Precise comparison of experimental and theoretical SNRs in CO 2 laser heterodyne systems R. Foord, R. Jones, J. M. Vaughan, and D. V. Willetts A detailed comparison of experimental and theoretical SNR in an IR laser heterodyne system has been made with three different signal analyzers. Good agreement, considerably better than a factor of 1.5, is reported. Accurate allowance was made for transmission in the receiver optics, the effective quantum efficiency of the detector due to shot noise domination by the local oscillator, and for coherent speckle effects across the col- lection aperture. The evaluation of SNR with a surface acoustic wave spectrum analyzer and digital inte- grator is described in some detail. As an illustration an absolute measurement of backscattering strength in the atmosphere from an airborne equipment at altitudes up to 13.1 km is provided. 1. Introduction The SNR expected theoretically in coherent IR het- erodyne systems has been well documented for many years, e.g. Refs. 1-7. However, to our knowledge no comparison of theoretical and experimental values has been reported that shows close agreement. Indeed in practice it is usually accepted as a general rule that up to an order of magnitude degradation of experiment with theory will be found. At a recent conference on coherent laser radar systems, 8 the closest agreement claimed by any worker in the field was a loss factor of -4. Just such a loss factor was reported in an early paper by Brandewie and Davis 5 with a cw system. More recently the calibration by Schwiesow and Cupp 9 of their 30-cm cw equipment showed a lidar response 5 + 1 dB less than the ideal limit. Similarly the cali- bration of the NOAA pulsed lidar by Post et al. 10 showed a 7-dB discrepancy. This figure was corrobo- rated by operating the equipment in a cw mode which gave a similar result. Post et al. suggested that part of this discrepancy, between predicted and observed SNRs, might be due to assumption of an over optimistic value for the quantum efficiency of their detector to- gether with various optical mismatching and beam ab- errations which could reduce heterodyning efficiency. An additional problem, which also renders difficult the comparison between the results of different groups, is the lack of reproducible well-calibrated scattering The authors are with Royal Signals & Radar Establishment, Great Malvern, Worcs. WR14 3PS, U.K. Received 2 June 1983. 0003-6935/83/233787-09$01.00/0. targets. We have recently had available surfaces of cut polystyrene and flame sprayed aluminum for which the scattering characteristics have been well established. 1 " Accordingly we have conducted a detailed examination of all the parameters entering the evaluation of the SNR with a twofold aim: first, to determine whether a dis- crepancy did exist that could be explained by factors not presently taken account of in the theoretical treatment and, second, to provide an absolute calibration of our C0 2 laser heterodyne system so that the measured SNRs could be translated into absolute backscattering strengths. A detailed evaluation has been made of factors such as transmission in the receiver optics, the effective quantum efficiencyof the detector due to shot noise domination by the local oscillator, and the effect of laser speckle on coherent integration across the col- lection aperture. Different estimates have been em- ployed for the measured noise, both the noise power and the observed variance of the noise, and different spec- trum analyzer systems have been used. As a guide to effective operation these scanning and multichannel instruments are considered in some detail. The final result is that the experimental SNRs are found to be within a factor of 1.2 0.3 of the theoretical values. Allowing for the cumulative uncertainties of the mea- surements it is concluded that there is no residual dis- crepancy. However, the study does emphasize the prime necessity of very careful matching of wave fronts of local oscillator and signal beams; it is suggested that this, together with effects of speckle, largely explain previous poorer agreement. The result is illustrated with a recording of atmospheric backscattering taken with an airborne CO 2 laser heterodyne velocimeter at altitudes up to 13.1-km ASL (above sea level). For this equipment the minimum detectable atmospheric backscattering coefficient is 2 X 10-11 sr- 1 m- 1 in a 1 December 1983 / Vol. 22, No. 23 / APPLIED OPTICS 3787
Transcript
Precise comparison of experimental and theoretical SNRsin CO2 laser heterodyne systems
R. Foord, R. Jones, J. M. Vaughan, and D. V. Willetts
A detailed comparison of experimental and theoretical SNR in an IR laser heterodyne system has been madewith three different signal analyzers. Good agreement, considerably better than a factor of 1.5, is reported.Accurate allowance was made for transmission in the receiver optics, the effective quantum efficiency of thedetector due to shot noise domination by the local oscillator, and for coherent speckle effects across the col-lection aperture. The evaluation of SNR with a surface acoustic wave spectrum analyzer and digital inte-grator is described in some detail. As an illustration an absolute measurement of backscattering strengthin the atmosphere from an airborne equipment at altitudes up to 13.1 km is provided.
