RF Spectrum of Lightning.pdf

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NASA Technical Memorandum 87788

Review of Measurements of the RF Spectrumof Radiation from Lightning

David M. L e Vinej N B S A - T M - 8 7 7 8 & ) REVIEW OF MEASUREHENTS OFTHE RP SI-ECTRUM CF RACI AT I CE i F G B L I G H T N I N G( N A S B ) 20 p CSCL 04B

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NASA Technical Memorandum 87788

Review of Measurements of the RF Spectrumof Radiation from Lightning

David M. L e VineLaboratoy for OceansMicrowave Sensors and Data Acquisition Systems BranchGoddard Space Flight Center

National Aeronautics andSpace AdministrationGwaara Space Fiigni CenterGreenbelt, Maryland207711986


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I . INTRODUCTIONTh e electrom agnetic radiation from lightning at microw ave frequencies and below is generally referred to asthe rad io frequ ency o r R F portion of the spectrum. Radiation from lightning in this portion of the spectrumis impo rtant both for scientific investigations of lightning and for enginee ring assessm ents of the interference env i-

ronment during thunderstorms. Measurem ents have been reported from frequencies below a kilohertz to frequen-cies above a gigahertz.Th e word spe ctrum is generally used in the literature on lightning in this frequency range to mean themagnitude of the Fourier transform of the electric field E(t) radiated during the discharge. In applications to light-ning, two methods have traditionally been employed to measure this spectrum. In o ne, the spectrum is obtainedfrom the electric field waveform itself by Fourier transforming. T he electric field waveform is recorded first usinginstrumentation such as a fast field change system and wide bandwidth recorder, and the spectrum is obtainednumerically from this record by means of a Fourier transform. This technique has the appeal of being straight for-ward, but it requires wide bandwidth recording devices with large dynam ic range because the power at high fre-quencies tends to decrease rapidly with frequency. The second technique is to measure the energy radiated at aparticular frequency directly using a filter and detector system tuned to the frequency of interest. Standard radio

receivers suitable for this purpose are available in this frequency range. Th e major difficulty with measurementsof this type has been in identifying the element of the lightning flash (e.g., leader, return stroke, etc.) which is thesource of the radiation. Each of these techniques will be described in this report and a summary given of the dataobtained with each.

11. SPECTR A OBTA INED BY FOURIER TRANSFORMA lightning flash is not a single event but, rather, is a sequence of many discrete events. S om e, such as returnstrokes and the leader steps preceeding a first return stroke, have received m uch attention and are reasonably wellkno wn , w hereas others such as the many different events which take place in the cloud are only recently beginningto be studied. T he electric fields radiated from these events tend to have characteristic (time dom ain) shapes which

permit the events to be identified and spectra to be obtained separately for the various events.For exa mp le, at the top in Figure 1 is show n an electric field w aveform E(t) recorded by the author in Floridaduring the Thunderstorm Research International Project (TRIP; Pierce, 1976). This waveform has a shap e charac-teristic of those observed during first return strokes (Um an and Krider, 1982;Weidm an and Krider, 1978). It be-gins abruptly with a rapid rise to peak and then decays irregularly toward zero. Frequently, the abrupt beginning ispreceede d by a string of sm all pulses associated with the stepped leade r (Krid er et al., 1977; Weidman and Krider,1980). The last such step is visible in this record just before the beginning of the return stroke. The graph at thebottom of Figure 1 is the magnitude of the Fourier transform of the waveform, E(t), shown at the top. T he Fouriertransform was o btained numerically after the waveform at the top was digitized.Figure 2 shows the results of averaging waveforms from 20 return strokes. The data was recorded in Florida

during a short period of strong activity associated with a nearby thunderstorm on July 20, 1976. At the top is theaverage of the time dom ain electric field waveforms, E (t), and at the bottom is the average of the magnitude of theFourier transform (the magnitudes were averaged). This data was collected by the author during TR IP-7 6.On e of the earliest applications of the Fou rier transform approach w as made by W att and Maxw ell (1957). Inthe course of describing measurements of low frequency attenuation due to the earth, they m ade Fourier transformsof electric field w aveform s recorded by Norinder (1954) and Florman (1955) and from them formed a compositespectrum; however few details were given regarding the w a v e f ~ m - s . zter, T z y ! ~ 1963) reperted a cxcfti!!ydocumented measurement of the spectrum in which the lightning discharges were located and propagation losses

