Necessary Conditions for a Maximum Likelihood Estimate to Become Asymptotically Unbiased and Attain the Cramer–Rao Lower Bound

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Necessary conditions for a maximum likelihood estimate to become asymptotically unbiased and attain the Cramer–Rao Lower Bound. Part I. General approach with an application to time-delay and Doppler shift estimation Eran Naftali and Nicholas C. Makris Massachusetts Institute of Technology, Room 5-222, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 ͑Received 17 October 2000; revised 20 April 2001; accepted 29 May 2001͒ Analytic expressions for the first order bias and second order covariance of a general maximum likelihood estimate ͑MLE͒ are presented. These expressions are used to determine general analytic conditions on sample size, or signal-to-noise ratio ͑SNR͒, that are necessary for a MLE to become asymptotically unbiased and attain minimum variance as expressed by the Cramer–Rao lower bound ͑CRLB͒. The expressions are then evaluated for multivariate Gaussian data. The results can be used to determine asymptotic biases, variances, and conditions for estimator optimality in a wide range of inverse problems encountered in ocean acoustics and many other disciplines. The results are then applied to rigorously determine conditions on SNR necessary for the MLE to become unbiased and attain minimum variance in the classical active sonar and radar time-delay and Doppler-shift estimation problems. The time-delay MLE is the time lag at the peak value of a matched filter output. It is shown that the matched filter estimate attains the CRLB for the signal’s position when the SNR is much larger than the kurtosis of the expected signal’s energy spectrum. The Doppler-shift MLE exhibits dual behavior for narrow band analytic signals. In a companion paper, the general theory presented here is applied to the problem of estimating the range and depth of an acoustic source submerged in an ocean waveguide. © 2001 Acoustical Society of America. ͓DOI: 10.1121/1.1387091͔ PACS numbers: 43.30.Pc, 43.60.Gk, 43.60.Pt ͓DLB͔

I. INTRODUCTION The purpose of the present paper is not to derive a new parameter resolution bound, but rather to determine, within In many practical problems in ocean acoustics, geophys- the framework of classical estimation theory1,6,7 the condi- ics, statistical signal processing, and other disciplines, non- tions on sample size, or SNR, necessary for the MLE to linear inversions are required to estimate parameters from measured data that undergo random fluctuations. The nonlin- become asymptotically unbiased and attain minimum vari- ear inversion of random data often leads to estimates that are ance. The approach is to apply the tools of higher order biased and do not attain minimum variance, namely the asymptotic inference, which rely heavily on tensor analysis, Cramer–Rao lower bound ͑CRLB͒, for small sample sizes or to expand the MLE as a series in inverse orders of sample equivalently low signal-to-noise ratio ͑SNR͒. The maximum size or equivalently inverse orders of SNR.7 From this series likelihood estimator ͑MLE͒ is widely used because if an as- analytic expressions for the first order bias, second order ymptotically unbiased and minimum variance estimator ex- covariance and second order error correlation of a general ists for large sample sizes, it is guaranteed to be the MLE.1 MLE are presented in terms of joint moments of the log- Since exact expressions for the bias, variance, and error cor- likelihood function and its derivatives with respect to the relation of the MLE are often difficult or impractical to de- parameters to be estimated. Since the first order error corre- rive analytically, it has become popular in ocean acoustics lation is shown to be the CRLB, which is only valid for and many other areas to simply neglect potential biases and unbiased estimates, the second order error correlation can to compute limiting bounds on the mean square error, such as provide a tighter error approximation to the MLE than the the CRLB, since these bounds are usually much easier to CRLB that is applicable in relatively low SNR even when obtain. The CRLB, however, typically provides an unrealis- the MLE is biased to first order. These expressions are then tically optimistic approximation to the MLE error correlation in many nonlinear inverse problems when the sample size is used to determine general analytic requirements on sample small, or equivalently the SNR is low. A number of bounds size, or SNR, that are necessary for an MLE to become as- on the error correlation exist that are tighter than the ymptotically unbiased and attain minimum variance. This is CRLB.1–5 Some of these bounds are based on Bayesian done by showing when the first order bias becomes negli- assumptions4,5 and so require the a priori probability density gible compared to the true value of the parameter and when of the parameters to be estimated, which can be problematic the second order covariance term becomes negligible com- when the a priori probability density is not known.6 pared to the CRLB.

J. Acoust. Soc. Am. 110 (4), October 2001 0001-4966/2001/110(4)/1917/14/$18.00 © 2001 Acoustical Society of America 1917 The first order bias is evaluated for general multivariate dard linear frequency modulated ͑LFM͒, hyperbolic fre- Gaussian data. The second order covariance and error corre- quency modulated ͑HFM͒, and canonical waveforms are pro- lation terms are evaluated for two special cases of Gaussian vided for typical low-frequency active-sonar scenarios in data that are of great practical value in ocean acoustics, geo- ocean acoustics. physics, and statistical signal processing. The first is for a deterministic signal vector embedded in additive noise and II. GENERAL ASYMPTOTIC EXPANSIONS OF THE the second is for a fully randomized signal vector with zero MLE AND ITS MOMENTS mean in additive noise. These cases have been widely used Suppose the random data vector X, given m-dimensional in ocean acoustic inversions, spectral estimation, beamform- parameter vector ␪, obeys the conditional probability density ing, sonar and radar detection, and localization problems, as function ͑pdf͒ p(X;␪). The log-likelihood function l(␪)is well as statistical optics.8–10 In a companion paper, each of defined as l(␪)ϭln(p(X;␪)) when evaluated at measured these cases is applied to determine the asymptotic bias and values of X. Let the rth component of ␪ be denoted by ␪r. covariance of maximum likelihood range and depth esti- The first log-likelihood derivative with respect to ␪r is then mates of a sound source submerged in an ocean waveguide ␪r.IfR ϭr ...r ,.....,Rץ/(l(␪ץdefined as l ϭ from measured hydrophone array data as well as necessary r 1 11 1n1 m 11 ϭr are sets of coordinate indices, joint moments of conditions for the estimates to attain the CRLB. m1...mnm the log-likelihood derivatives can be defined by v In the present paper, these expressions are applied to the R1 ,...,Rm active sonar and radar time-delay and Doppler-shift estima- ϭEbl ...l c, where, for example, v ϭE͓l l ͔ and Rl Rm s,tu s tu ϭ tion problems, where time delay is used for target range es- va,b,c,de E͓lalblclde͔. timation and Doppler shift is used for target velocity estima- The expected information, known as the Fisher informa- ϭ tion. Attention is focused on the commonly encountered tion, is defined by irs E͓lrls͔ where the indices r, s are scenario of a deterministic signal with unknown spatial or arbitrary. Lifting the indices produces quantities that are de- R ,...,R r s r s temporal delay received together with additive white noise. noted by v 1 mϭi 11 11...i mnm mnmv , s11 ...s1n ,...,sm1 ...smn The time-delay MLE is then the time lag at the peak value of Ϫ 1 Ϫ m where irsϭ͓i 1͔ is the r, s component of the inverse i 1 of a matched filter output. The matched filter estimate for a rs the expected information matrix i. The inverse of the Fisher signal’s time delay or position is widely used in many appli- Ϫ information matrix i 1 is also known as the Cramer–Rao cations of statistical pattern recognition in sonar, radar, and lower bound ͑CRLB͒. Here, as elsewhere, the Einstein sum- optical image processing. This is because it has long been mation convention is used. That is, if an index occurs twice known that the matched filter estimate attains the CRLB in in a term, once in the subscript and once in the superscript, high SNR. Necessary analytic conditions on how high the summation over the index is implied. SNR must be for the matched filter estimate to attain the The MLE ␪ˆ, the value of ␪ that maximizes l(␪) for the CRLB have not been previously obtained but are derived 1,6,7 here using the general asymptotic approach developed in given data X, can now be expressed as an asymptotic ␪ Secs. II–IV. expansion around in increasing orders of inverse sample size nϪ1 or equivalently SNR. Following the derivation of A number of authors have derived tighter bounds than 7 the CRLB for the time-delay estimation problem to help Barndorff-Nielsen and Cox, the component lr is first ex- ␪ evaluate performance at low SNR where the CRLB is not panded around as attained by the MLE, as, for example, in Refs. 5, 12, 13. The present paper follows a different approach by providing ex- ˆ ϭ ϩ ͑␪ˆ Ϫ␪͒sϩ 1 ͑␪ˆ Ϫ␪͒s͑␪ˆ Ϫ␪͒t l r lr lrs 2lrst plicit expressions for the second order variance of the time- delay and Doppler-shift MLEs that are attained in lower ϩ 1 ͑␪ˆ Ϫ␪͒s͑␪ˆ Ϫ␪͒t͑␪ˆ Ϫ␪͒uϩ ͑ ͒ 6lrstu , 1 SNR than the CRLB. The first order bias is also derived. ¯ These expressions are then used to provide analytic conditions on SNR necessary for the time-delay MLE, namely the where (␪ˆ Ϫ␪)rϭ␪ˆ rϪ␪r. Equation ͑1͒ is then inverted to ob- matched filter estimate, and Doppler-shift MLE to become tain an asymptotic expansion for (␪ˆ Ϫ␪)r, as shown in Ap- unbiased and attain minimum variance in terms of properties pendix D. After collecting terms of the same asymptotic or- of the signal and its spectrum. Illustrative examples for stan- der, this can be expressed as7