1. Introduction
The SNR expected theoretically in coherent IR het-erodyne systems has been well documented for manyyears, e.g. Refs. 1-7. However, to our knowledge nocomparison of theoretical and experimental values hasbeen reported that shows close agreement. Indeed inpractice it is usually accepted as a general rule that upto an order of magnitude degradation of experimentwith theory will be found. At a recent conference oncoherent laser radar systems,8 the closest agreementclaimed by any worker in the field was a loss factor of-4. Just such a loss factor was reported in an earlypaper by Brandewie and Davis5 with a cw system.More recently the calibration by Schwiesow and Cupp9
of their 30-cm cw equipment showed a lidar response5 + 1 dB less than the ideal limit. Similarly the cali-bration of the NOAA pulsed lidar by Post et al.1 0
showed a 7-dB discrepancy. This figure was corrobo-rated by operating the equipment in a cw mode whichgave a similar result. Post et al. suggested that part ofthis discrepancy, between predicted and observedSNRs, might be due to assumption of an over optimisticvalue for the quantum efficiency of their detector to-gether with various optical mismatching and beam ab-errations which could reduce heterodyning efficiency.
An additional problem, which also renders difficultthe comparison between the results of different groups,is the lack of reproducible well-calibrated scattering
The authors are with Royal Signals & Radar Establishment, GreatMalvern, Worcs. WR14 3PS, U.K.
Received 2 June 1983.0003-6935/83/233787-09$01.00/0.
targets. We have recently had available surfaces of cutpolystyrene and flame sprayed aluminum for which thescattering characteristics have been well established. 1"Accordingly we have conducted a detailed examinationof all the parameters entering the evaluation of the SNRwith a twofold aim: first, to determine whether a dis-crepancy did exist that could be explained by factors notpresently taken account of in the theoretical treatmentand, second, to provide an absolute calibration of ourC0 2 laser heterodyne system so that the measured SNRscould be translated into absolute backscatteringstrengths. A detailed evaluation has been made offactors such as transmission in the receiver optics, theeffective quantum efficiency of the detector due to shotnoise domination by the local oscillator, and the effectof laser speckle on coherent integration across the col-lection aperture. Different estimates have been em-ployed for the measured noise, both the noise power andthe observed variance of the noise, and different spec-trum analyzer systems have been used. As a guide toeffective operation these scanning and multichannelinstruments are considered in some detail. The finalresult is that the experimental SNRs are found to bewithin a factor of 1.2 0.3 of the theoretical values.Allowing for the cumulative uncertainties of the mea-surements it is concluded that there is no residual dis-crepancy. However, the study does emphasize theprime necessity of very careful matching of wave frontsof local oscillator and signal beams; it is suggested thatthis, together with effects of speckle, largely explainprevious poorer agreement. The result is illustratedwith a recording of atmospheric backscattering takenwith an airborne CO2 laser heterodyne velocimeter ataltitudes up to 13.1-km ASL (above sea level). For thisequipment the minimum detectable atmosphericbackscattering coefficient is 2 X 10-11 sr-1 m- 1 in a
measurement lasting 0.6 sec. Detailed presentation ofa series of measurements in the atmosphere at differentaltitudes will be given in following papers. Severalarticles describing backscattering measurements in theatmosphere have recently been presented by workersat the Wave Propagation Laboratory.10 1213 Charac-teristics of coherent lidar returns from calibration tar-gets and aerosols have also been described.9 14
11. General Considerations of Signal to NoiseThese investigations have been conducted with the
CO2 laser heterodyne velocimeter shown in Fig. 1, whichhas been outlined in previous publications.15 16 Thisequipment has been designed for airborne use; the air-borne model is currently installed in an HS125 aircraftin the Flight Systems Department at RAE Bedford,while the optically identical prototype is available forground tests and the work described here. Three typesof signal processing spectrum analyzer have been em-ployed: (a) a 0-25-MHz bandwidth surface acousticwave (SAW) spectrum analyzer, with analog-to-digitalconversion (ADC) of the output spectrum and subse-quent integration; (b) a 0-6.3-MHz bandwidth SAWspectrum analyzer also with ADC and integrator'7 ; and(c) a scanning spectrum analyzer, Tektronix type 7L5with L3 module. The first two equipments provideefficient multichannel operation and are commonlyused for airborne and ground-based measurement ofwind speed, while the latter is a single-channel scanningdevice with wide dynamic and calibrated range.Measurements with each of these equipments are de-scribed in turn.
In a laser heterodyne system the theoretical SNR inits simplest form is given by
SNRpower = 7effP8/h&B, (1)
where the SNRpower is appropriate to the power spec-trum of the current or voltage signal derived from thedetector, 'qeff is an effective quantum efficiency of thedetector, P, is the power in the scattered laser signaldelivered to the detector, hw is the energy per photon,and B is the appropriate spectral bandwidth. The termPjIhw is thus the number of signal photons incident persecond on the detector, and the whole expression is ef-fectively the number of such photons detected per in-verse bandwidth period.