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Figure 1 , Electric field waveform (top) and its Fourier transform (bottom) for a firstreturn stroke. Recorded in Florida during TRIP -76.2


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FREQUENCY (kHz)Figure 2.strokes in Florida during TRIP-76.

Average electric field waveform (top) and average magnitude of the Fouriertr2nsfcl-TE cf the indisridua! ssra~.lef%mScttcm). Data reccrded frcm frst retiiiz

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taken into accou nt. Taylor recorded return stroke waveforms and made Fo urier transforms over a frequency rangefrom 1 - 100 kHz. T he spectrum he ob tained is shown in Figure 3 (op en circles) normalized to a range of 50 km .The norm alization has been done assuming a (distance)-' dependence for the am plitude. The spectrum obtained byTaylor peaks near 5 kHz and then decay s roughly as (frequency)-' to 100 kH z, the limit of the measurements. Inobtaining this spectrum, Taylor used the recently verified theory for the effe cts of attenuation due to propagationover the earth to correct for the loss of signal at low frequencies. He had both this theory and tools for locating thelightning discharge at his dispo sal.

Th e Fourier transform approach was largely ignored until Serhan et. a l., (1980) again reported spectra of radi-ation from return strokes. Using m odem techniques for locating nearby return strok es (Krider et. a l., 1976; Krideret. a]., 1980), Serhan et. al. were able to separate first and subsequent return stroke waveforms and computespectra for each . Th eir data for first return strokes at 50km are shown in Figure 3. Thes e data fit well with T aylor's(1963) measurements and extend the (frequency)-' trend to several hundred kHz . T he spectra obtained by Serhanet. a l. (198 0) for-subsequent return strokes has substantially the sam e shape but is somewhat lower in am plitude.Weidman et. al., (1981) using substantially similar techniques also reported spectra of first return strokes(squares in Figure 3). Their data, collected from lightning over the ocean to minimize effec ts of propa gation, ex-tend the spectrum to 1 MHz and continue to show a decrease proportional to (frequency).'.Weidman et. a l., (1981) also reported first return stroke spectra at higher frequencies which they obtained byrecording the derivative, dE/dt, of the electric field waveform rather than E(t) itself. Recording the derivative im-proves the sensitivity of the measurement to higher frequencies (because the spectrum of the derivative is the spec-trum of the waveform m ultiplied by frequ ency). The data obtained by W eidman e t. al ., (1981) in this manner areshown w ith solid triangles in Figure 3 . Notice that the data show a decrea se with frequency which is greater thanthe (frequency)" decrease characteristic of the data at lower frequencies. The Weidman et. a l., (1981 ) results wereobtained using only the first few microseconds of the radiation waveform. Sinc e, this portion of the waveform ismost strongly affected by p ropagation losses, it is not clear whether the rapid dec rease evident in the data is a realcharacteristic of first return strokes or an artifact of the measurements. On the other hand, there is some theoreticaleviden ce to suggest that a decrease in the spectrum as (frequency)-2 should manifest itself at high frequ encies (LeVine, 1980).A few measurements using the Fourier transform approach have been made of the spectrum of events otherthan return strokes. All of these have been reported recently by Weidman, et. al., (1981). Figure 4 shows thespectra obtained for positive (top) and negative (bottom) intracloud events. T he curve s show the spectra of the in-tracloud events superimposed on the spectrum of first return strokes (solid line). On the left in Fig ure 4 re exam-ples of positive and negative intracloud events recorded by the author (Le Vine) at the Goddard Space FlightCente r, in Green belt, Maryland in 1982 . Notice that at the high frequencies the spectra for these events tend tocoincide with the spectrum of return stro kes, but at low frequen cies the spectrum is smaller than the spectrum of re-turn strokes and decreasing. This is the behaviour one would expect of a discharge with the sa me general physicalcharacteristics as a return stroke but of shorter length (e.g. L e Vine, 1980).

