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About SETI range and sensitivity
by Lieven Philips

In this short paper I would like to make an additional comment on the discussion of the achievable range of the Arecibo radio telescope, in SearchLites Vol. 11, No 3 (Summer 2005).

When Arecibo is mapping the hydrogen distribution in the Milky Way, then the radio telescope's receiver is piling up energy in a relatively narrow band centered around a certain frequency. Only the energy of the signal is important; there's no phase information recovered (it doesn't make sense). When Arecibo was receiving the beacon signal from Pioneer 10, it was also about recovery of energy in a very narrow band. But when there is data communication with a space probe (e.g. compressed images from Cassini-Huygens) then recovery of the data involves data demodulation, and data demodulation implies coherent tracking, i.e. carrier phase recovery (for a PSK signal). The range that can be achieved depends on the transmission power and the bandwidth. We can increase the range by reducing the data rate (which determines the bandwidth), at constant transmission power.

In most SETI searches, we are looking for very narrowband (CW) signals, which we try to detect with multi-million points FFT's. The accumulation of energy in each frequency bin (each possible channel) is performed over a certain limited time (say 100 seconds), in order to average out fluctuations of the noise power. The time interval is limited because Doppler effects and interstellar scintillation (fading) gradually influence the frequency and the amplitude of the signal. Instead of accumulating energy, however, we could also attempt - in principle - to track the CW ETI beacon over an extended duration. If this was possible, then we could detect CW signals which are deeply buried in the noise, because the coherent FFT gain is proportional to the tracking interval (i.e. the duration of the coherent correlation). This means that - if it was possible to track the CW signal - we could extend the recovery range far beyond the typical range non-coherent energy detection or modulated signals.

Of course it is not possible to actually track the CW signal, because we cannot phase synchronize to it (because the CW signal is assumed to be buried in the noise). But what we can do is to run a vast amount of hypotheses on the phase evolution on our interval (of say 1 hour duration). These hypotheses are similar to the Doppler compensation hypotheses that are applied e.g. in SETI@home. This leads to an explosion of calculations, but at the benefit of significantly extending the range. Moore's law and advances in grid computing would eventually allow this to be a practical approach.


Consider a 5 kHz wide frequency bin with a sine waveform (CW signal) at 1500 kHz at power = 1, accumulated on a noise signal (random variable) with variance = 36. This is hence a signal deeply buried in the noise. Figure 1 shows the signal, Figure 2 the signal in the noise. Now we perform FFT's with different lengths: Figure 3 shows the spectrum resulting from a 512-points FFT. The signal is not detected. Figure 4 shows the spectral analysis at an 8 times longer time interval, using a 4096-points FFT: the signal is present, but parasitic noise peaks prevent clear discrimination. With 8192 points (Figure 5), the FFT results in a high S/N ratio. This means that with a sufficiently long time window (i.e. FFT length) when can detect every CW signal, no matter how weak it is. All figures obtained using Matlab.

This reasoning is not in contradiction with what we normally understand as sensitivity, or with what the Shannon theorem tells us. Sensitivity is typically defined as the signal level above noise that is still detectable by a receiver. However, with the coherent detection we can go arbitrary far below the noise + interference floor, because we have defined our level of detectability not in function of a telecommunications link, but only in function of the detectability of the presence of a signal, not the signal content. Similarly, there is no violation of the Shannon theorem: we can consider the narrowband CW signal as a single bit, smeared out infinitely, over a very narrow bandwidth; as a consequence, the S/N can go arbitrarily low, or the range can be arbitrarily extended, at the expense of computation time.

In principle, this kind of search would be applicable to check out globular clusters or galaxies. The gigantic amount of computation would be compensated by the number of stars that can be scrutinized simultaneously: it is sufficient that one civilization on one planet of one star in the Andromeda Galaxy has detected a life bearing planet in our Galaxy (e.g. spectroscopically), and decided to install an eternal beacon directed to us. This search strategy would comply with the hypothesis from Cohen and Hohlfeld that life in the universe is quite rare, and hence we have to search for a beacon which is "very powerful, but very far away".

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