laser radar detection statistics a comparison of coherent and direct-detection receivers

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Laser Radar Detection Statistics: A Comparison of Coherent and Direct Detection Receivers Philip Gatt and Sammy W. Henderson Coherent Technologies, Inc. ABSTRACT Detection statistics for a coherent laser radar are substantially different from those of a direct detection laser radar. Direct detection ladar detection statistics vary depending upon the detection mode (i.e., photon-counting vs. continuous direct detection). Speckle noise also impacts the detection statistics. For a single-pixel single-frequency single-polarization coherent detection transceiver, speckle noise can only be suppressed through temporal averaging. Some degree of speckle averaging can also be achieved in coherent detection systems by using a multiple frequencies or dual polarizations. In addition to these, a direct detection receiver can exploit spatial diversity to suppress the effects of speckle. This paper develops theory useful for describing the performance of these three receiver architectures against diffuse and glint targets and provides example performance comparisons. We show that a photon-counting direct detection receiver can, in principle, provide superior performance, however practical limitations of current detector technology particularly in the near IR spectral region reduces the performance margin and for many applications a coherent detection receiver provides superior performance. Keywords: Laser radar detection statistics 1. Introduction There are many applications, which require precision measurements of laser radar signal intensity. Some of these include, range detection for target detection and ranging, differential absorption (DIAL) lidar for trace gas detection, differential scattering (DISC) for particle size discrimination, and polarametric imaging for discrimination of manmade and natural targets. In this paper we develop intensity detection statistics theory for both coherent and direct detection receivers and compare and contrast their performance in terms of the minimum number of accumulated photoelectrons required to achieve a given performance level. We consider both diffuse and glint targets and performance in terms of the number of temporal averages. Targets with glint characteristics are often encountered in the mid to far IR. Glint targets represent the diffuse target signatures under high levels of speckle diversity (spatial, temporal, polarization and/or frequency). The effect of spatial, frequency and polarization speckle diversity is not specifically addressed by this work. However, a direct analogy between these types of diversity and the number of independent pulses averaged (temporal diversity) exists and is discussed in Section 4. This work ignores the impact of refractive turbulence, which can produces slightly higher intensity fluctuations than those produced by a diffuse target. The literature contains numerous publications covering focused aspects of this generalized topic. For example, Goodman's work published by Bachman1 derives the intensity statistics for a heterodyne and photon-counting laser radar sensors for diffuse and glint targets with single pulse averaging. In another work2, Goodman takes into account the effect of aperture averaging of speckles for a photon-counting direct detection receiver. Youmans3 has published works on the performance of an avalanche photodiode direct detection receiver with single pulse averaging assuming a diffuse target and aperture averaging of speckle. Many others have contributed to the field but none provide a complete comparison of the three receiver (coherent, continuous direct and photon-counting direct detection) architectures described herein. This work provides, for the first time in the published literature, a unified presentation of the material, covering all three receiver types for both diffuse and glint targets as a function of the number of temporal averages. Relative performance is compared in Section 3 and conclusions are drawn in Section 4. 2. Receiver Theory In this section we develop the theory which describes the performance of the three intensity receiver architectures. The performance is characterized by the probability of detection as a function of the signal strength, the false alarm probability, the number of search bins, and the level of averaging or accumulation. 2.1. Coherent Detection Intensity Statistics A block diagram of the coherent detection intensity receiver is shown in Figure 1. Laser Radar Technology and Applications VI, Gary W. Kamerman, Editor, Proceedings of SPIE Vol. 4377 (2001) © 2001 SPIE · 0277-786X/01/$15.00 251

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Laser Radar Detection Statistics a Comparison of Coherent and Direct-Detection Receivers

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  • Laser Radar Detection Statistics:A Comparison of Coherent and Direct Detection Receivers

    Philip Gatt and Sammy W. HendersonCoherent Technologies, Inc.

