1959 Ya'coub: Probability Density Function 19

Delay a aperture and an appropriate corrective network offerszo better characteristics.

Ri g R= Zo The corrective networks will be found especially use-+k |ful when an aperture small enough to provide essentially

flat response up to a desired frequency is not technicallyfeasible. When a choice is possible of having the desired

C systematic fidelity (flatness of frequency response)Fig. 34-Corrective network for an exponential aperture. either by makiing the aperture small, or by utilizing a

corrective network with a larger aperture, the optimiza-be pointed out that one has even to avoid making u«<1 tion of the system (to be discussed in a later paper) de-siice for small u, pends on the signal spectrum, noise spectrum, and the

1 optimization criterion.1 - C-(u+P)a and

1 - e-(u+P)a VI. ACKNOWLEDGMENT

that appear in the transfer function of the aperture and The author wishes to express his gratitude to G. Hokof the corrective system, respectively, form a ratio of for his generous encouragement and guidance. Thesmall quantities in the transfer function of the system author feels especially indebted to L. L. Rauch, whosefor X = 0, 2r/a. Thuis any small inaccuracies in the value original suggestion of a corrective network consistingof the components cause relatively large distortion. ideally of a perfect differentiator, followed by an infinite

delay line tapped at regular intervals and giving a trans-V. CONCLUSIONS form function exactly inverse to that of a rectangular

Corrective networks have been described which come aperture, prompted the author to the development ofto offset a smoothing effect due to a finite size aperture. networks discussed herein. Several interesting discus-Certain forms of apertures (especially an exponential sions with E. G. Gilbert, who independently conceivedaperture) are to be preferred, if technically feasible, using delay linies in the feedback loop of an amplifier,sinice the instrumentation system consisting of such an are gratefully acknowledged.

Application 0f Quasi-Peak Detectorto the Measurement of Probability

Density Function*KAMAL YA'COUBt

INTRODUCTION ing of these several detectors can furnish some insightlr iRESENT interference-measuring instruments are into the type of interference being measured; this is es-t built mainly on the principle of a superheterodyne sentially the reason for incorporating more than one

receiver with the IF stage being followed by dif- detector circuit.ferent detectors which measure the average, rms, peak, Recently it has become evident that better use canand quasi-peak values of the time function.' be made of experimental data concerning noise if theseThe proper use of this equipment demands some un- data describe the statistical qualities of the noise. The

derstanding of its response to the various kinds of inter- reading of any of these detectors is dependent on theference likely to be encountered. Conversely, the read- statistical characteristics of the noise being measured,

* Manuscript received by the PGI June 30, 1958. Presented at the bttesmo nomto bandi hswyifeIRE-URSI Spring Meeting, Washington, D.C.; April 24-26, 1958. not sufficient for the solutionl of interference problems.This work was sponsored by the Bureau of Ships of the U. S. Navy. While a very detailed set of statistical data is imprac-

t Moore School of Elec. Eng., University of Pennsylvania, Phila- tial it is treta nmn ae,tefrtodrpoadelphia, Pa. tcl tI reta nmn ae,tefrtodrpoa

1 The quasi-peak detector, which is described in detail later, gives bility density function is adequate for the calculation ofa reading less than peak, the value depending on the circuit timeinefrc.constants. nefrc.

20 IRE TRANSACTIONS ON INSTRUMENTATION March

However, it requires more information to find this Differentiating this relation we get a third formfunction than can be obtained with the existing detec- dtors; we proposed to find the added data by varying the p(e > v.) --(yv,) (3)time constants of the quasi-peak detector. dvq

RELATION BETWEEN QUASI-PEAK VALUE AND where p(e>vq) is the probability that the inlput exceedsPROBABILITY DENSITY FUNCTION a value vq. This last form suggests a direct approach to

A quasi-peak detector circuit, comprised of a rectifier get the curve p(e>v,) by tabulating Vq, 7, 7Vq for a con-feeding a capacitor through a charging resistance, R., stant incremental change in v. If we are interested inand having a discharge resistance Rd is shown in Fig. 1. one point, as in the probability of the noise exceeding

the signal level s then, with two measurements givingVq and vq+AVq, we have from (3)

DIODE .ilVq p(n > s) = - Vq - y, where vq = s.

C Rdi This probability is in maniy cases related to the prob-ability of error as in a teletype system where:

Fig. 1-Quasi-peak detector.prob. of error = -p(n > s).