͑2͒

1918 J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB ϵ Ϫ where HR lR vR . The terms are organized in decreasing The ﬁrst order bias of the MLE is then the expected asymptotic order. The drops occur in asymptotic orders of value of Eq. ͑2͒, as derived by Barndorff-Nielsen and Cox,7 nϪ1/2 under ordinary repeated sampling, which is equivalent to an asymptotic drop of (SNR)Ϫ1/2. The asymptotic orders of each set of terms are indicated by symbols such as ͑3͒ Ϫm OA(n ) which denotes a polynomial that will be of order nϪm when n is large but may contain higher order terms, i.e., Ϫ(mϩ1) ͑ ͒ O p(n ), that can be signiﬁcant when n is small. Here Here we take a further step and use Eq. 2 to derive the Ϫm the symbol O p(n ) denotes a polynomial of exactly order error correlation of the MLE to second order as given by the nϪm for all values of n. order-separated expression

͑4͒

2 2 where notation such as vbce,d, f ,s(n ) means the n order terms of the joint moment vbce,d, f ,s . Using the identity Cov(␪ˆ r,␪ˆ a)ϭCor(␪ˆ r,␪ˆ a)Ϫb(␪r)b(␪a), we obtain the following expression for the covariance of the MLE to second order:

͑5͒

The ﬁrst order covariance term ira is the r, a component of Bhattacharyya.2 While it involves derivatives of the likeli- the inverse of the Fisher information, or the r, a component hood function, it is quite different from the multivariate co- of the CRLB. A bound on the lowest possible mean square variance derived in Eq. ͑5͒ that is valid for multivariate es- error of an unbiased scalar estimate that involves inverse timates that may be biased. For discrete random variables, sample size orders higher than nϪ1 was introduced by expressions equivalent to Eqs. ͑3͒–͑5͒ have been obtained in

J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB 1919 a significantly different form via a different approach by statistic that contains all measurement information about the Bowman and Shenton.14 parameters to be estimated.1,16 A necessary condition for the MLE to become asymp- The assumption of Gaussian data is valid, by virtue of totically unbiased. This is for the first order bias of Eq. ͑1͒ to the central limit theorem even for small n and N, when the become much smaller than the true value of the parameter total received field is the sum of a large number of statisti- ␪r. Similarly, a necessary condition for the MLE to asymp- cally independent contributions. In the case of a determinis- totically attain minimum variance is for the sum of second tic signal in additive noise, the additive noise typically arises order terms in Eq. ͑5͒ to become much smaller than the first from a large number of independent sources distributed over order term, which is the CRLB. the sea surface.17 These noise sources may be either caused by the natural action of wind and waves on the sea surface, III. ASYMPTOTIC BIAS, ERROR CORRELATION AND or they may be generated by ocean-going vessels.18 COVARIANCE OF THE MLE FOR GAUSSIAN A particular fully randomized Gaussian signal model DATA that is very widely used and enjoys a long history in acous- 19,20 The asymptotic expressions presented for the bias, error tics, optics, and radar is the circular complex Gaussian correlation, and covariance of the MLE in Sec. II are now random ͑CCGR͒ model. The basic assumption in this model evaluated for real multivariate Gaussian data. General multi- is that at any time instant, the received signal field is a 9,19 variate Gaussian data can be described by the conditional CCGR variable. This means that the real and imaginary probability density parts of the instantaneous field are independent and identi- 1 1 1 n cally distributed zero-mean Gaussian random variables. In ͑ ␪͒ϭ ͭ Ϫ ͚ ͑ Ϫ␮͑␪͒͒T p X; nN/2 n/2 exp Xi active detection and imaging problems, this model is typi- ͑2␲͒ ͉C͑␪͉͒ 2 ϭ i 1 cally used to describe scattering from fluctuating targets21,22 and surfaces with wavelength scale roughness.9 When the ϫ Ϫ1͑␪͒͑ Ϫ␮͑␪͒͒ͮ ͑ ͒ C Xi , 6 target or resolved surface patch is large compared to the wavelength, the total received field can be thought of as aris- ϭ͓ T T T T͔T where the data X X1 X2 X3 ...Xn are comprised of n in- ing from the sum of a large number of independent scatters dependent and identically distributed N-dimensional data so that the central limit theorem applies. Since World War II, vectors Xi to show an explicit dependence under normal re- the CCGR signal model has been used to describe ocean- peated sampling for convenient reference. It is noteworthy acoustic transmission scintillation in what is known as the that the CRLB is always proportional to 1/n but may be saturated region of multi-modal propagation.19,23,24 In this proportional to a more complicated function of the length of regime, natural disturbances in the waveguide, such as un- the data vector N. derwater turbulence and passing surface or internal gravity We begin by deriving the first order bias for the general waves, lead to such randomness in the medium that the multivariate Gaussian case where the data covariance C and waveguide modes at the receiver can be treated as statisti- the data mean ␮ depend on the parameter vector ␪. The joint cally independent entities. The central limit theorem can then moments required to evaluate both the error correlation and be invoked for the total received field, which behaves as a covariance for the general case are quite complicated but not CCGR process in time.16,19 In passive source localization of great relevance in most standard ocean acoustic and signal problems in ocean-acoustics, the source signal is typically processing problems.8 They are not derived in this paper, but mechanical noise that is accidentally radiated into the ocean are the subject of another work where the second order bias by a vessel. This noise typically has both narrow and broad- is also derived.15 We instead define two special cases that band components that arise from a broad distribution of in- have great practical value, since they describe a deterministic dependent mechanical interactions that lead to a signal that signal in additive noise and a fully randomized signal in can be represented as a CCGR process in time. The CCGR noise, respectively. In the former the data covariance C is signal model has become a very standard model in ocean- independent of the parameter vector ␪, while the mean ␮ acoustic matched field processing.16,25,26 depends on ␪ which is the subject of the estimation problem. A. The general multivariate Gaussian case In the latter, the data mean ␮ is zero while the covariance C depends on the parameter vector ␪ to be estimated. In the We obtain the following expression for the first order latter case, the sample covariance of the data is a sufficient bias of the MLE given general multivariate Gaussian data

