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which might result from synchronous detection of pulses
of the form
sin rrB( t-t i)
gi(t) = A i rrB(t-ti) cos 2 rrVot (14)
The spectrum of a single detected pulse is constant
from 0 to B/2 Hz while that of a single pulse prior to
detection is fiat from Vo - (B/2) to v0 + (B/2) Hz. Both
spectra are zero outside these limits. The matched filter
is either a predetection ideal bandpass filter of width B
centered at uo or a postdetection ideal low pass filter of
cutoff frequency B/2. (In this case, because the two in
tandem are idempotent, both may be used,) Since in this
case the matched filter does not affect the signal
spectrum the signal shape is unchanged.
If ti = i/B in equations (13) and (14) the peak pulse amplitudes
will be independent. Thus independently detectable
pulses may be sent at a separation of r = 1/tt sec,
and we may consider r to be the effective pulse
duration. We thus have
Br = 1 (15)
a relation that is approximately true for any matched
filter where B is the RF bandwidth and r is the pulse
duration, both appropriately defined.
With white noise, the signal to noise ratio from
equation (11) or (12) is
S 2E i 2,4i2r A i2
N _o _o _o(B/2)
If a long train of such pulses is sent with a constant
(16)
Ai=A, the transmitted signal amplitude will be constant at
the value A representing a signal power A 2 and the
received signal will show fluctuations about this amplitude
of mean square value ¢/o (B/2) representing the
noise power of spectral density 4o in the band B/2.
We see that the combination of synchronous detection
and matched filtering gives an output signal-to-noise
ratio that is twice the received signal-to-noise ratio if the
latter is defined as the ratio of the signal power to the
total noise power in the RF band.
and optical frequencies the radiation may be allowed to
fall directly onto a photon counter. A photocell, like a
square-law detector, gives a response (current) proportional
to the incident instantaneous power. In principle,
the limiting noise performance of both types of detector
is the same; however, at radio frequencies most of the
output noise is produced by fluctuation in the random
noise input and shot noise is usually negligible, whereas
at optical frequencies the reverse may be true.
If a steady coherent signal of power Pr and an
incoherent (black-body radiation or thermal noise)
background of power Po are both applied to a square
law detector or photocell the output signal-to-noise
power ratio is shown in Appendix D to be
where
S
rPr2
N (hvln)(Pr + Po) ÷ (eolB)[(2/m)Pr ÷ eo I
r = integration time
B = predetection bandwidth
m = 1,2 = number of orthogonal polarizations
reaching
detector
r/ = quantum efficiency of detector
(17)
The ratio of Po]B to hv]ri determines the relative
importance of the ratio of fluctuation noise to shot in
the detector output. If we let _brepresent this ratio, then
Po/B _ spectral density of fluctuation noise
hv/'O spectral density of shot noise
(_/'o )/B
hv
In the last form we 'see that _ is the expected number
(18)
of background photons counted in the time 1lB. If _ >> 1
fluctuation noise predominates; if ¢