## A note about non-matched pulse filtering

This is a short note about the losses cause by non-matched pulse filtering in the demodulation of a PAM waveform. Recently I’ve needed to come back to these calculations several times, and I’ve found that even though the calculations are simple, sometimes I make silly mistakes on my first try. This post will serve me as a reference in the future to save some time. I have also been slightly surprised when I noticed that if we have two pulse shapes, let’s call them A and B, the losses of demodulating waveform A using pulse shape B are the same as the losses of demodulating waveform B using pulse shape A. I wanted to understand better why this happens.

Recall that if $$p(t)$$ denotes the pulse shape of a PAM waveform and $$h(t)$$ is a filter function, then in AWGN the SNR at the output of the demodulator is equal to the input SNR (with an appropriate normalization factor) times the factor$$$\tag{1}\frac{\left|\int_{-\infty}^{+\infty} p(t) \overline{h(t)}\, dt\right|^2}{\int_{-\infty}^{+\infty} |h(t)|^2\, dt}.$$$This factor describes the losses caused by filtering. As a consequence of the Cauchy-Schwarz inequality, we see that the output SNR is maximized when a matched filter $$h = p$$ is used.

To derive this expression, we assume that we receive the waveform$y(t) = ap(t) + n(t)$with $$a \in \mathbb{C}$$ and $$n(t)$$ a circularly symmetric stationary Gaussian process with covariance $$\mathbb{E}[n(t)\overline{n(s)}] = \delta(t-s)$$. The demodulator output is$T(y) = \int_{-\infty}^{+\infty} y(t) \overline{h(t)}\, dt.$The output SNR is defined as $$|\mathbb{E}[T(y)]|^2/V(T(y))$$. Since $$\mathbb{E}[n(t)] = 0$$ due to the circular symmetry, we have$\mathbb{E}[T(y)] = a\int_{-\infty}^{+\infty} p(t)\overline{h(t)}\,dt.$Additionally,$\begin{split}V(T(y)) &= \mathbb{E}[|T(y) – \mathbb{E}[T(y)]|^2] = \mathbb{E}\left[\left|\int_{-\infty}^{+\infty} n(t)\overline{h(t)}\,dt\right|^2\right] \\ &= \mathbb{E}\left[\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} n(t)\overline{n(s)}\overline{h(t)}h(s)\,dtds\right] \\ &= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \mathbb{E}\left[n(t)\overline{n(s)}\right]\overline{h(t)}h(s)\,dtds \\ &= \int_{-\infty}^{+\infty} |h(t)|^2\, dt. \end{split}$Therefore, we see that the output SNR equals$\frac{|a|^2\left|\int_{-\infty}^{+\infty} p(t) \overline{h(t)}\, dt\right|^2}{\int_{-\infty}^{+\infty} |h(t)|^2 dt.}.$

The losses caused by using a non-matched filter $$h$$, in comparison to using a matched filter, can be computed as the quotient of the quantity (1) divided by the same quantity where $$h$$ is replaced by $$p$$. This gives$\frac{\frac{\left|\int_{-\infty}^{+\infty} p(t) \overline{h(t)}\, dt\right|^2}{\int_{-\infty}^{+\infty} |h(t)|^2\, dt}}{\frac{\left|\int_{-\infty}^{+\infty} |p(t)|^2\, dt\right|^2}{\int_{-\infty}^{+\infty} |p(t)|^2\, dt}}=\frac{\left|\int_{-\infty}^{+\infty} p(t) \overline{h(t)}\, dt\right|^2}{\int_{-\infty}^{+\infty} |p(t)|^2\, dt\cdot \int_{-\infty}^{+\infty} |h(t)|^2\, dt}.$

We notice that this expression is symmetric in $$p$$ and $$h$$, in the sense that if we interchange $$p$$ and $$h$$ we obtain the same quantity. This shows that, as I mentioned above, the losses obtained when filtering waveform A with pulse B coincide with the losses obtained when filtering waveform B with pulse A. This is a clear consequence of these calculations, but I haven’t found a way to understand this more intuitively. We can say that the losses are equal to the cosine squared of the angle between the pulse shape vectors in $$L^2(\mathbb{R})$$. This remark makes the symmetry clear, but I’m not sure if I’m satisfied by this as an intuitive explanation.