͑␪ˆ r͒ϭ 1 rs tu͑ ϩ ͒ϩ ͑ Ϫ3/2͒ b 2i i vstu 2vst,u O p n

␮ץ Cץ ␮ Tץ ␮ץ 2␮ Tץ Cץ 2Cץ m m m 2 1 ϭ ͚ ͚ ͚ ͓iϪ1͔ ͓iϪ1͔ ͭ trͩ CϪ1 CϪ1 ͪ ϩͩ ͪ CϪ1ͩ ͪ ϩͩ ͪ CϪ1ͩ ͪ CϪ1ͩ ͪ rs tu s t u s t u s u t ␪ץ ␪ץ ␪ץ ␪ץ ␪ץ ␪ץ ␪ץ ␪ץ ␪ץ sϭ1 tϭ1 uϭ1 n 2

Cץ 2Cץ ␮ 1ץ Cץ ␮ Tץ ␮ץ 2␮ Tץ Ϫ͚ ͩ ͪ CϪ1ͩ ͪ Ϫ͚ ͩ ͪ CϪ1ͩ ͪ CϪ1ͩ ͪ Ϫ ͚ trͩ CϪ1 CϪ1 ͪͮ , ͑7͒ s u t s t u s u t ␪ץ ␪ץ ␪ץ ␪ 2 s,tץ ␪ץ ␪ץ ␪ s,tץ ␪ץ ␪ץ s,t

1920 J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB by substituting Eqs. ͑A1͒–͑A3͒ for the relevant joint mo- C. Random signal in noise: Zero-mean and ͑ ͒ ͚ ments into Eq. 3 for the first order bias, where s,t indi- parameter-dependent covariance cates a sum over all possible permutations of s, t orderings, a Similarly, the error correlation and covariance of the ͚ ϭ ϩ total of two. For example, s,tvst vst vts . MLE can be obtained to second order for a zero-mean It should be noted that the expression contains both ten- Gaussian random signal vector in Gaussian noise by substi- sor notation, denoted by the indices s, t, and u, and vector- tuting Eqs. ͑C1͒–͑C10͒ into Eqs. ͑4͒ and ͑5͒, respectively. In matrix notation. For the first order bias, only first and second this case, ␮ is zero in Eq. ͑6͒. For scalar parameter and data, order parameter derivatives are required of the mean and the following simple expressions for the mean-square error covariance. and variance expressions are obtained: Suppose, for example, the bias of the vector

␮ˆ ␪ˆ ϭͫ ͬ Cˆ

͑10͒ is desired, where ␮ˆ and Cˆ are the maximum likelihood estimates of the mean and variance, respectively, from a set of n independent and identically distributed Gaussian random ͑ ͒ variables xi . The bias obtained from Eq. 6 is zero for the mean component and Ϫ(C/n) for the variance component. ͑ ͒ This result can be readily verified by taking expectation val- 11 ues directly.27 It is noteworthy that the first order bias of a scalar pa- Suppose, for example, that the bias, mean-square error, and rameter estimate always vanishes for general Gaussian data variance of the MLE of the parameter ␪ϭC2 are desired, ͑ ͒ as can be seen by inspection of Eq. 7 . where the xi are n independent and identically distributed Gaussian random variables with zero-mean. It can be readily B. Deterministic signal in additive noise, parameter- 2 independent covariance shown that the corresponding bias 2C /n, mean-square error 8C4/nϩ44C4/n2, and variance 8C4/nϩ40C4/n2, obtained The multivariate error correlation and covariance of the using Eqs. ͑3͒–͑5͒, correspond to those obtained by taking MLE can be obtained to second order for a deterministic expectation values directly.27 signal vector in additive Gaussian noise by substituting Eqs. ͑B1͒–͑B10͒ into Eqs. ͑4͒ and ͑5͒, respectively. In this case, C is independent of ␪ in Eq. ͑6͒. For scalar parameter and data, the following simple expressions for the mean-square IV. CONTINUOUS GAUSSIAN DATA: SIGNAL error and variance are obtained EMBEDDED IN WHITE GAUSSIAN NOISE Let a real signal ␮(t;␪) that depends on parameter ␪ be received together with uncorrelated white Gaussian noise of power spectral density N /2 that is independent of ␪. Sup- ͑ ͒ 0 8 pose the real signal has Fourier transform ␮(t;␪)↔⌿( f ;␪). The complex analytic signal and its Fourier transform ␮˜ (t;␪)↔⌿˜ ( f ;␪) are conventionally defined such that ⌿˜ ( f ;␪)ϭ2⌿( f ;␪) for f Ͼ0, ⌿˜ ( f ;␪)ϭ0 for f Ͻ0, and ⌿˜ ( f ;␪)ϭ⌿( f ;␪) for f ϭ0, so that ␮(t;␪)ϭRe͕␮˜ (t;␪)͖. The ␸ ͑9͒ total received analytic signal, ˜ i(t), then follows the conditional probability density28

1 T p͑˜␸ ͑t͒;␪͒ϭk expͭ Ϫ ͵ ͉͑˜␸ ͑t͒Ϫ␮˜ ͑t;␪͉͒͒2 dtͮ , ͑12͒ Suppose, for example, that the bias, mean-square error and i i 2N 0 variance of the MLE of the parameter ␪ϭ␮2 are desired, 0 where the xi are again n independent and identically distributed Gaussian random variables, and C is independent of ␪. where k is a normalization constant. The bias, the mean- 2 The corresponding bias C/n, mean-square error 4C␮ /n square error, and the variance of the MLE ␪ˆ are obtained 2 2 2 2 2 ϩ3C /n , and variance 4C␮ /nϩ2C /n , obtained using from Eqs. ͑3͒–͑5͒ as Eqs. ͑3͒–͑5͒ can be readily shown to correspond to those obtained by taking expectation values directly.27 Since the MLE for ␪ϭ␮2 is biased, the Bhattacharyya bound does not ˜ N0 Re͕I 2͖ hold for this example and in fact can exceed the actual vari- b͑␪ˆ ͒ϭϪ , ͑13͒ 27 2 ˜2 ance of the MLE. I 1

J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB 1921 ␮˜ 3͑tϪ␶͒ץ * ␮˜ ͑tϪ␶͒ץ ͑ ͒ ˜ ϭ ͵ ͩ ͪ ͩ ͪ ␶3 dtץ ␶ץ I 3 14

