PROCEEDINGS OF THE IEEE

VOL.

54, NO. 1

JANUARY,

1966

T h e Spectrum of Clipped NoiseJ. H. VAN VLECR AND DAVID MIDDLETON, FELLOW, IEEE

HISTORICAL INTRODUCTIONT IS RISKING anengineering platitude to observe engineering community a t large, a s i t remained unpubthat noise, whether i t be electronic, acoustical, lished because of post-war adjustments and changes in

the original authors area of research. Also, he did not seismic,optical,etc.,inorigin, is acriticaland ultimatelylimitingfactorinallcommunicationprorealize that the report would excite so much interest. A cesses. The study of suchnoise and its mathematical short summary of some of the principal results were, description have becomean integral part of the sophisti- however, presented in volume of the M.I.T. Radiation 24 cated statistical communication theory of the present Laboratory Series, but in most cases this was not enoug era. As is well known, an important element in the anfor detailed application. During the period from 1946 alytical description of noise is its power, or intensity, to the present, many requests for the original report is in- were received. spectrum.Inmanypracticalapplicationsone terested in what happens to the noise spectrum when Because of this continuing interest, because this work the noise is passed through both linear and nonlinear is an illustrative forerunner of the now more highly dedevices. One such problem is the subject of the histori- veloped approaches in current use, because a considercal paper presented below, namely, the calculation of ableportion of thematerialisstillnew, i.e., unpubthe power spectrum of normal, or Gaussian, noise after lished, and finally because the work itself appears to i t has beenrectifiedbyaclipper,orzero-memory have some intrinsic historical importance, the authors rectifier, that chops off the extreme values of the noise have submitted it in its entirety to the readership, feelwave. In particular, it was desired to out how much ing (with apologies to Dumas) a t t h e same time a little find of the total available powerin the noise wave was redis- like the aging dArtagnan and wondering whether this tributed into spectral components away from the orig- paper should perhaps better be entitled, The Spectrum inal, central region of the input spectrum. The applica- of ClippedNoise-Twenty(three)YearsAfter! The tion at the time was to enhance and effect the interfer- authors note, in conclusion, that apart from very minor ence with radar and communication systems over broad editorial changes and the inclusionof references to subspectral regions. sequent work, the present paper is the original Report, The Spectrum of ClippedNoisewaswrittenby with the addition of a short Appendix of pertinent maJ . H. Van Vleck and presented as Report No. 5 1 on terial (prepared in 1944 by the second writer, who was the July 21, 1943, by theRadio Research Laboratory of Har- principal authors research assistant a t Radio Research vard University, operating under the supervisionof the Laboratory during the war years). A certain amount of Office of Scientific Research Development. and This technical editing, in the form footnotes, has also been of workwasa part of amuchlargerexperimentaland carried out, in order to relate this earlier work to later theoretical study carried out at the time, during World developments and to embed it more effectively in the War 11, of thefactorswhichgovernedtheelectronic technicalhistory of the field. All footnotes(andthe jamming of the radar and communication systems of Appendix),accordingly,representadditionsmadeby that period. Although completely declassified t t h eend the second author at thepresent time. The original Rea of the War, this report was not readily available to the port gives new material not yet published in the literature, viz., the details of the calculations for superclipping of normal, narrow-band noise with a rectangular power spectrum, numerical results for the Gaussian and Manuscript received October 6, 1965. J. H. Van Vleck is with theDepartment of Physics; Harvard optical(Lorentzian)spectraldistributions,andthe University, Cambridge, Mass. treatment of Case (B), e.g., carrier amplitude-modulated David Middleton is a Consulting Physicist located at 23 Park by superclipped noise. Lane, Concord, Mass.2

VAN VLECK AND MIDDLETON: SPECTRUM THE

OF CLIPPED NOISE

3

Abstract-The present report calculates in some detail the ( -the center of this range. T h e distinction between (A) i tensity) spectrum to be expected for clipped (also called limited) and (B) is illustrated in Fig. 1, in which the unclipped noise. Two cases areconsidered: (A), the clipping of an unmodulated disturbanceisindicatedbydotted lines. In studying noise band (DINA) and @) a carrier modulated by clipped noise. (B) we assume throughout that the phase of the carrier The computations are made for various shapes of noise bands before clipping, viz., 1) a uniform or rectangular structure, 2) a Gaussian is undisturbed by the interruptions; this point is disdistribution, 3) an optical (Lorentzian) shape factor of the type cussed more fully a t t h eend of Section 11. 1 / [ ( ~ - ~ ~ ) 2 + 6 2 ]The simplest type of calculation to make is that for . I t is often illuminating to study particularly what we what we term extreme clipping, wherein the limiting amplitude shalltermextremeclipping. By this we mean that is very small compared to the nns amplitude before clipping. The in mathematical theory for this is given in Section III, while Section all but a very small portion of the dotted curve Fig. 1 N develops the theory for clipping at an arbitrary level. The basic isshaved off. T h e resultingwaveisthenpractically mathematical method, which is rather general and is useful, we berectangular in (A), and its envelope rectangular in (B), lieve, foravariety of noise problems, is presented inSection 1 as shown in Fig. 2. Usually we are interested in an (in1 and consists in utilizing a relation between the correlation function tensity)spectrum whose frequencybandwidthbefore and the normal surface, along lines suggested by Rice [l]. clipping is small compared to the carrier or mean freThe results of the calculation are discussed in Section I and are displayed in Figures 4-10 and Tables I and II. I the clipping is not quency. Then in (A) the instantaneous frequency will f down to more than the rms level before limiting (equivalent to c l i p deviateonlyslightlyfromthat at thecenter of the ping at about 1.4 times the rms level after clipping), there is practiband. On the other hand, in (B) the noise does not discally no distortion of the spectrum. Even in the case of extreme c l i p turb the carrier but gives very irregular discontinuities ping the wastage of power due to spoiling of the spectrums uniforin the envelope.Correspondingly,inFig. 2 we have mity is small, amounting only31 percent in(A)and 24 percent in@). to drawn the graph for (A) with more or less equal segOf the 31 percent loss in (A), 19 percent is due to production of harmonics of the centralfrequency. Corresponding harmonics are absent ments, but that for (B) with a more varied distribution in (B). Clipping is beneficial for jamming purposes in either ( A ) or of lengths. I n a practical case the deviations from equal( B ) since it reduces thepeak#ower requirements.In addition, in ( B ) it materially diminishes the wastage ofpower the cmrier in frequency. These facts are demonstrated particularly clearly by Tables I and II. For instance, Table 1 shows us that in (B) the ratio of the energy in 1 the noise sideband to that the carrier i only 0.23 when the clipping in s level i twice the original rms noise level, but increases to0.52 when sthese two levels are the same, and 1.0 for extreme clipping. to It is to be cautioned that the present report calculates only the (intensity) spectrum of the clipped noise, and does not deal with its effectiveness on a receiver, which we hope to discuss later from a quantitative standpoint.* We can, however, say qualitatively that i f the receiver breadth is very small compared to the noise band, the received disturbance will have the sametype of Gaussian fluctuation, and hence the same effectiveness as unclipped noise with the same spectral distribution. On the other hand, if the receiver is comparable with the noise band in width, there will be, due to the clipping, a tendency for a ceiling in the resultant deflection of the recording device, and under these conditions the utility of clipped noise for jamming is materially diminished.

ity in (A) would be even smaller than in Fig. 2, as we have exaggerated them here to make them visible. ( I t should be mentioned that the two parts of Fig. 1 and also of 2 are drawn to different scales.) The frequency of occurrence of the teeth in (A) is approximately the same as the central frequency the DINA noise band; of the carrier frequency in (B) is of the same orderas this, whereas the interruptions in(B) represent a modulation frequency which is much lower and which is of the order of the noise bandwidth.