In heterodyne operation in the IR the detected signalis given by an oscillating current i proportional to theproduct of the scattered signal field E, and local oscil-lator field ELO:
i9 . ELO X ES. (2)
The power spectrum of this continuous input signal isgiven by
P(W) = lim IY 2 iI (t) exp(-iwt)dt2, (3)
where Y is the duration of the data. For a given databatch the SNR is defined as
SNRpower mean signal - mean background
Fig. 1. Schematic optical arrangement of the airborne monostaticCO2 laser velocimeter. Polarization techniques promote good effi-ciency. The laser is locked to the P20 transition with the narrowband
filter in the local oscillator beam.
where the signal and background (the noise) are mea-sured in the power spectrum.
An evaluation of mean and variance for widebandrandom signals has been made by several authors, e.g.,Refs. 18 and 19. If the probability distribution of i isGaussian, the variance of the estimator P() of thespectrum derived from a finite data batch is given by thesquare of the mean value of P(w), that is, by
varP(co) = (p(W))2 (5)
In the absence of signal the standard deviation in thenoise background is thus given by [var PM background]which is equal to (P(W)background), and substituting inEq. (4) we have
SNRpower = mean signal - mean background (6)[var P(O)background] /2
mean signal - mean background
(P(CO)background)
mean signal power
mean noise power(7)
which is the usual formulation. Thus in practice theSNR may be evaluated by measurement of either noisepower in the vicinity of the signal or the square root ofthe variance on the noise background. Both techniqueshave been employed in the present work. The experi-ments have thus consisted essentially of a comparisonof measured SNRs determined via Eq. (6) or (7) fromobserved spectra, with calculated SNRs determined viaEq. (1).
111. Calculated SNR
Returning to Eq. (1) we see that SNRcalc requiresevaluation of three important terms: 71eff, the effectivequantum efficiency of the detector; B, the appropriatespectral bandwidth; and P, the scattered laser powerdelivered to the detector. Of these B is most simplydealt with and is determined by either the scatteredsignal or the analyzing instrument. Whichever has thelargest bandwidth gives the value of B. In much of thiswork surface acoustic wave spectral analyzers have been
used which accept a primary data sample of 25-Alseclength. For narrowband signals, for example, solidtargets in uniform motion or aerosols in laminar flow,the operational inverse bandwidth period 1/B is thusequal to 25 sec.
In our equipment the detector is a reverse biasedphotovoltaic cadmium mercury telluride device cooledto 77 K by a miniature Joule-Thomson air liquefier andincorporating a matched preamplifier and automaticcontrol of reverse bias on the p-n junction irrespectiveof local oscillator levels. The quantum efficiency(typically 0.5) was deduced from (a) the short circuitcurrent when viewing a 300 K blackbody, and (b) acombination of the voltage responsivity measured usinga 500 K blackbody and the zero bias slope impedance.In general IR heterodyne operation, one seeks to dom-inate detector noise with local oscillator shot noise. Theeffect of incomplete domination is to reduce the oper-ational quantum efficiency from its infinite local oscil-lator value . The appropriate expression is given by
Jleff = (M 1 ) (8)
where M is the ratio of detector noise powers with andwithout a local oscillator present. The values of 77eff forone of our detectors plotted against the dc detectorcurrent (which provides a convenient reproduciblemeasure of local oscillator level) is shown plotted in Fig.2.
The signal power P, delivered to the detector is de-termined by several factors and for a hard or solid targetmay be written as
P. = P( - Tatm (0,7i) Q Kpt Kspec* Khet.
laser power Po was measured on emergence from thetelescope at 0.90 W. Over the 1240-m return path Tatmwas estimated, depending on weather conditions, atvalues between 0.80 and 0.92. The collection solid angleQ is readily calculated from the geometry with a rangeof 620 m and collection aperture of 14.8 cm.
In the equipment shown schematically in Fig. 1 thereceived signal traverses twelve optical surfaces.Measurement of several of the transmitting surfacesgave a rather disappointing loss of 0.04/surface. Witha reflectivity of 0.78 at the germanium plate and 0.98and 0.90 for the mirror and beam splitter, respectively,together with a small loss at a spatial filtering aperture,a final value for the transmission around the receiverpath was found to be Kopt = 0.34 ± 0.04, where the errorindicates an estimated limit.