ϱ ϭϪ͑2␲͒4 ͵ f 4͉⌿˜ ͑ f ͉͒2 df. ͑21͒ 0 Noting that the first order bias of Eq. ͑13͒ is directly ˜ ˜ ϭ ͑ ͒ ͑15͒ proportional to Re͕I2͖, where Re͕I2͖ 0 from Eq. 19 ,we find that the first order bias for the maximum likelihood time- delay estimation problem is identically zero, as expected by inspection of Eq. ͑7͒. To evaluate the mean-square error and the variance, all after evaluating the joint moments for the parameter- ˜ ϭ ˜ three integrals are needed. Noting that Re͕I2͖ 0 and Re͕I3͖ independent covariance, where ˜I , ˜I , ˜I are defined as fol- ϭ˜ 1 2 3 I3 the following results are obtained: lows:

␮˜ ͑t;␪͒ץ * ␮˜ ͑t;␪͒ץ ˜ ϭ ͵ ͩ ͪ ͩ ͪ ͑ ͒ ␪ dt, 16ץ ␪ץ I 1 ͑22͒ ␮˜ 2͑t;␪͒ץ * ␮˜ ͑t;␪͒ץ ˜I ϭ ͵ ͩ ͪ ͩ ͪ dt, ͑17͒ ͒ ͑ ͒ ͑ ˜ ˜ ␪2ץ ␪ץ 2 where I 1 , I 3 are evaluated in Eqs. 19 and 21 , respectively. The mean-square error can also be expressed explic- ␮˜ 3͑t;␪͒ itly in terms of SNRϭ2E/N and signal parameters viaץ * ␮˜ ͑t;␪͒ץ ˜ ϭ ͵ ͩ ͪ ͩ ͪ ͑ ͒ 0 ␪3 dt. 18ץ ␪ץ I 3

There are two important issues to note. First, we are now working with continuously measured data as opposed to the discrete data vectors of Sec. III. Second, the fact that we are only estimating a scalar rather than a vector parameter greatly simpliﬁes the evaluation of the joint moments.

͑ ͒ V. TIME-DELAY ESTIMATION 23

Suppose ␮˜ (t;␪)ϭ␮˜ (tϪ␶) in Eq. ͑12͒ so that the scalar where f ϭ␻ /(2␲) is the carrier frequency, E is the total ␪ϭ␶ ␪ˆ ϭ␶ c c time delay is to be estimated. The MLE ˆ of time- energy of the real signal, ␤ is commonly defined as the sig- delay ␶ corresponds to the peak output of a matched filter for ͑ ͒ ␥4 8 nal’s root mean square rms bandwidth, and is the fourth a signal received in additive Gaussian noise. Estimates of moment of the expected signal’s energy spectrum. the time delay between transmitted and received signal ϱ waveforms are typically used in active-sonar and radar ap- ϭ ͵ ͉⌿˜ ͑ ϩ ͉͒2 ͑ ͒ 2E v f c dv, 24 plications to determine the range of a target in a nondisper- Ϫ f c sive medium. The asymptotic bias, mean-square error, and 2 ϱ 2 2 variance of ␶ˆ are obtained by substituting ␶ for ␪ in Eqs. ͑2␲͒ ͐Ϫ v ͉⌿˜ ͑vϩ f ͉͒ dv f c c ͑13͒–͑18͒. ␤2ϭ , ͑25͒ 2E The following alternative expressions are obtained for Eqs. ͑16͒–͑18͒ by applying Parseval’s Theorem 4 ϱ 4 2 ͑2␲͒ ͐Ϫ v ͉⌿˜ ͑vϩ f ͉͒ dv f c c ␥4ϭ . ͑26͒ ␮˜ ͑tϪ␶͒ 2Eץ * ␮˜ ͑tϪ␶͒ץ ˜I ϭ ͵ ͩ ͪ ͩ ͪ dt -␶ Equation ͑23͒ explicitly shows the asymptotic depenץ ␶ץ 1 ϱ dence of the MLE time-delay variance on increasing orders 2 2 2 Ϫ1 ␻ ϭ ϭ͑2␲͒ ͵ f ͉⌿˜ ͑ f ͉͒ df, ͑19͒ of (SNR) . For a base-banded signal, where c 0, the 0 first order variance term of Eq. ͑23͒ is proportional to the inverse of ␤2, while the second term is proportional to the -␮˜ 2͑tϪ␶͒ ratio ␥4/␤6. While it is well known that the first order variץ * ␮˜ ͑tϪ␶͒ץ ˜I ϭ ͵ ͩ ͪ ͩ ͪ dt ␶2 ance or CRLB decreases with increasing rms bandwidth atץ ␶ץ 2 fixed SNR, the behavior of the second order variance term ϱ ϭ j͑2␲͒3 ͵ f 3͉⌿˜ ͑ f ͉͒2 df, ͑20͒ has a more complicated interpretation since it involves both 0 ␥ and ␤.

1922 J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB The time-delay MLE asymptotically attains the CRLB Theorem, the following expressions are obtained: ␥4ϩ ␻2␤2ϩ ␻4 ␤2ϩ␻2 2Ӷ ϭ ͒ ␮˜ ͑t; fץ * ͒ ␮˜ ͑t; fץ when ( 8 c 3 c )/( c ) 2E/N0 SNR. For a base-banded signal, this condition means that the SNR ˜ ϭ ͵ ͩ D ͪ ͩ D ͪ dt ץ ץ I 1 must be much larger than the kurtosis ␥4/␤4 of the expected f D f D signal’s energy spectrum. This can be interpreted as meaning T/2 that as the signal’s energy spectrum becomes more peaked, ϭ͑2␲͒2 ͵ t2͉˜g͑t͉͒2 dt, ͑28͒ ϪT/2 higher SNR is necessary to attain the CRLB. ͒ ␮2ץ ͒ ␮ץ ˜ ͑t; f D * ˜ ͑t; f D Example 1 ˜I ϭ ͵ ͩ ͪ ͩ ͪ dt f 2 ץ f ץ 2 Assume a real Gaussian base-banded signal with a con- D D stant energy. Its waveform can be represented as T/2 ϭ j͑2␲͒3 ͵ t3͉˜g͑t͉͒2 dt, ͑29͒ 1 ϪT/2 ͑ ͒ϭ ͑Ϫ␲͑ 2 ␶2͒͒ ͉ ͉р ͒ ␮3ץ ͒ ␮ץ ,h t exp t / s , t T/2 ͱ␶ ˜ ͑t; f D * ˜ ͑t; f D s ˜I ϭ ͵ ͩ ͪ ͩ ͪ dt 3 ץ f ץ 3 where h↔H. For real signals, 2E is replaced by E/2, and ⌿ D f D replaces ⌿˜ in Eqs. ͑24͒–͑26͒, where in the present case ⌿ T/2 ϭϪ͑ ␲͒4 4͉ ͑ ͉͒2 ͑ ͒ ϭH. Under the assumption that ␶ /T is sufficiently small 2 ͵ t ˜g t dt. 30 s ϪT/2 that the limits of integration ͓ϪT/2,T/2͔ can be well ap- ˜ ˜ proximated as ͓Ϫϱ, ϱ͔, Eq. ͑23͒ for the variance of the time- To evaluate the bias, I 1 and Re͕I2͖ are substituted into ͑ ͒ ˜ ϭ delay MLE can be written to second order as Eq. 13 . Noting that Re͕I2͖ 0, we find that the first order ˆ bias for the Doppler-shift MLE f D is identically zero,as expected by inspection of Eq. ͑7͒. Equations ͑28͒–͑30͒ are then substituted into Eqs. ͑14͒– ͑15͒ to evaluate the mean-square error and the variance to second order. The resulting relations for the second order mean-square error and variance of the Doppler-shift MLE Since the signal’s energy is 1/&, the first and second order are similar to those obtained for the time-delay MLE: terms equalize when the SNR is 3, which is the kurtosis of a ϭ Gaussian density, where the SNR 2E/N0 . This makes sense because a Gaussian signal has a Gaussian energy spectrum by the convolution theorem. For SNRs less than 3, or in ͑31͒ decibels for 10 log SNRϽ5 dB, the second order term is higher than the first and the CRLB is a poor estimate of the ␶ true mean-square error. Moreover, since 1/ s is a measure of ␶ the signal’s bandwidth, decreasing s , or increasing the signal’s bandwidth, will decrease both first and second order since the two problems are related through the time- ˜ ˜ variance terms, and so improve the time-delay estimate. frequency duality principle. Expressing I 1 , I 3 in terms of SNR and signal parameters then explicitly yields the Doppler-shift MLE mean-square error in terms of increasing VI. DOPPLER SHIFT ESTIMATION orders of (SNR)Ϫ1 as Suppose now that a narrow band signal waveform is transmitted in a nondispersive medium and measured with additive Gaussian noise at a receiver that is moving relative to the source at low Mach number u/cӶ1, where u is the speed of relative motion and c the speed of wave propagation. The expected analytic signal waveform at the receiver ͑32͒ ␮ ˜ (t; f D) is then frequency shifted with Doppler-shift param- where ϭϪ eter f D 2u/c. The total signal and noise measured at the T/2 receiver will then obey the conditional probability density of ϭ ͵ ͉ ͑ ͉͒2 ͑ ͒ ͑ ͒ ␪ϭ 2E ˜g t dt, 33 Eq. 12 with f D . The goal now is to examine the ϪT/2 ˆ asymptotic statistics of the MLE f D for the Doppler-shift ͑ ␲͒2͐T/2 2͉ ͑ ͉͒2 2 ϪT/2t ˜g t dt parameter. ␣2ϭ , ͑34͒ With the given assumptions, the signal waveform can be 2E represented as 4 T/2 4 2 ͑2␲͒ ͐Ϫ t ͉˜g͑t͉͒ dt 4 T/2 ␲ ͑ ϩ ͒ ␦ ϭ . ͑35͒ ␮ ͒ϭ ͒ j2 t f c f D ͑ ͒ ˜ ͑t; f D ˜g͑t e , 27 2E where the complex envelope ˜g(t) is known and is zero out- The Doppler-shift MLE then asymptotically attains the Ϫ р р ␦4 ␣4Ӷ ϭ side the interval T/2 t T/2. By applying Parseval’s CRLB when / 2E/N0 SNR. For an analytic signal

