Full report

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• I . - , -- I . I TECHNICAL MEMORANDUM WO. 44 THE AVERAGE NUMBER OF ZERO CROSSINGS PER SECOND OF A SINUSOIDAL SIGNAL PLUS NOISE 1A B ORA T 0 RY- FOR ROSCIENCE RESEARC H ELECT D E P A R T M E N T OF E L E C T R I C A L E N G I N E E R I N G SCHOOL OF ENGINEERING A N D SCJENCE N E W YORK U N I V E R S I T Y N e w York 53, N e w York

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1 I c * ' . In addition to published papers, the School of Engineering and Science reports the results of its research i n the form of reports to sponsors of research projects, Technical Remrts. and Technical Notes. The latter are normally l i m i t e d to distribution within the School. Information regarding the availability of reprints of journal articles and Technical Reports may r be obtained by writing to the Director of the Research Division, School of Engineering and Science, New York University, New York 53, N.Y. 2

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"ECRNICAL MEMORANDUM NO. 44 TBE AVERAGE NUMBER OF ZERO CROSSINGS PER SECOND OF A SINtGOlIXL SIGNAL PLUS NOISE I Gerald Weiss January 1965 Contract No. NAS 3-9809 Prepared by NEW YOFX UMNERSITY SCHOOL OF ENGINEERING AND SCIENCE AEPARTMENT OF ELEX'JXICAL ENGINEXRING Laboratory for Electroscience Research University Heights Bronx, New York 10453 for NATIONAL AERONAUTICS AND SPACE AlXENISTRATION GODIIARD SPACE FLIGRT CENTER GREZNIBLT, MARYLAND

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The measurement of the average frequency of a sinusoidal voltage source may be implemented by counting the average number of zero crossings which have accrued i n a given counting period. If the source is present in a noiseless background, then the uncertainty in the measure of the average frequency is limited by the uncertainty of measuring the number of zero crossings and the observation time. If, i n addition, the sinusoidal voltage i s associated with random noise, then the uncertainty in the measure of the source frequency depends upon the noise characteristics. This report considers two models which show the effect of noise on the average number of zero crossings per second. One m o d e l u t i l i z e s as a signal source a quasi-harmonic sinusoidal voltage which I ' is characteristic of a propagated signal subdect t o fading. The other m o d e l assumes a fixed sinusoidal voltage which may be the output of a signal generator. In both of these models, the signal source voltage is added t o a random voltage which is Gaussian distributed, and the nmiber of zero crossings is averaged over an infinite period of t i m e . These results, then, establish an upper l i m i t on the error i n measuring frequency by a zero count technique.

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iii TAELE OF CONTENTS I. Tm QWI-HARMOMC SIMLSOIDAL 11. THE FIXED SINUSOIDAL SOURCE Page SOURCE 1 7 For the no-signal case ( Q = 0) 8 0) 8 For the no-noise case (% = For the case where fo is set equal t o -&/2 f o r any p 9 111. CONCLIBION 13 APPENDIX: Average mber of Zero Crossings of S i g n a l Plus N o i s e F i l t e r 14 Shaped by a Bandpass REPEZEZCES mwTRAT1oNs 18 19 TAHLE 1: AVERAGE FREQUENCY MEASUREDVS. S.N.R. 44

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TEE AVERAGE muMBw QF ZERO CRCXSSIKGS PER SECOND OF A SINUSOIDAL SIGNAL FL'B NOISE I. THE QUASI-FWWNIC SSNlBOSaAL SOURCE The average nmber of zero crossings per second for a narrow band sine wave process in randomnoise has been studied by many imresti gat or^.'^^,^ In this model, a sine wave of fixed frequency o)o(&do) and randarm amplitude and phase is considered. The bandwidth of the s i g n a l process is assumed to be narrow compared to the frequency q,. A randun noise process with a no& amplitude distribution is added to the randoim s i g n a l process resulting in perturbations in the average nunher of zero crossings compared to 2fo. The noise is represented by its autocorrelation function: R(T) = /G(w) COS a~ da, , 0 where G(w) is the power spectrum of the noise. The quasi-hammic sine wave process and mdcun noise process are assumed mutually independent, and the caposite process given by: where Q and & are the random amplitude and phase functions associated with the randun s i g n a l process. I

