COMPLEMENTARY DOCUMENT 3 PARAMETRIC RELATIONS IN A FULL SKY SEARCH Prepared by: Bernard M. Oliver Vice-President for Research PARAMETRIC RELATIONS IN A FULL SKY SEARCH It has been argued that in addition to searching likely main sequence stars for signals of intelligent origin, we should also search the entire sky (see Section II-5). While this cannot be done to as low a flux level in the same time, our uncertainty as to the highest flux level we might find is very large and we cannot rule out the possibility that such a search would discover a signal. We shall derive expressions relating antenna size, sensitivity, bandwidth, and search time in systems that are designed to optimize the cost-performance ratio. We find, not surprisingly, that to conduct full sky searches to flux levels of 1024 to 1026 W/m2 requires long times and large antennas. However, just as there are good reasons for conducting a targeted search at sensitivities less than that of a full scale Cyclops system, so also there are good reasons to search the full sky with existing antennas at lower sensitivities than we consider here. To be significant, a full-sky search should be capable of detecting coherent signals at least one or two orders of magnitude weaker than would have been detected by past radio astronomy sky surveys. This is not as difficult as it first seems because radio astronomy surveys have actually discriminated against coherent signals of the type we are seeking and are generally quite restricted in the frequency coverage. As in the case of a targeted search, we shall assume a monochromatic signal. If there is modulation we ignore it, initially, and try to detect only the strong CW carrier that may be present. DETECTABILITY OF PULSES Whether we point the antenna in a succession of sidereally fixed directions until we have tessellated the whole sky, or sweep the beam along some path that paints the entire sky, a CW signal will be received as a pulse embedded in random noise. In the first instance the pulse will have a constant amplitude and a duration, 7, equal to the dwell time per direction. In the second, the pulse envelope will be determined by the antenna beam pattern as it sweeps past the source. The best possible signal-to-noise ratio (SNR) for the detected pulse is 2W/V, where W is the received pulse energy and V (=kT in the microwave window) is the noise power spectral density. This optimum can be achieved in several ways, such as with a matched filter and a synchronous detector (Cyclops report, appendix C (ref. 1)). Lacking a priori frequency and phase information we cannot use synchronous detection and can only achieve The important point is that the SNR depends only on the pulse energy and not at all upon the way that energy is distributed in time. NUMBER OF POINTING DIRECTIONS An antenna that radiated uniformly into a hemisphere would have a gain of 2 and would have to be pointed in two directions to cover the sky. Similarly, one that radiated uniformly into an octant would have a gain of 8 and would require eight directions. In principle, if n is the number of pointing directions and g is the gain we have n = g. However, practical antennas do not radiate uniformly into a certain solid angle; instead, the gain falls off smoothly as an analytic function of the off-axis angle. In the Cyclops report it was asserted that to cover the sky with a maximum off-axis pointing loss of 1 dB at the periphery of each elemental patch of sky would require n≈ 4 g. This is true so long as no record is kept of cases that fail to exceed the threshold but are nevertheless strong. If this is done, the record can be used to confirm the presence of a signal as soon as an adjacent pointing direction also shows a strong level at nearly the same frequency. Assume that an antenna is being pointed successively in a set of directions separated by one half-power full beamwidth and forming a hexagonal lattice as shown in figure 1. Assume that there is a signal at A. When the antenna is pointed at a; the received signal is 3 dB below threshold, and at a¡+, it is again 3 dB below threshold. But suppose we observe that both signals are strong and record them both. If we average the two results and use a threshold appropriate for two independent samples, then we find from figures 11-14 of the Cyclops report that we will have improved our sensitivity by ~2.7 dB. Thus, we will lose only ~0.3 dB for a signal from A as against one from a¡ or a¡+1. The same is true for a signal from B if the observations a; and b¡ are averaged. For a signal from C we might average the three strong signals from observations aj, aj+1, and b. Using the appropriate threshold we find from figures 11-14 of the Cyclops report an improvement over one observation of ~4.1 dB and this is exactly the off-axis loss at C. Hence for signals from C we have no loss in detectability as compared with on-axis signals. We could probably do as well at B as at C by including in some way the observations at a¿+1 and bj- 1 · We need to analyze and optimize some set of rules such as: 1. If a sample received at b; is above a threshold x1, sound an alarm. 2. If the sample at b¡ is less than x1 but greater than x3 where x3 <<x1, record it. 3. If the sample at b¡ is greater than some threshold x2, where x3 <x2 <x1 and this is also true for a sample at bi-1, aj or ai+1, add the two and compare against the appropriate 2-observation threshold. 4. If the sample at b; and two others at b¡-1 and a¡ or a¡ and a¿+1 are all greater than x3, add and make a 3-observation test. 5. Record bit1 and erase a¡. The technique of tessellating the sky with an array of fixed pointing direction must be used if the multi-channel spectral analyzer (MCSA) speed limits us to one observation per beamwidth. If, as with an optical processor, continuous spectra can be formed, then the antenna can be scanned continuously. Off-axis signals are simple to detect in this case, since the maximum found on two successive scans will occur at the same value of the coordinate along the scan direction and will be caused by a signal at this same coordinate value. As shown in the appendix, if the hexagons of figure 1, or the strips of a continuous scan are a full half-power beamwidth wide, then If the received flux is W/m2 at wavelength λ, and the antenna area is A, the gain will be g 4A/2 and the (on axis) received power will be Ap. The effective duration of the received = where t, is the time required to search the entire sky in one frequency band. We thus find 247-831 - 78 - 10 (3) (4) |