    ABSTRACTDetection statistics for a coherent laser radar are substantially different from those of a direct detection laser radar. Directdetection ladar detection statistics vary depending upon the detection mode (i.e., photon-counting vs. continuous directdetection). Speckle noise also impacts the detection statistics. For a single-pixel single-frequency single-polarizationcoherent detection transceiver, speckle noise can only be suppressed through temporal averaging. Some degree of speckleaveraging can also be achieved in coherent detection systems by using a multiple frequencies or dual polarizations. Inaddition to these, a direct detection receiver can exploit spatial diversity to suppress the effects of speckle. This paperdevelops theory useful for describing the performance of these three receiver architectures against diffuse and glint targetsand provides example performance comparisons. We show that a photon-counting direct detection receiver can, in principle,provide superior performance, however practical limitations of current detector technology particularly in the near IR spectralregion reduces the performance margin and for many applications a coherent detection receiver provides superiorperformance.Keywords: Laser radar detection statistics

    1. IntroductionThere are many applications, which require precision measurements of laser radar signal intensity. Some of these include,range detection for target detection and ranging, differential absorption (DIAL) lidar for trace gas detection, differentialscattering (DISC) for particle size discrimination, and polarametric imaging for discrimination of manmade and naturaltargets. In this paper we develop intensity detection statistics theory for both coherent and direct detection receivers andcompare and contrast their performance in terms of the minimum number of accumulated photoelectrons required to achievea given performance level. We consider both diffuse and glint targets and performance in terms of the number of temporalaverages. Targets with glint characteristics are often encountered in the mid to far IR. Glint targets represent the diffusetarget signatures under high levels of speckle diversity (spatial, temporal, polarization and/or frequency). The effect ofspatial, frequency and polarization speckle diversity is not specifically addressed by this work. However, a direct analogybetween these types of diversity and the number of independent pulses averaged (temporal diversity) exists and is discussedin Section 4. This work ignores the impact of refractive turbulence, which can produces slightly higher intensity fluctuationsthan those produced by a diffuse target.The literature contains numerous publications covering focused aspects of this generalized topic. For example, Goodman'swork published by Bachman1 derives the intensity statistics for a heterodyne and photon-counting laser radar sensors fordiffuse and glint targets with single pulse averaging. In another work2, Goodman takes into account the effect of apertureaveraging of speckles for a photon-counting direct detection receiver. Youmans3 has published works on the performance ofan avalanche photodiode direct detection receiver with single pulse averaging assuming a diffuse target and apertureaveraging of speckle. Many others have contributed to the field but none provide a complete comparison of the three receiver(coherent, continuous direct and photon-counting direct detection) architectures described herein. This work provides, for thefirst time in the published literature, a unified presentation of the material, covering all three receiver types for both diffuseand glint targets as a function of the number of temporal averages. Relative performance is compared in Section 3 andconclusions are drawn in Section 4.

    2. Receiver TheoryIn this section we develop the theory which describes the performance of the three intensity receiver architectures. Theperformance is characterized by the probability of detection as a function of the signal strength, the false alarm probability,the number of search bins, and the level of averaging or accumulation.

    2.1. Coherent Detection Intensity StatisticsA block diagram of the coherent detection intensity receiver is shown in Figure 1.

    Laser Radar Technology and Applications VI, Gary W. Kamerman, Editor,Proceedings of SPIE Vol. 4377 (2001) 2001 SPIE 0277-786X/01/$15.00 251

  • PSis

    +ii;r:o1vs+ vn

    Figure 1 . Coherent detection intensity processor block diagram. The detected photocurrent is narrowband filtered using amatched filter bank (or FFT processor) then passed through a square modulus operator to obtain a signal proportional to theintensity of the return field. This signal is then averaged and compared to a prescribed threshold level. The threshold level isadjusted to achieve a desired probability offalse alarm.

    In this receiver, the signal return strength is characterized by the received optical power Ps. This return signal is mixed witha continuous wave optical local oscillator beam with power PLO. A non-multiplying (i.e., p-i-n) detector senses the intensityof the combined beams. The detector photocurrent consists of four terms.

    1d (t) = RPLO +RP5 (t) +2RJ7PLoP5 (t) cos(cnt + 0(t))+ i, (t). (1)In this expression, ci is the carrier frequency, 0(t) is the signal phase and R is the detector responsivity(R = iiqq/hV), 11q is the detector quantum efficiency, q is the electron charge, h is Planck's constant and V is the opticalfrequency. The heterodyne mixing efficiency, is the efficiency with which the local oscillator and signal fields interfere onthe detector. The first two terms are direct detection terms. They are baseband currents, which are proportional to the LOand signal power or intensity. The third term is an intermediate frequency (IF) photocurrent, corresponding to the mixing ofthe local oscillator field and the signal field. This term is the coherent detection signal current, which contains the targetamplitude and phase information. The last term corresponds to the noise current. In a direct detection receiver the signalintensity is linearly related to the baseband photocurrent (2nd term in Eq. 1). In a coherent detection receiver, the signalintensity is related to the square of the IF photocurrent envelope.