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I 2 ~, . In one developnent2 an incremental length dat is considered for a representative sine wave in the ensemble. The t i m e t o cross the interval dais related t o the slope of the waveform: The probability over the ensemble that y(t) is i n the interval (a,Wda) while its derivative i(t)is between (By s+d#3) is: This may also be interpreted as the amount of the time per unit time t h a t y ( t ) spends i n the interval (a,Wda) with velocity (By @+as) w B as illustrated below:

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, da j I B 1 B 1 P 1 P 1 -set* _1 The number of crossings per unit time through a level awith velocity 13 is "he average nmiber of zero crossings for all B where Do = 1 CU~G(W)h i d 0

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I . = r G ( u ) du, . 0 Consider the application of these results t o the case where the noise is derived from the cutput of an ideal rectangular f i l t e r with a response which fncludes the sine wave frequency %. The parer spectrum G(u) is obtained for the rectangular f i l t e r : ly(ja) I

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5 " e Gn(u) = K/2n volts2/cps. , D~ = J- v2n co2 d~ = k ( 2 ~ ) ~(s-fi:) . r L J0 J

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6 r . where p is the signal-to-noise power ratio: As the quantity p increases, the average number of zero crossings approaches 20. For p = 0, the average number of zero crossings is K, the expected numjber for noise alone. Both of these results are w h a t would be expected. Expression (20) may be used t o determine the frac- , tional deviation in 2f, when fo l i e s between fa and fb: where S The above results may also be applied t o noise shaped by an RLC bandpass filter (see appendix) centered on q by defining r = %/u)o. Figure 1is a plot of (21) f o r r C 1and Figure 2 for r 1.

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1. THE FMED SINUSOIDAL SOURCE 7 The average number of zero crossings per second f o r a sinusoidal source of fixed amplitude, frequency and phase plus normal noise has been studied by S.O. Rice.3 For the composite process y(t) = Q sin mot + n ( t ) , the resulting expression is: where a2 + b"? a = ¶ 4 Defining the following quent,,ies a2 - b2 8 = 4 p =- Q2 = signal-to-noise parer ratio, and =0 *fO 6 =- signal frequency offset relative Nn to the no-signal case, then the parameters of (22) may be written as: . = P ( 1 +2 b2 ) 9

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8 r . B = P ( l - & 2 )2 9 - =b2 2tj2 2a 1 + b2 In addition to the above Quantities Ie(k, X) = - e'UIo(ku) du . s 0 As an example of the above, consider the following cases: For the no-s(rmnl case (Q = Ob For the no-noise case (% = 0):

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9 then and For the case where f, is set equal t o Nn/2 f o r any p: P = O , a = p : IJO, x) = 1 - e-x = 1 - E - Q = 1 - E-P then, The above results satisfy w h a t would be intuitively expected i n the extreme cases. For a rectangular filter with lower limit fa ajld upper limit fb shaping the noise: For an RLC bandpass f i l t e r centered on fc:

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10 Using an IBM: 1620 computer, the average number of zero crossper second f o r a rectangular noise f i l t e r w a s computed. A copy ings of the cmputer shown in Figure 25. The parameters chosen program is were: fa = KC , fb = 15102 - % = 10408.333 crossings per second The center frequency of the sinusoid fo is varied between 5 I - and l5KC in lKC steps. The average number of zero crossings per second with p and fo as parameters istabulated i n Table 1and the data plotted in Figures 3 through 13. F r m this data, curves of constant error in and fo as parameters. C w w s corresponding t o cps were plotted with p errors i n f'requency of 5 , 10, 20, 50, and 100 cps are plotted in Figures 14 through 18. These cyclic errors correspond t o percent errors in measuring frequency of O.O$, 0.16, 0.26, O . f S @ , and 1.Nrespectively. It is seen from these curves that for a given error, the requisite signal-to-noise r a t i o decreases sharply as the sinusoidal frequency fo decreases ficun fa. As the frequency &/2 = fo is approached from the r a t i o approaches zero. For frequencies greater l e f t , the signal-to-noise - than fo = Nn/2, the requisite signal-to-noise ratio increases wain. "he required signal-to-noise r a t i o is maxhrum at fa and represents a vorst case design point. Since the signal-to-noise r a t i o faUs so sharply