111. SPECTRA FROM DIRECT MEASUREMENTIn addition to the approach described abov e, it is also possible to determine the spectrum directly by m easur-ing the power incident at a particular frequency . The procedure is to use a filter to accept signals only in a narrowband of frequencies near the one of interest and then to use a detector to measure the power being radiated in thisfrequency band. The precise relationship between the spectrum at the nominal frequency, v,, to which the filter istuned and the output from this system de pends on the specific filter and detector employ ed. An important specialcase occurs w hen the filter is very narrow (a small percentage of the center frequency, v,) and the detector is an

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envelope de tector of the type employed in conventional AM radio receivers. In this case, if the input is a singleimpulse, the spectrum S(v,) is proportional to the peak value, ep, of the output. This relationship is derived inAppendix A wh ere it is shown that w ith an ideal bandpass filter of bandwidth B and system gain G o ne obtains:eP

2G BS(v,) =-Data obtained using this technique are illustrated in Figure 5. The amplitude of the spectra have been nor-malized to lightning at 50 km using (distance).' as the rule for scaling amplitude and (bandw idth)'' as the rule forconverting from peak field measurements to spectra when necessary (Equation 1). Among the earliest measure-ments of this type are those of Schafer and Goodall (1939) who were interested in assessing the potential interfer-ence lightning presented for transmission of television signals. Their data at 139 MHz is shown with a solidtriangle in Figure 5. Extensive measurements over a wide range of freque ncies were made by H orner and Bradley(1964). T hese data are shown w ith the solid line in Figure 5. Additional measurements by Kosarev et. a l., (197 0)and in 1963 by H allgren and McD onald (adapted from Cianos, Oetzel and Pierce, 1972) are also shown in Figure5. T he spectrum w hich emerges from these measurements is one which peaks near 5kHz nd then decreases as l/fto the limits of the measurements (1 GH z).The preceeding are only some of the measurements which have been made with this technique. A largenumber of measurements have been made, but under a great variety of circumstances (e.g., distance, type of re-ceiving equipm ent and bandwidth). It is often difficult to reduce the measurements to com mon units. N evertheless,several attempts have been made to combine the measurements by normalizing the data to comm on bandwidth anddistance (e.g., Ho me r, 1 964,; O h, 1969; Kim para, 1965; Cianos, O etzel and Pierce, 1972). Additional datagathered from these reviews are plotted in Figure 6. The relationship (distance)" has been used to convert to light-ning 50 km away and the relationship (bandwidth)-' has been used to convert peak field measurements to spectra(Equation 1). Shown with open circles are data by Takagi and Takeuti (1963) as reported by Kimpara (1965). D ataby Iwata and Kanada (1967) as reported by Cian os, Oetzel and Pierce (1972) are shown with X's. Finally, mea-surements of several researchers using radar receivers (Atlas, 1959; Hew itt, 1957; Pawse y, 1957) as reported byO h (1969) and C ianos, Oetzel and P ierce (1972) are shown with open triangles. N otice that these additional data

tend to fall somewhat below the l /f spectrum suggested in Figure 5 and with a significact spread of the data, espe-cially at the higher frequencies.A m ajor problem fac ed in interpretating spectra such as presented in Figures 5 and 6 is that the data do not rep-resent radiation from single events (e.g. retu rn strokes) but rather represent some genera lly unknown collection ofevents in the flash. To illustrate the nature of the problem, the output from several AM rad io receivers as seen dur-ing a representative cloud-to-ground lightning flash is shown in Figure 7. The records at 3 , 30 and 300 MHz aredata from real lightning recorded by the author in Florida during the Thunderstorm Research Project (TRIP-76;Pierce, 1976) using standard AM radio receivers with a comm on bandwidth of 300 kHz. The records at 30 and 300kHz are the author's impression of what radiation from the flash would look like based on reports in the literature(e.g ., H omer and Bradley, 1964; Malan, 1958). The slow electric field change for this flash is shown at the bottomfor reference. N otice that the radiation consists of many discrete impulses. Som e of the impulses correlate with

