I963 Correspondence 49 1

only odd terms in both P and Q develop and that the ring modulator is formally identical to the other double-balanced circuit.

In practice, the nonlinear tubes of the double-balanced modulator have more desir- able characteristics than the ring-modulator diodes. Over a wide voltage range the tubes give linear modulation along with closely sinusoidal carrier waveform, and such opera- tion is difficult to obtain with the sharp cur- vatures of the semiconductor diodes of the ring modulator.

P. LY. DIPPOLITO L. B. ARGUIMBAU

Grason-Stadler Co., Inc. \Vest Concord, Mass.

Comment on “Parametric Behavior of an Ideal Two-Frequency Varac tor *

In private correspondence P. Penfield, Jr. and W. P. N. Court have drawn my at- tention to defects in the theory of a recent paper of mine.’ An “Ohm’s Law” argu- ment was used to select those terms in the varactor power expansion that contribute to the output signal. While this argument identities all frequency conversions that are self-starting (yielding a signal immediately in response to application of the pump), it neglects possible signal contributions from terms in E,‘ with z greater than 2, where E. is the signal amplitude. The results must therefore be qualified. They are strictly valid only for a signal amplitude E, small in com- parison with the pump amplitude Ep (al- though both amplitudes may be large). Further, rational frequency conversions of a form other than M / 2 may be possible, al- though they will not be self-starting. (The paper also contains a misprint; the exponent of E, in (9) should be n - k + l . )

Using the suggestions of my correspond- ents, the coupling of a linear signal load termination with the varactor network can be viewed as follows. In Fig. 1 the signal cur- rent ( - I a ) delivered by the network (oarec- tor) in response to signal amplitude E. and the load current ( load) for a given load conductance are plotted against E,. Fig. 1 represents, for small E., a constant-current source, independent of E, and corresponding to my nonregenerative case. The intersec- tion of the two curves yields the operating signal amplitude Eo. This operating point is stable since the magnitude of the (negative) source conductance (dI,/dE.) is less than the load conductance.

Fig. 2 represents an output current pro- portional to the first power of E,, correspond- ing to my regenerative case. The network here is a true oscillator, operation being ob- tained when the magnitudes of (negative) source and load conductances are equal. For

* Received November 27, 1962. 1 G. F. Montgomery. “Paynetric behavior of an

ideal two-frequency varactor. PROC. IRE, vol. 50, PD. 78-80; J a n ~ a r ~ . 1962.

Fig. 3.

Fig. 2. Fig. 4.

the ideal assumptions defined in the paper, no particular amplitude En is predicted, and the output is stable for any E,.

If the contributions of higher-order Eaa terms are properly included, the network output currents can be expected to behave generally as shown by the dashed extensions in these two figures. Thus, for sufficiently large E,, the load current may differ from that predicted by the theory.

Fig. 3 illustrates a case where neither constant nor first-order currents exist. An example is a divide-by-N network with N other than 2. The pseudo operating point indicated here is not stable since the magni- tude of the (negative) source conductance (dI . /dE, ) exceeds the load conductance. Even though unstable, the network would require “priming” with a temporary signal source of amplitude En in order for oscilla- tion to begin.

Dr. Penfield suggests tentatively that a varactor current of the form shown in Fig. 4 may be possible. While the lower of the two indicated operating points is unstable, a stable point is achieved a t EO. Such a charac- teristic may be typical of dividers and multi- pliers based on power terms in Eaz with z greater than 2. If they exist these networks will require priming to start, but this idio- syncrasy can be useful; it is presently a fail- safe feature in regenerative frequency di- viders of the Miller type.

Since it is evident that the published theory’ is less general than had been in- tended, it is appropriate to summarize its limitations:

1) Formulas (12 ) and (14) for output power are valid only for E,<<E,. Without this condition the indicated power is in error by the contribution of terms in E,. with z greater than 2. \\’hen E, is large, the current I(E,) in (15 ) and the conductance G(E,) in (16) are not independent of E..

2 ) The theory correctly predicts all pos- sible self-starting frequency conver- sions, which are of the form s /p = M / 2 where A4 is an integer.

3) The theory ignores rational frequency conversions of the form s / p # M / 2 , which may also be possible, although they will not be self-starting.

G. FRANKLIN MONTGOMERY National Bureau of Standards

IYashington, D. C .