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w i t h frequency, then a departure from the worst case design may be made with a resulting canpromise i n the error in a small fraction of the data points. As an example, from Figure 15, it is seen that i f p = 10, then all the data points between 6-15KC are within 0.1%. A reduction of p t o 7.9 (Wb) or a decrement of ldb results in a 1 6 loss of data points that are within 0.1%. The effect of a change i n bandwidth is shown i n Figures 19 through 23. In F L m s 19 and 20, the lower frequency fa is reduced to 3KIc and in Figures 21 through 23 to d.c. The increase i n error due t o a change i n bandwidth is m i n i m a l a t high signal-to-noise ratio. For example, for p = 6, fa = 5KC and fo = 15KC an error of -6 cps results in the measured frequency based an the average number of zero crossings. If fa is reduced t o 3KC, the error i n frequency is essentially unchaaged. However, when fa is reduced t o zero frequency the error is -7 cps which represents a minor increase. Cansequently, the effect of "roll-off 'I characteristics of the noise shaping f i l t e r on the error i n measuring any one particular frequency is negligible. The effect of heterodyning on the error in frequencymay be determined from Figure 23. In this case, the lOKC bandwidth centered lOKC i s translated t o a center frequency of b K C . In on a frequency of the curve, the cyclic error versus input frequency is plotted f o r a As an example, a t the 1OKC center signal-to-noise r a t i o of 6 and 8. frequency with fo = 6.9KC and p = 8 , the error i n measuring the fre- Quency is 10 cps. Heterodyning this data bandwidth t o 40KI:with

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I 2 I . f, = 36.9KC and p = 8 results in an error of 0.7 cps i n the measured frequency. This decrease in frequency error is due t o the narrow band effect of heterodyning. Since the noise bandwidth upon frequency translation represents a smaller h c t i o n of the center f’requency, the effect of noise perturbing this frequency is reduced.

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13 111. CONCLWION Two models of a signal. source have been considered for the effect of noise on the measurement of the frequency of the source. One model assumes a quasi-harmonic source representing a propagated s i g n d , the other a fixed source representing a signal generator. In either model, when the frequency of the source is measured by counting the average number of zero crossings per second, the s i g n a l - to-noise r a t i o over the entire data spectrum for a fixed error is not constant. The worst case (largest signal-to-noise ratio) occurs at the lower band edge, decreases t o zero a t the approximate arithmetic mean of the data band and then increases t o a threshold value a t the uppr band edge. When the data band is heterodyned up i n f’requency, the narrowband effect of the noise on the frequency of the signal results in a reduced error in measuring frequency. The measured frequency, however, assumes an infinite averaging t b e . Sampling Over f i n i t e time intervals w i l l result i n a spread of the measured frequency about the average value. If the process of heterodyning does not result i n an increased Spread in measured frequency, then it would appear that frequency translation will reduce measurement errors.

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14 APPEIVDXX: Average Number of Zero Crossjngs of S i g n a l Plus Noise Shaped by a Bmdpass F i l t e r The. average number of zero crossings i s given by expression (7): -+ Ro > Consider the voltage transfer f'unction of an RLC bandpass f i l t e r which shapes the noise power spectrum. Y ( s ) = IC R s2 i- s + - L 1 Lc (A-3)

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I I I t - I See Reference 4, Table 19, No. 7. 1 . 2 2 2 2 4 15 (A-4) (A-5) (A-6) (A-7) 2 2 4 b C - 0 1 + 4gw = a, + (4B-20,) I 3 + oc 2 2 4 (P +Q 1 = wc 2 2 2 ( P -Q = 48 - ag 2 2 p + q2 = wc P2 - s2 = 8/2 - cDc 2

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- , Do = 2P See Reference 4, Table 1gYNo. 6. 1 dx (A-9) (A-10) (A-12) --- 1 x , (A-13) J (p2+q2l2 i-2(p2-q2) x2 + x4 4P P2 + s2 0 p2 = $14 i 2 p2 + q2 = Uc (A- 14) Defining the signal-to-noise power ratio as:

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and frasn the previous results: (A-16) The above results indicate that the average number of crossings of the axis of signal plus noise is independent of the shape characteri s t i c s of the spectrum of the noise, but does depend upon the relative displacement of the center of the noise spectrum and s i g n a l .