identifiable portions of the discharge . For exam ple, the initial impulses in this record occu r where one ex pects tofind the steppe d leader, and the return strokes (which occur at the abrupt chang es in the slow electrical field chang erecord) are associated with large impulses of R F radiation. How ever, these are only a few of the impulses seen dur-ing the ilash. The others are probably associated with intracloud portions of the discharge about which we haveonly begun to learn. The discrete nature of the radiation is characteristic of the data recorded by this author fromboth cloud-to-ground and intracloud discharges (e.g., Le Vine, 1976). Another important characteristic of the datais that the radiation do es not appear to change from a series of discrete impulses a t the low er frequencies to a con-tinuum at the higher frequencies as reported in some investigatinns ( e . g . , Hcmer s ~ : !radley, 1964; Mi?iaii,1958). Rather the radiation consists of a sequence of discrete impulses at all frequencies in the range (3 - 300MHz) investigated by this author. In addition, the impulses tend to correlate well among the frequencies. That is, a7


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particular impulse tends to be present at all frequencies. Th e behaviour is consistent with a lightning flash consist-ing of a sequence of individual discharges (leader steps, return strokes, K -changes, etc .) each radiating over a verybroad band of frequ encie s. Certainly the return stroke is a discharge w ith this characteristic (Figure 3 ) .Th e problem w hich occurs with spectral measurements using the filter-detector approach is distinguishing be-tween impulses. The impulse response of the measuring system is inversely proportional to the system bandwidth;consequently there exists a fundamental co ntradiction between the requirement for a bandwidth narrow enou gh to

be close to the nominal frequency of interest and the ability of the measurement to distinguish between closelyspaced impulses. For exam ple, with a bandwidth of 250 Hz, the impulse response of the system is on the order of0.25 seconds. This would include many impulses in a typical lightning discharge (Figure 7). Very few inves-tigators have attempted to distinguish between events with measurements of this type and those who have triedhave had difficulty (e.g., Takagi and Takeuti, 1963). The spectra which have been reported using this methodgenerally must be regarded as an integral (average) over many events in the flash. T he exception is at VLF fre-quencies where, because of the low frequency, return strokes tend to be the dominant so urce of radiation.

IV . DISCUSSIONNotice the similarity between the spectrum of th e first return stroke (Figure 3 ) and the composite spectrumobtained from all the filter-detector measurements (Figures 5 and 6). To facilitate the comparison, the spectrahave been plotted together in Figures 8 and 9. Figure 8 show s the return stroke spectrum (Figure 3 ) and the spec-trum in Figure 5 together, and Figure 9 shows the return stroke spectrum and all the filter-detector measurements(Figure 6 ) ogether. C onsidering all the variables that enter such m easurements, the spectra are very similar. Thisis especially so at frequenices below 1 M Hz, but even at higher frequencies where the spread in the d ata is great,the two spectra overlap.The similarity between the spectra is not surprising because of any fundamental difference between the twotechniques. In fact, in principle the two techniques for measuring the spectrum of lightning discharges ought toyield identical results. In practice, they should be complimentary, the Fourier transform approach havin g advan-

tages at low frequencies and the direct (filter-detector) approach having advantages at higher frequencies. How-ever, because of the relatively narrow bandwidth employed, the filter-detector technique does not measure radia-tion from a single event, but rather is an average over many even ts in the flash. Furtherm ore, the data in Figures5 and 6 were obtained from measurements of different bandwidth and without any attempt at identifying the por-tion of the flash monitored. C onsequen tly, it would seem reasonable to assume that Figures 5 and 6 are more rep-resentative of the composite flash rather than of any particular event. What is surprising, then, is that the spec-trum of the composite flash and the spectrum of one particular even t, the return stroke, are so similar.

