We believe that requirement 2 is satisfied by any spatially and temporally coherent electromagnetic wave. Modulation of such a wave in order to transmit information need not destroy the coherence to such an extent that it would be mistaken for a natural signal. Further, the modulation will very likely contain regularities and repetitions of complex patterns that are inexplicable from natural sources.

THE OPTIMUM SPECTRAL REGION

Electromagnetic radiation covers a practically unlimited frequency range. Unless we can find some reason to prefer a relatively narrow portion of the spectrum, one of the dimensions of the search space remains unbounded. Here again we can appeal to energy minimization.

Let us examine critically what is implied by requirement 1 above (that the number of particles received exceed significantly the natural background count). First let us assume that we are operating in the "quantum" region of the spectrum (i.e., hv/kT> 1), where direct photon detection is easy, and that there is no background radiation. In principle, we require at least one received photon in the observing time 7 in order to detect a signal at all. This requires that the received power

so, for a constant collecting area, the equivalent isotropic radiated power (EIRP) must be proportional to frequency. In practice the reception of a single photon would hardly convince us that we had detected extraterrestrial intelligence. In order to determine the coherence and other properties of the signal we would need to receive some number, n, of photons. But n depends only upon the sophistication of our data processing and upon our assigned a priori improbability and does not depend upon the operating frequency. Thus the proportionality implied by equation (1) still holds, as long as there is no background.

If the radiated signal is coherent and of constant amplitude, the photon arrivals will be Poisson distributed. If the expected number of arrivals in the observing time 7 is n, the mean square fluctuation will also be n. The detected signal-to-noise power ratio (DSNR) is the ratio of the square of the mean to the mean square fluctuations; that is,

where W is the total energy received during the observation and P is the instantaneous power. In communication systems 7 is the Nyquist interval and is the reciprocal of the RF bandwidth B. Then equation (2) becomes

and the photon shot noise behaves as if it had a spectral power density hv.

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(3)

The above considerations suggest that the lower the operating frequency the less will be the required EIRP and this is true until we reach the "thermal" region (i.e., hv/kT < 1). In this region direct photon detection becomes difficult but the "radio" techniques of linear amplification, mixing, and spectral analysis are applicable; in fact, these have by now been extended into the optical region. Suppose we Fourier transform a block of the received IF signal of duration 7. In effect, a constant amplitude monochromatic signal is then detected as a rectangular pulse of duration 7 to which noise has been added. The effective selectivity curve of the Fourier transform will be the transform of the time window or

and this is the matched filter through which a CW pulse of frequency w。 and duration 7 should be passed to give the greatest ratio of peak output signal power to mean square fluctuation in the peak. For this case, or for any equivalent optimum detection process, the detected signal-to-noise ratio is

Three contributions to are unavoidable and are essentially the same anywhere in the Galaxy; these are shown in figure 1 (divided by k to give their equivalent noise temperature). The first is the synchrotron radiation of the Galaxy itself, given by

where the coefficient T varies from about 1 to 2.5 depending upon galactic latitude. This noise dominates below 1 GHz but, above this frequency, rapidly becomes less than the relict cosmic background radiation:

where T~2.76 K. Finally, for v>kT/h ~ 60 GHz the spontaneous emission noise

(7)

(8)

The sum of these noise contributions defines a broad quiet region extending from about 1 to 60 GHz known as the free-space microwave window. On Earth (or on any Earth-like planet) the absorption lines of the water vapor and oxygen in the atmosphere re-radiate noise with broad

peaks at 22 and 60 GHz that degrade the window above about 10 GHz, as shown in figure 2. Thus the terrestrial microwave window extends from ~1 to ~10 GHz, and is clearly a preferred region of the spectrum for interstellar search using ground-based low-noise receivers.

g

The suggestion is sometimes made that one might avoid and b by going to very high frequencies and then eliminate o by employing direct photon detection. At frequencies for which dominates, equation (5) becomes

and, since B can, in principle, be made as narrow as 1/7, we find

exactly as given by equation (2). Thus, ignoring technological limitations, linear amplification is just as good as direct photon detection. Spontaneous emission noise does not make linear amplifiers noisier than photon detectors, it merely prevents them from being quieter, which would violate the principle of complementarity (ref. 3).

DEGREES KELVIN

100

10

1

.1

100

1000

Figure 2.- Terrestrial microwave window.

We may be unable, at optical frequencies, to realize bandwidths as narrow as 1/T. The point is that, even if we could, the linear amplifier, which would then be no noisier than the photon detector, would still be far noisier than a linear amplifier operating in the microwave window.

Not only is the minimum detectable received power least in the microwave window, but also the cost per unit of collecting area is less there than for higher frequencies. This latter consideration also favors the low end of the microwave window over the high end. Additional factors favoring the low end are:

Greater collecting area for the sharpest beam that can be used effectively

Greater freedom from H2O and O2 absorption, which may be stronger on many planets than on Earth if ground-based systems are used

Higher beacon powers are probably easier to achieve

Narrower bandwidths are possible

EFFECT OF DOPPLER DRIFT

The relative motion of transmitter and receiver caused by planetary rotation and revolution produces a frequency modulation of the received signal. A frequency drifting at a rate i will drift clear through a band B in a time T = B/v. If T <1/B the receiver will not respond fully and if T>1/B the bandwidth is larger than necessary and admits more noise per channel. Setting 7 = 1/B we find B = v1. Since i is proportional to v it follows that the optimum bandwidth increases as 12. The total receiver noise per channel is therefore proportional to v12(v).

If the ordinates of the curves of figures 1 and 2 are multiplied by 12 when v is in gigahertz, we obtain figures 3 and 4, respectively. The ordinates are now proportional to the noise contributed per channel in a receiver optimized at all frequencies for the Doppler rate produced by a given radial (line-of-sight) acceleration. We see that the upper end of the free space window has become relatively noisy and that the H and OH lines are at the quietest part of the spectrum.

The doppler drift caused by motion of the receiver can be corrected by drifting the local oscillator. This is also true for the transmitter when the signal is radiated directively. It does not appear possible to correct a coherent omnidirectional beacon for all directions simultaneously.