    The coherent detection photocurrent is obtained by AC coupling the total current. This signal contains the IF signalphotocurrent and a noise photocurrent

    i(t) = i (t) + n (t) (2)Where the signal current is given by

    i(t) = 2R.J FLOPS (t) cos(ot + 0(t)). (3)In a properly designed coherent detection receiver the dominant noise source is the shot-noise generated by the local-oscillator beam. This noise is well modeled as a zero-mean Gaussian random process whose variance is given by

    Y=c(ifl(t))=2qRPLOB. (4)

    Where, is the expected value operator andB is the receiver bandwidth. One of the more important performance metricsof a coherent detection receiver is the carrier-to-noise power ratio (CNR) given by.

    CNR = (is(t)2)/((t)2) = Y'flqPS /h'B. (5)For a matched filter receiver PIB is the energy of the return signal, E, and E5/hv is the number of detector-plane signalphotons. Therefore, the matched filter CNR is the number coherently detected signal photoelectrons (K =y11qEsIhV). If wedefine the CNR as the number of signal photoelectrons divided by the effective number of noise electrons, we see that theeffective number of noise electrons in an integration time ('c =1/B) is unity.

    Proc. SPIE Vol. 4377252

  • 2.1.1. Noise Model and Probability of False AlarmThe coherent detection photocurrent, i(t) is a narrowband random process. As such, it is can be modeled by its analytic signalrepresentation i(t) (i.e. a complex photocurrent).

    i(t) = (t) +i,. (t))exp(jot) . (6)Where i is and i are the complex amplitudes ofthe signal and noise photocurrents andj= 'NJ-i . In this formalism, the noisecan be written as a complex sum of inphase and quadrature noise components (n, and nq), each a zero mean Gaussian randomprocess with variance (i.e., power) equal to

    in (t) = fl1 (t) + Jflq (t). (7)Therefore, the joint statistics of the noise currents are circular complex Gaussian

    1 1 2 2 22 exp(z +q )/2). (8)2it

    Wherep(i,q) is the joint probability density function (PDF) of the two noise components. For a coherent detection receiver,it is the square magnitude of the photocurrent that is proportional to the optical field intensity, (J = I2). Therefore theintensity noise PDF is an exponential density function4.

    (9)

    Where b = 22 is the total noise power. When averaging is included in the receiver, the noise statistics change fromexponential to gamma (or Chi22N) provided the averaged signals are independent5. For averaging as defined in the blockdiagram of Figure 1 (i.e., sum/N) the appropriate form of the Gamma density function is

    p (I) =F(N) '(NIb)N 1N -l exp( NI/b) (10)Where N is the number of signals averaged and F(z) is the gamma function. The mean and variance of the gamma densityfunction, as defined above, are

    (I)_b and v[I]_((I_(I))2)b2/N. (11)The probability of false alarm depends on the threshold level and the noise statistics. For Gamma distributed noise, it isgiven by

    PFA = p (I)dI =]T(N,NIth /2)/F(N). (12)Where 'th is the threshold level, set to achieve a specified PFA, and F(z,x) is the incomplete gamma function defined by

    F(a,x) = je_tta_ldt. (13)For a single pulse averaging the false alarm probability reduces to PFA = exp(-Ih/22).2.1.2. Target Detection StatisticsThe probability of detection, PD, is the probability that the detected signal (signal plus noise) exceeds the threshold leveldefined by the specified PFA. The primary factors, which impact the probability of detection, are the PFA, the CNR, thenumber of independent signals averaged and the target model. In this section we develop expressions for the PD for a fullydiffuse (i.e., rough) and perfectly specular (i.e., smooth or glint) target.Diffuse TargetFor a diffuse target, the complex signal photocurrent is, because of laser speckle, a circular complex Gaussian process5.Therefore, the joint PDF of the signal is given by Eq. 8 where the variance of each quadrature component is Y2 and the totalvariance (i.e. signal power) is Thus the signal plus noise is also a circular complex Gaussian random process. Thevariance of each component is the sum of the noise and signal variances. Hence the intensity PDF of the signal plus noise,