Th e similarity between the spectrum of one event, the return stroke, on the one han d, and w hat probably rep-resents the spectrum of the composite flash on the other hand, suggests to this author a physical process commonto all the various individual discharges which m ake up the lightning flash. There is som e additional evidence sup-porting this view . For exam ple, Weidm an, et. al ., (1981) measured spectra of intracloud processes and steppedleaders, and these tend to fall on the spectra for return strokes (Figure 4). The d ifference is a roll- off at the lowfrequencies which is to be expected for identical discharges of shorter length. Also, Le V ine (1976, 1980) hasbeen able to predict spectra for the return stroke and composite flash which agree with measurements by assuminga transmission line model common to all events. (The events differ in such parameters as channel length, peakcurrent and tortuosity.)

A critical test of this hypothesis occurs at high frequen cies where all even ts should behave similarly. But thisis where the data is most scattered. Consequen tly, it would seem important to obtain reliable spectral measure-ments of individual events at frequencies above a few MH z. This w ould resolve the ambiguities in the spectralshape apparent in Figures 5 and 6 and would add insight into the physics of the lightning disch arge.11


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REFERENCESAtlas, D. Rad ar Lightning Echoes and Atmospherics in Vertical Cross Sectio n, in Recent Advances inAtmospheric Electricity, L. G . Sm ith, ed. New Y ork: Pergamon Press, 1959, pp. 44 1-458.Born, M., and E. W olf, Principles of Optics, Pergamon Press, 1959.Cianos, N., G. N. Oetzel and E. T . Pierce, Structure of Lightning Noise -Especially Above H F, Lightningand Static Electricity Conferen ce, Wright Patterson AFB ,December 1972.Dennis, A. S . , and E. T . Pierce, T he Return Stroke of the Lightning Flash to Earth as a Source of VLFAtmo spherics, Radio Science, 68D (No. 7), pp. 777-794, 1964.Florrnan, E. F., National Bureau of Standards Report #355 8, No vem ber, 1955 .Hallgren, R. E ., and R . B. McD onald, Atmospherics from Lightning from 100 to 600 MHz, R ep. No. 63-538-89 . IBM Federal Systems Division, 1963.Hew itt, F. J . , Rad ar Echoes from Interstroke Process in Lightnin g, Proc. Phys. SO C.London,-0, pp. 122-204, 1957.Ho rner, F., Rad io Noise from Thunderstorm s, in Advances in Radio Research , Vol. 2, Academic Press,

J. A. Saxton, ed. , pp. 122-204, 1964.Horner, F ., and P. A . B radley, Spectra of Atmospherics from Near Lightning, J . Atmos, Terr. Phys.,-6, pp. 1155-1166,1964.

Kim para, A ., Electromagnetic Energy Radiated from Lig htning, in Problems in Atmospheric and SpaceElectricity, S. C . Coroniti, ed., Elsevier Pub. Co ., pp. 352-365, 1965.Kosarev, E. L ., V . G . Zatsepin and A. V . M itrofanov, Ultrahigh Frequency R adiation from Lightning,

J. Geophys. Res., 75 (36), pp. 7524-7530, 1970.Kraus, J . D . ,Radio Astronomy, McGraw-Hill Book Co., 1966.Krider, E. P ., C . D. Weidman, and R . C. No ggle, T he E lectric Fields Produced by Lightning Stepped Leaders.

J. Geophys. Re s.,-2, pp. 951-960, 1977 .Krider, E. P., R . C. Noggle and M . A. Um an, A Gated Wideband Magnetic Direction-Finder for LightningReturn Strokes, J. Appl. M eteorol.,-5, pp. 302-306, 1976.