J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB 1923 FIG. 1. Gaussian signal time-delay variance terms as a function of SNR.

with symmetric magnitude, this can be interpreted as meaning that the SNR must be large compared to the kurtosis of the signal’s squared magnitude.

Example 2 ␮ ϭ ␲ ϩ For real signals where (t; f D) g(t)cos 2 t(fc fD), The first and second order terms equalize when the SNR is 6, ͑ ͒ Eq. 32 can be used when 2E is replaced by 4E in Eqs. twice the Gaussian kurtosis as expected, where SNR ͑ ͒ ͑ ͒ 34 – 35 , and the real signal envelope g(t) replaces ˜g(t)in ϭ2E/N . For SNRs less than 6, or in decibels when Eqs. ͑33͒–͑35͒. For the CRLB to be attained in this real 0 10 log SNRϽ7.8 dB, the second order term is higher than the signal case, the SNR must be large compared to twice the first, and the CRLB provides a poor estimate of the true kurtosis of the squared magnitude of the real signal enve- ␶ lope, assuming a symmetric magnitude. Computing E, ␣2, MSH. Increasing s decreases the signal’s bandwidth and so and ␦4 for the real signal envelope g(t)ϭh(t) of example 1, also decreases both first and second order variance terms, by Eq. ͑32͒, the variance of the Doppler-shift MLE can be which improves the Doppler-shift estimate. written to second order as

FIG. 2. LFM signal time-delay variance terms as a function of SNR.

1924 J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB FIG. 3. HFM signal time-delay variance terms as a function of SNR.

ϭ␻ ␲ VII. ASYMPTOTIC OPTIMALITY OF GAUSSIAN, LFM where f 0 0/2 is the carrier frequency and the bandwidth AND HFM WAVEFORMS IN MAXIMUM is given by bT/2␲. The signal is demodulated when multi- LIKELIHOOD TIME-DELAY AND DOPPLER-SHIFT ␻ plied by cos( 0t) and low-pass filtered. The HFM signal is ESTIMATION defined by The general expressions for the second order variance for both the MLE time-delay and the Doppler-shift estima- ␮͑t͒ϭsin͑␣ log͑1Ϫk͑tϩT/2͒͒͒, ͉t͉рT/2, ͑37͒ tors, Eqs. ͑22͒ and ͑31͒, are now implemented for the Gauss- ian linear frequency modulated ͑LFM͒ and hyperbolic fre- ϭ Ϫ ␣ϭϪ ␲ where k ( f 2 f 1)/f 2T, 2 f 1 /k, and f 1 and f 2 are quency modulated ͑HFM͒ waveforms. All waveforms are the frequencies that bound the signal’s spectrum. This signal demodulated. ␻ is demodulated when multiplied by cos( 0t), where f 0 The Gaussian signal is described in examples 1 and 2 ϭͱ f 1 f 2, and is low-pass filtered. To control sidelobes in the above. The LFM signal is defined by frequency domain, the signal is often multiplied by a temporal window function, or taper. We use the modified Tukey ␮͑ ͒ϭ ͑␻ ϩ 1 2͒ ͉ ͉р ͑ ͒ t cos 0t 2bt , t T/2, 36 window that has the form

FIG. 4. LFM signal Doppler-shift variance terms as a function of SNR.

J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB 1925 FIG. 5. Gaussian signal Doppler-shift variance terms as a function of SNR.

w͑t͒ MLE and in Figs. 4–6 for the Doppler-shift MLE for the three signals. The bandwidth is fixed at 100 Hz in Figs. 1–6, for a typical low-frequency active-sonar scenario.29 The fig- tϩT/2 ϩ͑ Ϫ ͒ 2ͩ ␲ ͪ р р ures illustrate some important characteristics of the variance p 1 p sin for 0 t Tw 2Tw terms for both the time-delay and the Doppler-shift MLE. As р р Ϫ 1 for Tw t T Tw SNR increases, the first order variance exhibits the expected ϭ , ͑tϩT/2͒Ϫ͑TϪ2T ͒ linear fall-off and the second order variance falls off with the 2 w Ά pϩ͑1Ϫp͒sin ͩ ␲ ͪ expected second order power law as can be seen more gen- 2Tw erally in Eqs. ͑23͒ and ͑32͒ where the second order term is for TϪT рtрT w Ϫ proportional to 20 log10(N0/2E), and the first to ͑38͒ Ϫ 10 log10(N0/2E). The value of either term at a specific ϭ ϭ where Tw 0.125T is the window duration and p 0.1 is the bandwidth and SNR can then be used to determine its value pedestal used. at the same bandwidth for all SNRs. The dependence of the first and second order variance Table I specifies the SNR’s values beyond which the terms on SNR is presented in Figs. 1–3 for the time-delay second order variance can be neglected relative to the first by

FIG. 6. HFM signal Doppler-shift variance terms as a function of SNR.