    Proc. SPIE Vol. 4377 253

  • Psn('), is a gamma distribution (c.f. Eq. 10) with a mean, b = (2r2 + 2o,2) or b = 2o,2 (CNR +1) since theCNR = 22/2o2.For an intensity threshold detector, the probability of detection is defined as

    PD=J7pSfl(I)dI (14)The detection probability is, from Eq. 12, given by

    PD = F(N,NIth I2i(1+ CNR))/F(N). (15)For single pulse averaging the probability of detection reduces PD = exp(-Ih/2fl2(1+CNR)) and the following simple relationapplies

    PD=PFA1. (16)Glint TargetFor glint targets, the signal photocurrent complex amplitude, A, is assumed to be a constant and is arbitrarily chosen to havezero phase. Therefore, the photocurrent complex amplitude is described by

    i(t)A+fl,(t)+Jflq(t). (17)Where the quadrature noise components are the same as they were for the diffuse target case and thus the statistics of theintensity noise and the probability of false alarm remain unchanged. However, the signal plus noise statistics are different.For single pulse averaging, a Rician-Square or noncentral Chi22 density function6 applies.

    p5(I) =(1/2)exp(_(A2 +I)I2)Io(A-/7R). (18)Where as before, 2 is the noise power of each quadrature component. The density function of the averaged intensity isknown to follow the noncentral chi2N2 density function6. The correct form of this density function, which applies to anaverage rather than accumulation, is given by

    = (2& )(I/ A2)_2 exp(_N(A2 + I)/2 )IN_l(NA/) (19)Its mean and variance are given by

    (i)= A2 +2 2(CNR+1) and var[I]= 4o(A2 +)/N. (20)Since, for a constant amplitude signal, CNR = A2/2o2. This form of the noncentral chi2N2 distribution is directly related tothe form often found in the literature (ch2)

    p(x)=(1/2)(xIX)_24exp(_(x+)I2))Iv,2_l() (21)Where v is the number of degrees of freedom and X is the noncentral parameter. It can be shown that these two distributionsare the same with the following substitutions, v = 2N, ?= A2N/o2 and x = I NI2. The form presented in Eq. 19 isintuitively advantageous since it provides direct access to the constituents to the random process (i.e., signal amplitude andnoise power and the number of averages). The probability of detection is, from Eq. 14, given by

    PD = f (x)dx = J(1/2)(x/ )(v_2) '4exp ((x + X) I2))I ,2_1()dx = QN (,) (22)Xth Xth

    Where QN(a,13) is the generalized Marcum Q-function7'8 defined as

    QN(0c,1 Jz(zi a)/V_l exp(_(z2 +a2)/2)IN_l(otz)dz (23)

    Proc. SPIE Vol. 4377254

  • Eq 23 follows from Eq. 22 with the substitutions v =2N,x = z2, 7t = a2 and Xth = P2 Returning back to first form of thenoncentral chi2N2 distribution (i.e., Eq. 19), the probability ofdetection is given by

    PD= QM(INA2 2 ,INIth n) QM(I2NCNRdNIth n) (24)The Marcum Q-function can be evaluated using the series expansion published by Robertson9.

    2.2. Continuous Direct Detection Intensity StatisticsAblock diagram ofthe continuous direct detection intensity receiver is shown inFigure 2.

    PS linear APD I Ndetector 7 j::1

    '

    Figure 2. Continuous direct detection intensity processor block diagram. The detected photocurrent is lowpass filtered to removeexcessive noise bandwidth, then averaged, and compared to a threshold level set to produce a specified PFA.

    The adjective "continuous" is used to distinguish this receiver from a "photon-counting" direct detection receiver. In thisreceiver, the detector is either a p-i-n or a linear-gain detector like a conventional avalanche photo-diode with photoelectrongain M. The signal photocurrent is therefore given by

    i(t)=RP(t)f. (25)This photocurrent is a baseband random process, which is directly proportional to the optical field intensity. In this type ofreceiver the dominant noise term is typically preamplifier thermal noise (referred to its input as an equivalent noisephotocurrent i(t)). This thermal noise is modeled as a zero-mean Gaussian random process whose variance is given by

    =NB. (26)Where N0 is the "white" noise spectral density (Amp 2IHz). For high levels of signal power, (i.e. short range) signal shotnoise can dominate preamplifier thermal noise, however for this to be the case the number of signal photoelectrons must belarge and the noise is still well approximated by a zero mean Gaussian random process, with a larger N0.