The quasi-peak value of a signal is the output of this It is often desirable, for instance in the analysis anddetector when fed by that signal. Although the output design of noise nmeter circuits, to be able to predict themay fluctuate as the capacitor charges and discharges, quasi-peak reading which would be obtained with a par-with proper choice of the capacitor these fluctuations ticular set of time constants and a given type of noisecould be made small compared to the dc value, and a probability function.steady voltmeter indication would be obtained. We This result can be found by solving (2a) graphically.notice that this value is not a fixed one but depends on An example is shown in Fig. 2 where the curve on thethe charge-discharge time constants we choose. Thus, right is the probability that the envelope of thermalwith a variable time constant we can get discrete values noise exceeds a certain level.3 The second curve is theor even continuous function of quasi-peak values of the integral of the first from oo to v. The latter curve cor-input. A continuous function gives more detailed infor- responds to the right-hand side of (2a). The straightmation about the input signal from which we will try lines erected on this same curve correspond to the left-to deduce the probability density function. hand side of (2a), for each of several values of y. The

In a paper by Burgess,2 the equation relating the q p points of intersection are the required quasi-peak values.value to the pdf is given. We can justify this equation We may note that for the standard time constants inby inspecting the net charge entering and leaving the use in present day noise meters, 1:600 and 1:160, thecondenser as the average values of the currents both readings are respectively 43iv and 37,tv for a randomways are equal. The equation obtained is: noise of 20,v. Also, in order to get equal quasi-peak

v , X e-z reading and rms reading, y must be 1/7.- q~~p(e)de (1)

Rd Jq R, APPLICATION FOR NARROW-BAND SYSTEMS

where v, is the quasi-peak value and p(e) is the prob- THROUGH THE EDGEWORTH SERIESability density function of the variable e. So far we have discussed the possibility of extracting

It is necessary to develop this relation to simplify the pdf from the qp values in a discrete manner. Anotherconverting qp values into pdf and vice versa. We sub- possibility of interest is that of dealing with the proba-stitute for the ratio of the resistance R,/Rd, the variable bility function as a continuous function, where they, and integrating by parts, we are left with: question is: what type of a function?

(3 Is This problem is approached through the fact that ino'vq = Iwj p(e)de. dv (2) a system of infinitesimal bandwidth, the output due to

Vq J a nonperiodic noise is the sum of a large number ofor periodics with random amplitude, each of which is a

result of the previous history of the wave. The centralvrv 2) limit theorem states that such an output will have a pdf

7vq = pe vd.(2a) given by a gaussian distribution. For bandwidths in

3A. D. Watt, R. M. Coon, E. L. Maxwell and R. NAT. Plush,2 R. E. Burgess, "The Measurement of Fluctuation Noise by "Performance of Some Radio Systems in the Presence of Thermal

Means of a Diode Voltmeter," American Standard Assoc. Paper and Atmospheric Noise," Natl. Bur. of Standards Rep. No. 5088;RRB/C 92; March 8, 1945. June 12, 1957. See Fig. 6.

1959 Ya'coub: Probability Density Function 21

__r_ l_ ___- l5 of the variable.4 As the bandwidth approaches zero, the

f l Ov dv -R lI higher order terms in the series vanish, leaving us withvq RVD the limit of a gaussian distribution. However, in anI

RC ODl \ - < _ _ °v ordinary narrow-band system, as encountered in typicalR0q 0jOvfP(e)de-D l =Jv P(e) e | communications systems, the pdf will depart from the

4 e 40 gaussian distribution, and to express this departure weinclude the first few terms in the series. We now consider

___ __: ____ ___ the possibility of utilizing the different quasi-peak read-ings for calculating the coefficient of these terms.A simplification in the original Edgeworth series

3 _ _- 30 could be obtained from the fact that the output of anarrow-band system is a modulated wave with carrier

___\ /___ ___ roughly equal to the mid-band frequency. The averagevalue is zero; therefore the variable x becomes simply

4XY { (\ t N 1,0 1 v/a. Also the voltage v or x will have an even pdf, but2 -- 20 upon examination, we find that the function f(m) is even

,. '1 / \ \ /or odd, depending on whether m is an even or odd num-___ ___ ___/ __ ___ ber; thus the coefficient of the odd terms C3 and C5

should vanish. The first several terms of the resultingseries are then:101

p(X) = f(X) + - C4f(4)(X) + C6f(6)(x). (4)

The rms value of the function described by this pdf0 k£_ = 0 is easily calculated, and is found to be equal to the rms0 10 20 :30 40 50 60 70 80 .v of the function described by a pdf equal to the first

Fig. 2-Graphical computation of quasi-peak values from probability term. That is rms = -, which fits the definition of u.distribution function. The relation between the qp values and these co-

efficients is developed from (2) by passing to normalizedwhich the output is :not gaussian, the pdf can be approxi- values and integrating twice. We get:mated by the Edgeworth series given by:

exp'exp(x) =f(x) YXq p 2 -2 x(1 -erf xq)1 C4 C6- C3f(3)(x) + f (2)(Xq) + f (4)(Xq). (5)