As an example, let us compute the losses caused by receiving a square pulse shape, defined by $$p(t) = 1$$ for $$0 \leq t \leq \pi$$ and $$p(t) = 0$$ elsewhere, with a half-sine pulse shape filter, defined by $$h(t) = \sin t$$ for $$0 \leq t \leq \pi$$ and $$h(t) = 0$$ elsewhere. This case shows up in many different situations. We can compute the losses as indicated above, obtaining$\frac{\left(\int_0^\pi \sin t \, dt\right)^2}{\int_0^\pi \sin^2t\,dt\cdot \int_0^\pi dt} = \frac{2^2}{\frac{\pi}{2}\cdot\pi}= \frac{8}{\pi^2}\approx -0.91\,\mathrm{dB}.$

## Trying to observe the Vega-C MEO cubesats

On July 13, the Vega-C maiden flight delivered the LARES-2 passive laser reflector satellite and the following six cubesats to a 5900 km MEO orbit: AstroBio Cubesat, Greencube, ALPHA, Trisat-R, MTCube-2, and CELESTA. This is the first time that cubesats have been put in a MEO orbit (see slide 8 in this presentation). The six cubesats are very similar to those launched in LEO orbits, and use the 435 MHz amateur satellite band for their telemetry downlink (although ALPHA and Trisat-R have been declined IARU coordination, since IARU considers that these missions do not meet the definition of the amateur satellite service).

Communications from this MEO orbit are challenging for small satellites because the slant range compared to a 500 km LEO orbit is about 10 times larger at the closest point of the orbit and 4 times larger near the horizon, giving path losses which are 20 to 12 dB higher than in LEO.

I wanted to try to observe these satellites with my small station: a 7 element UHF yagi from Arrow antennas in a noisy urban location. The nice thing about this MEO orbit is that the passes last some 50 minutes, instead of the 10 to 12 minutes of a LEO pass. This means that I could set the antenna on a tripod and move it infrequently.

As part of the observation, I wanted to perform an absolute power calibration of my SDR (a USRP B205mini) in order to be able to measure the noise power at my location and also the power of the satellite signals power, if I was able to detect them.

On January 13, the SpaceX Transporter-3 mission launched many small satellites into a 540 km sun-synchronous orbit. Among these satellites were DELFI-PQ, a 3U PocketQube from TU Delft (Netherlands), which will serve for education and research, and EASAT-2 and HADES, two 1.5U PocketQubes from AMSAT-EA (Spain), which have FM repeaters for amateur radio. The three satellites were deployed close together with an Albapod deployer from Alba orbital.

While DELFI-PQ worked well, neither AMSAT-EA nor other amateur operators were able to receive signals from EASAT-2 or HADES during the first days after launch. Because of this, I decided to help AMSAT-EA and use some antennas from the Allen Telescope Array over the weekend to observe these satellites and try to find more information about their health status. I conducted an observation on Saturday 15 and another on Sunday 16, both during daytime passes. Fortunately, I was able to detect EASAT-2 and HADES in both observations. AMSAT-EA could decode some telemetry from EASAT-2 using the recordings of these observations, although the signals from HADES were too weak to be decoded. After my ATA observations, some amateur operators having sensitive stations have reported receiving weak signals from EASAT-2.

AMSAT-EA suspects that the antennas of their satellites haven’t been able to deploy, and this is what causes the signals to be much weaker than expected. However, it is not trivial to see what is exactly the status of the antennas and whether this is the only failure that has happened to the RF transmitter.

Readers are probably familiar with the concept of telemetry, which involves sensing several parameters on board the spacecraft and sending this data with a digital RF signal. A related concept is radiometry, where the physical properties of the RF signal, such as its power, frequency (including Doppler) and polarization, are directly used to measure parameters of the spacecraft. Here I will perform a radiometric analysis of the recordings I did with the ATA.

## More data from Voyager 1

Back in September, I showed how to decode the telemetry signal from Voyager 1 using a recording made with the Green Bank Telescope in 2015 by the Breakthrough Listen project. The recording was only 22.57 seconds long, so it didn’t even contain a complete telemetry frame. To study the contents of the telemetry, more data would be needed. Often we can learn things about the structure of the telemetry frames by comparing several consecutive frames. Fields whose contents don’t change, counters, and other features become apparent.

Some time after writing that post, Steve Croft, from BSRC, pointed me to another set of recordings of Voyager 1 from 16 July 2020 (MJD 59046.8). They were also made by Breakthrough Listen with the Green Bank Telescope, but they are longer. This post is an analysis of this set of recordings.

## 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.

Published
Categorised as Maths Tagged ,

## Voyager-1 single dish detection at Allen Telescope Array

This post has been delayed by several months, as some other things (like Chang’e 5) kept getting in the way. As part of the GNU Radio activities in Allen Telescope Array, on 14 November 2020 we tried to detect the X-band signal of Voyager-1, which at that time was at a distance of 151.72 au (22697 millions of km) from Earth. After analysing the recorded IQ data to carefully correct for Doppler and stack up all the signal power, I published in Twitter the news that the signal could clearly be seen in some of the recordings.