    Continuous direct detection receiver performance is characterized by its signal-to-noise ratio (SNR), which is defined as

    SNR = (i5(t))2/(i(t)2 )= (RP(t)M)2/N0B. (27)The notation SNR is used here rather than CNR because the signal photocurrent in direct detection receiver is a basebandsignal and there is no carrier frequencyper Se. The numerator of the SNR is defined as the square of the mean signal, unlikethe coherent detection CNR whose numerator is defined as the mean of the squared signal, because for the direct detectionreceiver the signal is not a zero mean process.

    2.2.1. Noise Model and Probability of False AlarmThe PDF of the averaged noise is Gaussian because the sum of a series of Gaussian random variables remains Gaussian.However the noise power (variance) is reduced by a factor of N.

    (I) = 1 exp(12N/ (28)J2ico/N

    For the general case of a non-zero mean Gaussian random process, the probability that x > x0 is given by

    Pr(x > x0)= exp((xu)2 /22) dx =ec((xo _u)I)/2 (29)J2ic

    Proc. SPIE Vol. 4377 255

  • Where u is the mean and is the standard deviation and erfc is the complementary error function. Therefore, for zero meanGaussian noise, the probability of a false alarm is given by

    PFA= Jp(I)dI =erfc(IthTN/2Y )12. (30)

    2.2.2. Detection StatisticsDiffuse TargetFor the case of single pulse averaging and unit speckle diversity, the density function of the signal intensity follows anexponential density function with mean =b (c.f. Eq 9). Hence, the signal power is b2, the SNR is b2/o2 and 1> =iSNR. The impact of higher levels of speckle diversity will be addressed in Section 4. The corresponding densityfunction for the signal plus noise can be evaluated from the convolution integral'0. That is, the signal plus noise PDF is theconvolution the Gaussian noise PDF with the exponential signal PDF. The resultant PDF can be shown to be given by

    Psn (I) =exp(( 2b1) I2b2)erfc[( bI)/-/i1x]/2b. (31)This expression reduces to the exponential distribution when the noise variance is zero. The averaged intensity, againassuming unit speckle diversity, is Gamma distributed (c.f., Eq 10) with mean b and variance of b2/N. The density functionof the averaged signal plus noise can be evaluated from the convolution integral. This results in the following expression

    (I) = 2_+12(b/ N)_N exp(N12 I2 )(N /N)"32N 1 l(RY J_2

    Cj)lFl 2'2'2t b J . (32)- 2(2N_l)/2(2

    _bI)) lFl[;;![ ]2 }(br(N))Where, ,F,(a;b;x) is the Hypergeometric function. From a numerical standpoint, this expression is unstable for large valuesof the noise variance o,2 or small values of the mean signal current or equivalently for SNRs

  • PD=erfc((Ith SNR)JN/2))/2.(35)

    Therefore for a glint target, the impact ofaveraging is equivalent to an increase in the SNR by a factor ofN, since for a fixedPFA, 'th scales by 1/\JN. That is to say, the PD obtained for a given PFA, SNR and N is the identical to the single pulseaveraging PD when the SNR is increased by a factor of N. This conclusion is demonstrated by the performance curvespresented in Section 3.

    2.3. Photon-counting Direct Detection Intensity StatisticsThe structure of the photon-counting direct detection intensity receiver is shown in Figure 3.

    n(t) _________ _________

    Figure 3. Photon-counting direct detection intensity processor block diagram. The detected photocurrent triggers a high speedgated counter or alternatively an accumulated charge is sampled by the gated counter and converted to a discrete count level. Thecounts are then accumulated over N counting intervals and compared to a prescribed threshold level.

