MASSACHUSETTS INSTITUTE OF TECHNOLOGY LINCOLN LABORATORY

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP67B00341R000400080001-8
• Release Decision: RIFPUB
• Original Classification: R
• Document Page Count: 7
• Document Creation Date: December 23, 2016
• Document Release Date: April 10, 2014
• Sequence Number: 1
• Publication Date: July 30, 1953
• Content Type: REPORT

Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
RESTRICTED
This document consists
of 7 pages. No. 50
of 50 copies.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LINCOLN LABORATORY
THE RELATIVE SENSITIVITIES OF PRE- AND POST-DETECTION INTEGRATORS
F. A. Rodgers
Technical Memorandum No. 35 30 July 1953
This document contains information affecting the
national defense of the United States within the
meaning of the Espionage Laws, Title 18, U.S.C.
Sections 793 and 794. The transmission or the
revelation of its contents in any manner to an un-
authorized person is prohibited by law.
CAMBRIDGE MASSACHUSETTS
RESTRICTED
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
RESTRICTED
THE RELATIVE SENSITIVITIES OF PRE- AND POST-DETECTION INTEGRATORS*
INTRODUCTION
When a signal is known to lie somewhere within a given band of frequencies, there
are two means available for increasing the probability of detection of that signal for a given false-
alarm rate. The first method, called pre-detection, or IF integration, involves the use of a
bank of narrow-band filters connected in parallel and staggered in frequency so as to cover the
entire band of possible frequencies. Each of these filters then feeds its own biased-diode alarm
circuit.
In the second method, called post-detection, or video integration, the band in which
the signal is known to lie is isolated by a bandpass filter, detected to beat the unknown frequency
down to zero frequency; the signal is then passed through a low-pass filter, the cutoff frequency
of which will usually be equal to the half-width of the narrow-band filters used in the other meth-
od. These widths will be determined by the possible frequency spread of the signal. Schematic
diagrams for these two systems are shown in Fig. 1.
If it is necessary to retain information as to the frequency of the signal, obviously
one must use pre-detection integration; but if this information is not required, then the choice
between the two systems must be based on a consideration of complexity and sensitivity. We
shall compare the two systems by determining the input signal-to-noise ratios required to give
the same per cent detectabilities and the same false-alarm rates. Since the results are prac-
tically independent of the choice of false-alarm probabilities in the range 10-5 < PN < 10-14 and
per cent detectabilities in the range 50% < Ps 10). Thus, by setting the alarm level four
standard deviations above the mean noise voltage, we shall realize a false-alarm probability of
X 10-5.
To match this performance in the pre-detection system, the false-alarm probability
for each of the n alarms must be given by PN-?, (3 x 10-5)/n. The probability distribution for the
envelope of a sine wave in narrow-band noise is given by Rice** as
dR R
2 + P2) /Zcr2f I (-rill) dR
a-' a-cr 2 _--exp[?(R [ (R o a-2
(1)
where (r.2 is the mean-square noise'voltage, P is the amplitude of the signal, and Io is the mod-
ified Bessel function of the first kind and zero order. The moments of the distribution are
given by
*This paper
approach is
RM-753; D.
**
S.O.Rice,
TM-35
contains results previously given by othe
the only attribute of the presentation. Cf
Middleton, Proc. IRE 36, 1467 (1948).?
Bell Sys. Tech. Jour. 25, 151 (1945).
1
r authors. The simplicity of the present
. Sperry Report No. 5223-1109; RAND,
RESTRICTED
Declassified and Approved For Release @ 50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
RESTRI GTED
R n = (20- 2)n/2 r + n/2) 1F1 (?n/2; 1;?E0) ,
(2)
in which Eo = P2/20-2 and is thus the output signal-to-noise power ratio. If the level of an enve-
lope detector is adjusted to Vo volts, the false-alarm probability will be given by
cooo
PM=/ P(R) dR =i
Vo V
R2 exp[.../12/20.2] d
= exp [? V02/20-2] .
Thus we may make a table of alarm levels as a function of the number of filters.
TABLE I
Alarm Levels for Pre-Detectior. Integration
----------..?_.&
1
32
64
128
256
512
V02/20.2
10.3*
13.8
14.5
15.2
15.9
16.6
(3)
The signal-to-noise power ratio E0 required to give 50 per cent detection probability
may be found by choosing Eo such that the first moment of the envelope distribution will be
equal to Vo. By using the asymptotic expansion of the confluent hypergeometric function
k
Xk 2 k2(K ? 1)2
1F1(?k; 1;?X) [1 + + , (4)
Ilk + 1) 11X
21X2
we obtain
or
1
Vo = rs/20-2 Eo [1 + ? ? ?]
zo vo2/2,2
(5)
S/N at input to narrow-band filters:- It is now easy to deduce the required signal-
to noise-ratios El at the input to the pre-detection system. If we assume n equal rectangular
narrow-band filters to cover the full band, then E1 =
? For the practical case in which LCR (simple, single-tuned) filters are used, the
input signal-to-noise ratios would have to be 3 db greater than those given in Table II for
rectangular filters. (Noise bandwidth = Tr/2 )< 3 db band width.) This is because of the increased
noise passed by the optical filter (a factor of Tr/2) and the small average decrease in signal
power due to the fact that the signal may not fall at the center of a filter (a factor of 4/0.