3 ! ~~~4! 6!1 10

+ - C4f(4)(x) + C32f(6)(X) By measuring the rms value of the input, the normalized4! 6! voltage scale becomes known and the only unknowns

1 35 280 in (5) are C4 and C6. Therefore, with two qp measure-- C5f(5)(x) - C3C4f(7)(x) - C33f(9)(x) ments we may find them by solving the pair of simul-3! 7! 9! taneous equations. We may simplify the work of finding

1 C4 and C6 if we note from Fig. 3 that each of the two+ - C6f(6)(x) + * - . right-hand terms in (2) are zero at certain values of the

6! variable. The fact thatf(l) and f(4) have roots which oc-The first term of this series is a gaussian function; the cur for xq 1 and 0.74, respectively, means that varyingothers as f(m) are the mth derivative of the gaussian y, such that the normalized qp voltage is equal to onefunction of these roots, rids us of one function. Thus, the pair of

simultaneous equations reduce to two inldependentf(m)(X) = _ r_[ exp--l.] equations, each involving only one coefficienlt:

The variable x is simply a normalized voltage 4!C4=5(.8-)(6

v-__ - C6 = 2.06(.085 - 72) (7)

wvhere a- is the standatrd deviation of the variable u. The 4H. Cramer: "Mathematical Methods of Statistics," Prinlcetoncoefficients Cm are functionls of the high order moments Univrersity Press, Princeton, N. J., p. 223; 1951.

22 IRE TRANSACTIONS ON INSTRUMENTATION March0.4 where y, and 'Y2 are the values of y needed to give quasi-

0.2- 2Lexp(-X) X(4-erfX) peak voltages equal to 0.74o and o- respectively. Thus0.2- ffi1r 2 / with such direct relations, the variable resistance might

be calibrated to indicate the values of these coefficientso- J 1L directly and individually.

Q 30.2 3CONCLUSIONS

References in the past to the quasi-peak detector have

04 viewed the device as a weighted circuit chosen for itsability to reflect the subjective effect of certain kinds of

f (X) interference on specific communication systems. Un-0°2- certainty as to what electrical quantities are being- N measured has led to differing views on the merit of the

o 1 1 1.2device.._0.3 This work was aimed at establishing more firmly the

connection between the reading of such a device withthe probability density function being measured; thatis, showing its objective side. On the assumption that ahalf-wave linear diode can be realized, it was shown that,

0.8- > f141X) by some variation in the circuit used in the past, the pdfcan be determined from its readings.

ACKNOWLEDGMENTThe author wishes to thank Professor F. Haber for

fruitful discussions and advice, and Dr. R. M. ShowersFig. 3-Components of -jx vs normalized voltage, x. who suggested the work.

Relative Voltmeter for VHF/UHF Signal GeneratorAttenuator Calibration *

B. 0. WEINSCHELt, G. U. SORGORt, AND A. L. HEDRICHt

INTRODUCTION other than the calibration of signal generators. It is

flF! HE standard signal generator has become one of basically an insertion loss test set and, as such, can bethe most useful tools in the modern electronics used to measure insertion loss over a very large rangelaboratory. These generators usually consist of a with high accuracies.

stable generator, a level monitoring device and an at- A frequency range of from 100 to 1000 mc is coveredtenuator. The level into the attenuator is set, and then by the instrument as it now exists but its upper fre-the attenuator is relied on to reduce this to the required quency limit is restricted only by the availability of thelevel for the measurement being made. proper local oscillators and mixer assemblies. The volt-The need for a means of checking and calibrating the age range (in a 50 ohm system) of from 20 my to 20 pv

output level of such generators has long been felt, and (-20 dbm to -82 dbm) is covered with an accuracy ofthe equipmenlt described in this paper was developed to 0.02 db/1O db. This range can be extended by 6 db onfill this need. The instrument has, of course, many uses both the upper and lower ends with an accuracy of 0.1

db/10 db in the extended portions. It should be pointedout here that this instrument is not an absolute volt-

* Manuscript received by the PGI, June 18, 1958. Presented at meter, but rather is a relative voltmeter and indicatesthe IRE-URSI Spring Meeting, Washington, D.C.; April 24-26,'1958. This work was supported by the Air Force under Contract No. voltage ratios-thus, the expression of accuracies inAF .33(600)-25238 with Wright Air Development Center, W\right- terms of decibels. The reproducibility of a measurementPatterson Air Force Base, Dayton, Ohio.

t XVeinschel Engineering, Kensington, Md. iS in the order of 0.01 db over the entire range.