H E N WE SPEAK of a clipped noise we may mean either of two things: (A) a disturbance due entirely tonoise (DINA), which we may suppose confined t o some frequency band and whose maximum amplitude is limited,or (B) a carrierwave whichis noise modulated and whose envelope of modulation is limited.Effect (A) isproducedbyfilteringandthen limiting a pure noise source, while(B) may be generated, for instance, by impressing a biased, clipped noise on the grid of a vacuum tubewhose plate circuit containsa sinusoidal RF. The Case (B) will also be produced even if the noise on- thegrid is unclipped, provided the tube amplifies linearly over a certain range of grid voltage, with perfect blocking and saturation a t t h e lower and upper limits of this range, and provided the bias is a tThis is an acronym for direct noise amplification; DINA is usually produced physically as amplified photomultiplier noise. * This analysis was never carried out.

w

I. DESCRIPTIVE SURVEY RESULTS AND

(A) (B) Fig. 1. The general case of clipping at an arbitrary levelschematic illustration of the difference between Cases (A) and (B).

Fig. 2.

(A) (B) Examples of extreme clipping for Cases (A) and (B).

It may be objected that the introduction of extreme clipping is too much of an idealization, sincei t can never be reached in practice. However, the hypothesisof complete clipping greatly simplifies the analytical work, as comparison of Section I11 (Mathematical Theory of Extreme Clipping) and Section IV (Clipping at a n Arbitrary Level) will show.As theextrememodeloveraccentuates clipping any effects introducing by the maximum possible, it shows in a nutshell what is the general effect of clipping. I t t u r n s o u t t h a t even ex-

4

IEEE PROCEEDINGS OF THE

JANUARY

treme clipping does not have a great deal of effect on the fundamental angular frequency are, respectively, we the spectrum, and so we may feel confident that in no 8/91r2, 8/251r2, 8/497r2, of thetotal. All told, a fraccase does clipping do much harm as far as spectral dis- tion 1-8/a2 of the total energy is converted into hartribution isconcerned. The present report does not, how- monics, and so the energy wasted in harmonicsis 19 ever,attempttotreatthemoredelicatequestion of percent of the total. In practical cases, the wastage is whether or not clipped and unclipped energies of the probably less than this estimate would imply, as the same spectral distribution are equally effective in jam- resonant amplifying devices of the transmitter are ming a receiver. We hope to treat this later.a Without usually not tuned to the harmonics andso dissipate but any quantitative analysis it is apparent that the relittle energy in the latter. However, it should be mensponse of a receiver to clipped noise will have a ceiling, tioned that the filtering of the harmonics increases the above which a pip is easily spotted, unless the bandfluctuations in the energy, and so does not particularly width of the receiver is very narrow compared with the help if avoidance of high peak power is the important spectral width of the noise. With a very narrow receiver criterion. For instance, it will destroy the equality of the noise pattern on the receiver will fluctuate in the rms and peak amplitude characteristic of extreme clipsame fashion as without clipping, the reason being that ping, as this behavior is secured only when the fundathe limitation in amplitude due to clipping applies only mentalandharmonicFouriercomponentsaresuperwhen all Fourier components are added, and when the posed. I n general, removal of the harmonics will make receiver selects only a small portion of the noise spec- the ratios of peak to average power somewhat greater trum this limitation is lost. The quantitative question than the values to beshownin Table I, though not which we reserve for a later paper is the study of ex- nearly as large as before clipping. actly how the ceiling disappears when the receiver width The different behavior of (A) and (B) as regards haris graduallynarrowed.Instead, we shallexamine at monicsiswhatonemightguessfromonesphysical present simply the spectrum emitted by a transmitter intuition. Namely, if the noise band is exceedingly narequipped with a clipper, and see how much energy is row, then the disturbance portrayedin Fig. 2(a) is very lost by being spilled outside the frequency range which closely a square wave, as the caterpillar effect, it isdesired to cover with the barrage. whereby the successive segments varya little in length, The mathematical theory the spectrumof a clipped of noise5 will be developed in Sections11-IV; i t is based on a statistical method suggested by Rice We are much [l indebted to him for stimulating discussions. In mathe- E, matical terms, the effect of extreme clipping turns out to be to replace the original correlationfunction r(t) before clipping with a new correlation function which has the value (2/7r)sin- r ( t ) [cf., our later (17)]. By A the correlation function is meant the mean valueof the Fig. 3. Intensity spectra of clipped noise; Cases (A) and (B). product of the amplitude at times t o and h+t, as explained more fully in Section 11. There is an important difference between clipping in Case (A), (DINA),andin(B),(noise-modulatedcarTABLE I rier). In (A) part of the energy is located after clipping CASE(A) (DINA) in bandswhichareharmonics of the original funda(Extreme (No mental band present before clipping, while in (B) the clipping) clipping) energy is confined to the fundamental. This distinction is illustrated in Fig. 3. The amount of energy located in Clippingamplitude =b 0 1 . 00 . 5 2.0 00 each of the harmonics in(A) is precisely the same as that original rms amplitude in a square wave devoid any noise effects.The Fourier of analysis of such a wave is well known, viz., even har- Clipping amplitude 1 1 . 1 7 1.39 2 . 0 9 00 monics are wanting, and the fractions of the total enrms amplitudeafterclipping ergy associated the withharmonics he, 7we * . of Bottleneck factor 5we, 0.69 0.86 0.95 0.97 1

3.

I

.

Subsequent work by Middleton [2], [3], Davenport [4], and Price [SI has extended the theory. For a short account, with associated experimental results, see also Lawson and Uhlenbeck [ 6 ] .An extensive treatment of these and relata topics involving noise through nonlinear devices may be found in Middleton [7]. We remark that the studies in [2]-[5] are concerned with Case (A) only. A theory for Case (B), including vth-law rectifier and an additive background noise, is developed in 171, 13.4-4, pp. 591-594.6

This was never written.

* This was never done.

oe Fraction of p w r in harminicz = 7Effective power Total rms power Effective power

0.19 0.69

0.08 0.03 0.0050.86 0.95 0.97

01

0.69 Peak power

0.63 0.49

0.22

0

Sec.

1966

VAN VLECK AND MIDDLETOS: THE SPECTRUM OF CLIPPED NOISE

5

is small. Hence, under these conditions one is tempted to use the ordinary Fourier analysis of a square wave to determine the ratios of the energies in the various harmonics and the more exact treatment of Section I11 shows that these ratios are not altered as long as the bandwidth of the noisesmall is compared with its central frequency. On the other hand, in (B) the interruptions do nothing to the carrier, and harmonics of so the latter do not occur. Neither in Case (A) nor (B) can the spectrum of the fundamental after clipping, which is the thing of main interest, be determined without statistical and mathematical analysis, which is given in Sections 11-IV. Our mathematical method would also enable us to compute -50 -4.0 -30 -20 -10 0 IO 20 30 1 4 0 the shapes of the harmonic bands in (A) but this does Fig. 4. Intensity spectrum of DINA (i.e., narrow-band normal noise) after extreme clipping, when the original spectrum is not really interest us, as the harmonics represent specrectangular [Case (A)]. tral domains of far higher frequency than the barrage to be covered. Hencewe have not calculated the shapes of the harmonics in detail; without detailed numerical computationsthemathematicalanalysisinPart I11 shows that the shapes the harmonics are not the same of as the fundamental, and that the diffuseness becomes greater the higher the degree the harmonic6 of The spectral curves for the fundamental which are furnished by the calculations of Section I11 are shown UNCLIF?EO in Figs. 4 and 5 for extreme clipping of Types (A) and (B), respectively,applied t o a noisebandwhichwas rectangularbeforeclipping. Theordinate scale is in each caseso chosen that the total power in the spectrum is normalized to unity bothbefore and after clippingso as to enable us to compare spectra of equal power content. We use the letter X for the dimensionless ratio X = (w -wc)/ua, where wc is the central angular frequency in (A) or carrier in (B) and wc-wa, coc+wa are the limits of the noise spectrum before clipping. Reference to Figs. 4 and 5 shows that the effect of clippingis t o Fig. 5. Intensity spectrum after extreme clipping of a carrier modulated by narrow-band normal noise with a rectangular spill a certain amount of energy outside the limits power spectrum [ C a s e (B)]. X = & 1, and also to make the spectrum cease to be quite uniform within these limits.I t is rather remarkable that even with clipping there is a finite discontinuity at the original boundary X = f 1. The amountof discontinuity in either (A) or (B) amounts to 2/a of the height of the curve before clipping. The energy spilled outside X = 1 is practically wasted, as i t is too diffuse to be of much value; a little may be picked up by receivers near the sides of the barrage, but not much. The fact that the curve for (A) is lower than that for (B) is easily understandable, as in (A) 19 percent of the energy is located in harmonics not shown in thefigure. I t should be mentioned that the curves for (A) and (B) have somewhat CLlRlNE LEVEL R M 5 MISE LEVEL B W E E different shapes, i.e., they do not differ merely in ordix;w-wc CLIPPINS nates; physically this is because the clipping processes wa are not quite the same. Apart from the harmonic losses, rather less energy is spilled outside the original band in (A) than in (B). -50 -3.0 -4.0 -2.0 -1.0 0 10 2.0 3.0 x 4.0 5.0