The alarm level for one filter was included in order to show the small effect on sensitivity that
results from knowing in which filter the signal will appear. If this information is known in a
system consisting of 32 filters, all but one of the filters can be turned off and the alarm level
lowered by only 1.3 db to realize the same false-alarm probability.
TM-35 2
RESTRICTED
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
RESTRICTED
TABLE II
El vs n for Pre-Detection Integration
n
32
64
128
256
512
E1 (rect) db
?3.7
?6.4
?9.2
?12.1
?15
El (LRC)db
?0.7
?3.4
?6.2
? 9.1
?12
The spectral distribution from a full-wave square-law detector:- In order to deter-
mine El for the post-detector integration system, we must analyze the action of the detector in
some detail. It can be shown in a rigorous fashion that, if a rectangular band of white noise
constitutes the input to a full-wave square-law detector, the output spectrum is as shown in
Fig. 2, where the low-frequency components consist of a DC component plus a triangular distri-
bution of low-frequency components. We shall present a rather naive derivation of these low-
frequency components in order to bring out more clearly the physics of the problem.
A pure sine wave will be converted into a sine-squared wave by the detector and
hence will contribute a DC component of amplitude equal to a2/2 where a is the amplitude of the
original sine wave. Now we may consider an infinitesimally narrow band of the noise spectrum
In which the power is given by Adf. The amplitude of the wave, which in the limit is a pure sine
wave, isNiTArlf and this is converted by the detector into a sine4squared wave of amplitude
2Adf. The DC voltage contributed by the entire band of noise is thus (2AB/2) and the DC power
Is A2132. This DC component will not interest us directly, but it will help make clear the
conditions under which the output of the rectifier plus low-pass filter will be essentially Gaussian.
We shall designate this term (NXN).
The low-frequency components are given by the mixing of two different components
of noise and we shall designate such terms by (NXN'). If two frequency components X(t) = a sin
cult + b sin co2t are passed through a square-law detector, we get the output
a2 + 12 2
b2
y(t) = X2 (t) = 2 + ab cos WI ? (...)2)t ? (-4T cos 2?), t + cos 22t
+ ab cos (031 + 6)2)0
(6)
The first term in this expression gives the DC contribution from (NXN) already discussed. The
second term gives the low-frequency components from (NXN'). Thus the power contributed to
the noise in a frequency interval between f and (f + 6,f) by the mixing between an infinitesimal
band of width df' and a finite band of width tif separated by a frequency f is proportional to
2A.O.f ? P TAT1 ) = 2 A2 Af df' .
The power is now summed over f' holding f and tif constant, giving
W(f)Lif =f 2A 6,f df' = 2A2(B? f)M (f < B) ,
*W.
B. Davenport, Jr., unpublished notes on Noise Theory.
TM-35
RESTRICTED
(7)
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
RESTRICTED
where W(f) is the power density in the detected signal.
Since the power in the low-frequency rectified noise band is as great as the power in
the rectified DC noise component while the detected signal can never go negative, the unfiltered
output of the detector cannot possibly be Gaussian. However, if we filter out all but the very
lowest frequencies, the power in the low-frequency rectified noise that is passed by the filter
can be made much smaller than the DC component and, further, is almost white. Such an
argument makes it appear plausible, at least, that the noise now approximates a Gaussian distri-
bution. More rigorous treatments prove that such is the case.
Input S/N for the post-detection system:- If we add a pure sine-wave signal to the
noise, we have two more terms to add to the output. The signal can beat against itself to give a
DC signal term (SXS) in the output power proportional to P4/4. The signal can also beat against
the noise to give additional noise (SXN), and the amount of this contribution will depend on how
near one edge of the broad frequency band the signal lies. Let the distance in frequency from
the signal to the nearer edge of the band be designated by fo. For frequencies in the output less
than f the power due to (SXN) is proportional to (PAM); for frequencies in the range
(fo< f < B ? fo), the contribution is (1/2 P2AAf); while for higher frequencies, it is zero.