Since then, I have been intending to write a post explaining in detail the signal processing and publishing the recorded data. I must add that detecting Voyager-1 with ATA was a significant feat. Since November, we have attempted to detect Voyager-1 again on another occasion, using the same signal processing pipeline, without any luck. Since in the optimal conditions the signal is already very weak, it has to be ensured that all the equipment is working properly. Problems are difficult to debug, because any issue will typically impede successful detection, without giving an indication of what went wrong.

I have published the IQ recordings of this observation in the following datasets in Zenodo:

## Second Moon observation with my QO-100 station.

In May 25, the Moon passed through the beam of my QO-100 groundstation and I took the opportunity to measure the Moon noise and receive the Moonbounce 10GHz beacon DL0SHF. A few days ago, in July 22, the Moon passed again through the beam of the dish. This is interesting because, in contrast to the opportunity in May, where the Moon only got within 0.5º of the dish pointing, in July 22 the Moon passed almost through the nominal dish pointing. Also, incidentally this occasion has almost coincided with the 50th anniversary of the arrival to the Moon of Apollo 11, and all the activities organized worldwide to celebrate this event.

The figure below shows the noise measurement at 10366.5GHz with 1MHz and a 1.2m offset dish, compared with the angular separation between the Moon and the nominal pointing of the dish (defined as the direction from my station to Es’hail 2). The same recording settings as in the first observation were used here.

The first thing to note is that I made a mistake when programming the recording. I intended to make a 30 minute recording centred at the moment of closest approach, but instead I programmed the recording to start at the moment of closest approach. The LimeSDR used to make the recording was started to stream one hour before the recording, in order to achieve a stable temperature (this was one lesson I learned from my first observation).

The second comment is that the maximum noise doesn’t coincide with the moment when the Moon is closest to the nominal pointing. Luckily, this makes all the noise hump fit into the recording interval, but it means that my dish pointing is off. Indeed, the maximum happens when the Moon is 1.5º away from the nominal pointing, so my dish pointing error is at least 1.5º. I will try adjust the dish soon by peaking on the QO-100 beacon signal.

The noise hump is approximately 0.085dB, which is much better than the 0.05dB hump that I obtained in the first observation. It may not seem like much, but assuming the same noise in both observations, this is a difference of 2.32dB in the signal. This difference can be explained by the dish pointing error.

The recording I have made also covers the 10GHz Amateur EME band, but I have not been able to detect the signal of the DL0SHF beacon. Perhaps it was not transmitting when the recording was made. I have also arrived to the conclusion that the recording for my first observation had severe sample loss, as it was made on a mechanical hard drive. This explains the odd timing I detected in the DL0SHF signal.

The next observation is planned for October 11, but before this there is the Sun outage season between September 6 and 11, in which the Sun passes through the beam of the dish, so that Sun noise measurements can be performed.

## First Moon observation with my QO-100 station

A month ago, I spoke about planning the passes of the Moon through the beam of my QO-100 station. These give an occasion to observe the Moon without moving a dish that is pointing to Es’hail 2. The next opportunity after writing that post was on May 16 at 20:16 UTC.

Since I wasn’t going to be at home at that time, I programmed my computer to make a recording for later analysis. I recorded 4MHz of spectrum centred at 10367.5MHz using a LimeSDR connected to the LNB that I use to receive QO-100. The recording was planned to be 30 minutes long starting at 20:01 UTC, but for some reason only approximately 27 minutes were recorded.

This kind of events can be used to measure Moon noise and receive 10GHz EME signals. This post is an analysis of my recording, looking at these two things.

## Changes to the QO-100 NB transponder settings

Yesterday, AMSAT-DL announced that the narrowband transponder of QO-100 was under maintenance and that some changes to its settings would be made. This was also announced by the messages of the 400baud BPSK beacon. Not much information was given at first, but then they mentioned that the transponder gain was reduced by 6dB and a few hours later the beacon power was increased by 5dB.

Since I am currently doing continuous power measurements of the transponder noise and the beacons, when I arrived home I could examine the changes and determine using my measurements that the transponder gain was reduced by 5dB (not 6dB) at around 15:30 UTC, and then the uplink power of the beacons was increased by 5dB at around 21:00 UTC, thus bringing the beacons to the same downlink power as before. In what follows, I do a detailed analysis of my measurements.

## New dish for Es’hail 2 reception

I have replaced the dish I had for receiving Es’hail 2 by a new one. The former dish was a 95cm offset from diesl.es which was a few years old. I had previously used this dish for portable experiments, and it had been lying on an open balcony for many months until I finally installed it in my garden, so it wasn’t in very good shape.

Comparing with other stations in Spain, I received less transponder noise from the narrowband transponder of QO-100 than other stations. Doing some tests, I found out that the dish was off focus. I could get an improvement of 4dB or so by placing the LNB a bit farther from the dish. This was probably caused by a few hits that the dish got while using it portable. Rather than trying to fix this by modifying the arm (as the LNB couldn’t be held in this position), I decided to buy a new dish.