Krider, E. P ., R . C. Noggle, A . E. P ifer and D. L . Vanc e, L ightning D irection-Finding System s for Forest FireDetection, Bull. Amer. Meteorol., SO C., 1(9), pp. 980-986, 1980.-Le Vine, D . M ., et. a l., T he Structure of Lightning Flashes HF-UHF: September 12 , 1975, Atlanta, Georgia,NA SA X -953-76-176.1976.Le Vine , D. M ., T he E ffect of Pulse Interval Statistics on the Spectrum of Radiation from Lig htnin g, JGeophys. Res. 82 (12), pp. 1773-1777, 1977.

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Le V ine, D. M ., The Spectrum of Radiation from Lightning, Proc. IEEE International Symposium onElectromagfietic Com patibility, pp. 249-25 3, October 198 0.M alan, D . J. , Radiation from Lightning Discharges and its Relation to the Discha rge Process, in RecentAd vances in Atm ospheric Electricity, Proceedings of the 2nd Con ference on Atmospheric. Electricity,

pp. 557-563, 1958.Norinder, H., Th e Waveform s of the Electric Field in Atmospherics Reco rded Simu ltaneously at two DistantStations, Arkiv for Geofysik,-(9), pp. 161-195, November 1954.Oh , L. L., Measured and Calculated Spectral Amplitude Distribution of Lightning Sferics, IEEE T rans.,

EMC-11 (4), pp. 2125-130, 196 9.Pawsey, J. L . , Radar Observations of Lightning, J. Atmospheric Terr. Phys., 11, pp. 289-290, 1957-Pierce, E. T ., The Thunderstorm Research International Program (TRIP) - 1976, Bull . Amer. Meteorol. SO C.,57, pp. 1214-1216, 1976.-Serhan, G. I . , M. A. Uman, D. G . Childers, and Y . T. Lin, T he R F Spectra of First and Subsequent LightningReturn Stroke s in the 1-200 km Range, Radio Sci., 15, pp. 1089-1094, 1980.-Schafer, J.P., and W.M . Goodall, Peak Field Strengths of Atmospherics due to Local Thunderstorms at 150Megacycles, Proc. IRE, 27, pp. 202-207, 1939.-Takagi, M . and T. T akeuti, Atmospherics Radiation from Lightning Discharge, Proc. Res. Inst. A tmos.,Nagoya Univ ., 10, 1963.-Taylo r, W. L ., Rad iation Field Characteristics of Lightning Discharges in the Band 1 kc/s to 1 00 kc/s, J. R es.Nat. Bur. Stand ., 67D, pp. 539-550, 1963.-Um an, M . and E . P. K rider, A Review of Natural Lightning: Experimental Data and Modelling, IEEE Trans.on Electromagnetic Com patibility, EMC -24 (2), pp. 79-1 12, 1982.Watt, A. D., and E. L. M axw ell, Cha racteristics of Atmospherics Noise From 1 to 100 kc. ,Proc. Inst. RadioEng ., 45, pp. 787-794, 1957.-Weidman, C. D ., and E. P. Krider, T he Fine Structure of Lightning Return Strokes Waveform s, J. G eophys.Res.-3, pp. 6239-6247, 1978.Weidm an, C. D. and E. P. K rider, Submicrosecond Rise Times in Lightning Return Stroke Fields, Geophys.Res. Lettrs., 7, pp. 955-958, 1980.-Weidman, C. D., E . P. Krider, and M . A. Um an, Lightning A mplitude Spectra in the Interval From 100kHzto 20 M Hz, Geophys. Res. Lett., 8 , pp. 931-934, 1981.-

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APPENDIX ARADIO RECEIVER RESPONSE.

