The situation below is obviously fictitious, but should be equivalent to the level of information available to an alien radio astronomer viewing any messages that we may send out. If you find places where we have included physically impossible values, we will be quite happy - because that will mean that you have decoded the message. To begin the scenario:
The following message has been transcribed from the modulated radio emission observed from the direction of the star 55 Cnc over the past forty years. This emission consists of unpolarized pulses with 0.113 Hz bandwidth and 8.85 s duration, occurring in sets of eight with 7670 s between the start of one pulse and the start of the next. There is then a longer pause, followed by the next set. The first pulse of the each set begins 107350 s after the first pulse of the previous one.
The frequency of the pulses is variable, but oscillates about 1420.27639 MHz (measured in a reference frame equivalent to the center of mass of the solar system if the Sun had zero mass).
Fourier analysis of the pulse frequencies shows a number of peaks. With a period of 259.9431 days, the frequencies change with an amplitude of plus and minus 145.47 kHz. Within each set, the frequencies are well fit by a 107350 s period sinusoid with an amplitude of 25.83 kHz. The remaining frequency shifts and the overall strength of the pulses are consistent with the source being a 300 KW constant-frequency transmitter in orbit about the planet 55 Cnc f that is being Doppler-shifted due to changes in its radial velocity. However, there are residuals in this fit.
The residuals have constant values during each pulse: -3, -1, +1, or +3 Hz relative to the assumed constant-frequency reference. The pulses are also modulated in polarization, either completely left-hand-circular or completely right-hand-circular. The eight possible states have been transcribed below according to this scheme:
-3 Hz, LCP => 0
-3 Hz, RCP => 1
-1 Hz, LCP => 2
-1 Hz, RCP => 3
+1 Hz, LCP => 4
+1 Hz, RCP => 5
+3 Hz, LCP => 6
+3 Hz, RCP => 7
There are some inevitable gaps, particularly soon after the signal was discovered, where no one was observing 55 Cnc when a pulse was due to occur. These are marked by "X". The full message is available here.
So: what are they trying to tell us?