The term Kspec, while recognized, has not commonlybeen included quantitatively in the past. It arises sincethe scattered beam in the IR, just as is well known in thevisible, must give a characteristic speckle pattern, e.g.,Refs. 20 and 21. The relative importance of the phe-nomenon depends on the size of the collecting aperturerelative to a typical speckle blob, which itself is deter-mined by the size of the illuminated region at the target.One thus requires the correction factor Kspec to takeaccount of the fact that the scattered energy is con-centrated in the speckle blobs and not distributed withuniform amplitude and phase over the collection aper-ture. An expression for Kspec given by
Kspec =
(9)
In this expressionPo represents the laser power from the trans-
mitter directed to the target,Tatm is the atmospheric transmission on the path
to and from the target,(0,,7r) is the backscattering per unit solid angle where
the angle of incidence on the target is 0; for aLambertian scatterer this is equal to [e(0) -cosOlbr, where e(0) is the diffuse reflectance,is the collection solid angle,
K.pt is the transmission efficiency of the opticalreceiving system,
Kspec is a term due to the speckle characteristics ofthe scattered signal, and
Khet is a heterodyning efficiency, which is less thanunity due to mismatch of the local oscillatorwave front and signal beam.
Of these the first four terms are specific to individualexperiments, while the final three terms are more gen-eral in character and common to all the experimentalarrangements.
In the first series of experiments a large plate wasmounted on the side of a vehicle which was driven atsteady speed with the laser beam incident at an angleof -35° and a range of -620 m. At this angle the flamesprayed aluminum surface was taken"1 to have a Lam-bertian diffuse reflectance of 0.68. The transmitted
(heterodyne detectionsoutput power)
ensemble averaged with speckle
(heterodyne detections same mean intensity butoutput power I constant amplitude and phase
(10)
has been evaluated 2 2 for different receiver apertures ofradius R relative to an adjacently (or concentrically)transmitted laser beam with an untruncated Gaussianwave front of radius w0 (e-2 intensity), supposedly fo-
Fig. 2. Typical effective detector quantum efficiency Jeff plottedagainst detector current at different frequencies. Detector currentprovides a convenient reproducible measure of the applied local
Fig. 3. Speckle factor Kapec (see text) plotted against the apertureparameter R2/2w2 to provide a quantitative measure of the effect oflaser speckle on the scattered signal [from G. Parry, Royal Signals and
Radar Establishment (1980), unpublished work].
cused onto the far-field target. Results of the evalua-tion are given in Fig. 3 and show clearly that when thecollection aperture and hence Q in Eq. (9) is increased,Kspec decreases. Thus, as is well kown in coherentsystems, there is little point in making the collectionaperture very much larger than the transmitted beam.In the present work the size of the transmitted beam atthe single lens used as transmitter and receiver waschosen experimentally to give the best quality of focuswith minimal truncation. In this uniaxial system thereceiver and transmitter apertures are thus of radius R= 7.4 cm; the outgoing laser beam has a measured e-2radius of 5.2 cm, which at 0.7 of the full aperture issomewhat smaller than has been previously employed.Detailed examination of the focused beam was carriedout with a pyroelectric array at ranges of 20-40 m. Thespot was found to be very close to Gaussian in form but-20% larger than due to the diffraction limit of theuntruncated beam, presumably due to minor opticalaberrations. The size of the beam at the target deter-mines the size of the speckle at the receiver. Accord-ingly the correspondingly reduced value of wo (appro-priate to the larger beam size at the target) of 4.3 cm waschosen as the effective size of the Gaussian beam for theevaluation. Thus for R/(Woeffective) of 1.72 the apertureparameter R2 /2W2 is 1.48, and a value of Kspec = 0.38 isfound and has been employed in this work. It shouldbe noted that this value would be appropriate only forgood optical propagation. In a turbulent atmosphereadditional speckle effects will occur, and in consequencea smaller value of Kspec would be required, e.g., Refs.23-26.
In consideration of the final term Khet the local os-cillator beam should ideally be of uniform amplitudeacross the wave front, matching in size the beam due tothe signal, and also coplanar with it. In practice thelocal oscillator is usually derived by truncating thecentral portion of a weak beam derived from the laserand in consequence has a Gaussian peak profile. Thisproblem has been considered by Cohen who providesa convenient tabulation. 2 7 Our optical arrangement has
been designed to optimize the available heterodyningefficiency, and with the dimensions chosen a value ofKhet equal to 0.80 is found. The importance of carefulalignment of signal beam and local oscillator beamcannot be overemphasized. It is most important toensure the colinearity and coplanarity of the beams asprecisely as possible, using adjustable apertures andevery available optical check. Such precision is notunduly difficult with a large table mounted system inwhich the optical elements are well separated. How-ever, with our compact miniaturized system up to a10-dB gain in signal was often achieved in practice byfurther very precise adjustment from an initial, ap-parently good, alignment.
With all the quantities derived as outlined above,numerical values of SNRcalc were obtained from Eqs.(1) and (9) and are shown plotted as the abscissa of Fig.4. Calculations were made both for the two multi-channel SAW spectrum analyzers and the scanningspectrum analyzer. Different values for the SNRcalc areobtained taking account of different ranges, scatteringsurfaces, incidence angles, transmitted laser power,levels of local oscillator, and atmospheric transmission.For the SAW analyzers the calculated values are ap-propriate to a measurement of duration one inversebandwidth period, which for the SAW analyzers was 25iAsec or one integration period. As an example, withTatm 0.92 and a rather small local oscillator giving Jleff
= 0.12, a value of (SNRcac)poweri = 10.8 X 104 is found,where the indices show this is in the power spectrum forone integration or sampling period.
