In article <0Mtzb.32209$_g4.16482@newssvr29.news.prodigy.com>,
Alfred A. Aburto Jr. <aburto@sbcglobal.net> wrote:

Another point to be raised is that the SNR need not be so "high"
as shown in the FAQ. In fact, an SNR of 3 might be ok in many
cases. I note that Kraus in the Radio Astronomy reference, used

3:1 will produce totally unacceptable false positive rates.

One of the problems with low SNRs is that there is not a simple cut off 
threshold for detecting signals; the number of false negatives varies
fairly slowly with threshold (strictly speaking, the S@H cut off isn't
22:1 SNR, but signal + noise exceeds 22 times mean noise).  For that reason,
the following calculations are based on a threshold of 4 times mean noise,
rather than on a particular signal to noise ratio.

The false positive rate is exp (-threshold/mean noise power), which
is about 1 in 55.  However, with modern system, one examines a very
large number of channels (without chirping or alternative bandwidths,
S@H examines more than 1,000,000 - and that takes under a second on
modern machines), which means that one actually expects a large number
of detections for one physical observation.

S@H prunes this down by requiring a coincident detection at a different
time.  However that doesn't help that much.  The probability of a false
positive for two observations, at one chirp rate and bandwidth and
requiring an exact match on frequency is an almost certainty, as follows
(done with Unix dc command):

2.71828 4 ^
p
54.59791
sa
30000 k
1 1 1 la la * / - 1024 ^ 1024 ^ - p
99999999999999999999999999999999999999999999999999999999999999999999\
999999999999999999999999999999999999999999999999999999999999999999999\
999999999999999839142176....

Many S@H targets don't get even two observations, but even with three,
one is still well over 99% certain of a false positive:

10000 k
1 1 1 la 3 ^ / - 1024 ^ 1024 ^ - p
9984080225.....

In reality, one has to allow some fuzz in the the frequency, so the false
positive probability is even higher than this.

Using such low SNR values is alright for demonstrating an already
confirmed signal to the press, but it is not suitable for initial
detection, without a large number of repeat observations, and the need
to carry over large lists of detections from observation to observation
(when you add bandwidths and chirps to S@H, the data for the candidates
would exceed the work unit size by several orders of magnitude - Berkeley
currently can't afford to keep work unit data online).

The way you work with this sort of raw signal to noise ratio is by
using time bandwidth products which are greater than one (i.e. averaging
over several consecutive FFTs).  This technique is already covered by
some cases in the FAQ.  Doing so effectively increases the SNR by the
square root of the number of samples averaged.   You can only do this
for wide bandwidths, wide beamwidths or targetted SETI.  In my previous
article, I discussed this in terms of detecting full bandwidth TV,
and in averaging between frames to recover programme content, once the
signal has been found.

The Gaussian and pulse detection processes in S@H also effectively do
this.  I assume that the Gaussian sensitivity is higher than the spike
sensitivity for the same bandwdith.

I consider such averaging as effectively changing the signal to noise
ratio.  For earth-moon-earth transmissions, people do operate below the
noise floor (SNR < 1) by averaging.  Detecting the 3 degrees microwave
background requires working at SNRs of less than 1.

PS.  Don't rely too much on the 10 % illumination efficiency.  I think
it is probably somewhere between 10 and 20%, though - the effective
illumination from the line feed is about 100m across and in the form
of a ring.