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la REFERENCES 1. H. Steinberg, P.M. Schultheiss, C.A. Wagrin, andF. Zweiz. "Short- Time Frequency Measurements of IJarrow-Band Random S i g n a l s by Means of Zero Counting Process,It J. Appl. Phys., vol. 26 (February 1%3), pp. 195-201. 2. J.S. Bendat. "Principles and Applications of Randan Noise Theory." 3. S.O. Rice. "Statistical Properties of a Sine Wave Plus Randm Noise," Bell System Technical Journal (194.8)q,1, 109. 4. Bierens DeHasn. Nauvelles Tables, Hafher Publishing Co. (197).

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p * S I G N A L - T O - NOISE POWER R A T I O : r c I

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. I .6 I .5 I .4 1.2 f . 1 I.o PLOT OF - N p + r 2 ,+ 2 = ( 2 f. P + I p - S I G N A L - T O - N O I S E POWER R A T I O r e 1

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22

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0 W a lL W (3 a a W > a

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31 a i v) v) > D W a 3 W (3 a a W > a

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33

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35 'CI ' N ' S

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W

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37 0, a . 2 v) Q) v) > b a W a 3 v) W n (3 4 a W > N a

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38

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40 . a z v) v) > 3 0 W a Ir, W (3 a a W > Q

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41 0 a 0 W a D IL W 0 t a a W > a r) J

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42 J F R E Q U E N C Y E R R O R V S FREQUENCY W I T H CONSTANT S.N.R.

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I 0 (L wz 0N mm Y m C , - n c 0 c 5 < C< C C1 L C P1 : L c3 U C K a V ) .C U 0 c I L < < m mm C 0 " m 4 U " I . 0 U 04 LI n , c1 4 0 . z K Y 4o cc C I I 43 z 0 I Y " a

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44 TABLE 1 AVERAGE FREBuENCY MEASURED VS, S.N.R. 5 1 7647.6042 9 1 2 6368.6252 2 3 5747 86-72 3 4 5430.7860 4 5 5260.0673 5 6 5163 3365 6 7 5105 9638 7 8 5070 5673 8 9 5047.9578 9 10 5033 1329 10 6 1 8041.3441 10 1 13 10153.2070 14 2 6982 7636 2 lOO57*99aO 3 649 8553 3 10021.5940 4 6257 4367 4 1ooC81060 5 6139.7461 5 6 10001.160 6 6078.5476 7 609.5067 7 8 6@7 0453 a 9 6016.4204 9 10 10000 0120 10 6010.1464 10003.04io 10000.4glO lOOOO.l~g0 1ooo0.~10 7 1 11 1 10788.3290 15 2 2 10924.2100 3 3 10972.8470 4 4 10990.2530 5 5 lOgg6.4940 6 6 10988.730 7 7 10999.5320 8 8 10999.822 9 9 10999.925 10 10 10999.962 8 1 9OOo. 7068 12 1 11455.876 2 8421.1804 2 11812.92l 3 8179.7196 3 11935.338 4 8077 7951 4 11977.537 5 803b.1791 5 11992-185 6 SOl’j.2413 6 11997.242 7 8006.8941 7 1199g.019 8 8003.1623 8 13999.642 9 8OO1.47E 9 11999.864 10 8000.6876 io 11999.952

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The Research Division of the School of Engineering and Science is a n integral part of the educational program of the School. The faculty of the School takes part in the work of the Research Division, often serving a s coordinators or project directors or a s technical specialists on the projects. This research activity enriches the educational experience of their students since it enables the faculty to be practicing scientists and engineers, in close touch with developments and current problems in their field of specialization. At the same time, this arrangement makes available to industrial and governmental sponsors the wealth of experience and special training represented by the faculty of a malor engineering school. The staff of the Division is drawn from m n y areas of engineering and research. It inciudes men formeriy with the research divisions of industry, governmental and public agencies, and independent research organizations. Following a r e the areas represented in the research program: Aeronautical Engineering, Chemical Engineering, Civil Engineering, Electrical Engineering, Engineering Mechanics, Industrial and Management Engineering, Mechanical Engineering, Metallurgical Engineering, Mathematics, Meteorology and Oceanography, and Physics. In addition, a n interdisciplinary research group is responsible for studies which embrace several disciplines. Inquiries regarding specific areas of research may be addressed t o the Director, Research Division, for forwarding to the apprnprlate research group.

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