    An extremely high-gain detector capable of detecting single photon events is assumed. This detector is followed by a gatedcounter, which reports the number of events (counts) generated during a counting interval (range gate). Counts fromsuccessive lidar pulse returns are accumulated. The accumulated count is then compared to a threshold level, adjusted toproduce a specified probability of false alarm.For a photon-counting direct detection receiver, the number of signal counts is given by

    ksTIqEs/hV. (36)Where E is the signal energy or integrated signal power, which is directly proportional to the optical field intensity. Addedto the signal are noise counts, k. The additive noise counts arise from several sources, which include background light anddark current. The mean number of additive noise counts are not of direct concern because it can be characterized andsubtracted from the number of detected counts. It is the fluctuation of the noise and signal counts that results in receivernoise. This noise is called shot-noise and is assumed to be Poisson distributed. An interesting property of Poisson randomvariable is that its variance is equal to its mean. Therefore, we define the SNR for a photon-counting direct detection receiveras

    SNR = K /(K + Ku). (37)Where K and K are the mean number of signal and dark counts accumulated in the counting interval. When K =0, thereceiver is signal photon-limited and, like a coherent detection receiver, the SNR is proportional to the number of signalphotoelectrons (SNR Ks). When K >> K, the receiver is dominated by dark noise and, like the continuous direct detectionreceiver, the SNR is proportional to the signal power squared (SNR K2/K).2.3.1. Noise Model and Probability of False AlarmFor a photon-counting receiver the noise signal (i.e. the number of noise photoelectron counts in an integration time) is adiscrete random variable. The distribution that describes the number of photoelectron counts k, in a given time interval for asignal with a stationary mean, is the Poisson Distribution' .

    p,. (k) = K exp(K )/ k! (38)Where, K is the mean number of noise counts in the integration interval. The variance of a Poisson process is equal to itsmean. Like a Gaussian process, the Poisson distribution is preserved under accumulation. That is to say that the sum ofindependent Poisson processes results in a random variable, which is also Poisson. The mean of the accumulated process isthe sum of the constituent means or simply NK for the sum of N identically distributed Poisson Processes. Here the symbolN represents the order of accumulation rather than the number of averages. The distinction is that an averaged quantity is the

    Proc. SPIE Vol. 4377 257

  • accumulated quantity divided by the order of accumulation N. Thus the PDF of the accumulated noise in a photon-countingreceiver is given by

    p(k) =(NK) exp(NK)/k! . (39)The probability that the Poisson-distributed noise exceeds a threshold represents the PFA. This probability is given by

    00 Kth-lPFA = (k) = 1 p (k) = 1 T(Kth ,NK,. ) /(Kth 1)! (40)

    k=Kth k=0

    When the threshold is set to its minimum (i.e., Kth = 1) this probability reduces to

    PFA=1exp(NK). (41)Because of the discrete nature of this counting receiver's threshold level, precise PFAs can not be realized. For examplewhen NK = 0.1 the PFA is [0.095, 0.0047, and 0.00015] for Kth = [0, 1, and 2].

    2.3.2. Detection StatisticsDiffuse TargetThe probability density function for a discrete random process can be calculated from its continuous counterpart using thePoisson Transform6.

    p(k)=f00(xkexp(_x)/k!)p(x)dx. (42)Where, p(x) is the continuous density function. The Poisson transform of a continuous distribution is the distribution of adiscrete Poisson random variable whose mean, x, is conditioned on the statistics of the continuous distribution p(x). Forexample for a glint target p(x) is a delta function and the discrete PDF is Poisson. For a diffuse target, the averaged signalintensity follows a gamma distribution (Eq. 10) and the resultant discrete signal PDF is known as the negative-binomialdistribution

    p(k) (N+k 1)Kk I(K5 + 1)N+k (43)Where K is the mean number of signal counts in a single accumulation interval. The mean of this distribution is NK and itsvariance is NKS(KS +1). If the order of accumulation is unity the negative binomial distribution reduces to the geometric orBose-Einstein distribution.

    The PDF of the signal plus noise can be calculated from any one of a number of techniques (e.g. a convolution of the noisePDF with the signal PDF, or by inverse transforming the product of their characteristic functions, or by the Poisson transformintegral using the continuous counterpart forp(x)).Goodman2 derives the appropriate expression using the conditional probability integral. In his expression he assumes aspatial diversity (averaging) parameter M, whereas here we assume a temporal accumulation N. Therefore, we obtain ourexpression by substituting N for Goodman's M parameter, and NK and NK for his and parameters. Using thesesubstitutions we arrive at

    p (k)= exp(NK) (NK)3 (k+Nf1)!( K5 k-JSn(K +1)N(N 1)! (kf)! K5 +1)

    The probability of detection is just the infinite summation of the discrete density function starting from Kth. Which is equal to1 minus the finite sum from zero to the 1-Kth.

