Prepared by:

Bernard M. Oliver

Vice-President for Research and Development

Hewlett-Packard Corporation

[128] Blank page

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 10^-24 to 10^-26 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.

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,, 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, where W is the received pulse energy and(=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

(1)

[130] 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.

An antenna that radiated uniformIy 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

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_i the received signal is 3 dB below threshold, and at a_i+1 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_i or a_i+1 . The same is true for a signal from B if the observations a_i and b_i are averaged. For a signal from C we might average the three strong signals from observations a_i, a_i+1,

[131] ....and b_i. 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_i+1 and b_i-1.

We need to analyze and optimize some set of rules such as:

1. If a sample received at b_i is above a threshold x₁, sound an alarm.

2. If the sample at b_i is less than x₁ but greater than x₃ where x₃ << x₁, record it.

3. If the sample at b_i is greater than some threshold x₂, where x₃ < x₂ < x₁ and this is also true for a sample at b_i-1, ai or a_i+1, add the two and compare against the appropriate 2-observation threshold.

4. If the sample at bi and two others at b_i-1 and a_i or a_i and a_i+1 are all greater then x₃, add and make a 3-observation test.

5. Record b_i+1 and erase a_i.

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

(2)

If the received flux is at wavelength, and the antenna area is A, the gain will beand the (on axis) received power will be. The effective duration of the received

pulse is

(3)

where t_s is the time required to search the entire sky in one frequency band. We thus find

(4)

[132] The received pulse energy is independent of the antenna area and is the energy that would be received by an isotropic antenna in the full sky search time.

We must set our detection threshold high enough that noise alone rarely exceeds the threshold. This will require a SNR = m where . Thus

(5)

or conversely,

(6)

Figure 2 shows as a function of ts for variouswith, T= 10 K and m = 25.

Although for a fixed search time the sensitivity does not depend upon antenna area, there are reasons to prefer a highly directive receiver. If a signal is discovered we want to know its direction of arrival. We need to discriminate against interference. Finally, the larger the antenna the shorter will be the duration of of the received pulse and the wider will be the bandwidth of the matched filter. If we are using a multi-channel spectrum analyzer this permits us to search a wider band, , for the same number, N , of channels or, if the receiver bandwidth is the limit, to use fewer channels.

Since

we have

(7)

Assuming that

t_s = 1 year = 3 x 10⁷ sec

= 6 Hz

T = 10K

m = 25

we obtain from (5) and (7) the limiting flux levels and antenna diameters shown by the solid lines of figure 3. The sensitivity of 3X 10^-26 W/m² at the water hole is quite good: it is only about 43 dB poorer than a full Cyclops. However, the antenna sizes are Cyclopean too. The 1-km diameter at 1.3 GHz would require one hundred 100-m dishes.

There are only two ways to reduce the antenna diameter given by equation (7). One is to reduce t_s, which decreases the sensitivity. The other is to reduce, which requires N to be increased.

If we assume the receiver bandwidth B = 300 MHz then, for the case just computed, N = 300 x 10⁶/6 = 5 x 10⁷ . But we are considering N = 10⁹, this would permit = 0.3 Hz and gives the dashed line in figure 3 for antenna sizes.

Obviously, as long as the receiver bandwidth is limiting we can trade-off antenna area, A, against the number of channels, N, in the MCSA. Let us find the optimum. At threshold

thus

(8)

[135] Further, the total cost, C, of the antenna and data processor we take to be

where K_a and K_c are the dollars per square meter and per channel, respectively. Substituting (8) and (9)? and differentiating with respect to A (or N) we find

(10)

(11)

(12)

Let us estimate some values for K_a and K_c and see what sort of costs are involved. If we can build a 100-m dish for $10 million, K_a = $1273/m². If we can build a 106 channel MSCA for 5500,000, K_c = $0.5. Assuming that T = 10 K and B = 300 MHz:

(13)

(14)

(15)

The preceding designs assume a threshold flux that is independent of . Thus, the search times as given by equation (6) will vary as or v² Rewriting equation (6) in terms of frequency

(16)

The total time required to search from v₁ to v₂ where v₂ - v₁ << B is approximately 1/B times the integral of equation (16)

(17)

Taking m = 25, and T and B as before, we find from equation (6) that the time to search the water hole is

[136] (18)

and to search the microwave window from v₁ = 10⁹ Hz to v₂ = 10¹⁰ Hz

Based upon the above relations (13) through (19) we find the parameters given in table I for systems of various capabilities.

A few comments are in order:

1. The cost is that of antenna and processor only - it does not include buildings, auxiliary equipment, operating costs, etc.

2. The times assume continuous operation in search mode.

3. Because no Earth-based antenna can cover the entire sky, the possibility of two systems at ±45° latitude should be considered. This would double the costs and halve the times.

4. If arrays are used for the larger size antennas the solid angle of the beam is reduced by the filling factor so the times shown must be divided by the same factor. Two arrays designed to cover the sky from ±45° latitude could have filling factors of 0.9 at 45° elevation and 0.64 at the zenith.

For comparison, the detection sensitivity with 1000 sec observation time per star, vs the number of 100-m antennas and the corresponding costs, assuming N = 10⁹, are listed in table 2.

The time required to search the water hole for all F, G, and K stars within

The time required to search the entire sky for CW signals is inversely proportional to the minimum detectable flux level and, for large relative bandwidth, increases as the cube of the highest frequency. By replicating optimum systems, dollars can be traded for time. If enough systems are built to keep the search time constant the cost varies as the 3/2 power of the sensitivity.

A full-sky search of the water hole could be made to a flux level of 10^-24 W/m² in only 12 days with an optimum system. To search the entire 1 to 10 GHz region to the same sensitivity would require 17 years, half of which would be spent searching between 8 and 10 GHz. Evidently if the entire microwave window is to be searched, the sensitivity should be reduced as the operating frequency increases. If the minimum detectable flux level is made proportional to v² the search time is proportional to the frequency range covered rather than the cube. This would permit covering the 1 to 10 GHz region at from 5 x 10^-25 to 5 x 10^-23 W/m², respectively, in about 1 year. This assumption ofproportional to v² is consistent with the assumption that the signal we will detect is being beamed at us, for then . Hence the full sky search covers the cases of beamed or very strong omnidirectional signals originating at great distances: the "giant" or "supergiant" transmissions of very advanced cultures.

Assume that the amplitude of the illumination of a circular antenna of radius b is

(Al)

These are the so-called Sonine functions. The effective area is

(A2)

and the on axis gain is

The decreasing effective area and gain with increasing v are the result of illumination being confined more and more to the central region. We can compensate for this by letting

(A3)

where a is independent of v. The clear aperture case corresponds to v= 1. The scaling given by equation (A3) gives every case the same effective area and gain as a clear aperture of radius a. If we now let , equation (A1) approaches

(A4)

The far field amplitude patterns are the Hankel transforms of (Al) and (A4) and, for finite v,

are

(A5)

where and . For the Gaussian case

(A6)

where is the on-axis gain. The half-power beamwidth,, is twice the value of that makes . We find

[139]

If we tessellate the sky with hexagons of width B between opposite sides, each will have a solid angle so the required number is . For the cases listed

If we scan the sky in strips of width, the effective dwell angle per direction is

(A7)

For the Sonine cases this gives

(A8)

For the Gaussian case

(A9)

The effective number of directions in the scanning mode is

(A10)

[140] Thus for various v we find

and the assertion that seems justified.

1. Oliver, Bernard M.; and Billingham, John: Project Cyclops, A Design Study of a System for Detecting Extraterrestrial Intelligent Life. NASA CR 114445, 1972.

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