Posts – Page 12 – Daniel Estévez

Computing the symbol error rate of m-FSK

How to compute the symbol error rate rate of an m-FSK modulation is something that comes up in a variety of situations, since the math is the same in any setting in which the symbols are orthogonal (so it also applies to some spread spectrum modulations). I guess this must appear somewhere in the literature, but I can never find this result when I need it, so I have decided to write this post explaining the math.

Here I show an approach that I first learned from Wei Mingchuan BG2BHC two years ago during the Longjiang-2 lunar orbiter mission. While writing our paper about the mission, we wanted to compute a closed expression for the BER of the LRTC modulation used in the uplink (which is related to \(m\)-FSK). Using a clever idea, Wei was able to find a formula that involved an integral of CDFs and PDFs of chi-squared distributions. Even though this wasn’t really a closed formula, evaluating the integral numerically was much faster than doing simulations, specially for high \(E_b/N_0\).

Recently, I came again to the same idea independently. I was trying to compute the symbol error rate of \(m\)-FSK and even though I remembered that the problem about LRTC was related, I had forgotten about Wei’s formula and the trick used to obtain it. So I thought of something on my own. Later, digging through my emails I found the messages Wei and I exchanged about this and saw that I had arrived to the same idea and formula. Maybe the trick was in the back of my mind all the time.

Due to space constraints, the BER formula for LRTC and its mathematical derivation didn’t make it into the Longjiang-2 paper. Therefore, I include a small section below with the details.

Hermes-Lite 2 external 10 MHz reference

Interested by the forthcoming HamSci December 2021 eclipse festival of frequency measurement, I have decided to enable and test the external 10 MHz input of my Hermes-Lite 2 DDC/DUC HF transceiver. This will allow me to use a GPSDO (the Vectron MD-011 which has appeared in other posts) to reference the Hermes-Lite 2 in order to measure frequency accurately.

Decoding DART

DART, the Double Asteroid Redirection Test, is a NASA mission that launched last Wednesday from Vandenberg. The goal of this mission is to crash the spacecraft into the small asteroid Dimorphos, allowing us to measure the small change in the orbit of the asteroid caused by the impact.

From the communications perspective, this spacecraft is the first to use a Spiral Radial Line Slot Array (RLSA) as high-gain antenna. Details about the antenna design can be seen in this paper. The paper shows that antenna polarization is LHCP. Most DSN communications use RHCP, although there are a few notable exceptions (for instance Emirates Mars Mission), and the DSN stations are equipped to handle both polarizations. I’m not sure if DART is indeed using LHCP or if this is just a matter of the convention in the definition of the polarization used in the paper (there are actually two opposite conventions to define the sense of circular polarization).

A few hours after launch, as the spacecraft passed over Europe, Miguel CT1BYM and Iban EB3FRN recorded the X-band telemetry signal from DART at 8421.79 MHz. This post is a first analysis of the signal.

Tianwen-1 remote sensing orbit

On November 8, the Tianwen-1 orbiter made a manoeuvre to move itself to the remote sensing orbit, as reported by Chinese media. This orbit is the final orbit in the mission, as depicted in this figure from Wikipedia. The main goal of this orbit is to study the geophysical properties of Mars with all the orbiter instruments (see this paper) and to continue acting as a communications relay for the rover Zhurong.

As usual, AMSAT-DL has been collecting telemetry from Tianwen-1 with the 20 metre antenna at Bochum observatory, including spacecraft state vectors. This allows us to study the orbit change manoeuvre and the properties of the remote sensing orbit. This post is a first look at the state vector data.

An update about Lucy’s telemetry

In my previous post I presented a description of the X-band telemetry of Lucy, including a GNU Radio decoder, some recordings from the Allen Telecope Array, and an analysis of some of the telemetry received in the two days following the launch. This is a short post with some updates about the telemetry analysis of the Lucy mission.

Decoding Lucy

Lucy is a spacecraft that will study the Trojan asteroids, during a twelve year mission. It was launched last Saturday at 9:34 UTC from Cape Canaveral on an Atlas V rocket. Its telemetry downlink is on X-band, at a frequency of 8445.768 MHz.