    00 Kth-lPD = p (k) =1 (k). (45)kKth k=0

    When the threshold is a minimum (i.e., NK

  • PD = 1 exp(NK ) /(K + 1)N (46)

    Furthermore, when NK

  • approximately 4 dB. For a glint-target, as the level of averaging increases the performance curves shift to lower SNRsaccording to SNRIN. Additionally, the diffuse target performance converges to glint target performance of high levels ofaveraging.

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    Figure 5. Same as Figure 4 except for continuous direct detection. For the glint target, the effect of averaging is an equivalent toan increase in the single-pulse SNR by a factor equal to the number of pulses averaged. For example, 0 dB performance at 10pulse averaging is the same as 1 0 dB performance for single pulse averaging.

    Similar plots for a photon-counting direct detection receiver are shown in Figure 6. These curves demonstrate the superiorperformance of a photon-counting direct detection receiver. Because the threshold level is discrete, each curve correspondsto a different false alarm probability, which is less than the prescribed values of 0.1%. This explains the peculiar horizontalspacing between the curves as the order of accumulation is increased.

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    Figure 6. Same as Figure 4 except for a photon-counting direct detection parametric in the order of accumulation, N, with themean number of noise electrons equal to 0.1. The threshold level, for which the PFA is less than the prescribed PFA, and thecorresponding actual PFA is indicated on the plot.

    A useful figure of merit for comparing the three receiver models is the total number of accumulated photoelectrons Wpe)required to achieve a given detection probability. For a matched filter coherent receiver (E = Ps/B), this figure of merit isgiven by

    Nped = NCNR = NYTIqPS/hVB. (50)

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  • Whereas, for a matched filter continuous direct detection receiver (E =P/2B) the number ofphotoelectrons required isN ISNRNo

    Npedd (51)2qlqMV BThis result demonstrates the well know result the non-photon-limited performance improves with shorter laser pulse widths(i.e., increased bandwidth). For a photon-counting direct detection receiver we have

    Npedd NK . (52)Comparison plots of this figure of merit are presented in Figure 7 for both diffuse and glint targets. For the coherentdetection receiver, the FOM is minimized for levels of averaging between 5 and 1 0. The curves suggest that a total of 26photoelectrons distributed over 6 independent returns (-4 photoelectrons/shot) achieves the desired performance (90% PDwith a 0.1 % PFA). At lower levels of averaging speckle noise mandates a higher number of photoelectrons. At higherlevels of averaging performance is degraded because the noise standard deviation, and hence the threshold level to achieve agiven PFA, decreases at a rate ofonly 1I'JN. In this regime, the FOM asymptotically converges to a square-root dependenceon the number of pulses averaged. For a glint target, there is no speckle noise and optimal performance is achieved with noaveraging. Here the single-shot number ofrequired photoelectrons is approximately 12.For the continuous direct detection receiver, the performance is substantially worse than a coherent detection receiver. Here

    298 (86 for a glint target) photoelectrons are required to achieve the desired detection statistics. This analysis assumed a 2ns pulsewidth ('-250 MHz BW) a near-JR (i.e., InGaAs) avalanche photodiode receiver in with an APD gain of 10, and anoptimistic preamplifier noise spectral density of 1 pA/\JHz. The analysis ignored the effects of signal shot-noise and APDexcess noise, which would degrade performance even further.The photon-counting performance can be superior to both coherent detection and continuous direct detection. Photon-noiselimited (K 0) performance requires only 2.3 accumulated distributed across many ('100 or more) returns. When K is ashigh as 0.1, only 9.4 photoelectrons are required over 7 returns. When K is approximately equal to 1, the photon-countingperformance is almost the same as a coherent detection receiver. This results from the fact that in a coherent detectionreceiver the equivalent number of dark counts, due to shot-noise, is unity. The jagged nature of the non-photon limited (K >0) receiver is a result of the discrete threshold levels required to achieve a given PFA.

    Total Number of Photoelectrons-

    ml 10 ---------

    t 2I CIo 0t 0

    .o 0o 010I 10

    io0 io1 10 100 1 10 100

    Navg Navg

    Figure 7. Total number of accumulated photoelectrons required to achieve a 90% PD with a 0.1 % PFA versus the number ofaverages for a diffuse (left) and glint (right) target. The curves conespond to a coherent detection receiver (solid), a continuousdirect detection receiver (dot) and three photon-counting receivers with varying number of dark-counts per sample interval (1dot-dot-dot-dash), (0.1 dot-dash) and (0 dash). The classical direct detection receiver assumes 250 MHz BW an APD gain of 10,and an optimistic preamplifier noise spectral density of 1 pAt'JHz.