IV. Observed SNR
Scattering from the calibrated targets is very large,and the calculated value of SNR is also large-of theorder of 105 in the previous section. However, the laserheterodyne system has been designed for the utmostsensitivity in analysis of very weak signals. Perfor-mance evaluation requires very precise control of signallevels (both optical and electrical) throughout the sys-tem to ensure that the dynamic range of reliable oper-ation is not exceeded. Calibrated optical and electrical
Fig. 4. Comparison of observed and calculated SNR with differentSAW spectrum analyzers and a scanning analyzer. Good agreementwith the points close to the 450 line is apparent. The error bars are
Fig. 5. Signal analysis system with the SAW spectrum analyzer anddigital integrator. The amplifiers and attenuators are required to
keep the signal within the dynamic range of the components.
filters have been employed as described in the followingmeasurements. The experiments thus consist essen-tially of measuring the noise in the observed spectra atgreatest sensitivity and then introducing calibratedattenuators for accurate measurement of the muchstronger signal.
A. Measurements with the 0-25-MHz SAW SpectrumAnalyzer
This signal processing system has been outlined inprevious publications1 5 16; analysis of the Doppler signalis performed with a 25-MHz bandwidth surface acousticwave spectrum analyzer with 70-kHz resolution builtby Microwave and Electronic Systems (Racal) Ltd.After 4-bit digitization individual spectra from each25-Msec sample of analyzed signal are accumulated ina 30-MHz integrator (Cambridge Consultants, Ltd.)employing 833 channels each of 30-kHz width. TheSAW spectrum analyzer may be set to operate in one offour switchable overlapping ranges up to a maximumfrequency of 62.5 MHz. The equipment is operated sothat the digitized record of a single sample is added tothe sum of previous records until a final integrated sumis recorded. The complete signal processing system isshown in Fig. 5. Appropriate attenuation is employedto guard against exceeding the dynamic range of thecomponents and to ensure proper statistical averagingof recorded data. In this equipment as a matter ofconvenience and for compressing effectively the dy-namic range we have chosen to take the modulus ratherthan the square modulus of the SAW signal. The in-tegrated spectrum thus has the form of the square rootof the power spectrum and is thus referred to as a cur-rent or voltage spectrum rather than a power spectrum;the directly observed SNR is accordingly a SNRvoitagerather than SNRpower. This distinction has to becarefully considered in the eventual comparison ofcalculated and observed values.
Measurements were carried out with the scatteringsurface on a large plate mounted on the side of a vehicle.Observation of the heterodyne signal showed typicalcoherence times on the -1.5-MHz Doppler shiftedsignal of -1 msec so that the corresponding signalbandwidth of -1 kHz was much less than the instru-mental bandwidth of -70 kHz. This justifies the choiceof the sampling time of 25 gsec for the inverse band-
width 1/B in Eq. (1). The settings of the attenuatorsshown in Fig. 5 were selected to establish linear regionsof operation in relation to the observed signal and noisecharacteristics. Typically, for noise measurements,attenuations of 20 dB at A and 6-16 dB at B were re-quired. For measurement of signal an additional 40-50dB was required at A. Establishing the values of signaland noise may be carried out in such digital systems ina number of different but equivalent ways: the contentof the integrator can be output to a computer for de-tailed numerical analysis or can be displayed on an os-cilloscope for rapid visual appreciation. In addition,the store content of a particular channel after N inte-grations can be expressed in a number of ways:
(1) The actual number 2N=1 n, where n is thenumber of levels tripped or recorded in the ithsample;
(2) as a mean level n = 1/N VN=1 ni; and(3) as a percentage signal y = (100 ii/15)%, corre-
sponding to the 15 digitization levels.Options (2) and (3) have been found to be a conve-
nient choice and have been widely used in this work.Measurement of the signal was carried out with an
adequate number of integrations to provide a good es-timate. Typically N = 1600 was used. In the inte-grator the Doppler signal is distributed over the mchannels of the effective instrumental profile. Thus thecomplete measured signal is given by
Sm = L niXm
(11)
where the summation extends over the m channels(usually 3 or 4) of the distributed signal. If the mea-sured signal is reduced by the attenuation factor A dB,the full signal that would be measured without atten-uation in a system of adequately large dynamic rangeis given by
eyvolt,1 = (A/10)1/2 Sm, (12)
where the index 1/2 is due to measurement of the voltagerather than power spectrum. The measured noise
2variance ameasured in a digital system is derived from anumber of different sources and may be written gen-erally as
2nesd| = Uf N2 l 2et| 1 + I dig| 1]I arneasured N -Uynch N 1U (13)
where the suffices N and 1 indicate measurements overN and 1 integrations, respectively. Synchronous noiseon the digitization levels contributes the term osynch,and det is the genuine quantum-derived hererodynenoise arising at the detector. The final term adig is ef-fectively the digitization noise introduced by the ana-log-to-digital converter28; the effect of digitization canbe shown to be equivalent to a pure white noise signalwith an rms level equal to 12-1/2 of the digitization in-terval. Thus 0dig = 0.29 of an interval between levelsand is usually small compared with 0 det. Examinationof Eq. (13) shows that at very large numbers of inte-grations N, the measured noise is dominated by thesynchronous component. The greatest system sensi-tivity is achieved by arranging for the heterodyne con-
tribution a 2et to be as large as possible-by amplifyingthe noise distribution to cover a large number of digi-tization levels. If, however, this is made too big, thelarger fluctuations are limited at the top and clipped bythe 4-bit digitizer; the noise would in consequence beunderestimated. A more detailed analysis of signal-to-noise optimization will be presented in a later article.For the moment we note that examination of the noisecharacteristics in our system establishes that Eq. (13)is obeyed to a good approximation.
