 315 deg,

eInterference = 60 deg, aInterference = 120 deg

Figure 7 LCMV beamformer, circular trajectory

radius = 0.2 m with eLOS 53 deg, aLOS = 315 deg,

eInterfernce = 60 deg, aInterference = 120 deg, additional 4 nulls with +/- 5 deg from eLOS and aLOS


radius = 1 m with eLOS = 53 deg, aLOS = 315 deg


radius = 1 m with eLOS = 53 deg, aLOS = 315 deg,

eInterference = 60 deg, aInterference = 120 deg


Figure 10 LCMV beamformer, circular trajectory

radius = 1 m with eLOS 53 deg, aLOS = 315 deg,

eInterfernce = 60 deg, aInterference = 120 deg, additional 4 nulls with +/- 5 deg from eLOS and aLOS

From the beam patterns shown above, the larger the

radius, meaning the larger the antenna aperture, the better

the directivity exhibited. Assuming no AOA mismatch on

the array manifold vector, the motion which synthesizes a

large antenna aperture will more effectively mitigate the

undesired signals (interference and multipath) from other

directions. For some applications (e.g. a handheld GNSS

receiver), the size of the antenna aperture may be limited.

Therefore placing additional nulls in the spatial domain

becomes important for strong interference mitigation.

Given a precise estimate of the correlation matrix, the

MVDR beamformer can null out the interference signals

without estimating the interference AOAs. The null size

changes as the antenna aperture size changes. The larger

the antenna aperture, the smaller the null size is observed.

In practice, the estimate of the correlation matrix contains

errors. Usually, additional null points around the main

interference direction are placed to enlarge the null space

to count for the inaccurate estimation of the interference

AOA and the correlation matrix. In general, the spacing

between the null points around the interference direction

is proportional to the standard deviation of the

interference AOA estimate.

Linear Synthetic Array

In this case, an antenna is simulated moving with a linear

trajectory and a speed of 1 m/s towards the North. The

virtual antenna array is synthetic based on the linear

motion. Since the nulling effect will be similar to the

results in the circular synthetic array, only the noise

channel is considered here. The beam patterns with a

small and a large antenna aperture are plotted in Figure 11

and Figure 12 respectively. Similar to the observation in

the circular array case, the array with a larger antenna

aperture provides a better directivity thus attenuates

interference and multipath more severely.

Figure 11 Small aperture, linear trajectory, eLOS = 80

deg, aLOS = 60 deg

Figure 12 Large aperture, linear trajectory, eLOS = 80

deg, aLOS = 60 deg

Rectangular Synthetic Array

In (Soloviev et al 2009), the combination of a synthetic

array and a physical real array is used to construct a

synthetic rectangular synthetic array. The beam patterns

with a small and a large antenna aperture are shown

below to illustrate the concepts. In this case, a physical

antenna array is linearly moving towards the North with a

speed of 20 km/h. The virtual antenna array is synthetic

based on the linear motion along the motion, while the

physical array is used to provide the spatial information

perpendicular to the motion. A similar conclusion can be

made from the beam pattern shown below.


Figure 13 Rectangular array, small aperture on the

moving direction, eLOS = 75 deg, aLOS = 120 deg

Figure 14 Rectangular array, large aperture on the

moving direction, eLOS = 75 deg, aLOS = 120 deg

AOA ESTIMATION ALGORITHMS

As shown in previous sections, beamforming with a

synthetic array can enhance the directivity of the antenna

thus mitigating interference and multipath. One of the

major requirements of beamforming is the array manifold

vector. In other words, the trajectory of the motion and

the AOA of the GNSS signals should be known. In this

section, three algorithms for AOA estimation will be

discussed briefly.

