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Mutual Coupling Estimation for GPS Antenna Arrays in the Presence of Multipath
Zili Xu, Matthew Trinkle
School of Electrical and Electronic Engineering
University of Adelaide
PACal 2012
Adelaide
27/09/2012
1. Introduction
2. Maximum Likelihood DOA Estimation
3. Calibration Algorithm
4. Simulation results
5. Experiment results
Need for Calibration
Antenna arrays can be used for GPSinterference DOA estimation
DOA estimation accuracy affected by
– Channel Gain & phase errors
– Mutual coupling
– Receiver position errors
Errors To Be Calibrated
Calibrating source: GPS signals (supply DOA information,used as disjoint calibrating sources)
GPS signals are 15 dB below the noise
Apply matched filter to the particular GPS signal wewant to use for calibration => processing gain > 30 dB
Errors to be estimated:
1. Array orientation error
2. Gain/phase errors in each channel
3. Mutual coupling between antennas
Multipath Effect of Calibrating Source
• Multipath may be present on calibration sources.
• Causes errors in DOA and calibration parameters
• Likely to be correlated with direct path (only 2 MHz BW)
• DOA difficult to estimate
• Requires
• Spatial smoothing => would still allow MUSIC to work
• Maximum Likelihood Techniques => multi-dimensional
search requiring high computational load.
Maximum Likelihood Hood
Measured Phase & Amplitude of Signal
Modelled Phase & Amplitude from
direct and multipath signal
Amplitude and phase factor of
direct and multipath signal
Steering vectors based on
DOA of direct and multipath signal
Multi-dimensional search over all direct and multipath DOAs amplitude and phase factors
Assume fully correlated direct and multipath signal & M antenna elements
Full Cost functionUse N disjoint calibration sourcesInclude Mutual coupling and gain phase errors in model
Amplitude and phase of multipath
Amplitude and phase of direct path
Array orientation error
Steering vector of multi path
Measured phase and amplitude
Unknowns – shown in red
For each Calibration Source: DOA of multipath
Common: Orientation Error, Mutual Coupling Parameters, Gain Phase Errors
Gain Phase errors in each channel
Steering vector of direct signal
Solution Existence Condition
Consider current array configuration with N disjoint calibrating sources
(assume there is 1 multipath signal in each source & 8 antennas)
Array uncertainties:
Multipath DOA: 2N
Relative Gain/phase error: 2*7
Mutual Coupling parameters: 2(8/2)
Orientation error: 1
Least squares equations given by N disjoint sources: 2N(8-1)
2N+14+2(8/2)+1 ≤ 2N(8-1)
So N ≥ 3
3. Calibration Algorithm
Two Key steps:
Estimate phase and amplitude of N disjoint GPS signals at each antenna
Minimise:
Estimate gain & phase of the n’th GPS signal at each antenna
Column vector with Gain and phase of GPS signal, also includes highly correlated Multipath signals
Reduce dimensionality by replacing sn by its least squares estimate
Use Alternating Projections: Maximise with respect to one parameter while keeping others fixed.
Multipath DOA Estimation
Array Orientation Estimation
Algorithm Flow ChartInitialisation
Generate R for each satellite using matched filter
Use DOA of each satellite
Orientation error estimation
Multipath signal DoA estimation
Complex number estimation
Mutual coupling estimation
Repeat until
converge
Alternating Projection
Gain/phase estimation
Minimise
Convergence Check
If the difference in the previous and current cost function is larger than a
preset threshold ε,
Then another iteration of the algorithm is performed, otherwise the
calibration algorithm will stop and the current array orientation error,
mutual coupling matrix are taken to be the estimated results.
The Number of Multipath Estimation (possible solution)
Least squares based Wiener filter is applied to estimate the multipath signals in
the time domain
70 C/A code periods are used in the Wiener
filter and the filter has 60 taps
Multipath is one sample delayed
SNR of direct path signal: -24dB
SNR of multipath signal: -32dB
Simulation Scenario
7 elements circular array with 1 additional antenna in the centre (Radius = 1.25λ).