t

See

171,

Secs. 5.1,

5.2.

Fig. 6. Same as Fig. 4, now with various levels of clipping.

6

PROCEEDINGS OF THE IEEE

JANUARY

T h e effective bottleneck that determines the amount the noise pattern can be reproduced symmetrically only 100 percent of power needed is the height of the curve at the edges by using a bias. Weassumethroughout X = & 1, since a certain minimum power is required t o modulation, so as t o minimize the carrier wastage. (In jam any station, and this will be less effective at X = 1 other words, we presuppose the minimum bias necessary in order for the tube to function even when the than at X = O because the spectral curve slopes away from its maximum achieved a t X = 0. (Apart from this, noise voltageonthegridhasitsmaximumnegative value.) Because of the bias, instead of having to deal there is the fact that because of the finite bandwidth with the spectrum of a functionwhichoscillatesbeof the receiver, the barrage generated by the sending station must be somewhat wider than that which it is tween the two values - l, l, we have really a function f whose two possible values are 0, ~ 2(the norwished t o cover, since one must have power available over most of the bandwidth of the receiver. However, malization in each case we take such that = 1). T h e amount of energy in the carrier is (T>z and that in the correctionsforthis effect will beapproximatelythe hence, if f spendsontheaverage same with and without clipping since the band is so sidebandsis 7f so nearly rectangular in either case.) If we call unity the equal times atf=O and = 4 2 , one has @)*= (1/2)7, that the energies in the sidebands and in the carrier are height of the rectangle before clipping, then the height of of the curve at = 1, which we shall call the bottleneck just equal. With clipping not an extreme type, an even X factor, is 0.69 in Fig. 4 and 0.76 in Fig. 5 . Hence, we greater portion of the energy is wasted in the carrier. see that in (A) 31 percent of the energy is wasted, due to The formula for this wastage is given in the final parathe nonrectangular shape of the curve, and 24 percent graph of Part IV. With unclipped noise, the amount of in (B). Of the 31 percent in (A), 1 percent is due to har- energyinthecarrierwould,strictlyspeaking,bein9 that in the noise sidebands, monics and, as already mentioned, this kind loss can finitely large compared with of t oftenbeavoidedthroughfiltering,though at the ex- as an infinitely large bias would be requiredo cope with the largest conceivable fluctuation in a Gaussian law. pense of greater fluctuation. The mathematical theoryfor clipping at an arbitrary From a practical standpoint, however, clippingat twice the rms level has only a very small effect on the speclevel, instead of extreme clipping, is developed in Section IV. Figure6 shows the resulting curves for Case trum. (A) Table I compares the results of clipping at various (DINA)forclipping a t amplitudes 0.5, 1.0,and 2 0 . times theoriginal rms level. As previously, it is assumed levels for Case (A) (DINA), while Table I1 shows analogousfiguresforCase(B)(noise-modulatedcarrier). that the noise bandisuniformorrectangularbefore clipping,andthatthetotalpowerisnormalized t o In thefirst and second rows of the table we give, respectively, the ratio of the limiting amplitude to the rms unity after clipping (as well as before). We have not drawnthecorrespondingcurvesforCase (B) (noise- amplitude before and after clipping; the latter is obvimodulated carrier),as the general trend one goes from ously the greater of the two ratios, as clipping reduces as mean amplitude. column The labeled with zero no clipping to extreme clipping is the same in (B) as in the (A) : namely, the distortions in the spectrum occasioned ratio of clipping t o original rms amplitude corresponds for The bottleneck factor by clipping vary in an exponential fashion, as they are to the theory extreme clipping. is the ordinate of the curves in Figs. 4-6 a t t h e edges given b y formulas whose most significant factor is edf, X = & 1, as previously mentioned. The ratio effective where b is the ratio of the clipping to rms amplitude. This mode of dependence on the amount of limiting, power/total power is obtained by dividing this factor taken in conjunction with the fact that the distortion is by 1 + p , where p is the ratio of carrier to sideband energy. I t is the proper measureof efficiency if the average not great even with extreme clipping, shows that if, say, we clip at twice the rms value, the disturbance of the power generated after clipping is the critical factor govspectrum will be almost negligible, and evena t t h e r m s erningtheexpense of the apparatus. If inCase(A), however, the harmonics can be suppressed b y filtering, level it is slight. In comparing the results with clipping various lev- then the estimate of the effective power should be inat ( els, it is essential in Case (B) t o include the energy dissi- creased b y 1 / 1 -y), or approximately 1 +y, where y is pated in the carrier, which has not been included when the fraction of energy in harmonics. Often peak power in each case we normalize the output to unity. With ex- is the restricting factor in the designof apparatus, and then, clearly, the ratio effective t o peak power is that of treme clipping, the amount of power in the carrier is of interest. I t is seen t h a t a high degree of clipping imjust equal to that in the noise sidebands or, in other proves the efficiency quite materially in either (A) or words, theareaunderthedelta-functionindicated (B), schematically by the vertical straight lines a t X = 0 in (B) if the peak criterion is used, and also in even if, as a Fig. 5 is just equal to the area under the rest of the instead,thetotalgeneratedpowerisemployed curve: namely, unless one goes to the troubleof having measure. I t is t o be cautioned that all the figures in the a balanced modulator, the transmitting apparatus will tablesmakenoallowanceforthedifferencebetween have an output only the grid potential is positive and clippedandunclipped noise as regards the confusion if

+

F

1966

VAN VLECK AND MIDDLETON: THE SPECTRUM OF CLIPPED NOISE TABLE I1 CASE(B) (NOISE-MODULATED CARRIER)

7

Clipping amplitude

=bOriginal nns amplitude Clipping amplitude

0

0.5

1.0

2.0

0

1

2.09 1.39 1.17 0.95 0.74 0.98

I2%E,CARRIER,,UNCLIPPEO'OPTICAL'SPECTRUM SPECTRUM 'OPTICAL' EXTREME

00

rms amplitude Bottleneck factor

clipping0.88 0.761.001

AFTER

Ratio of sideband to carrier power = 1/p Effective power Total rms power Effective power

0 . 5 2 0.23

0 0-50

0.18 0.32 0.37 0.38

-40

-30

-2.0

-1.0

0

1.0

2.0

3.0

x

4.0

!i.0

0.19

0.16

0.12 0.06

0

Peak power

Fig. 9. Intensity spectrum after extreme clipping of a carrier modulated by narrow-band normal noise with an optical (Le., Lorentzian) spectrum [ C a s e (B)].