Beam Scan Algorithm The Beam Scan algorithm is one of the most classical

AOA estimation methods. It estimates AOA by scanning

the signal power at each possible angle of arrival and

selects the one with the maximum power. The power from

a particular direction is measured by first forming a beam

in that direction and setting the beamformer weights equal

to the array manifold vector corresponding to that

particular direction. The mathematical formula for this

method is as follows (Van Trees, 2002):

( )

( )( ){ }( )( ) ( )( ){ }

( )

( )( ){ }( )( ) ( )( ){ }

2

2

ˆ arg max ,

arg max ,

arg max , ,

ˆ arg max ,

arg max ,

arg max , ,

Ha

Ha a

Ha

Ha a

P

E k

k k

P

E k

k k

θ

θ

θ

ϕ

ϕ

ϕ

θ θ ϕ

θ ϕ

θ ϕ θ ϕ

ϕ θ ϕ

θ ϕ

θ ϕ θ ϕ

=

=

=

=

=

=

x

x

v x

v R v

v x

v R v

(17)

where θ̂ is the estimated elevation angle, ϕ̂ is the

estimated azimuth angle, ( )( ),ak θ ϕv is the assumed

array manifold vector at elevation θ and azimuth ϕ ,

( ),P θ ϕ is the power at elevation θ and azimuth ϕ , x is

the vector of received samples (raw samples or

compensated correlator outputs) and xR is the sample

correlation matrix.

Capon’s Algorithm

Similar to the previous algorithm, Capon’s algorithm also

estimates AOA by measuring the power of received

signals in all possible directions. The major difference is

the weight vector. Instead of using the assumed array

manifold vector, it uses the weight from the MPDR

beamformer based on the assumed direction. In other

words, the power from the AOA estimate is measured by

constraining the signal power gain to be unity from the

assumed direction of arrivals and using the remaining

degrees of freedom to minimize the contribution to the

output power from all other directions. The mathematical

formula is as follows (Van Trees, 2002):

( )( )( )

( )( ) ( )( )

( ){ }( ) ( ){ }

( ){ }( ) ( ){ }

2

2

,,

, ,

ˆ arg max ,

arg max , ,

ˆ arg max ,

arg max , ,

a

Ha a

H

H

H

H

k

k k

E

E

θ

θ

ϕ

ϕ

θ ϕθ ϕ

θ ϕ θ ϕ

θ θ ϕ

θ ϕ θ ϕ

ϕ θ ϕ

θ ϕ θ ϕ

=

=

=

=

=

-1x

-1x

x

x

R vw

v R v

w x

w R w

w x

w R w

(18)




array manifold vector , x is the vector of received samples

and xR is the sample correlation matrix.

MUSIC Algorithm

The MUSIC algorithm belongs to the family of subspace

algorithms, which can provide AOA or frequency

estimates with super-resolution. It can be easily shown

that the array manifold vectors corresponding to the

incoming signals lie in the signal subspace and are

therefore orthogonal to the noise subspace. One way to

estimate AOAs is to search through the set of all possible

directions and find those that are orthogonal to the noise

subspace. Given the array manifold vector of the

incoming signals ( )( ),sk θ ϕv , and the matrix

corresponding to the noise subspace nQ , then the

following should hold

( )( ),H

s nk θ ϕ =v Q 0 (19)

Based on the equation above, the mathematical formula

for MUSIC is (Van Trees, 2002):

( )

( )( ) ( )( )

( )

( )( ) ( )( )

ˆ arg max ,

1arg max

, ,

ˆ arg max ,

1arg max

, ,

MUSIC

H Ha n n a

MUSIC

H Ha n n a

P

k k

P

k k

θ

θ

ϕ

ϕ

θ θ ϕ

θ ϕ θ ϕ

ϕ θ ϕ

θ ϕ θ ϕ

=

=

=

=

v Q Q v

v Q Q v

(20)



array manifold vector, ( ),P θ ϕ is the signal power, nQ is

the noise subspace matrix, which can be extracted from

the sample correlation matrix.

Although MUSIC algorithm theoretically provides AOA

estimates with super-resolution, the correlation matrix and

the number of the desired signals should be known. In

practice, these parameters need to be estimated. The

accuracy of these estimates will strongly affect the

performance of MUSIC.

