Several weeks ago, in an AMSAT EA informal meeting, Eduardo EA3GHS wondered about the possibility of using WSJT-X modes through linear transponder satellites in low Earth orbit. Of course, computer Doppler correction is a must, but even under the best circumstances we cannot assume a perfect Doppler correction. First, there are errors in the Doppler computation because the TLEs used are always measured at an earlier time and do not reflect exactly the current state of the satellite. This was the aspect that Eduardo was studying. Second, there are also errors because the computer clock is not perfect. Even a 10ms error in the computer clock can produce a noticeable error in the Doppler computation. Also, usually there is a delay between the time that the RF signal reaches the antenna and the time that the Doppler correction is computed for and applied to the signal, especially if using SDR hardware, which can have large buffers for the signal. This delay can be measured and compensated in the Doppler calculation, but this is usually not done.

Here we look at errors of the second kind. We denote by \(D(t)\) the function describing the Doppler frequency, where \(t\) is the time when the signal arrives at the antenna. We assume that the correction is not done using \(D(t)\), but rather \(D(t – \delta)\), where \(\delta\) is a small constant. Thus, a residual Doppler \(D(t)-D(t-\delta)\) is still present in the received signal. We will study this residual Doppler and how tolerant to it are several WSJT-X modes, depending on the value of \(\delta\).

The dependence of Doppler on the age of the TLEs will be studied in a later post, but it is worthy to note that the largest error made by using old TLEs is in the along-track position of the satellite, and that this effect is well modelled by offsetting the Doppler curve in time. This justifies the study of the residual Doppler \(D(t)-D(t-\delta)\).