1926 J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB TABLE I. Minimum signal-to-noise ratios ͑SNRs͒ necessary for the MLE to APPENDIX A: JOINT MOMENTS FOR ASYMPTOTIC asymptotically attain the CRLB. The given values are the minimum SNRs GAUSSIAN INFERENCE: GENERAL MULTIVARIATE needed for the CRLB to exceed the second order MLE variance by 10 dB. GAUSSIAN DATA All signals have 100 Hz bandwidth. ͑ ͒ Gaussian signal LFM signal HFM signal For the general multivariate Gaussian case of Eq. 6 , both the mean ␮ and the covariance matrix C depend on the Time Delay 15 dB 32 dB 18 dB parameter vector ␪. The joint moments required to evaluate Doppler Shift 18 dB 16 dB 15 dB the first order bias are n i ϭ tr͑CϪ1C CϪ1C ͒ϩn␮TCϪ1␮ , ͑A1͒ showing where the former is an order of magnitude less than ab 2 a b a b the latter. Table I then provides conditions necessary for the MLE to attain the CRLB in time-delay and Doppler-shift n ϭ Ϫ1 Ϫ1 Ϫ1 ͒ vabc ͚ tr͑C CaC CbC Cc estimation for the given signals. It also specifies conditions 3 a,b,c necessary for the MLE to be approximated as a linear func- n n tion of the measured data. Ϫ ͑ Ϫ1 Ϫ1 ͒Ϫ ␮T Ϫ1␮ ͚ tr C CabC Cc ͚ abC c 4 a,b,c 2 a,b,c VIII. CONCLUSION n ϩ ͚ ␮TCϪ1C CϪ1␮ , ͑A2͒ By employing an asymptotic expansion of the likelihood 2 a b c function, expressions for the first order bias, as well as the a,b,c second order covariance and error correlation of a general n ϭϪ ͑ Ϫ1 Ϫ1 Ϫ1 ͒ϩ ␮T Ϫ1␮ MLE, are derived. These expressions are used to determine vab,c ͚ tr C CaC CbC Cc n abC c 2 a,b conditions necessary for the MLE to become asymptotically unbiased and attain the CRLB. The approach is then applied n ϩ tr͑CϪ1C CϪ1C ͒ to parameter estimation with multivariate Gaussian data. 2 ab c Analytic expressions for the general first order bias of the multivariate Gaussian MLE and the second order error cova- Ϫ ␮T Ϫ1 Ϫ1␮ ͑ ͒ n͚ a C CbC c , A3 riance and correlation of the MLE for two special cases of a,b multivariate Gaussian data that are of great practical signifi- where, for example, ͚ indicates a sum over all possible cance in acoustics, optics, radar, seismology, and signal pro- a,b,c permutations of a, b and c orderings, leading to a total of six cessing. The first is where the data covariance matrix is in- terms. Terms such as C and ␮ represent the derivatives dependent of the parameters to be estimated, the standard ab ab of the covariance matrix C and the mean vector ␮ with re- deterministic signal in additive noise scenario. The second is spect to ␪a and ␪b, respectively. where the data mean is zero and the signal as well as the noise undergo circular complex Gaussian random fluctuations. In a companion paper, the expressions derived here are applied to determine the asymptotic bias, covariance, and APPENDIX B: JOINT MOMENTS FOR ASYMPTOTIC mean-square error of maximum likelihood range and depth GAUSSIAN INFERENCE: MULTIVARIATE estimates of a sound source submerged in an ocean wave- GAUSSIAN DATA WITH PARAMETER-INDEPENDENT guide from measured hydrophone array data.11 Necessary COVARIANCE: DETERMINISTIC SIGNAL IN conditions for these source localization estimates to attain INDEPENDENT ADDITIVE NOISE 11 the CRLB are also obtained. For this case the covariance matrix of Eq. ͑6͒ is inde- ␪iϭ0ץ/Cץ ,.In the present paper, general expressions for the first pendent of the parameters to be estimated, i.e order bias, second order mean-square error, and variance of for all i. The joint moments required to evaluate the first scalar maximum likelihood time-delay and Doppler-shift es- order bias, as well as the second order error correlation and timates are obtained for deterministic signals in additive covariance are Gaussian noise. The time-delay MLE is the peak value of a ϭ ␮T Ϫ1␮ ͑ ͒ matched filter output. Both time-delay and Doppler-shift iab n a C b , B1 MLEs are shown to be unbiased to first order. Analytic con- n ditions on SNR necessary for the time-delay and Doppler- ͑ 1͒ϭϪ ␮T Ϫ1␮ ͑ ͒ vabc n ͚ abC c , B2 shift MLEs to attain the CRLB are provided in terms of 2 a,b,c moments of the expected signal’s squared magnitude and v ͑n1͒ϭ0, ͑B3͒ energy spectrum. For base-banded signals, the time-delay a,b,c ͑ 1͒ϭ ␮T Ϫ1␮ ͑ ͒ MLE, namely the matched filter estimate, attains the CRLB vab,c n n abC c , B4 when the kurtosis of the expected signal’s energy spectrum is n much smaller than the SNR. This can be interpreted as mean- ͑ 1͒ϭϪ ␮T Ϫ1␮ vabcd n ͚ abC cd ing that higher SNR is necessary to attain the CRLB as a 8 a,b,c,d demodulated signal’s energy spectrum becomes more n peaked. The Doppler-shift MLE is found to have dual behav- Ϫ ␮T Ϫ1␮ ͑ ͒ ͚ abcC d , B5 ior for narrow band analytic signals. 6 a,b,c,d

J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB 1927 n2 3n ͑ 2͒ϭ ␮T Ϫ1␮ ␮T Ϫ1␮ ͑ ͒ ͑ 1͒ϭϪ ͑ Ϫ1 Ϫ1 Ϫ1 Ϫ1 ͒ va,b,c,d n ͚ a C b c C d , B6 vabcd n ͚ tr C CaC CbC CcC Cd 8 a,b,c,d 8 a,b,c,d

v ͑n1͒ϭ0, ͑B7͒ n a,b,cd ϩ Ϫ1 Ϫ1 Ϫ1 ͒ ͚ tr͑C CabC CcC Cd n2 2 a,b,c,d ͑ 2͒ϭ ␮T Ϫ1␮ ␮T Ϫ1␮ ͑ ͒ va,b,c,de n ͚ a C b c C de , B8 2 a,b,c n Ϫ Ϫ1 Ϫ1 ͒ ͚ tr͑C CabC Ccd n2 16 a,b,c,d ͑ 2͒ϭ ␮T Ϫ1␮ ␮T Ϫ1␮ va,b,cd,ef n ͚ a C cd b C ef 2 ͑a,b͒ϫ͑cd,ef ͒ n Ϫ ͚ tr͑CϪ1C CϪ1C ͒, ͑C5͒ ϩ 2␮T Ϫ1␮ ␮T Ϫ1␮ ͑ ͒ abc d n cdC ef a C b , B9 12 a,b,c,d n2 ͑ 2͒ϭ ␮T Ϫ1␮ ␮T Ϫ1␮ ͑ ͒ va,b,c,def n ͚ a C b c C def , B10 2 a,b,c n2 2 Ϫ1 Ϫ1 Ϫ1 Ϫ1 where the notation ͚ ϫ indicates a sum over all v ͑n ͒ϭ ͚ tr͑C C C C ͒tr͑C C C C ͒, (a,b) (cd,ef ) a,b,c,d 32 a b c d possible permutations of a and b orderings combined with a,b,c,d ͑C6͒ permutations of cd and ef orderings, leading to a total of four terms.