DEMONSTRATION FOR AOA ESTIMATION

This section uses the simulated datasets generated from a

Spirent GSS7700 advanced hardware GNSS signal

simulator and the experimental datasets collected in the

field to demonstrate the AOA estimation of GPS L1 C/A

signal with a circular synthetic array. The AOA estimates

from the methods presented in the previous section will

be compared to the elevation and azimuth estimated from

the GPS navigation solution. Since the AOA estimate

from the navigation solution is used as the reference, an

open sky environment is chosen for the testing. As

mentioned earlier, the carrier phase due to satellite motion

and clock drifts need to be compensated before the

synthetic spatial processing. To compensate the satellite

motion, the location of the antenna or the Doppler

information from a nearby static reference antenna is

needed. However, this information will not be available in

some applications. As mentioned previously, the linear

curve fitting method does not require this information.

The main objectives of this test is to demonstrate the GPS

satellite AOA estimation with a synthetic array and

compare the performance from the linear curve fitting

method and the correlator based differential processing,

which provides the compensated correlators with the

phase information due to only the antenna motion .

Results with Simulated Data from the Hardware

Simulator

In the simulated scenario, an antenna was moving

circularly with a speed of 2 revolutions per seconds. The

radius was 1 m. A National Instruments

downconverter/digitizer was used to capture the raw IF

GPS L1 signal samples at a sampling rate of 10 MHz. The

AOAs (elevation and azimuth angles) of the GNSS

signals were estimated at the post-correlation level after

PLL. The coherent integration time was 20 ms.

Correlator-Based Differential Method

In the Spirent hardware simulator, it was configured that a

static antenna was placed at the centre of the circular

motion to provide the Doppler caused by the satellite

motion for the synthetic array processing. The two

antenna output ports were connected to the dual-channel

NI front-end. It should be noted that the same clock was

used inside the NI front-end to drive both channels.

Therefore the clock drifts from two channels will be

exactly the same. In other words, the correlator-based

differential processing will completely remove the carrier

phase due to the net effect of the satellite motion and the

front-end clock drift. The AOA spectrum from Beam

Scan, Capon and MUSIC are shown in Figure 15 to

Figure 17 for PRN 3. The estimated elevation and

azimuth angles were compared to the ones from the

navigation solution as shown in Figure 18 . The result for


PRN 5 is shown in Figure 19. From the results above,

MUSIC and Capon algorithms as expected provide higher

resolution on the AOA estimation than the Beam Scan

algorithm. The estimated elevation and azimuth angles

match to those from the navigation solution.

Figure 15 AOA spectrum from BeamScan with

correlation based differential processing on PRN3

Figure 16 AOA spectrum from Capon with correlation

based differential processing on PRN3

Figure 17 AOA spectrum from MUSIC with

correlation based differential processing on PRN3

Figure 18 Azimuth and Elevation estimates from

MUSIC, Capon and BeamScan with correlation based

differential processing compared to navigation solution on PRN3


MUSIC, Capon and BeamScan with correlation based

differential processing compared to navigation solution on PRN5

Linear Curve Fitting Method

In this case, the satellite motion was assumed to be linear,

and the Doppler due to the antenna motion was

approximately extracted by de-trending the Doppler. The

results for PRN 3 and PRN 5 are shown below.

Comparing these AOA spectrum plots with the ones from

the correlator based differential processing, the AOA

spectrum and the estimated AOA are similar for PRN 3

(high elevation) but different for PRN 5 (relatively low

elevation). In Figure 22 the AOA spectrum for PRN 5 is

noisy. The noise floor increases in Figure 23. Also a 1.5

degree bias on the estimated elevation from all three

methods compared to the elevation from the navigation. It

is suspected that these errors are due to the linear

approximation of the satellite motion and clock drifts, and

the relatively lower C/N0 observed at low elevations.