If we now pass the output through a square low-pass filter which passes frequencies
out to B/2n we have the following contributions to the noise power:
(NXN') cc A2B2/n ,
(SXN) cc 132AB/2n .
The last term is correct unless the signal lies within B/2n of the edge of the band, a situation
that we shall ignore.
The signal-to-noise relations are somewhat confused in the post-detection system by
the fact that the noise at the output of the detector is increased when a signal is introduced. Let
us designate the mean voltage and standard deviation in the absence of signal by Fri(0) and Cr (0),
respectively. The same quantities with signal present we shall designate by m(X) and cr. (X).
The signal voltage we shall designate by [iii(X) ? r-Ti(0)].
If, in the absence of signal, the alarm level (Vo) is set four standard deviations
above the mean noise level so that Vo = 4 cr (0), the false-alarm probability will be given by
PN 3 x 10-5. If signal is now introduced, the mean voltage and standard deviation will both
increase. For the general case, the input signal-to-noise ratio required to give a certain per
cent detection probability would be dependent on (SXS), (NXN') and (SXN). However, for the
special case of 50 per cent detection probability, the criterion is simply that the mean voltage
in the presence of signal must increase to the alarm level. Thus we have that
m(X) ? = Vo = 4o- (0).
We see that the criterion for 50 per cent detection probability can be given in terms
of the ratio of signal power to zero signal noise power (EO) as
V2
o -8
20,2(o)
and hence is not dependent on the noise contributed by (SXN) terms.
TM-35 4
RESTRICTED
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
RESTRICTED
to can be expressed in turn by
z , _ (SXS) _ p4
n
= nE12 (rect. filter) (8)
o - (NXI\V) - -4- ' A2B2
for rectangular filters of width I3/2n. We now can form a table of Zi vs n for the case of a post-
detection system using a full-wave square-law detector.
TABLE III
El vs n for a Post-Detection System
n
32
64
128
256
512
E (rect) db
1
E (RC) db
1
-3
-1.0
-4.5
-2.5
-6
-4.0
-7.5
-5.5
-9
-7.0
The loss due to the use of an RC filter in this case only involves the additional noise passed by
the filter since the signal always appears at zero frequency.
Thus for the RC filters we have
2n z
(RC filter) ,
o 7F
where the half-power point of the RC filter falls at a frequency given by B/2n.
A comparison between Tables III and II will now show the relative sensitivities of
pre- and post-detection systems. The results of the comparison depend upon the particular
problem. For example, let us assume a signal can be expected to last a given length of time.
This time will determine the optimum width for the single filter in the post-detection system.
If the target can be expected to have no radial component of acceleration, then n will be the same
for the pre-detection as for the post-detection system, and a comparison of input signal-to-
noise ratios for the same value of n is called for. One then concludes that the pre-detection
system is better by from 0 to 5 db for the range of n included in the tables. However, if the
target is accelerated, it may spend appreciably less than full time in any given filter in the pre-
detection system. In addition, propeller modulation and variations in aspect could conceivably
broaden the spectrum more than scanning. Hence the sensitivity in the pre-detection system in
practical cases could drop to the point where the two systems are essentially equivalent.
In some cases, the optimum number of filters is so large and the expectation of
accelerated targets sufficiently high that one may use less than the optimum number of filters
and not sacrifice much in the average sensitivity. In such cases, however, the post-detection
system can still be made optimum, and hence the comparison will be for different values of n
in the two cases. Thus if one uses 32 filters in the pre-detection system but can use n = 128
in the post-detection system, the post-detection system is better than the pre-detection system
by 3 db. At least some, and maybe all, of this loss could be regained by using a hybrid system
that consisted of the pre-detection filter system followed by a post-detector integrator in each
channel to give an over-all integration time equal to the time on-target.
(9)
TM-35 5
RESTRICTED
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8
RESTRICTED
Effect of detector law:- The above calculations have been based on the full-wave
square law detector. Middleton* has shown that the sensitivity of systems using half-wave square-
law detectors is identical with that given for full-wave square-law detectors, and the half-wave
linear detector is never better or worse by as much as 0.2 db. Thus we may conclude that the
exact nature of the detector is not important in the present considerations.
BROAD- BAND
FILTER
DETECTOR
LOW-PASS
FILTER
ALARM
( PRE-DETECTION SYSTEM IA POST-DETECTION SYSTEM
Fig. 1. Block diagrams for the two systems to be compared.
2A28
Fig.2(a). Input spectrum into Fig.2(b). Output spectrum from full wave square law
detector.
full wave square law detector.
*D. Middleton, Proc. IRE 36, 1467 (1948).
TM-35 6
RESTRI CTED
Declassified and Approved For Release @50-Yr 2014/04/10: CIA-RDP67B00341R000400080001-8