Fig. 7 . Intensity spectrum after extreme clipping of a carrier modulated by narrow-band normal noise with Gaussian a spectrum [Case (B)].

2% Ew1.0

.I

,-4.0

/:'I'-3.0

'OPTICAL'SPECTRUMUNCLIPPEO'OPTICAL'SPECTRUM. CLIPPED DIM

DIM

EXTREMELY

,-20-IO

,20

,30

~

0

'

IO

x

4.0

1

Fig. 10. Same as Fig. 9, but for narrow-band noise (DINA) alone [Case (A)].

,

UWCLIPPEU GAUSSIAN SPECTRUM,DINA GAUSSlAllSPECTRUM,DINA EXTREME CLIPPING

AFTER

-50

-40

-30

-20

-10

0

10

24

3.0

x

40

Clipping Abrupt

Groduo I Clipping

Fig. 8.

Same as Fig. 7, but for narrow-band noise (DINA) alone [Case (A)].

Fig. 11. Dynamiccharacteristics of amplitude limiters with abrupt and gradual clipping.

8

JANUARY

IEEE PROCEEDINGS OF THE

produced in the receiver pattern. Hence, unless the receiver width is very small compared with t h a t of the where f ( t ) is the function being studied. The meaning barrage,theadvantage of going toextreme clipping of the bars requires particular comment. I t means an will not be as great asis indicated by the tables. average over all the fluctuations the system, or, more of We have also made calculations for the effect of extechnically, an average over an ensemble of similarly tremeclippingon noise bands whichbeforeclipping prepared systems differing only in phase.9 Under these have a Gaussian shape e - - ( w - * ) z ~ ~ : 2 and on bands which , circumstances, R(t) is b y definition independent of the have a shapefactor l/[62+(w-wc)2] like t h a t of a n initial time t o . Also R(t) will be even in t. If there is a opticalabsorptionlineor a simple LRC circuit [2]. carrier frequency uc, then the correlation function will The results appearin Figs. 7, 8 for clipping of types (B) be of the form ~(t)cos27rvct. factor The cos of, and (A) for the Gaussian case, andin Figs. 9 and 10 for (w, = 27rvc), is of no interest, as it will be ironed o u t in the optical one. I t is seen t h a t in each case the clipping the video part of thedetector.Clearly, R(t), orits tends to diffuse out the spectrum and lower i t in the envelope in case there is a carrier, is a measure of how center, although in the aggregate the distortion is not rapidly the disturbance tends to fluctuate on the screen large. The lowering is greater for (A) than for (B), due of a (radar) receiver.I0 t o dissipation in theharmonicsin(A).Theenergy T h e use of the correlation function as the measure of wastedintheharmonicsin (A) hasthesamevalue, the quality of receiver response, however, requires cau19 percent, as in the case of a uniform band, since this tion. The value of t at which R(t) drops to say half its fraction is independent of the shape of the band,I promaximumvaluecanbeemployed a s a gauge of the vided only t h a t i t is narrow compared with thecentral width of the blades of grass only if a), there is no frequency. periodicity to the disturbance and b), it does not have Hitherto we have assumed that the clipping sets in any holes through which the signal can be seen. abruptly atsome given level. Another type distortion of Incasea) is not satisfied, the physical reception is what we may term gradual clipping,inwhich the screen does not really average over the different phases, saturation commences gradually rather than suddenly. i.e., R(t) is a periodic function of t , and the disturbance The distinction between the two isshowninFig. 11. will have a discernible periodicity.For instance, consider A convenient analytical representationof weak clipping a uniform square wave, for which f is zero half the time is obtained by assuming that the response curve of the and unity the rest. The uniformity means that all the amplifier, instead of being linear, is of the forms tanh segmentsin Fig. 2 are to be considered of, the same ( V / C ) ,where V is the grid voltage measured relative t o length. Then R(t) is a periodic function t and does not of some appropriately chosen bias, and is the ratioof the C approach a definite limit t = m . If the reciprocal of the, at amplitude at extreme saturation to the rms amplitude bandwidth of the receiver corresponds toa time interval which would result were there no saturation corrections. shortcomparedwiththelength of thesegment,the The (intensity) spectrum can be readily computed, as pattern on the receiverwill still be a well-defined square is shown at the end of Section 11, provided C is large wavewith holes andpeaksalternatingregularly.If, compared with the original rms noise level Vo. The dehowever, the reciprocal of the bandwidth corresponds viations from uniformity, however, are small a s long a s t o a time interval long compared with that represented C is large enough to make the convergence adequate. by the segment length, then the result really will be a Forinstance, if C equals 4, thedeparture fromuniuniform smear. The fact that in this example the corformity amounts to less than one percent, so t h a t relation function does not go to zero t+ m shows that as gradual clipping of this typeis practically without effect there is a recurrent periodicity. In fact, the behavior of on the spectrum. The saturation does, however, reduce thecorrelationfunction at infinityenablesusto see the mean square amplitude even it does not distort when whether or not the disturbance is truly irregular. With the spectrum,lowering it about10 percent, for example, order, has one R( m ) # O , while with randomness in the case of C=4 (unless i t is renormalized to unity R( w ) =o. as in the procedure followed in Figs. 4-10). (1) is With regard to b), the average in overall phases, and so includes both holes and peaks. Therefore, thecor11. THECORRELATION FUNCTION AND relation function does not give (an) indication as to the THENORMAL SCRFACE existence of holes in the pattern. Even though the deis I t is perhaps well to begin the mathematical analysis cay timeof the correlation function so short that holes by discussing the role of the so-called correlation func- of corresponding length would be too brief t o be discernible, there can actually be holes of such great length tion. By the latter is meant the expression9 Strictly speaking, we have here an (auto-) covariance function for an assumed stationary process, which, of course, is an ensemble average. 10 For detailed a discussion of the correlation and covariance functions of a process, and the spectral concepts associated with them, see [7], Secs. 3.1 and 3.2.

See [l],Sec. 4.9, Eq. (4.9-19); I also [7], Eq. (5.29) and subsequent remarks. 8 Other rectifier laws, in particular the symmetrical vth-law devices, are discussed in [7], Sec. 13.42.

1966

MIDDLETON: SPECTRUM AND VLECK THE VAS

OF CLIPPED NOISE

9

1 +aa as to be troublesome. Let us suppose, for instance, that f ( t > = + - A (w)eiwtdo, A ( w ) = 2, f(t)e-iwdt, (3) the disturbance consists of a series of irregularly spaced -m pulses, so that the width of a pulse is small compared where A ( - w ) is the complex conjugate of A ( w ) , since with the spacing. The correlation function is then the f ( t ) is real. We use the imaginary exponential rather than same as that for one pulse considered separately, and yields an apparent fluctuation time corresponding to thethe trigonometric form of the integral, as it makes the easier. From Plancherels theorem we pulse width, but there will be obvious holes in the pat- manipulations have : tern. These will disappear when the interval between pulses becomescomparabletothe receiver width. A quanJ-Yj(t)2dt = 2 r A (a) 2 d W . (4) titativecriterionfordeterminingwhetherornotthe hole difficulty is serious may be established by examining whether F(t)4 greatly exceeds [ F ( t ) j 2 or is of the We shall now imaginethe disturbanceconfined to a very same order of magnitude as the latter: with a distur- long period 2 T such that this intervalis so long that the bance which functions only a small fraction of the time, cutoff a t t h e beginning and end has no bearing on the the former will be much the greater of the two. Here, shape of the Fourier spectrum, but does make the inteF ( t ) specifies thetimedependence of the receiverregrals converge. Then we see that sponse, and can be identified with the form f ( t ) of the impingingwaveonly if the receiver breadth iswide 1 J-, compared with the mean fluctuation frequency of the disturbance. The left side gives the mean energy, and the right side From the above it is seen that the correlation funcshows how the various frequencies contribute. Since E, tion must be used with some caution as a gauge of the is to be identified with both the terms in +wand - w , fluctuation width. The same remarks, however, apply we have immediately the result that evenmorestronglytotheusualattemptstojudge effectiveness by looking a t t h e Fourier components, as 2?r E, = - A ( w ) the latter obviously give no information concerning the T existence of holes. In the last analysis, it is rather futile to say whether the studyof Fourier components or cor- Now let us compute E , in another way. We can write relation function gives the greater information, because I r+= the two areso intimately connected that they should be taken in conjunction rather than separately. Namely, if we denote by E , dw the energy12 per unit time in the if we average angular frequency interval w , w+dw, then there are the For a chaotic (or random) disturbance, over all phases, f(t)f(t) is a function only of t =t- t, following very simple relations between E , and the corand is even in this variable. Also, we shall imagine t h e relation function : disturbance confined t o a finite, but very large interval 1 += 2 T . Then, on changing variables to t =t-t, we have E, = a R(t) cos wtdt,

J

J--

s-;

I

I

I

1.