Iban Cardona EB3FRN made a 30 minute recording of the telemetry downlink at 19:00 UTC on Saturday, as the spacecraft first appeared over Europe after launch. r00t.cz did a brief analysis of this recording overnight, and then published some more details about the telemetry data. On Sunday, at 8:52 UTC, I did a long recording with one of the dishes in the Allen Telescope Array. This recording lasts 3 hours 26 minutes, and ends when the spacecraft set below the 16 degree elevation mask of the ATA. In this post I give a first analysis of the telemetry data in both recordings.

The recording done at ATA can be downloaded from the following datasets in Zenodo:

GNU Radio 3.9 in Buildroot

Recently I’ve had to cross-compile GNU Radio for an ARM embedded system. I have decided to use Buildroot to build GNU Radio and its dependencies, since I’m fairly familiar with using Buildroot to generate embedded Linux images. Earlier this year, Jean-Michel Friedt and
Gwenhaël Goavec-Merou presented in FOSDEM their work about adding a GNU Radio package in buildroot. They gave a talk called “Never compile on the target!“.

Unfortunately, the version they used was GNU Radio 3.8, and the package hasn’t been updated to GNU Radio 3.9 yet. I wanted to use GNU Radio 3.9, so I decided to try to update the Buildroot package. After some assorted problems, I have managed to get GNU Radio 3.9 running on my ARM target. The fixes I’ve done are really horrible, so I’ve been quite tempted not to share my changes. I’ve finally decide to share this even though it’s far from perfect, because it might save someone from having to replicate this work, and because if anyone wants to do this properly and update the upstream package, this could be useful as a starting point.

Tianwen-1 communications relay orbit

As you may have seen in my last post, lately I have been reviewing some of data we have from Tianwen-1. In the days following the landing of Zhurong, back in May, we had so much data in our hands that I couldn’t post about it in a timely manner. We were wondering if we could use this data to plan for a number of experiments with the 20 meter antenna at Bochum observatory. These included trying to receive data from the rover relayed by the orbiter, and trying to detect the rover’s direct X-band link to Earth. We didn’t manage to do any of these, unfortunately, as they had a great deal of luck involved.

During the summer I’ve been involved in several activities such as collaborating with the SETI Institute and BSRC REU summer student programmes by teaching some GNU Radio lessons, and preparing material for GRCon21 (a talk, a workshop and paper). Now I have more time at hand, so it’s good to revisit this data. In this post I’ll look at Tianwen-1’s orbit after the release of the lander.

Tianwen-1 attitude in Mars orbit

It has been a while since the last time I wrote an update about Tianwen-1’s attitude. In that post I showed that Tianwen-1 had changed to a sun angle of 0 degrees with respect to the spacecraft’s X axis on 2020-10-22. Since then, nothing changed, even after the spacecraft arrived to Mars orbit on February 2021, so I have been sloppy and ignored the ADCS telemetry for quite a while.

As you may have heard, the 20 metre antenna at Bochum observatory had a failure at the beginning of June due to a thunderstorm. It has been under repair until the end of August, when it became operational again. As AMSAT-DL has now returned to receive telemetry from Tianwen-1 at Bochum, I am now reviewing the new data to see if anything interesting might have happened in the last months.

I have updated my attitude study Jupyter notebook, since the previous version assumed that the spacecraft’s state vectors were heliocentric coordinates, so it didn’t work in Mars orbit, due to the change to Mars-centric coordinates.

When plotting the new data I have learned that Tianwen-1 changed attitude during the release of the lander on May 14, and has maintained this new attitude until the present day.

Decoding Voyager 1

Today is the 44th anniversary of the launch of Voyager 1, so I want to celebrate by showing how to decode the Voyager 1 telemetry signal using GNU Radio and some Python. I will use a recording that was done back in 30 December 2015 with the Green Bank Telescope in the context of the Breakthrough Listen project. Most of the data from this project is open data and can be accessed through this portal.

In contrast to other posts about deep space probes in this blog, which are of a very specialized nature, I will try to keep this post accessible to a wider audience by giving more details about the basics. Those interested in learning further can refer to the workshop “Decoding Interplanetary Spacecraft” that I gave in GRCon 2020, and also take a look at other posts in this blog.