Analysis of Nonstationary Data Using an Airborne Spectrum Analyzer*

Spectrum analysis of missile or aircraft vibration signals carried out in real time using on-board swept spectrum analyzers has been discussed previously.‘ One aspect of this novel method of analysis which was not covered and which has since been the subject of considerable interest is the very pertinent question of stationarity; the con- ditions of flight affecting vibration levels often change significantly, even over a single sweep-period of the analyzer, short though it is.

A stationary time series is one in which the statistical properties are invariant with time. Truly stationary vibration signals do not exist in a missile or aircraft during flight

* Received November 23, 1962.

using airborne spectrum analyzers.” PROC. IRE, vol. 1 A. G. Ratz. “Telemetry bandwidth compression

48. pp. 694-702; April. 1960.

492 PROCEEDINGS O F THE IEEE March

and we think of the vibration record as cut up into separate time segments. For each individual segment, flight conditions are constant enough that it can be regarded as sliced out of a unique (and infinite) sta- tionary time series. The duration of a seg- ment depends on the rate of change of con- ditions in the vehicle. Sow, the statistical properties of interest for a stationary signal, x ( t ) , are the family of functions repre- sented by (l<n<= ): p , ( x l , X?, , x,; T ~ , 7 2 , . , ~ ~ - 1 ) . Selected values of x are xl, xz, . - - , x, and zj is a selected (fixed) value of the time between the Occurrence of x ( t ) at the value x1 andthat at the value xj+1. The probability density function is p l ( x l ) . The joint probability density func- tions are p2(xl , XZ; T ) , etc. These functions can only be measured with zero error if the record is of infinite length. The shorter the time slice, the more inaccurate their determi- nation and the greater the uncertainty that must be attached to any assessment of sta- tionarity.

For the purpose of spectral analysis, only p l ( x l ) and pl (x1 , x ? ; T ) are of inter- est, The power spectral density Pcf) is estab- lished by f i2(x l , xp; T ) :

p ( 9 L 1 +me-i*rf+ 2* -P

.dr J-r ~ - ~ x ~ . v z p ~ ( x , , XZ; T ) ~ x I ~ x ? ,

When X ~ = X Z , r=O, pz(x1, X Z ; T ) decays to pl (x1) . Hence, there are many forms of the function Pcf) for the same function p ( x 1 ) .

The efficiency of a fast-sweep airborne spectrum analyzer has been questioned in nonstationarity situations. The same ques- tion is not often applied to the usual ground- station swept spectrum analyzer. I t would seem useful to compare the two types of analyzers in both stationary and nonsta- tionary situations. In a stationary situation, successive sweeps of the fast-sweep analyzer can be overlayed and an average value of Pcf) obtained for each value of f . The sta- tistical error of this new estimate is de- creased over that of a single sweep by the factor of l /dN. (N is the number of over- lays.) Thus, the longer the period of “sta- tionarity,” the better the accuracy of the result. A conventional slow-swept analyzer operating on continuously recycled data usually has a statistical accuracy that is no greater than that of the airborne fast-sweep analyzer, since its longer averaging time is exchanged for somewhat higher resolution. For either type of analyzer the accuracy is more than adequate, considering the condi- tions of the test.

In a nonstationary situation, the fast- sweep analyzer can be considered as sam- pling the spectrum at successive frequencies. The data need be stationary only over the time taken to sweep one bandwidth. This time is short compared to that needed for a significant change in flight conditions. Thus, the accuracy of the samples is limited only by the statistical error of the analyzer. An excellent time history of the spectrum at any given frequency, f’, can be plotted using the values of Pcf‘) obtained from successive

the plot of PV) are separated in time by sweeps. Of course, the accurate points on

one sweep period. The values of Pcf’) at the times between the accurate points can only be assessed by interpolation; fortunately, the trend is usually very obvious.

Consider now the laboratory analyzer fed with nonstationary data recycled from a tape loop. The best it can give us is a sort of blurred estimate of the spectrum. There is no method of assessing the accuracy of this estimate at any point that does not involve a lot of additional analysis: the analyzer out- put has a reduced quantitative value. Hence, the fast-sweep airborne analyzer with its accurate spectral samples is to be preferred in a nonstationary situation. T o estimate the stationarity of a time slice, some workers have suggested using such quantities as rms or average level. These quantities are estab- lished by p l ( x ) . If they are changing, the relevant statistical properties must, indeed, be changing: but if they are not changing, it is still possible (and has been observed in practice) that Pcf) is actually changing. Something better than a device based on p l ( x ) is required.