APPENDIX C: JOINT MOMENTS FOR ASYMPTOTIC n GAUSSIAN INFERENCE; MULTIVARIATE v ͑n1͒ϭϪ a,b,cd 2 GAUSSIAN DATA WITH ZERO-MEAN: RANDOM SIGNAL IN NOISE ϫ Ϫ1 Ϫ1 Ϫ1 Ϫ1 ͒ ͚ tr͑C CaC CbC CcC Cd For this case the mean is zero in Eq. ͑6͒. The joint mo- ͑a,b͒ϫ͑c,d͒ ments required to evaluate the ﬁrst order bias, as well as the n second order error correlation and covariance are then ϩ Ϫ1 Ϫ1 Ϫ1 ͒ ͑ ͒ ͚ tr͑C CaC CbC Ccd , C7 2 a,b n i ϭ tr͑CϪ1C CϪ1C ͒, ͑C1͒ ab 2 a b

n 2 ͑ 1͒ϭ ͑ Ϫ1 Ϫ1 Ϫ1 ͒ n vabc n ͚ tr C CaC CbC Cc ͑ 2͒ϭϪ ͑ Ϫ1 Ϫ1 ͒ 3 a,b,c va,b,c,de n ͚ tr C CdC Ce 24 ͑a,b,c͒ϫ͑d,e͒ Ϫ Ϫ Ϫ n Ϫ Ϫ ϫ ͑ 1 1 1 ͒ Ϫ 1 1 ͒ ͑ ͒ tr C CaC CbC Cc ͚ tr͑C CabC Cc , C2 4 a,b,c n2 Ϫ ͑ Ϫ1 Ϫ1 ͒ n ͚ tr C CaC Cb 1͒ϭ Ϫ1 Ϫ1 Ϫ1 ͒ ͑ ͒ 8 ͑a,b,c͒ϫ͑d,e͒ va,b,c͑n ͚ tr͑C CaC CbC Cc , C3 6 a,b,c ϫ Ϫ1 Ϫ1 Ϫ1 ͒ tr͑C CcC CdC Ce n ͑ 1͒ϭϪ ͑ Ϫ1 Ϫ1 Ϫ1 ͒ 2 vab,c n ͚ tr C CaC CbC Cc n Ϫ Ϫ 2 a,b ϩ 1 1 ͒ ͚ tr͑C CaC Cb 8 a,b,c

n Ϫ Ϫ ϫ ͑ Ϫ1 Ϫ1 ͒ ͑ ͒ ϩ tr͑C 1C C 1C ͒, ͑C4͒ tr C CcC Cde , C8 2 ab c

n2 1 ͑ 2͒ϭ ͭ ͑ Ϫ1 Ϫ1 ͒ͫ ͑ Ϫ1 Ϫ1 Ϫ1 Ϫ1 ͒Ϫ ͑ Ϫ1 Ϫ1 Ϫ1 ͒ͬͮ va,b,cd,ef n ͚ tr C CcC Cd tr C CaC CbC CeC Cf tr C CaC CbC Cef 8 ͑cd,ef ͒ϫ͑a,b͒ϫ͑c,d͒ϫ͑e,f ͒ 2

n2 1 ϩ ͭ ͑ Ϫ1 Ϫ1 ͒ͫ ͑ Ϫ1 Ϫ1 Ϫ1 Ϫ1 ͒ ͚ tr C CaC Cb tr C CcC CdC CeC Cf 8 ͑cd,ef ͒ϫ͑a,b͒ϫ͑c,d͒ϫ͑e, f ͒ 2

1 Ϫ tr(CϪ1C CϪ1C CϪ1C )ϩ 1tr(CϪ1C CϪ1C )ͬͮ 2 c d ef 8 cd ef

n2 ϩ Ϫ1 Ϫ1 Ϫ1 ͒ Ϫ1 Ϫ1 Ϫ1 ͒ ͚ tr͑C CaC CcC Cd tr͑C CbC CeC Cf 8 ͑cd,ef ͒ϫ͑a,b͒ϫ͑c,d͒ϫ͑e, f ͒

1928 J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB n2 ϩ Ϫ1 Ϫ1 ͒ Ϫ1 Ϫ1 ͒ ͚ tr͑C CaC Ccd tr͑C CbC Cef 8 ͑cd,ef ͒ϫ͑a,b͒ n2 Ϫ Ϫ1 Ϫ1 ͒ Ϫ1 Ϫ1 Ϫ1 ͒ ͑ ͒ ͚ tr͑C CaC Ccd tr͑C CbC CeC Cf , C9 8 ͑cd,ef ͒ϫ͑a,b͒ϫ͑c,d͒ϫ͑e, f ͒ n2 2͒ϭ Ϫ1 Ϫ1 Ϫ1 ͒ Ϫ1 Ϫ1 Ϫ1 ͒ va,b,c,def͑n ͚ tr͑C CaC CbC Cc tr͑C CdC CeC Cf 18 ͑a,b,c͒ϫ͑d,e, f ͒ n2 Ϫ Ϫ1 Ϫ1 Ϫ1 ͒ Ϫ1 Ϫ1 ͒ ͚ tr͑C CaC CbC Cc tr͑C CdeC Cf 24 ͑a,b,c͒ϫ͑d,e, f ͒ n2 ϩ Ϫ1 Ϫ1 ͒ Ϫ1 Ϫ1 Ϫ1 Ϫ1 ͒ ͚ ͕tr͑C CaC Cb ͓tr͑C CcC CdC CeC Cf 8 ͑a,b,c͒ϫ͑d,e, f ͒ Ϫ ͑ Ϫ1 Ϫ1 Ϫ1 ͒ϩ 1 ͑ Ϫ1 Ϫ1 ͔͖͒ ͑ ͒ tr C CcC CdeC Cf 6tr C CcC Cdef . C10

