### Decoding LilacSat-2 telemetry

After having my first QSO through the Harbin Institute of Technology amateur radio satellite LilacSat-2, I decided to give a serious try to the telemetry decoding software. This is available as a GNURadio module. A Linux distribution with all the proper software installed and configured is provided, for an easy use. However, I have used GNURadio in the past, so I wanted to try to setup the GNURadio module directly on my machine.

The web page for LilacSat-2 gives also a description of the different telemetry formats. The satellite has an SDR radio transmitting on 437.200MHz. This radio is used for the FM amateur radio transponder and also to transmit several different telemetry formats. The satellite also transmits telemetry on 437.225MHz, presumably using a different (non-SDR) radio and a different antenna, so that the satellite can keep transmitting telemetry even if the SDR system fails. Typically, when the FM transponder is disabled, the satellite will transmit 9k6 BPSK telemetry on 437.200MHz and 4k8 GFSK telemetry on 437.225MHz. These can be seen in the picture above, which was made using my RF recording and baudline. The packet on the upper right is 4k8 GFSK and the packet on the lower left is 9k6 BPSK. Notice the slight slant due to Doppler.

### Another look at recursive quadrature oscillators

In a recent post, we looked at which $$2\times 2$$ Toeplitz real matrices $$T$$ gave useful quadrature oscillators by the recurrence $$x_{n+1}=T x_n$$. There, we computed their eigenvalues and solved the recurrence in terms of them. Of course, there are many other ways to approach this problem. Here we look at another approach that gives a good geometric picture of what happens, can be applied to general $$2\times 2$$ matrices, and may be used as a starting point for the $$n\times n$$ case.

### A look at a new digital quadrature oscillator

Two sinusoidal signals are said to be in quadrature if they have a constant phase difference of 90º. Quadrature signals are widely used in signal processing. A digital quadrature oscillator is just an algorithm that computes the sequence $$x_n = (\cos(\omega n), \sin(\omega n))$$, $$n\geq 0$$, or a similar sequence of sinusoids in quadrature. Here $$\omega$$ is the oscillator frequency in radians per sample. Direct computation of this sequence is very time consuming, because the trigonometric functions have to be evaluated for each sample. Therefore, it is a good idea to use a linear recurrence scheme to compute $$x_n$$. Using basic trigonometric identities, we see that$x_{n+1} = A x_n,\quad x_0=\begin{pmatrix}1\\0\end{pmatrix},$with$A = \begin{pmatrix}\alpha_1 & -\alpha_2\\\alpha_2 & \alpha_1\end{pmatrix},\quad \alpha_1 = \cos(\omega),\ \alpha_2=\sin(\omega).$

However, to actually perform these computations in a digital processor, one has to quantize $$\alpha_1,\alpha_2$$, meaning that one has to replace $$\alpha_1,\alpha_2$$ by approximations. It is easy to see that if one replaces $$\alpha_1,\alpha_2$$ by some perturbation, then the eigenvalues of $$A$$ are no longer in the unit circle, so the oscillation can grow or decay exponentially and one would need to apply an AGC scheme to keep this method stable.

Here we will look at a new quadrature oscillator by Martin Vicanek that has appeared recently and solves this problem.