Bendat2 has suggested that stationarity can be tested by breaking up the time slice of interest into a number of intervals and by intercomparing the statistical properties of these intervals. The airborne analyzer does this automatically. An overlay of successive sweeps quickly permits the experimenter to establish (as accurately as the statistics of the situation allow) whether or not P(f) is changing and, hence, whether or not the signal being analyzed is stationary. The use of a conventional tape-loop analyzer for this purpose is cumbersome and time consuming.

ALFRED G. RATZ Ortholog Division

Gulton Industries, Inc. Trenton, N. J.

A. G. Piersol “The Application of Statistics to the 2 J. S. Bendat. L. D. Enochson. G. H. Klein. and

Flight Vehicle‘ Vibration Problem.” ASD Tech. Rept. 61-123.

Dynamics of a Signal-Squelched Oscillating Limiter*

The oscillating limiter has been shown1-* to be an effective technique for enhancing the stronger-signal capture capability of an F M receiver, for providing automatic in- terstation noise squelch, and for reducing

tained in this communication w a s developed under * Received September 24, 1962. The analysis con-

the sponsorship of the National Aeronautics and Space Administration, George C. Marshall Space Flight Center Huntsville Ala.

1 E. J. Bag‘hdady. “Thkry of feedback around the limiter,” 1957 IRE NAIL. CONVENTION RECORD, pt. 8. pp. 176-202.

suppression properties of the oscillating limiter, 1959 2 E. J. Baghdady. “FM interference and Foise-

and IRE TRANS. ON VEHICULAR CO?~I&LXCATIONS. IRE NATL. CONVENTION RECORD, pt. 8. PP. 13-39;

vol. VC-13. pp. 27-63;aSe~tember, 1959. a E. J. Baghdady. A technique for lowering the

noise threshold of conve2tional frequency. phase and envelope demodulators, IRE T ~ u u s . ON C O M M ~ N I - CATIONS SYSTEMS, vol. CS-9. PP. 194-205; September. 1961.

the FM random-fluctuation noise threshold below that of a conventional FM demodu- lator. Important aspects of the mechanism of this technique, the effect upon signal modulation and other important properties of the dynamic response of a signal-squelched oscillating limiter, can be brought out by a simple analysis that we shall now present. This analysis shows that a signal-squelched oscillating limiter simulates a filter for phase or frequency fluctuations, whose selectivity depends directly upon the input signal strength, and inversely upon the feedback signal strength and open-loop delay. This provides a degree of flexibility and effective- ness, combined with inherent simplicity of implementation, that may exceed what is achievable by phase-lock or any other threshold reduction techniques.

The instantaneous phase fluctuations of the signal at the output of the signal- squelched oscillating limiter can be expressed in terms of the instantaneous phase fluctua- tions of the input signal as follows. Let the signal delivered by the IF amplifier be described by

eit(t) = E . COS [of + W ) ] (1) where $( t ) is a low-frequency function and E. is constant. If the limiter proper is as- sumed to introduce no phase change, the instantaneous phase w.t++,,t(t) of the sig- nal at the limiter output is identical with the instantaneous phase of+@in(t) of the re- sultant signal at the limiter input. Thus, with the aid of the phasor diagram of Fig. 1 relating the input and feedback signals at the limiter input under conditions of oscilla- tion squelch by the input signal, we can write

+out!t) = +in(O

In a circuit designed for noise threshold re- duction, I &b(Wi) I <0.775 radian, and we can write‘

(since W< = 0. + d+b(t)/dt).

Thus (2) can be simplified into the form

sin [+out(t) - W l - (-&/&)Td.&out(t)/dt. (3)

This is a nonlinear differential equation that can be manipulated in a number of ways to obtain solutions of varying degrees of ap- proximation. Approximation of the sine by its argument, which holds provided that I @,,&) - $ ( t ) 1 <0.775 almost all of the time (and hence I &*(wi)I E,,/E. <0.70 almost all of the time), reduces (3) to a linear differ- ential equation with constant coefficients whose general solution is given by the in- verse Laplace transform of

of oscillating limiters for noise threshold reductlon, 4 For all filters that one would use in the design

the phase shift near the center frequency of the filter varies approximately linearly with frequency devia- tion from the center.