J-

R(t) =

J =E, cos wtdw.0

(2)

We express our results throughout in terms the anguof lar frequencyw rather than the true frequency (w/27r) Y= as this will enable us t o avoid factors 27r in many places. T h e first relation of ( 2 ) follows immediately on combinT h e corresponding energy densities are connected by theing ( 6 ) and (8) and noting that R ( t ) is even; the second relation E , = 2?rE,. part of ( 2 ) is merely the Fourier transformof the first. Equation ( 2 ) shows t h a t except for a proportionality The normal surface. In order to compute the correlafactor, the correlation function is the Fourier transform of tion functionl5 in statistical problems, it convenient to is the spectral energy density, and vice versa.13 use a formula of probability theory known the as TOprove ( 2 ) we first write down the Fourier integralsnormalsurface. Let X and Y be two linear functionsThis term is used throughout in the sense of a covariance function, e.g., an ensemble average. If Or power, in the angular interval of width d w ; ( ~ = 2 ~ fthus ); E , is the power/unit bandwidth, or intensity density, of the process i n arbitrary units. The term energy density used here refers to energy per unit time, per (angular) frequency interval, or, equivalently, a power/(angular) c/s. This, of course, is the celebrated Wiener-Khintchine theorem (171, Sec. 3.2-2).14 The baron the left number of (7) has been added subsequently. The proof above is strongly heuristic. As is well known now, when f ( t ) is a random process rather subtle conditions (in addition to the usual ones for analytic functions) must be obeyed in the definition of the intensity spectrum and its relations to the covariance and correlation functions. See, in particular, 171 Secs. 3.2-1 and 3.2-2; also (3) of Sec. 3.2-3, and the footnote on page 152. However, since appropriate statistical averages have been taken in the subsequent treatment here, the results are correct. 16 E.g., the covariance function.

10N N

JANUARY IEEE PROCEEDINGS OF THE

X

=i-1

aixi

Y

=i 1 -

bixi

of a set of (random) variables XI, xp, . . , X N , each of which is distributed independently according tothe normal (Gaussian) error law with mean zero value (2, = 0) and unit-mean-square or standard deviation. I n other words, the probability that i falls in the interx ~ dx;. Then val x i , x i + d x i is taken to be ( 1 / 2 ~ ) e--zi2/2 X and Y each also have a Gaussian distribution, though not in general independent of eachother, and we shall assume that their amplitude factors are so normalized that each has a unit standard deviation. This supposition involves no loss of generality, and implies thatN

We are justified in applying statistical reasoning of the type underlying (9) t o noise problems because the total disturbance is a linear function of the amplitudesof the individual Fourier components, and the latter are known to obey a Gaussian distribution law for ideal noise. If we have a distorting apparatus, e.g., a clipper, such that after distortion the amplitude is f(X) rather than X,then the resulting correlation function is by :17 (9)

. , - ( ~ * + y ~ - 2 r ~ ~ ) / ~ ( 1 - r ~ ) d ~ d ~ .

(10)

N

ai2 =i- 1

biz = 1.i-1

I t is often convenient to have the distorted amplitude normalized to unity; in this case the right side of (10) should be divided by

The probability that X falls in X ,X+dX, and that Y also simultaneously falls in Y , Y+dY is16

Do not confuse the correlation function R(t) after distortion with that, r ( t ) , before. One readily verifies t h a t 2*(l - r2)+ for the linear amplifier, characterized by f(X) X , = the right side of (10) reduces tor ( t ) , as onewould expect. where I is the average value of the product XY. The As an exampleof the use of (lo), consider a n amplifier expression (9) isclosely relatedtowhatis called the It whose characteristic isf(X) = C tanh (X/C).is easiest normal surface in statistical theory; namely the equato compute the integral by using the coordinate system tionZ= [ 2 ~ 2 ( 1 - r 2 ) ] - 1 1 2 e x p [ - ( ~ 2 + ~ * - 2 r ~ ~ ) / 2 ( 1 - ~ 2 ) ] X,Xdescribed above in the proof of (9). In this system gives rise t o a locus called the normal surface in a threewe have dimensionalplotin which theprobabilitydensity is taken as the third variable. We owe t o S. 0. Rice [ l ] the suggestion of using (9) in problems of the typewhich we are considering. A derivation of (9) is found in many books on probability theory but, without consulting them, may see we t h a t (9) is readily established b y utilizing the fact that If C is large, i.e., if the saturation corrections are not the Gaussian distribution is left invariant under a unigreat at the rms level, then we may evaluate (11) by tary transformation. transformation The from XI, series expansion xg, * * * , X N to any set of variables inclusive of X, Y cannot be unitary, as X, Y are not orthogonal except inthetrivialcase r =O. However, thevariables X, Y are orthogonal and normalized to unity if we take Y=(Y-rX)/(1-r2)1/2, and we can, by a well-known of the hyperbolic tangent. We thus obtain procedure (cf., Courant-Hilbert, [8] p. 19), find N - 2 othervariables x i , x4, . . , xN suchthatthetransformation from XI, x2, x3, . . , X N t o X ,Y, X:, * * . , X X is unitary. Then the Gaussian distribution law holds in Results based on (12) have already been referred t o at the X, Y, x i , . . . , XN space, and if we integrate over the end of Section I. Xg, * * * , x ~ we simply get unity and so obtain the I n closing this section, it may be mentioned that the Gaussian distribution law ( 1 / 2 ~ ) exp[ - (X2+ Yt2)/2] problem of determining the intensity spectrumof a disapplied to coordinates the X, Y j u s t as though turbance subject to extreme clipping is closely related x i , . . . , x . ~ did not exist. On transforming from the to thatof finding the zeros (crossing points of the axis)ls X, Y to the X,Y space we immediately get (9). in Fig. 2(A) and of those of the envelope in Fig. 2(B). In noise problems we can take X and Y to be the un- If we can determine the distribution of the zeros it is clipped amplitude at, respectively, t = 0 and t = t. Then possible t o find the correlation function, and hence the r ( t ) is the correlation function for the unclipped noise. spectrum. However, it much easier t o find the correlais1,-(X2+yz-2rXY)/2(~--rP)d~dy 7

(9)

4

4

-

[7],

Set. 7.2, Eq. (7.13a), $ = l .

1718

See, for example, 171, Sec. 9.4.

171,

Sec. 13.1.