MUSIC, Capon and BeamScan with linear curve

fitting compared to navigation solution on PRN3

Figure 21 AOA spectrum from BeamScan method

with linear curve fitting on PRN5

Figure 22 AOA spectrum from Capon method with

linear curve fitting on PRN5

Figure 23 AOA spectrum from MUSIC method with





Results with experimental data collected in the field

Experimental data collected on the roof top of the CCIT

building at the University of Calgary was used to

demonstrate the AOA estimation with a synthetic array.

The antenna was mounted on a circular rotation table

rotating at a speed of 2 revolutions per second with a

radius of 66.5 cm. The NI front-end was again used to

capture the GPS L1 signals. The AOAs (elevation and

azimuth angles) of the GNSS signals were estimated at

the post-correlation level after the PLL. The coherent

integration time was 20 ms. A picture of the system is

shown in Figure 25. The linear curve fitting method was

used to remove the major part of carrier phase due to

satellite motion. The navigation data was wiped off based

on the sign estimates of the prompt correlator values in

the PLL. The AOA spectrums from the proposed methods

are plotted in the figures below. The results from the field

experimental data are similar to what we have observed

with the data generated from the Spirent Hardware

simulator. The linear curve fitting method seems to

perform better for high elevation satellites than low

elevation satellites. For satellites with low elevation, some


random biases (on the magnitude of 2 – 8 degrees) were

observed from the AOA estimates with the proposed

methods. It is suspected that these errors are due to the

linear approximation of the satellite motion and clock

drifts, and the lower C/N0 observed at low elevations.

Figure 25 Picture of the rotating antenna system

Figure 26 AOA spectrum from BeamScan method

with linear curve fitting on PRN3

Figure 27 AOA spectrum from Capon method with


Figure 28 AOA spectrum from MUSIC method with








BEAMFORMING WITH UNKNOWN AND

RANDOM TRAJECTORY


The materials presented in the previous sections are

mainly for an antenna moving in some regular trajectory

(e.g. circular motion). In reality, the trajectory of the

antenna is based on the motion of the user, which could

be unknown and irregular. The section will explain the

concepts of beamforming techniques utilizing the

dominant eigen-vector of the sample correlation matrix.

The sample correlation matrix based on the prompt

correlation values shown in (5), can be further

manipulated using eigen-decomposition.

( ) ( )2 2

2

1 1

Hp s s n

K NH H

i i i n i i

i i K

s k k σ

λ σ= = +

= +

= +∑ ∑

xR v v I

e e e e (21)

where K is the number of dominant eigen-values in the

correlation matrix (in the case of GNSS signal at the post-

correlation level in a noise only channel, K should be 1.),

N is the number of virtual antennas used in the synthetic

array, ie is the ith eigen-vector, iλ is the ith eigen-value,

and 2nσ is the noise variance.

Assuming there are K desired signals, the following can

be easily proven (Zoltowski & Gecan 1995).

( ) ( ) ( ){ }{ }

1 2

1 2

, ,

, ,

s s sK

K

span k k k

span=

v v v

e e e

L

L

(22)

In synthetic beamforming, both the trajectory and motion

of the antenna and the AOA of the desired signals are

required to construct the array manifold vector. However

since the vector space spanned by the array manifold

vector is the same as that spanned by the eigen-vectors

extracted from the sample correlation matrix,

beamforming can be performed based on the eigen-

vectors of the desired signals. In the previous section, it

was mentioned that the inverse operation of the sample

correlation 1−xR will tend to null out the interference and

multipath signals. This can be explained nicely with the

eigen-decomposition of the sample correlation matrix

(Zoltowski & Gecan 1995).

1

21 1

2

21

21

1 1

11

1

K NH H

i i i i

i ni i K

KHn

i i

in i

KH

i i

n i

λ σ

σ

λσ

σ

−

= = +

=

=

= +

= − −

≈ −

∑ ∑

∑

∑

xR e e e e

I e e

I e e

(23)

Based on (10), (22) and (23) it can be proved that the

weight vector w in MPDR will tend to null out strong

signal sources which are not the desired signals

(Zoltowski & Gecan 1995).

