Talk on KiwiSDR TDoA is available on YouTube

This is just a short note that my recent talk on KiwiSDR TDoA at the Software Defined Radio Academy 2019, which took place during the German Ham Convention in Friedrichshafen, Germany, is now available on YouTube.

For other talks from this meeting see http://youtube.sdra.io/.

KiwiSDR GNSS position solutions using an extended Kalman filter

This is an update to this blog post, see also https://github.com/jks-prv/Beagle_SDR_GPS/issues/147. References for using Kalman filters for GNSS positioning are, e.g., DST-Group-TR-3260.pdf and these proceedings from ION GPSGNSS 2003.

The plots below show time series of variations in ECEF X,Y,Z coordinates, the oscillator correction, and the number of available satellites.

SPP - single point position; EFK - extended Kalman filter

The process noise covariances are set to 5e-5 for the X,Y,Z coordinates (KiwiSDRs are not expected to move) and the covariances for the time and the oscillator correction are build from Allan variance parameters as described in a497248. It would be interesting to measure the Allan variance of the used oscillator.

The advantages of using an extended Kalman filter include:

• Once a position solution has been obtained, the Kalman filter can be updated with fewer than four satellites: this will improve the availability of those KiwiSDRs with poor GNSS reception for TDoA work.
• The residuals per link in the Kalman filter update are used to exclude satellites with bad time measurements.
• Large deviations from the nominal position due to bad satellite geometries are avoided.

Kalman filter position solution for the KiwiSDR

This is work in progress, updates can be found here on GitHub.

Using an extended Kalman filter for TDoA HF geo-location

In this post the precision of geo-location using TDoA on shortwave is explored using the data shown on this blog before.

The idea is to feed the time series of time differences into an extended Kalman filter (EKF) using two models for the propagation time from the transmitter to the receiver:

1) Simple ray-geometry for oblique propagation in the ionosphere:

• based on arXiv:1104.2248 which uses the Breit and Tuve + Martyn theorems to convert vertical to oblique incidence
• using vertical electron density profiles from the IRI2016 model as an input
• group refractive index: 1/sqrt(1-f_N²/f²), i.e., no magentoionic effects are taken into account
• when integrating over the group refractive index in the vertical direction, analytic integration of 1/sqrt(f(h)) is performed in each height interval in order to avoid the spurious numerical singularity at the reflection point
• spherical coordinates are used (r_E=6371km)

2) Alternatively, great-circle distances on the WGS84 ellipsoid are used:

• geographiclib adapted for octave

All receiver coordinates are assumed to be in WGS84, however in practice it is suspected that there are deviations from the true locations of up to 20 km, as the KiwiSDR positions are provided by the KiwiSDR owners and are not taken from the GPS position solutions used for time synchronization.

The plot below shows the output of the EKF

• implementation in octave following ADA285972
• state = [lat, lon] of the unknown position
• measurements = time differences between pairs of KiwiSDRs
• the measurements covariance matrix (constant) is build from the covariances of time differences (and is set to zero for time differences not sharing a common receiver)
• state covariance matrix = diag(0.01, 0.01)² deg² (constant)
• start covariance = diag(0.1, 0.1)² deg²
• there are about 60 measurements which are run 5 times sequentially through the EKF

Extended Kalman Filter (EKF) output; note that the range of the x(y) axis corresponds to approximately 17(67)km.

The simple ionosphere propagation model improves the match to the known position but there is still a remaining bias of about 30km North, 4km West. Clearly, more studies are needed in order to find out if this improvement is by chance or real.