It is the purpose of this appendix to derive an expression for the impulse response of an RF receiving systemtypical of the sort used to measure the spectrum of radiation from lightning. Th e system consists of an antenna,usually vertically polarized, connected to a standard AM radio receiver and followed with a post detection filter(Figure A l) . Assuming that the antennas are vertically polarized and isotropic over the bandwidth of the measure-ments, they can be assumed to deliver a voltage proportional to the incident (vertical) electric field. Except for aphase which is ultimately lost in the detector, this proportionally constant is d m L / k where k =COG2 dA, G is the gain of the antenna in the plane parallel to the grou nd, and A, is the effective receiving area of the an-tenna (Kraus, 1966). The receiver is a device for detecting and amplifying the envelope of an amplitude mod-ulated sinusoid (carrier) at a particular frequency. This is norma lly done by translating the input signal to an inter-mediate frequency where the actual processing is done. How ever, the frequency translation is don e for engineer-ing purposes to make the detection more efficient, and it is not necessary to do this in order to model the receiveroutput. The ideal device is a perfect envelope detector in series with a filter which represents the equivalentbandwidth and gain of the system.

For systems who se bandwidth 6 is small compared to the nominal frequen cy, vo, of the measurement, theseoperations can be written explicitly in terms of the Fourier transform of the incident radiation. To do so it is con-venient to write the vertical compone nt of incident radiation, E(t), in the form:E (t ) = Re f ? E (v)e-J27F"tv

where Re means "real part of" and where E (v) is the Fourier transform of E (t). The integral is called the complexanalytic representation of E (t) (e.g., Born and W olf, 1 959). Using this notation, the signal V(t) out of the antennaand filter and into the enve lope detector is:V(t) =Re 2a(v)H(v)E (u)e-J2n"t vKwhere H (v) is the Fourier transform of the filter h(t) and a(v) = (A/k)d?FT;TA, s the c om bined effect of the. an-tenna and an amplifier with gain, A. Although in practice the post detection filter is applied to the video output ofthe receiver, m athematically the effect of the post detection filter can also be included in H(v), This will be donehere and it will be assumed that the equivalent filter H(v) has a passband centered about frequency uo which isvery narrow comp ared to v,. That is:

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where

e (v,,ti =JI 2a(v, +S)E (v, +S)H(S)and where the assumption that H(v) is nonzero only in a narrow band about v, (Equation A3) has been used to for-mally extend the low er limit of the integration in Equation A 4 to infinity. From Equation A 4 it is clear that V (t) hasthe form of an amplitude modulated sinusoid, co s(2 .rrv 0t+ ~) , t frequency v,. The output of the detector is the en-velope, le (v,,t)I, of this carrier. Thu s, the outp ut, e,(t) can be written:

e,(t) = 1 JT,2a(v, +5) E (v, +6) H(S) deNow , sup pose that over the narrow band of frequen cies passed by the filter a(v, +6 ) a(v,) and E(v, + k)E(v,). Then,

where IE(vo)l is the magnitude of the spectrum of the electric field at freque ncy v, and whereh(t) =lI 2 a(v,)H(S) e-j2mStS

is the impulse response of the system. (Note: e2=e e*) Integrating both sides of Equ ation A7 and using Parsevalstheorem for Fourier transforms, one obtains:a,eg(t) dt = IE(v,)I2 Jm h2(t) dt

Finally, solving for the spectrum, one obtains:

whereA = 1 a(v,) H(O)/J I s SH2 ( 0 )

Th e integral in Equation A1 1 is commonly called the bandwidth (power bandwidth) of the system and la(v,) H(O)Iis the gain of the system. E quation A 10 states that the spectrum of the input signal can be ob tained by integratingthe output and dividing by twice the gain times the square root of the bandwidth of the system. T his result onlyA-2


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applies to individual input signals, E(t), w hose bandwidth is much larger than the bandwidth of the system (i.e.appear as impulses on the time scale of the impulse response of the system). This restriction is a consequenceof factoring E(vo) out of Equation A6. If the input were a sequence of such pulses rather than an individual eventE(v) in Equation A6 would be a sum of the form Z Ei(v) d2ntihere Ei(v) is the spectrum of the individualpulses and 4 is the time between pulses. In this case the spacing betw een.pulses can affect the spectral estimate(Le Vine, 1977 ; Dennis and Pierce, 1964).