A simulation is used to illustrate the concept of eigen-

based synthetic beamforming with a random trajectory.

The antenna was assumed to move randomly on a 2D

plane. The data collection points of the random trajectory

are plotted in Figure 31. Assuming three desired signals

are coming from azimuths 30o , 60o and 75o , the antenna

pattern of the beamforming based on the eigen-

decomposition of the sample correlation matrix is shown

in Figure 32.

Figure 31 Random trajectory

Figure 32 Beamforming at azimuths 30 deg, 60 deg

and 75 deg

This technique can be applied for interference mitigation

as well. In another simulation, with the same random

trajectory, the signals coming from azimuths 30o , 60o

and 75o are strong interference signals. The desired signal

is very weak and below the noise floor (GNSS signals

before dispreading). In this case, the dominant eigen-

vectors in the correlation matrix are from the interference

signals. Utilizing this property, the strong interference can

be blocked by applying the projection matrix formulation


with the dominant eigen-vectors from the correlation

matrix (Van Trees, 2002). The beam pattern is shown in

Figure 33. The beam response is flat in all directions

except those with interference signals, since the desired

signals are so weak (below noise floor) they can be

neglected in the correlation matrix, and the synthetic array

tries to form beams to noise, which is not directional.

Figure 33 Nulls at azimuths 30 deg, 60 deg and 75 deg

Although the AOAs of the incoming signals and the

trajectory are not required for eigen-based beamforming,

the knowledge of the correlation matrix is required. The

accuracy of the estimated sample correlation matrix is

critical for eigen-vector based beamforming. The

estimation of the correlation matrix will be investigated in

future work.

CONCLUSIONS

The generation of a GNSS synthetic phased array using a

moving antenna was described. Several classical

beamforming techniques previously developed for a

physical real array were investigated in the context of a

synthetic array based system. Their performance was

compared theoretically in terms of the antenna

beamforming beam-pattern. A novel approach based on a

linear curve fit of the received signal Doppler was

presented for the compensation of the effect of satellite

motion on the synthetic array. The AOA estimation for

GNSS signals was successfully demonstrated with

datasets from both an advanced hardware simulator and

field measurements. The linear curve fitting approach was

shown to provide results comparable with results where

the correlator compensation is achieved through the use of

a reference antenna. A potential technique for synthetic

array based spatial processing for the case of random

unknown trajectory was also introduced.

ACKNOWLEDGMENTS

The financial support of Research In Motion, the Natural

Science and Engineering Research Council of Canada,

Alberta Advanced Education and Technology and the

Western Economic Diversification Canada is

acknowledged.

REFERENCES

Broumandan, A., T. Lin, A.R.A. Moghaddam, D. Lu, J.

Nielsen and G. Lachapelle (2007) Direction of Arrival

Estimation of GNSS Signals Based on Synthetic Antenna

Array. Proceedings of GNSS07 (Forth Worth, 25-28 Sep,

Session C2), The Institute of Navigation, 11 pages

De Lorenzo, D (2007) Navigation Accuracy and

Interference Rejection for GPS Adaptive Antenna Arrays.

Ph.D. Dissertation, Stanford Uniersity, August, 2007.

Pany, T., M. Paonni, and B. Eissfeller (2008 a) Synthetic

Phased Array Antenna for Carrier/Code Multipath

Mitigation. Proceedings of ENC-GNSS08, Toulouse

Pany, T., B. Eissfeller (2008 b) Demonstration of

Synthetic Phased Array Antenna for Carrier/Code

Multipath Mitigation. Proceedings of GNSS08

Soloviev, A., F. van Graas, S. Gunawardena, M. Miller

(2009) Synthetic Aperture GPS Signal Processing:

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ITM09

Trees, H. L.V. (2002): Optimum Array Processing, part

IV, Detection, Estimation, and Modulation Theory, John

Wiley & Sons, Inc., New York, 2002.

Zoltowski, M. and A. S. Gecan, (1995) Advanced

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Documents

Robust Beamforming for GNSS Synthetic Antenna Arrays