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In general, if the receiver is not ideal, but generates
noise of its own, this noise may be included by
increasing T in equations (8) and (9). Strictly speaking,
this is not quite correct. Several noise sources at temperatures
T1, T2 . . • Tm do not have the same total blackbody
radiation as a single source at a temperature
T = TI + 7"2 +... Tm ; the latter has a higher frequency
quantum cutoff. However, the total noise at low frequencies
is proportional to the sum of the temperatures,
while at high frequencies the spontaneous emission term
dominates, so the overall result is not greatly in error.
COHERENT DETECTION AND MATCHED
FILTERING
To realize the maximum signal to noise ratio in a
coherent receiver a synchronous (or homodyne) detector
must be used; in addition, the transmission band must be
shaped to match the signal spectrum (Appendix C).
When both these techniques are employed, the effective
noise power spectral density is half that given by
equation (8) or (9). Neither technique can be used until
the signal has been discovered in the first place, so in the
search process we must rely on square-law (energy)
detectors. But once a coherent signal has been found, a
local oscillator may be locked to it, and the nature of
the signal spectrum can then be determined. Thus,
coherent detection and matched filtering would be
employed in any communication link once contact had
been established,
In a synchronous detector, the coherent RF or IF
signal is mixed with a local oscillator signal of the same
frequency and phase and any modulation of the former
is recovered by low pass filtering. Any noise wave
present may be resolved into two statistically independent
waves a(t) cos 2¢rvot and b(t) sin 2rrvot, where
cos 21rVot is the local oscillator wave. Both a(t) and b(t)
will, in general, be gaussian variables whose mean square
values each represent one-half the total noise power.
Only a(t) will produce an output after the low pass
filter, so that only half the noise power present at the
input appears in the output. But since (in double
sideband reception) the postdetection bandwidth is half
the predetection bandwidth, the noise power spectral
density is unaffected.
If we know in advance the waveform or spectrum of
the 'signal we are trying to receive (or if we can
determine either of these), we can design the receiver to
give the best estimate of some property of the signal
(e.g., its amplitude or phase or time of occurrence) in
spite of the added noise. In Appendix C we show that if
the signal consists of a series of pulses, the best ratio of
detected peak signal amplitude to the rms noise fluctuations
in that amplitude will be obtained if the receiver
band limiting filter has a complex amplitude transmission
K(p)
given by
K(v) = m -- (10)
_(v)
where m is an arbitrary real constant, F(u) is tile
complex conjugate of the signal amplitude spectrum,
and Jg(p) is the noise power spectral density. This
equation applies to filtering done either before or after
synchronous detection, provided the appropriate spectra
are used.
If _b(p) is a constant (white noise), the matched filter
has a transmission everywhere proportional to the
conjugate of the signal spectrum. The conjugacy aligns
all Fourier components of the signal so that they peak
simultaneously, while the proportionality weights each
component in proportion to its own signal-to-noise ratio.
With a matched filter the output peak-signal to noise
power ratio is
S If(w) l2
-- = 2 -- dv ¢11)
N
_v)
Again, if the noise is white, so that _A_v) has the
constant value qJo, equation (1 1) becomes simply
S 2E
-- =-- (_2)
N _o
where E is the energy per pulse. We see from this
expression that the ultimate detectability of a signal
depends on the ratio of the received energy to the
spectral density of the noise background. The energy of
the signal can be increased, of course, by increasing the
radiated power, but once a practical limit of power has
been reached, further increase is only possible by
increasing the signal duration. This narrows the signal
spectrum. In the limit, therefore, we would expect
interstellar contact signals to be highly monochromatic.
As a simple example of a matched filter assume that
the signal consists of a train of pulses of the form
sin lrB(t-ti)
fi(t) = A i 7rB(t-ti )
(13)
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