1966

VAN VLECK SPECTRUM MIDDLETON: AND THE

OF CLIPPED NOISE

11

f(X)= 1, ( X > 0); f ( X ) = - 1 ( X < 0). (13) tion (function) by means of (10) than to find the zeros, which is more intricate,lg and which has been examined We have here assumed a normalization such that after to some extent by Rice.20 Ones first guess might be that clipping the mean-square amplitude is unity, or in other the zeros of the envelope in Fig. 2(B) obey a random words that the ordinates of the horizontal straight lines law, like the emission of alpha particles from a radioacin Fig. 2 are L 1. T h e expression (10) becomes tive material. This model is exceedingly simple, and has beenstudiedby G. W.Kenrick [ 9 ] . A t onetime we R(t) = LWe-.dXd Y thought it might be a sufficient approximation to the 2*(1 - rZ)+ behavior of noise, but actually itis not. It gives a spectrum of type 1/[62+(w-wc)2] like the response curve s-:radxd Y for a circuit with LRC, and this is very materially different from our results in Figs. 4 and 5 , with their proe-adXd Y nounced discontinuities at X = l . Actually, as Rices a (normal) random theoryz1 shows, the zeros do not obey law, and instead, after one zero, another one is unlikely ime-adXdY ] t o occur for a while. Throughoutthereportit is assumed that in (B) where (modulation by clipped noise) the phase of the carrier a = (X2 Y2 - 2rXY)/2(1 - 7). persists or, in other words, is unaffected by the blocking of the transmission of the tubewhen the grid potential is We cansimplify (14) by using the relation unfavorable. assumption usually This is pretty well 1 +- +o warranted. The case that the phase of the carrier has e-adXd Y = (15) 1, 2*(1 - I z , + s - . no memory of its past and acquires new random value a each time the grid potential is favorable is much more which is readily verified mathematically and which is difficult to treat as the statistical formulas based on the also obvious from the fact that the correlation would be normal surface cannot be used. Without persistence in unity if instead of (13) we had f ( X ) = 1 for all values of phase, there will be no coherence between the contrithe argument. Also, it is convenient to introduce polar butions of the successive wave trains generated by the coordinates X = p cos 4$ Y = p sin 4. I t is thus found periods of favorable grid potentials, and the problem is that (14) can be written the same as that of determining the average spectrum of an ensemble of wavetrains whose distributionin R(t) = 4 L r d t $ 1 lengths is the same as thatof the periods between con2*(1 - r2)* secutive zeros. Wehavealreadymentionedthatthis .e-pz(~--r sin 2+)/2(1-rf) pdp - 1. (16) is not an easy problem, and we may note particularly that the determination of the distribution of the inter- Integration gives valsbetweenconsecutivezeros is moredifficult than *I2 & that of finding the probability that zero occura certain a -1 R(t) = 2(1 - I z ) + 1 - y sin 2t$ r time after a selected zero, as the two zeros correlated may not be consecutive. In fact no one appears yetz2 to 2 have found the distribution law for consecutive zeros. = - sin- ( I ) . (1 7) I Withoutquantitativeanalysisitcanbe said t h a t if there is no persistence in phase or, in other words, if the We thus have the rather simple and elegant result that carrier is reborn for each favorable grid sequence, then the effect of extreme clipping is to make the correlation the delta function or extreme concentration energy function 2/7r times the arc sine the original correlation of of at exactly the carrier frequency will be absent, but the function before clipping.= In case there is a carrier of spectrum will otherwise be less uniform than for the case angular frequency we [Case (B) of Fig. 21, the right side of persistence. I t will slope away sharply from a maxi- of (17) must be multiplied by cos w,t t o give the true mum at the center, and not have the discontinuity correlation function t o be used in (2) ; namely, the effect will a t X = 1 shown in Fig. 5. of the carrier is to introduce an extra factor B cos wct0 cos wc(t+to) whose mean value is (3)B cos w,t on aver111. EXTREME CLKPPING aging over all phasest o and the proportionality factorB With extreme clipping the functionf(X) involved in is 2 if R(0)= 1 , i.e., if the mean-square amplitude after (10) has the form clipping is taken equal to unity.

+

+

s-:J-1+

[

sorn

so s-1

s--

sorn

S,

Sec. 9.4-1. [ll, secs. 3.3, 3.4. *l Ill, secs. 3.3, 3.4. No precise analytictheory is available to this day. See [7], Sec. 9.4, for a discussion and further references.S e e , for example, [7],zo

and 13.1-1.

This is a special case in the theory of symmetrical limiters, rridc (71 S c 13.4-2(2) and Eq. (13.8 la,b). See also M.,Sets. 13.1 e.

12

IEEE PROCEEDINGS OF THE

JANUARY

The Form of the Correlation Function

Before proceeding further t is necessary t o specify the i correlation function. The simplest assumption to make Ja&,d, = 1. in Case (A), (DINA), is that before clipping there is a uniform noise band extending from wc-wa t o wc+wa. If the total energy24 beforeclippingisnormalized t o Thus, we have explicitly: unity, so t h a t r ( 0 ) = 1 , then the energy density before Gaussian distribution, clipping of Type (A) : clipping has the value , = 1 / 2 w a for we - w. < w < we+wa E and vanishes outside this interval. By) the correlation (2 function before clipping isnow

clipping, to be calculated later as our goal. We assume the energy (e.g., power) normalized to unity, so t h a t

T h e corresponding correlation function after clipping is by (17)2 R(t) = - sin-'a

Gaussian distribution, clipping of Type (B) :

[( ) 7.sin oat cos w , t ]

(19)

I n Case (B) (modulation by clipped noise), the uniform noise band is low frequency and is used only t o modulate the carrier. So if the noise band is assumed t o be uniform, it may be takenas extending from w = 0 t o w=w,, with an energy density l / w a . T h e original correlation function is then

~ ( t = - cos wet sin-l[e-t*'t']. )T

2

(23)

Optical distribution, clipping of Type (A) :

wa

J

o

Wot

while after clipping i t becomes

Optical distribution, clipping of Type (B) :(21)

R(t)

=

2 -cosa

wet sin-'

the factor cos wet being added for reasons previously given. From examination of the preceding equations, we see that the mathematical differencebetweenclipping of Type (A) and that of Type (B) is t h a t in the former the factor cos wf appears inside the argument of the arc sine, while in the latter it appears outside. We shall give also the correlation functions for the caseinwhich the noise band before clipping has aGaussianspectraldistribution,andforthecasein which it has what we may term an "optical"distribution. By the latter we mean that it has a structure factor ~ / [ W ~ ~ + ( W - W ~ ) of] the same form as an optical ~ (i.e. Lorentzian)absorptionbandor as the response curve of a simple circuit with L , R , C (provided% R / 2 L < < ( L C ) - 9 . The results are given in ( 2 2 ) - ( 2 5 ) . T h e expressions r ( t ) and R ( t ) are, respectively, the correlation function before and after clipping, and & is the , energy density or power spectrum before clipping, which is not to beconfused with the energy spectrum E , afterE.g. ywer/unit bandwidth; see remark,footnote following(2)." See [ 1, Types 3,4,p. 169.