͚ 1 ͑ The notation (a,b,c)ϫ(d,e, f ) indicates summation over all C. R. Rao, Linear Statistical Inference and its Applications Wiley, New possible permutations of d, e, f and a, b and c, leading to a York, 1966͒. 2 total of 36 terms. A. Battacharya, ‘‘On some analogues of the amount of information and their uses in statistical estimation,’’ Sonkhya 8, 1–14, 201–218, 315–328 ͑1946͒. 3 E. W. Barankin, ‘‘Locally best unbiased estimators,’’ Ann. Math. Stat. 20, 477–501 ͑1949͒. 4 D. Chazan, M. Zakai, and J. Ziv, ‘‘Improved lower bounds on signal APPENDIX D: DERIVATION OF THE ASYMPTOTIC parameter estimation,’’ IEEE Trans. Inf. Theory IT-21, 90–93 ͑1975͒. EXPANSION OF THE MAXIMUM LIKELIHOOD 5 A. J. Weiss and E. Weinstein, ‘‘Fundamental limitations in passive time ESTIMATE delay estimation,’’ IEEE Trans. Acoust., Speech, Signal Process. ASSP- 31, 472–485 ͑1983͒. Following Barndoff-Nielsen and Cox,7 Eq. ͑1͒ is first 6 R. A. Fisher, Statistical Methods and Scientific Inference ͑Hafner, New ͒ inverted for (␪ˆ Ϫ␪)r to obtain the expansion York, 1956 . 7 O. E. Barndorff-Nielsen and D. R. Cox, Inference and Asymptotics ͑Chap- ͒ ͑␪ˆ Ϫ␪͒rϭ rs ϩ 1 rs ͑␪ˆ Ϫ␪͒t͑␪ˆ Ϫ␪͒uϩ 1 rs man & Hall, London, 1994 . j ls 2 j lstu 6 j lstuv 8 H. L. Van Trees, Detection, Estimation, and Linear Modulation Theory— Part III ͑Wiley, New York, 1971͒. ϫ͑␪ˆ Ϫ␪͒t͑␪ˆ Ϫ␪͒u͑␪ˆ Ϫ␪͒vϩ ͑ ͒ 9 ¯ , D1 J. W. Goodman, Statistical Optics ͑Wiley, New York, 1985͒. 10 rs O. Diachok, A. Caiti, P. Gerstoft, and H. Schmidt, Eds., Full Field Inver- where j is the inverse of the observed information matrix sion Methods in Ocean and Seismic Acoustics ͑Kluwer, Dordrecht, 1995͒. ϭϪ 11 jrs lrs . Iterating this procedure leads to an expression for A. Thode, E. Naftali, Ian Ingram, P. Ratilal, and N. C. Makris, ‘‘Necessary (␪ˆ Ϫ␪)r that is solely in terms of the derivatives of the like- conditions for a maximum likelihood estimate to attain the Cramer-Rao lihood function lower bound. Part II: Range and depth localization of a sound source in an ocean waveguide,’’ J. Acoust. Soc. Am. ͑submitted͒. ͑␪ˆ Ϫ␪͒rϭ rs ϩ 1 rs tu vw ϩ 1 rs tu vw xy͑ 12 A. Zeira and P. Schultheiss, ‘‘Realizable lower bounds for time delay j ls 2 j j j lstvlulw 6 j j j j lsuyw estimation,’’ IEEE Trans. Signal Process. 41, 3102–3113 ͑1993͒. ϩ pq ͒ ϩ ͑ ͒ 13 A. B. Baggeroer, ‘‘Barankin bound on the variance of estimates of the 3lswpj lquy ltlvlx . D2 ¯ parameter of a Gaussian random process,’’ M.I.T. Res. Lab., Cambridge, rs The difficulty with this expression is that j is not well MA, Electron, Quart. Prog. Rep. 92, 324–333 ͑1969͒. 14 defined for all likelihood functions and all values of ␪ˆ . This L. R. Shenton and K. O. Bowman, Maximum Likelihood Estimation in ͑ ͒ problem is circumvented by expanding jrs in terms of well- Small Samples Charles Griffin, London, 1977 . 15 M. Zanolin, E. Naftali, and N. C. Makris, ‘‘Second order bias of a multi- defined quantities. First, note that variate Gaussian maximum likelihood estimates with a chain rule for ͑ ͒ ϭ ͕ Ϫ Ϫ1͑ Ϫ ͖͒ ͑ ͒ higher moments,’’ J. Roy. Statist. Soc. B submitted . j i I i i j , D3 16 N. C. Makris, ‘‘Parameter resolution bounds that depend on sample size,’’ where I is the identity matrix. The inverse is then J. Acoust. Soc. Am. 99, 2851–2861 ͑1996͒. The misstatement in this paper that the distribution for the sample covariance of a CCGR data vector is jϪ1ϭ͕IϪiϪ1͑iϪj͖͒Ϫ1 iϪ1. ͑D4͒ unknown is corrected in Ref. 17 below. The distribution in question is the complex Wishart distribution. which can be expanded as 17 N. C. Makris, ‘‘The statistics of ocean-acoustic ambient noise,’’ in Sea ͑ Ϫ1ϭ Ϫ1ϩ Ϫ1͑ Ϫ ͒ Ϫ1ϩ Ϫ1͑ Ϫ ͒ Ϫ1͑ Ϫ ͒ Ϫ1ϩ Surface Sound ’97, edited by T. Leighton Kluwer Academic, Dordrecht, j i i i j i i i j i i j i , ͒ ¯͑ ͒ 1997 . D5 18 B. R. Kerman, Ed., Sea Surface Sound, Natural Mechanisms of Surface or equivalently as Generated Noise in the Ocean ͑Kluwer, Boston, 1987͒. 19 N. C. Makris, ‘‘The effect of saturated transmission scintillation on ocean- rsϭ rsϩ rt su ϩ rt su vw ϩ ͑ ͒ j i i i Htu i i i HtvHuw ¯ , D6 acoustic intensity measurements,’’ J. Acoust. Soc. Am. 100, 769–783 ϭ Ϫ ͑1996͒. where HR lR vR for any set of coordinate indices R 20 ϭ ϭ Ϫ N. C. Makris, ‘‘A foundation for logarithmic measures of fluctuating in- r1 ...rm , where, for example, Htu ltu vtu . Inserting Eq. tensity in pattern recognition,’’ Opt. Lett. 20, 2012–2014 ͑1995͒. ͑D6͒ into Eq. ͑D2͒ then leads to Eq. ͑2͒. 21 N. C. Makris and P. Ratilal, ‘‘A unified model for reverberation and sub-

J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB 1929 merged target scattering in a stratified ocean waveguide,’’ J. Acoust. Soc. 26 A. B. Baggeroer, W. A. Kuperman, and P. H. Mikhalevsky, ‘‘An overview Am. 109, 909–941 ͑2001͒. of matched field methods in ocean acoustics,’’ IEEE J. Ocean Eng. 18, 22 P. Swerling, ‘‘Probability of detection for fluctuating targets,’’ Rand Re- 401–424 ͑1993͒. ͑ ͒ port RM 1217 1954 ; reissued in Trans. IRE Prof. Group Inf. Theory 27 Naftali, ‘‘First order bias and second order variance of the Maximum ͑ ͒ IT-6, 269–308 1960 . Likelihood Estimator with application to multivariate Gaussian data and 23 P. G. Bergmann, ‘‘Intensity fluctuations,’’ in The Physics of Sound in the time delay and Doppler shift estimation,’’ SM Thesis, MIT, Cambridge, Sea, Part I: Transmission ͑National Defense Research Committee, Wash- ͒ MA, 2000. ington, DC, 1946 . 28 24 I. Dyer, ‘‘Statistics of sound propagation in the ocean,’’ J. Acoust. Soc. P. M. Woodward, Probability and Information Theory, with Application to ͑ ͒ Am. 48, 337–345 ͑1970͒. Radar Pergamon, New York, 1964 . 29 25 A. B. Baggeroer, W. A. Kuperman, and H. Schmidt, ‘‘Matched field pro- N. C. Makris, C. S. Chia, and L. T. Fialkowski, ‘‘The bi-azimuthal scat- cessing: Source localization in correlated noise as an optimum parameter tering distribution of an abyssal hill,’’ J. Acoust. Soc. Am. 106,2491– estimation problem,’’ J. Acoust. Soc. Am. 83, 571–587 ͑1988͒. 2512 ͑1999͒.

1930 J. Acoust. Soc. Am., Vol. 110, No. 4, October 2001 E. Naftali and N. C. Makris: Necessary conditions for a MLE to attain CRLB