In the experiments the noise was typically measuredfor sixteen integrations so that little correction ofUrmeasured was required to derive 0
-det. This noise figureis appropriate to a 1-frequency channel, but as indicatedthe full signal is integrated over m channels; in conse-quence the appropriate noise for comparison mustsimilarly be integrated over m channels and is givenby
J.volt, = m1/2 det. (14)
The resultant SNRvoit,j is simply derived from Eqs. (12)and (14). Finally the observed power signal-to-noiseratio SNRpowerl may be found, since, for a large sig-nal-to-noise ratio only, this is equal to a good approxi-mation to the square of SNRvolt,1 . Cross checking withwell-calibrated attenuators has established the validityof this approximation.
The procedure is illustrated by a numerical examplethat compares with the value of SNRcalc derived in theprevious section. The measured root variance of thenoise for sixteen integrations was found to be 0.61 dig-itization intervals (d.i.). On a single sample (one inte-gration) this would give 0
det = 2.43 d.i. The peak signalwith 40 dB of attenuation amounted to 5.0 d.i. in thelargest channel and after integration over m = 4 chan-nels amounted to 15.2 d.i. From Eq. (12) the value of,Vvolt,l is thus 1.52 X 103 d.i. From Eq. (14), iVvoit,l is 4.86d.i. to give a measured SNRvoltl of 3.13 X 102. Fromthis a final [SNRpower,1]observed of 9.78 X 104 is found tocompare with the calculated value, in the previoussection, of 10.8 X 104. These results are plotted withothers in Fig. 4. The agreement is altogether very sat-isfactory, especially in view of the large number of termsentering the expressions.
B. Measurement with the 0-6.3-MHz SAW SpectrumAnalyzer
This equipment has been previously described withapplication to digital signals.17 It has also been usedextensively for ground-based CO2 laser velocimetry.The SAW analyzer takes a 25-Ausec length of input signalevery 51 Asec and outputs the spectral analysis of thissample as an amplitude against time (or related fre-quency) record. The spectral amplitude is digitized to4-bit accuracy by the ADC and recorded in an integratorwhich consists of a 10-MHz 16-bit wide shift register.Each channel in the integrator corresponds to 16 kHz,and the resolving width of the system is -60 kHz.
Experiments were conducted in a manner similar tothat for the broadband SAW analyzer. With the slowerintegrator synchronous noise was found to be very small,
L
A B
1 MHz 2 1.2 MHz 1.6
Fig. 6. Typical spectra recorded on the oscilloscope with the 6.3-MHz bandwidth SAW spectrum analyzer: (A) noise and (B) signalwith additional 50 dB of attenuation. The noise amplitude and
variance are readily estimated by expanding the scale of (A).
and integration was typically performed over 6000samples to provide full averaging. Measured (voltage)spectra are shown, directly traced from the oscilloscope,in Fig. 6. To keep the signal within the dynamic rangeof the ADC and comparable with the recording of thenoise, the Doppler shifted signal had to be recorded with50 dB of (power) attenuation on the input to the SAWanalyzer. In this case the signal and noise could easilybe measured off the oscilloscope display. The rootvariance of the noise was evaluated by establishing theamplitude distribution of the noise spectrum in the vi-cinity of the Doppler signal frequency. In the exampleshown in Fig. 6, the noise spread at -1.4 MHz is 0.036units; the root variance (1/6 of spread) is thus -0.006units; the peak signal was 0.36 units (in a single chan-nel). Processing these data in the manner indicated inthe previous section gave a SNRobs of 13.0 X 104 to becompared with a calculated value in this case of SNRcalcof 12.1 X 104. Other comparison data are given in Fig.4 and show very satisfactory agreement.