12 calibrating sources with the SNR of 20dB
250 snapshots
1st, 4th, 6th and 10th calibrating sources have 1 multipath signal with SNR of 10dB.
Orientation error: 10 degrees.
2. Gain/phase errors:
3. Mutual coupling matrix coefficients:
C 1 C 2 C 3 C 4
Gain (amplitude) 0.2 0.22 0.15 0.1
Phase (degrees) 174.3⁰ -133⁰ 277.9⁰ 240.7⁰
Ch1 Ch2 Ch3 Ch4 Ch5 Ch6 Ch7 Ch8
Gain 1 1.18 1.22 0.98 0.88 1.04 0.93 1.01
Phase 0⁰ 10.1⁰ 30.2⁰ -31.7⁰ 17.1⁰ -70.3⁰ 10.9⁰ 13.4⁰
Simulation Estimated Errors
The calibration algorithm converges after 25 iterations.
After the calibration, both of the gain/phase error and mutual coupling estimation are
quite accurate with the estimation errors below 1%.
Beampattern
Ideal beampattern (blue) and beampattern without
calibration (red), main beam steering direction:
180⁰, elevation angle = 0⁰
Ideal beampattern (blue) and beampattern with
calibration (green), main beam steering
direction: 180⁰, elevation angle = 0⁰
Experiment 1
1. Matched Filter Integration Time: 143ms
2. 5 satellites SV 11, 20, 23 30, 32 were obtained.3. SNR of about 18dB after integration.
4. Azimuth and Elevation angles of GPS signals
4. Mutual coupling model only considers coupling from adjacent
antennas.
5. Assume one multipath in each calibrating source (no number
of multipath estimation).
Azimuth Elevation
SV 11 314.3⁰ 48.1⁰
SV 20 226⁰ 44.4⁰
SV 23 279.5⁰ 35⁰
SV 30 52.6⁰ 10.1⁰
SV 32 194⁰ 64.8⁰
Antenna Array Geometry
8 elements monopole antenna array - 7 elements uniformly spaced
circular array with additional 1 element in the centre
Estimated Parameters
After 10 iterations, the calibration converges
Array orientation error
51.71⁰
Mutual coupling coefficients
C 1 C 2 C 3 C 4Gain
(amplitude)0.1085 0.0944 0 0
Phase
(degrees)-165.54 -15.72 0 0
The estimated results are:
The measured value was rotated to 52⁰ which was estimated
by a compass and the magnetic declination in Adelaide
The network analyser measurements of coupling
for C1and C2 are about -20dB
Multipath Estimation Results
SV 30:
The elevation angle of SV 30 is only 10.1⁰, so the
multipath could be caused by the ground reflection.
SV 20:
Elevation = 44.4°
Azimuth = 226°
Algorithm estimated multipath:
Azimuth = 93.5
Elevation = 48.5
Set north as x axis:
Azimuth = 93.5+51.7 = 145.2
SV 11 SV 20 SV 23 SV 30 SV 32
0.3536 0.3143 0.3088 0.2836 0.3043
0.0826 0.1502 0.1001 0.1324 0.1006
-12.6dB -6.4dB -9.8dB -6.6dB -9.6dB
Higher than others
Estimated orientation angle
Experiment 2 Smaller Aperture Array
Differences from the Experiment 1
1. The aperture of the antenna is reduced to 10cm (originally 25cm).
Expect more obvious and higher mutual coupling .
2. 18 calibrating sources – longer data collection time
3. Mutual coupling model includes the coupling from all the antennas - smaller aperture.
4. 300 snapshots (originally 143 snapshots) – better GPS Doppler frequency estimation
Experiment Result Comparing with CST Simulation
Simulation
Estimated mutual coupling coefficients
C 1 (S2,1) C 2 (S3,2) C 3 (S4,2) C 4 (S5,2)
Coupling Gain
(dB)-14dB -13.8dB -20dB -22.1dB
Estimation from the experiment
C1
C2
C4C3
C1 = -13 dB
C2 = -12 dB
C3=C4 = -24 dB
References
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