An impo rtant special case occurs when the system can be modelled as an ideal bandpass filter:

The n, from Equation A 6 one obtains:

J -1312=2(a(vo)E(vo)lHoBsinc(.rrBt)=epsinc (TBt)

where ep s the peak value of the signal out of the receiver. Now , squaring and integrating over all time , one obtains:1:(t) dt =ep s,sinc2 (n Bt) dt2=e,/B

Finally, putting this result into Equation A10 , on e obtains:

where G = la(vo) H(0) I is the gain of the system . Equation A15 applies only if the time between pulses is longcom pared to the response time of the system .Another important special case occurs when the input is a random process. If the process consists of a sequ-ence of identical pulses with random amplitude and/or arrival time, then the analysis proceeds as above w ith mod-ifications as indicated in the text and described in detail by Le Vine (1977). However, if the random process isnoise-like (Le. a continuou s, fluctuating signal), then the analysis must be modified. In this case , the appropriatedefinition of the spectrum, S(v), s the Fourier transform of the autocorrelation function, R(T) = of the inpu t signal. Assuming that the detector is an ideal s uare-law detector and that the input, E(t), is astationary, ergodic random process, and defining IE(v)I =+-S (v) , one obtainsA-3


21/22

value of the output signal by dividing by the square root of the bandwidth a nd twice the system gain. This resulthas been employed to compute spectra of lightning in some cases (e.g. Oh, 1969); however, as indicated in thetext, lightning is intrinsically imp ulsive in nature and as a result this formula m ust be used with caution.

>ILTER DETEC-- ILTER -0TOR-

n t

Figure A1 . Example of a radio receiver used to measure spectrum of radiation from lightning.

A - 4


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. Repor t No.NASA TM 87788

BIBLIOGRAPHIC D A T A SHEET2. Government Accession No.

L Perform ing Organ izat ion Name and AddressLaboratory for OceansGoddard Space Flight CenterGreenbelt, Maryland 2077 1

12. Sponsoring Agency Name and AddressNational Aeronautics andSpace AdministrationWashington, D.C.

15. Supplementary Notes

17. Key Words (Selected by Au t h o r ( s ) )Lightning SpectrumLightning Noise

3. Recipients Catalog No.

18. Dist r ibut ion StatementUnclassified-UnlimitedSubject C ategory - 47

5. Report Date

!9. ecnr i ty &ss i f . (ef this repnrt!Unclassified

6. Per forming Organizat ion CodeCode 675

2g. Secnrity C!assif. {cf hi; page) 21. \!= =f page; 22. p;ice*Unclassified

8. Per forming Organizat ion Repor t N o.

10 . Work Un i t No.11 . Cont ract or Grant No.

13. Ty pe of Repor t and Period Covered

14. Sponsoring Agency Code

16. Abst ractA review is presented of the measurements reported in the literature of the spectrum of elec-tromag netic radiation from lightning in the frequency range from 1 kHz to 1 GH z. Measurements

have been m ade either by m onitoring the power received at individual frequencies using a narrowbandwidth recording device tuned to the frequencies under investigation or by recording the trans-ient (tim e dependent) radiation with a w ide bandwidth dev ice and then Fourier transforming thewaveform to obtain a spectrum. Measurements of the first type were made extensively in the1950s an d 1960s and several composite spectra have been d educed by norm alizing the data ofdifferent investigators to comm on units of bandwidth and distance. Th e compo site spectra tend topeak near 5 kHz nd then decrease roughly as (frequency). upto nearly 100MH z where scatter inthe data make the behaviour uncertain. Measurements of the second type have been reported forreturn strok es, the stepped leader and for some intracloud processes. T he spectrum of first returnstrokes obtained in this manner is very similar to the c om posite spectra obtained from the narrow-band measurements.

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