C. Measurements with the Scanning SpectrumAnalyzer
Electrical measurements on the signal and noise wereperformed using a scanned IF spectrum analyzer Tek-tronix type 7L5 with L3 module. The scattering targetused in this experiment was a rotating expanded-poly-styrene cylinder of 240-mm diam. Typical spectra ofsignal and noise are shown in Fig. 7. Since the signalis distributed in frequency due to finite beam size on thecylindrical target, we chose to measure SNR at the peakof the signal and allow for finite signal bandwidth B, asfollows.
If P(co)s is the spectral power density of signal, eq. (1)becomes
SNRpower(w) = ?7eff[P(w))sBnoise]hwxBnoise
since P, = P(o)sBnoise for Bnoise << B,peak, P(wX max = Ps/Bs so that
At the signal
SNRpower max = f7effPs/hwBs, (15)
and the appropriate value of B to include in Eq. (1) issimply the signal bandwidth, provided it is much greaterthan the noise bandwidth that was selected for the an-alyzer. This condition was satisfied since B wasmeasured as several tens of kilohertz for the two dif-ferent configurations used, and Bnoise was 750 Hz.
Provided that the noise bandwidth of the analyzer issuitably small and satisfies the above inequality, itsprecise value need not be known.
Three sets of experiments were performed; in the firstset, the incidence angle on the rotating cylinder was 600at a range of 120 m from the instrument with a Dopplershift of 2.00 MHz. The signal bandwidth was found tobe 139 kHz, and the diffuse reflectance was measuredas 0.2 at this incidence angle. In the second and thirdsets, these parameters were an angle of 13.70, range of20.0 m, Doppler shift of 1.00 MHz, bandwidth of 52.2kHz, and diffuse reflectance of 0.62, respectively. Thepower on target P0 could be varied by interposinghigh-optical quality calibrated attenuators in the outputbeam; further control over signal level was achieved byplacing attenuators directly before the detector, whichalso served to control the local oscillator level, and hencethe factor M [Eq. (8)]. These attenuators were re-stricted to a factor of -100 to avoid multiplicative error.The other parameters needed to calculate the SNR,namely, detector quantum efficiency 7), receiver aper-ture R, and the three terms Khet, Kspec, and Kopt re-mained unchanged from the experiments of the previ-ous section.
We estimate SNRpower from Eq. (7); i.e., the meannoise power is set equal to the variance of the noisepower. This mean was measured with an accuracy ofa few percent by digital postdetection filtering with aneffective bandwidth of 2.5 Hz. The signal P(w) was
Fig. 7. Recordings with the scanning spectrum analyzer: (A) at-tenuated signal from the rotating expanded polystyrene wheel; (B)noise background with (upper) and without (lower) local oscillator.
Table 1. Comparison of Observed and Calculated SNRs Obtained with aScanned IF Spectrum Analyzer
SNRcaic(dB) 54.7 + 2 58.4 1 64.7 I 1SNRobS(dB) 55 1 58.3 1 64 I 1
similarly obtained; input mixer linearity was checkedat all levels by use of an input buffer, and analyzer lin-earity was checked over the dynamic range of 60 dB byuse of accurate attenuation of a signal source.
The results of the three sets of experiments aresummarized in Table I. Observed and calculated valueof the SNR agree to considerably better than 1 dB.These data are also shown plotted in Fig. 4.
V. Absolute Backscattering in the Atmosphere
In the atmosphere the scatterers are extended inspace, and the signal-to-noise expression is somewhatdifferent, e.g., Refs. 29-31, to that derived from Eqs. (1)and (9). In this case the effective detected power isgiven by
In this expression /3(7r) is now the backscattering coef-ficient per steradian per unit length of illuminated re-gion, X is the wavelength, and F is the range to focus.The second term in brackets is not greatly differentfrom 7r/2, and the most appropriate choice of D is theeffective Gaussian beam diameter of the laser beam atthe transmitter. Truncation effects31 do not affect thisexpression significantly for the parameters chosen.
