The present paper examines theoretically the relative sensitivity of the detection of signal pulses in the presence of noise, (a) by observation of an oscilloscope, (b) by aural perception, in which one listens to the fundamental or a low harmonic of the pulse repetition frequency (PRF), (c) and by a meter. The metering scheme may be either aperiodic, where the rectified current is fed directly to a meter with a long time‐constant, or periodic, where the rectified current is sent through an audio‐filter tuned to the PRF, given a supplementary rectification, and then passed through the meter. The dependence of the sensitivities of the different methods on various relevant parameters is studied in some detail. These parameters include the width and the shape of the IF response, the pulse length, the PRF, and in aural or meter reception, the duration of the gate, the width of the audio‐filter, and/or the time constant of the meter. The descriptive survey of the results is given in Part I and the mathematical analysis in Part II. Among the more important results are (I): The optimum IF filter is the conjugate of the Fourier transform of the pulse, not merely for visual reception, as was previously known, but also for aural or meter reception as well. (II): For very weak signals the linear detector requires only about 5 percent more input signal power than does the quadratic to achieve the same minimum detectable signal (same final signal‐to‐noise ratio). (III): The aperiodic meter has the advantage of not requiring knowledge of the PRF, and has potentially great sensitivity if spurious fluctuations in gain can be balanced out. (IV): Meter methods can be made more sensitive than the oscilloscope if long time‐constants are available. Gating is also necessary. (V): Although the best IF filter is the Fourier transform of the pulse, the best pulse is not the Fourier transform of the filter in aural reception (though it is in visual), for the best results in meter or audio‐detection are obtained by using long pulses. In visual work, the pulse length is immaterial, to a first approximation. Curves are given showing the power required to achieve a given signal‐to‐noise ratio as a function of pulse length, IF filter width, the PRF, gating time, and audio‐filter width, in some cases when the pulse and filter are not matched (i.e., are not related as Fourier transforms). Some numerical estimates of aural and meter performance relative to visual are also essayed.

1.
D. O. North, in an unpublished report PTR‐6C, entitled “Analysis of factors which determine signal‐noise discrimination in pulsed carrier systems.” (RCA, Princeton, June 25, 1943.)
2.
G. E. Uhlenbeck and M. C. Wang have carried through similar studies which it is hoped will be available soon at the Massachusetts Institute of Technology’s Radiation Laboratory. In this connection, very extensive experimental work has been done on all phases of the visual problem by J. W. Lawson, also at (M.I.T.) Radiation Laboratory. It is expected that this work will be published by 1947, appearing in the appropriate Handbook. (Book 24, Radiation Laboratory Series, “Threshold Signals,” by Lawson and Uhlenbeck.)
3.
P. J. Sutro, “Theoretical effect of integration on visibility of weak signals through noise.” (Harvard Radio Research Laboratory, Report 411–77, February 14, 1944.) Available at Harvard War Archives, Littauer Building, Harvard University.
4.
E. R. Brill, “Improvement in minimum detectable signal in noise through the use of the long afterglow tube and through photographic integration.” (Harvard Radio Research Laboratory, Report 411–84, February 8, 1944.) Available at Harvard War Archives, Littauer Building, Harvard University.
5.
H. Fletcher, Bell Lab. Monograph B‐1205 (Dec. 1938), cf. Fig. 17 of “Auditory Patterns as a Means of Studying the Hearing Process Taking Place when Sound is Sensed.”
6.
N.
Wiener
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Acta. Math.
55
,
117
(
1930
),
and later, independently,
A.
Khintchine
,
Math. Ann.
19
,
604
(
1934
).
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G. I.
Taylor
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Proc. Lond. Math. Soc. Sec.
2
,
20
,
196
(
1920
),
and
G. I.
Taylor
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Proc. Roy. Soc.
164
,
476
(
1938
).
8.
S. O.
Rice
,
Bell Sys. Tech. J.
23
,
282
(
1944
),
and
S. O.
Rice
,
24
,
46
(
1945
). These two papers contain an excellent and quite comprehensive treatment of a large number of noise problems.
9.
J. H. Van Vleck, “The Spectrum of Clipped Noise” (Harvard Radio Research Laboratory, Report 411–51, July 21, 1943). Available at Harvard War Archives, Littauer Building, Harvard University.
10.
D. Middleton, “Note on The Theory of Square Law Rectification of Modulated Carrier in Presence of Noise.” (Harvard Radio Research Laboratory, completed July 12, 1944.) Available at Harvard War Archives, Littauer Building, Harvard University.
11.
Equations (14) and (16) are derived on the assumption of small signal quadratic detection, arising from the square‐law character of the dynamic response in the neighborhood of the operating point. It often happens that “large‐signal” rectification occurs, where now the rectifier may have essentially a half‐wave quadratic response. However, the results for the low frequency correlation are proportional, Eq. (16) being 4 times as great as the corresponding expression for half‐wave detection. This is easy to see when we observe that only the envelope of the incoming, narrowband disturbance is reproduced, albeit squared, in the low frequency output, exclusive of d.c. The factor of proportionality follows since in the latter case but half the input wave is transmitted.
12.
The various forms of RLF(t) appear to have been derived independently by Fränz, Zeits. f. Hochfrequenztechnik, p. 140 (1941), who gave essentially the series expression, by D. O. North, who mentions it in a synopsis of a paper entitled “The Modification of Noise by Certain Non‐Linear Devices,” presented at the Winter Technical Meeting of the I. R. E., Jan. 28, 1944, and by G. E. Uhlenbeck in terms of the complete elliptic integrals E and K. in (M.I.T.) Radiation Laboratory Report 453, “Theory of Random Processes,” Oct. 15, 1943. These results may also be obtained from Rice’s expression (4.7‐6), reference 8, for the correlation function of the noise output after half‐wave linear rectification, with the help of (12) and the expansion of (cos ω0t)2n.
13.
See S. O. Rice, reference 8, Eq. (4.2‐3).
14.
S. A. Goudsmit, “Comparison Between Signal and Noise” (M.I.T. Radiation Laboratory, Report No. 193, January 29, 1943).
15.
W. H. Jordan, “Action of Linear Detector on Signals in the Pressure of Noise” (M.I.T. Radiation Laboratory, Report No. 305, July 6, 1943).
16.
Jahnke and Emde (Stechert, New York, 1938), pp. 275 et. seq.
17.
See, for example, E. C. Kemble, Fundamental Principles of Quantum Mechanics (McGraw‐Hill Book Company, Inc., New York, 1935), p. 36.
18.
“Realizability of Filters” (M.I.T. Radiation Laboratory Report No. 637, Dec. 8, 1944).
19.
J. W. Lawson and G. E. Uhlenbeck, Radiation Laboratory Series, book 24.
20.
See the first part of section 4.3, of Rice’s article (reference 8).
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