    4. Conclusions and SummaryDirect detection receivers can more easily exploit speckle diversity (aperture, frequency, and polarization averaging). Theeffective level of diversity is analogous to pulse averaging except for the fact that there is no reduction in noise variance.Approximate performance for a given level of speckle diversity can be estimated from the plots presented in this paper, by

    , Total Number of Photoelectrons:__ Coh Det, 25.2,8 Log(Pa3C)

    .Y..CDD,2.1,4- - - DDLog(Nn)'=-If,2.3, 100-. - OD Loq(Nn) =-1 ., 9.4,7DO Nn) .00, 20.3,4

    CDD Pamnieters:-

    -, -

    BW: 250 MHz, APD GaIn: 10, NolseDen: 1.00 pAirt(Hz)

    CohDet,11.9,1 Log(PFa.40O COD, 88.2,1- - - DD Log(Nn) =-Int,23,1

    DD LoNn)=-1.00, 5.2,1--.DDLg(Nn) =0.00,82, 1

    I GOPararae4erw - -BW: 250 MHz, APD GaIn: 10, NolseDen: 1.00 pAirt(Hz)

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  • rescaling the SNR axis appropriately to account for the lack of decreased noise power with averaging. For example, theperformance of a direct detection receiver with a speckle diversity of ten is approximately the same as that shown in thecurves with the ten pulse averaging and an SNR ten times larger. For very high levels of speckle diversity the performanceapproaches that of a glint target. For a coherent detection receiver, it is difficult to obtain speckle diversity levels greater thanunity unless array receivers or multiple polarizations or transmitted frequencies are employed'2.The results presented above suggest that the low-noise photon-counting receivers provide superior performance over coherentdetection receivers, in terms of the number of required photoelectrons. In general this is true, however the actual energy-aperture product required to achieve equal or greater performance may be significantly higher because of poor quantumefficiency of practical detectors. In the UV to visible regime, the quantum efficiency of PMTs is typically on the order of10%. In the near-to far IR PMT efficiency is substantially worse. New "photon-counting" devices for near IR operation areemerging. However, to date no commercial devices are available with substantial quantum efficiencies. Research gradeGeiger-mode APDs have problems with low quantum efficiency, after-pulsing and relatively long reset times associated withquenching. Thus before a conclusion can be drawn for a specific application, a detailed analysis needs to be conducted.This analysis needs to consider all aspects of the sensor including "effective" aperture diameter, energy, detector performanceand availability, pulse width, background light, etc.

    5. References

    Bachman, C. G., Laser Radar Systems and Techniques, Chapter 2, Artech, Wayland, Mass 19792 Goodman, J. W., "Some Effects of Target-Induced Scintillation on Optical Radar Performance," Proceedings of the IEEE,53,111965

    Youmans, D. "Avalanche Photodiode detection statistics for a direct detection laser radar," Proc. SPIE, 1633. 1992'Goodman, J. W., Statistical Optics, Chapter 2, Wiley and Sons, New York, 1985

    Goodman, J. W., "Statistical Properties of Laser Speckle Patterns," Chapter 2 of Laser Speckle and Related Phenomena,Springer-Verlag, New York, 1984

    6 B. Saleh, Photoelectron Statistics, Table 2.1 with ji =2o2/N and jt =A2, Springer-Verlag, New York, 1978Whalen, Anthony D., Detection of signals in Noise, Chapter 8, Academic Press, Inc., 1971

    8 Marcum, J. I. and P. Swerling, "A statistical theory of target detection by pulsed radar", IRE Transactions on InformationTheory, April, 1960

    Robertson, G. H., "Computation of the noncentral chi-square distribution", Bell Syst Tech Journal 48, No 1, 201-207, 1969' The PDF of a random process that is the sum of two independent random processes is the convolution of the constituentPDFs.

    ' Papoulis, A., Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, 198412 Gatt, P. et. al. "Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects" AppliedOptics, 35 No. 30, 5999-6009, 1995....

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