Preliminary accounts of our airborne laser radarequipment for backscattering and true airspeed mea-surement have been given.15 Nearly two years of flightexperience have now been accumulated, and good re-laibility is demonstrated. In one period of eightmonths, terminated only by a sudden laser failure, theoptical equipment was untouched in its housing.Regular performance checks throughout showed no lossof sensitivity with an average of one flight per week.Since the latest installation the equipment has per-formed faultlessly without further adjustment for tenmonths, including an extended flight program at theJoint Airport Weather Studies (JAWS) project in Col-orado and, more recently, trials in Gibraltar. A re-cording of backscattering with altitude made at theJAWS project is shown in Fig. 8. The backscatteringcoefficient has been derived from Eqs. (1) and (16). Foroperational use and low scattering the highest sensi-tivity is obtained with the following typical parameters:transmitted power Po = 1.5 W; range to focus F = 100m; effective quantum efficiency 3leff = 0.31; and numberof integrations N = 104. Typically, backscatter mea-surements (i.e., individual accumulations of 104 inte-grations lasting -0.6 sec) are made at 3-sec intervals to
Fig. 8. Atmospheric backscattering coefficient :l (r) sr- 1 m- 1 andstatic air temperature SAT vs indicated height in feet and kilometers(ASL). These recordings were made during flight 772 from JeffersonCounty Airport, Colo. on the morning of 2 July 1982 in clear summerconditions with visibility at the surface >130 km (80 miles). Thestrong return c at 10.4 km (34,000 ft) later in the day became a visiblelayer of cirrus cloud. It is very noticeable that the signal increasestoward the top of many small temperature inversion layers and de-creases suddenly as the aircraft passes through the top of the layer.
allow for recording onto tape of the full integrator store.Detailed calculation shows that 3(7r) is given by
f(7r) = [44.6 108]-1 X (SNR)power,,1
where (SNR)power,1 is evaluated in a single integration,after allowance for the synchronous noise contributionobserved in the measurement over a large number ofintegrations. The minimum observable value for(SNR)poweri is -0.1 [from a minimum (SNR)voltl04 of-5] so that the minimum backscatter that can presentlybe detected is 0(0)minimum 2 X 10-11 sr- 1 m- 1 .
As is evident from the recording (Fig. 8) the back-scatter was well above this level throughout the flight.Considerable fine scale structure, with rapid changesof scattering, is clearly shown. The strong level ofscattering above -12 km (40,000 ft) in this recording isalso evident. A detailed account of the evaluation andinterpretation of these atmospheric measurements,including results at the JAWS trial, will be made in laterpublications. It is worth noting that preliminarycomparison with simultaneous measurements made bythe vertically pointing pulsed CO2 lidar of the WavePropagation Laboratory shows very good agreement.
VI. Conclusions
With accurate allowance for the multiplicity of terms,particularly optical transmission, detector efficiency,beam propagation, and speckle, good agreement ofcalculated and observed SNR has been demon-strated.
This establishes that the theoretical treatment is ona sound basis and provides a good guide to the assess-ment of system performance provided realistic evalu-ations are made of the successive terms involved.Second, investigations show that our optical and signalprocessing is working at optimum efficiency; it is hopedthat the detailed description provided here will proveinstructive, particularly for SAW spectral analysis fol-lowed by digital integration. Finally, the study allowsan absolute calibration of backscatter without recourseto allowance for untabulated correction factors.
We are greatly indebted for discussion and assistancein this work with many colleagues at RSRE, particularlyR. Callan and A. Parkin. We thank G. Parry for accessto his speckle calculations. It is a great pleasure to ac-knowledge the close and effective collaboration on thisprogram over several years with our colleagues JohnCannel and Alan Woodfield at RAE Bedford.
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9. R. L. Schwiesow and R. E. Cupp, Appl. Opt. 19, 3168 (1980).10. M. J. Post, R. A. Richter, R. M. Hardesty, T. R. Lawrence, and
F. F. Hall, Proc. Soc. Photo-Opt. Instrum. Eng. 300, 60 (1981).11. It may be noted that to obtain surfaces of reproducible reflec-
tance, the scatterer should preferably be a Lambertian diffusereflector. Under this condition, differences in small scale surfacedetail do not cause a variation in reflectance. In a study at RSRE,measurements have been performed on a wide number of mate-rials, using a 10-Lm CO2 laser, direct detection of backscatteredradiation with pyroelectric detector, and phase sensitive detectionfor good SNR. Fresnel reflection from a KBr prism was used toprovide an accurate system calibration. Of the many surfacesexamined, few were found to be Lambertian; flame sprayed alu-minum was to a very good approximation, and hot-wire cut ex-panded polystyrene to a fair approximation. The measuredvalues of e(0) have strictly been obtained for unpolarized incidentand scattered radiation, whereas in this work circularly polarizedbeams are used. Studies with our equipment have shown <1%depolarization in the scattered beam for circularly polarized in-cident light. It may, however, be noted that if the quite highvalues of E(O) that we have adopted were too large by up to 20%the minor residual discrepancy found in this work for the observedand calculated values of SNR would be removed. However, theactual agreement of these values is well within experimentalerror.
12. R. L. Schwiesow, R. E. Cupp, V. E. Derr, E. W. Barrett, and R.F. Pueschel, J. Appl. Meteorol. 20, 184 (1981).
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15. R. Callan, J. Cannell, R. Foord, R. Jones, J. M. Vaughan, D. V.Willetts, and A. Woodfield, in Proceedings, Fifth National