Spectrum for Originally Rectangular Distribution Subject to Clipping of Type ( A )We now proceed to study the spectrum of unmodulated clipped noise (DINA) which before clipping had a uniform or rectangular distribution, so t h a t (18) and (19) are applicable. By ( 2 ) and (19) the energy distribution in the spectrum after clipping is given by the expressionE,=

-Lm4a 2

cos o sin-' t

[cos uta:"

w,t

]dt.(26)

The problem of finding the spectrum is hence reduced to the evaluation of an integral, Since the integral ( 2 6 ) is certainly not evaluable in closed form, one immediately tries expanding the arc sine in a Taylor's series

This gives

1966

VAN VLECK AND MIDDLETON: SPECTRUM THE

OF CLIPPED NOISE

13~

E, = a2

2

s," {

[cos (w

cos (w

-3

4

+ 3 cos (o - we)#4

cos (W - 5wJt40I-

+ 5 cos+7

(O

- 3wJt - sw,)t

+ 10 cos (w - u,)t+ 2164COS (w

16COS

5 [cos (w - 7 4 t

(w

-3 4

+ 35

COS (W

- we)q

'

112L

X

sin w,t (+ ) 7* } d t .' I*

Hereand elsewhere we omitterms in cos (w+w,)t, cos (w+3we)t, etc. which involve the sum rather than the difference of two highfrequencies, as such terms oscillate so rapidly that their contributions to the integrals are negligible. Ordinarily we are interested only in the distribution in the main band, rather than in the harmonics which centerabout soc, 50,, etc.Thenin (28) we need retainonlythetermshaving a factor cos ( w - w,)t. By expressing sinm wat cos ( w - w,)t as sums of terms of the type sin (w-wc+Rw.)t, - m S K l m , as canbedonebymeans of elementary trigonometric identities,andthenbyintegratingbyparts [cf. the Appendix], the expression (28) can be reduced t o t h e sums of integrals of the form

Zone III: (3 =

x ( I x I < b ) , f ( X >= 1 f ( X ) = - b, ( X < - b ) .

bl

(X

> b),(37)

On the other hand, for the portion E,,, where r(t) is nearly unity, we do not use the series (27) a t all, and instead develop arc sin r(r = sin w,t/w.t) as a Taylor's series in t. This is possible, for althoughd(sin-'r)/dr has a n infinite derivative at Y = 1, on the other hand dr/dt vanishes at Y = 1, and it turns out that d(sin-' r)/dt has a finite value a t t = O . One thus finds t h a t

R(t) = (2/7)cos= cosw,t[ie

w,t

sin-'

Y

+ 2r-1d3(-e/3 + e3/270 + 7.93 10-585 + 8.7. - 2.73.10-se9 + . . . )]. (e = w,t)Ie O'7 -

T h e linearity off for arguments of modulus less than b means that the disturbance is assumed unaffected by theclippinginstrumentunless i t reaches thecritical value b. With (37), theintegral (10) cannotbeconveniently evaluated by using polar coordinates, as was done in Section I11 for extreme clipping. Instead there is a n artifice which has been introduced by Bennett101, [ Rice [l 1, andothers [2], [3], and whichconsistsin representing the functionf defined by (37) as a contour integral.27 Namely, f ( X ) can be expressed as 1 ei(X-b)r - eiXz

(36)

f W

=

2u

-J+

dzc22ei(X+b)r

With this approximation for R(t) one may immediately evaluate in a n elementary manner

1

-eiX~dz1

(38)

'' cos wtR(t)dt.

For small values of X = (w - w c ) / w , it is easiest also t o expand COS ( w - w,)t in a series before integrating, as * S e full account,5.1, chaptersillustrations of the method. ' e a Secs.with It], 2.3, 12, 13 and references therein for cited, many exact formula the for

where C is a path of integration extending along the real axis from - m t o m except that it is indented downwards near the origin. The path C' is similar t o C b u t

16

PROCEEDINGS OF THE IEEE JANUARYY

is indented upwards near the origin. To prove the legiti- readily seen that no even powers of macy of the representation ( 3 8 ) for ( 3 7 ) we utilize the that fact that by theresidue theorem we have R(t) = U l Y U3Y3 *

are involved, so*

+Z

+

.we find

[z-2eikzdz =J

- 2rk

working On out the

coefficients explicitly, sin bze-z'I2d~y3+m

for a closed counterclockwise path including the origin, while

1'*

S

z sin bze-"'12dz

r2eikZdz 0 =

+

for a closed path not including the origin. The paths C and C' may be closed without affecting the value of the integral by introducing a semicircle of infinitely large radius in the upper or lower half plane, depending on the sign of the exponent. Now quite generally ifn n

(40)

Here we can omit the indentures near the origin, a s each integrand in (40) is finite at z = O . Nowe-z4/2dz

1=U

Lbs-m+m

cos bze-zz/2dzdb=@(b/dj),

where D and D' are arbitrary contours we have(1/2?r)(l Y2)-$

=

s,"

(+)'e-b'/zdb

where O b ) is the error functionz8(2/74

s0

'e-g'dy.

a s is readily established by interchanging the order integration. Hence

of

Also when n is a n odd positive integer other than unity we havez ~ sin ~ - bze-z'/2dz

= (-

(b),

where Hn(y) is the Hermitian polynomialz8

Hence The expression for the correlation function takes thus a simpler form in the zq than in the X,Y space, a t least regards asthe dependence on Y . the For case of extreme clipping, the use of the X,Y space was, to be sure, more straightforward, as the integration went through most simply in polar coordinates. For clipping at an arbitrary level, however, the polar coordinate scheme ceases to be feasible, and we resort t o a series integration which is most simply performed in z , 7 space. If we develop ( 3 9 ) with respect to I , it is

R(t) = r f O ( b / d j ) )

+

n=3,5,.

7[Hn-2(b)e-bZ'212 .. n .

In

= Y{O(b/&)j2

+ -[??r

2

r3b2erb'

+ - (b3 - 3b)2e-b* 120

y5

+504028

Y7

(b5

- lob3

+ 15b)2e-b2+ -

See [7], Appendix A.1.1, pp. 1071-1073.

1966

VAXSPECTRUM MIDDLETOS: AND VLECK THE

OF CLIPPED NOISE

17

As a check on the work it may be noted that reduces (41) properly in limiting cases; viz: for b = 30 it reduces to 7 , while for b very small i t becomes

R(t)=-

r3

2n- 2 [ b

r+-+-+-+

37540

577

6

112

...

1(42)

then proportional, respectively,t o 4b2,R(O),and b2 where R(0) is given by ( 4 3 ) . The ratio p of carrier to sideband power is thus b 2 / R ( 0 ) . In Case (X) there is no carrier, and the ratio of the peak power to the rms noise power is b2/R(0), whereas in (B) the ratio of the peak power to mean noise sideband power is 4 b 2 / R ( 0 ) . APPENDIX

=

- b2 sin-n-

2

Y.

EVALUATION DEFINITEINTEGRAL OF A

The result (42) agrees with ( 1 7 ) except that (42) conAn important definite integral needed in the calculatains an extra factor This is because in obtaining (41) tion of the spectrum of superclipped normal noise is b2. it wasassumed that the mean-square amplitude was normalized to unity before clipping, whereas in ( 1 7 ) it was supposed normalized to unity after clipping. Obviously, clipping will reduce the mean-square amplitude, = @(-A I U, b ) , (44) unless a new normalization is made. If, in general, we wish to renormalize the mean-square amplitude to unity where ( a , b) denotes various ranges of values of after clipping, we must divide (41) by a constant factor X( = ( w - w , ) / w , ) , a normalized angular frequency, and R(O),where n is a positive integer. The intervals, or zones ( a , b ) , are the various regions throughout which 9 assumes , R(0) = b2 - ( 2 / ~ ) ) b e - ~ * / (1 - b 2 ) 0 ( b / d j ) . ( 4 3 ) ~ different analytic forms. I n all (except cases for n = 0 and l ) , 9 is continuous over the boundaries of , This division removes the factor b2 in (42) for the limit adjacent zones. of small b. Several methods of evaluating (44) suggest themT h e integral selves: we may use contour integration, noting that

+

E,

=

(2/n) l m R ( t ) wtdt cos

may now beevaluatedinseriesbythesamegeneral where C extends along the real axis, with a downward method as described in Section 111, and so we shall not indentation about the origin. Or may employ a somewe give numerical details here. -1s explained in Sections I what more direct approach, using integration by parts. and 111, the situation is different for Cases (A) and (B), Let us consider the latter: we begin first by writing as in theformerharmonics of thecentralfrequency appear. In consequence, the convergence in (A) is improved when only the energy in the fundamental band is considered, and is then reasonably satisfactory. The convergence is only mediocre for (B). However,the and examining the case when n is even, observing that curves for (B) can be corrected by extrapolation from the case of extreme clipping, where an alternative method was available which was described at the end j=O I of Section 111, and which was more accurate than the n = 0, 2 , * * * , ( 4 7 ) power series in r . (A similar alternative method could probably be developed in the present case, but does not where ,Cj is the usual binomial coefficient n ! / ( n - j ) !j!. warrant the labor.) Curves based on ( 4 1 ) have already Integrating by parts, we find that been incorporated in Section I. The important thing is thevalue of b at which thedeviationsfromthe unclipped spectrum become appreciable. This clearly takes The effect place when b %ecomes comparable with unity. of clipping will hence fade out very rapidly when the n = 2, 4, * * , (48) limiting amplitude exceeds the rms noise level and, even when the two are equal, the effect is small. where @(e) is the(divergent)remainderas E+O. T h e If the tube conducts only for positive values of the symbol lx is a discontinuity factor which has the value grid potential, then in Case (B) a bias of b is necessary unity when X > 0 and - 1 when X < O . Substituting (47) if it is to amplify the noise disturbance throughout the into (46) we see that the sum of all the remainders a() amplitude range -b to +b. The peak power, the power vanishes (for --to), leaving the expected finite result. in the noise sidebands, and the power in the carrier are We obtain finally

-

18

PROCEEDINGS OF THE IEEE

JANUARYii

@n (X

I

i

i

i

n=l

n = 2, 4,

-

*

,

(49)

When n is odd we observe in a similar way that

sin" e =

(n-1)/2

21-n

jO =

(-l)t("-')+j,~~sin (nn = 1, 3, 5,

- 2j)e,

- ..*

(50).40-

We obtain, as before,(- 1)tb-l) x de =

(n - l ) !

---.X"'lk+

2

a, ()

n = 1, 3, 5 ,

-

.30

-

,

(51).x)-

on integrationbyparts,whereagain lim @(e) is the (divergent) remainder a t t h elower limit. Applying (So), (51) t o (46) now yields

10

-

0

Fig. 12. The function an@) O X and 11n18. for S

As in the case of even n, thedivergentremainders (as E+O) all sum to zero a t 8 =O. A number of recurrence relations may also be deduced on differentiation respect with to X. We for have, example,

%(X

I 0, 2)2,

=

1 - X/2;0.

w) =

(564

0, 1) = (6 - 2X2)/8;

1, 3)

=

(9 - 6X

+ X2)/8;(5 7)

3, w ) = 0 .0, 2)= = =

- - @-( ,1X2%(O)=

n

- l),@,+1(1);

n

1 1;n

(Sa)

2, 4)

+ 3X3)/48; (64 - 48X + 12X2 - X3)/48;(32 - 12X2

4, w )n

0 .

(58)

( F - 1) ) n=

2

2.

(53b)

0, 1) = (230 - 60X2

are tabuThe first nine (n=O included) functions lated below, and plotted in Fig. 12 (for n2.1). We have specifically

*,,

1, 3) 3, 5 )5,

+ 6X4)/384; = (220 + 40X - 120X2+ 40X3- 4X4)/384; = (625 - 5OOX + 150X2 - 20X3 + X4)/384;0.

w) =

(5 9)

@o(X)Also, we have@'(X

26(X

- 0).

(54)

0, 2) = (2112 - 480X2A6

+ 60X4- 10X5)/A6;

=

3840;

I 0, 1)

= 1;

2, 4) = (1632

+ 1200X - 1680X2+ 6OOX'

, a) = 0.

- 90X'

+ 5XS)/As;

1966

VANSPECTRUM MIDDLETON: AND VLECK THE

OF CLIPPED NOISE@8(X

19

a6(X1 4,

6)

= (7776 - 64801

+ 30X - X5)/Aa;

+ 2160X - 360X3(60)

1 6, 8)

= (2 097 152

- 1 835 0081 f

688128X2

- 143 360X

@6(X

@ 7 ( ~

1 6, a) = 0. I 0, 1) = (23 548 - 4620X2+ 420XA7 = 46 080;

- 1344X5 56X6 - X7)/A8;@s(X

+

+ 17 920X

- 2oX6)/A7;

1 8, a) = 0.REFERENCES

@,(X 1, 3)

1 1

= (23 548

+

- 210X - 4095X2 - 700X3 945X4 - 210X5 15X6)/A7;

+

a 7 ( h 3, 5 )

1 5 , 7)@7(X

+ 30 408X - 29610X2 + 10640X3 - 1890X + 168X5- 6X6)/A7; = (117 649 - 100 842X + 36 015Xz - 6860X3 + 7 3 3 - 42X5 + X6)/A7;= (8274

@8(x

1 7 , a ) = 0. 1 0 , 2) = (309 248 - 53760X- 280X6

(61)

+ 35X7)/A8;

+ 4480XA8 = 645 120;

@8(X

I 2, 4)

= (316 416

- 25 088X - 16 128X2

- 31 360X3

+ 20 160X4 - 4704X5

@8(X

1 4, 6)

+ 504x6 - 2 1 ~ 7 ) / ~ ~ ; = (-142 336 + 777 728X - 618 240X + 219 520X3 - 42 5601 + 4704X5 - 280X6+ 7X7)/A8;

S. 0. Rice, at that time (1943) unpublished notes, which appeared later as Mathematical analysis of random noise, Bell Sys. Tech. J.,vol. 23, p. 282, July 1944, vol. 24, p. 46, January 1945. See especi$ly Part IV. D. Middleton, The response of biased, saturated linear and quadratic rectifiers to random noise, J . AppZ.Phys., vol. 17, p. 778, 1946, cf., in particular, Sec. VI. D. Middleton, Some general results in the theory of noise through nonlinear devices, Quart. Appl. Math., vol. 5, p. U S , 1948, cf., in particular, Eq,. (7.15). I . B. Davenport, T Jr., Signal-to-noise ratios in bandpass limiters, J . AppZ. Phys., vol. 24, p. 720, 1953; cf., Section 111, B. R. Price, A note on the envelope and phase-modulated comI R E Trans. on Inponents of narrow-bandGaussiannoise, formation Theory, vol. IT-1, pp. 9-13, September 1955. Besides giving a short history of Case ( A ) here, this paper is particularly interesting for its numerical results and a study of the spectral behavior of clipped noise bands well away from the central frequency. J. L. Lawson and G. E. Uhlenbeck, Threshold Signals, vol. 24, MIT Radiation Laboratory Series, New York,McGraw-Hill, 1950. See pp. 57-59; also Sec. 12.5, pp. 354-358. D. Middleton, An Introduction to Statistical Communication Theory. New York: McGraw-Hill, 1960. See, in particular, Sec. 9.1-2, and pp. 405-408. For various generalizations, see also (9.3), (13.16), and Sec. 13.4-4 for Problems (9.1), (13.15), Case (B). R. Courant and D. Hilbert, Methoden Mathemdischen der Physik. Berlin: Springer, 1931. G. W. Kenrick, The analysis of irregular motions with applications to the energy frequency spectrum of static and of telegraph signals, Phil. Mag., vol. VIII, ser. 7, pp. 176-196, January 1929. W . R. Bennett and S. Rice, Note on methods of computing modulation products, Phil.Mag., vol. 18, ser. 7, pp. 422-424, September 1934.

2.

CorrectionR. N. Thurston,author of thepaper,Ultrasonic Data and the Thermodynamics of Solids, which appeared on pages 1320-1336 of the October, 1965, issue of these PROCEEDINGS, called the following to the has attention of the Editor. On page 1323, in (3.4), the first factor on the right should be u (upsilon), rather than Y (nu). On page 1323, last line, first column, the words particle displacement should be substituted for propagation. On page 1333, in the first of equations (11.2), the extra subscripts Ok should be ignored. On page 1333, in the fourth row of Table 11, the entry in the last column should be 614. Reference [27] on page 1336 should read,H. J. Maris, On the mean free path of low energy phonons in single crystal quartz, Phil. Mag., vol. 9, pp. 901-910, June 1964.