A Bound on Mean-Square Estimation Error With ... - Semantic Scholar
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 4, APRIL 2004
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A Bound on Mean-Square Estimation Error With Background Parameter Mismatch Wen Xu, Member, IEEE, Arthur B. Baggeroer, Fellow, IEEE, and Kristine L. Bell, Senior Member, IEEE
Abstract—In typical parameter estimation problems, the signal observation is a function of the parameter set to be estimated as well as some background (environmental/system) parameters assumed known. The assumed background could differ from the true one, leading to biased estimates even at high signal-to-noise ratio (SNR). To analyze this background mismatch problem, a Ziv–Zakai-type lower bound on the mean-square error (MSE) is developed based on the mismatched likelihood ratio test (MLRT). At high SNR, the bound incorporates the increase in MSE due to estimation bias; at low SNR, it includes the threshold effect due to estimation ambiguity. The kernel of the bound’s evaluation is the error probability associated with the MLRT. A closed-form expression for this error probability is derived under a random signal model typical of the bearing estimation/passive source localization problem. The mismatch is then analyzed in terms of the related ambiguity functions. Examples of bearing estimation with system (array shape) mismatch demonstrate that the developed bound describes the simulations of the maximum-likelihood estimate well, including the sidelobe-introduced threshold behavior and the bias at high SNR. Index Terms—Background mismatch, maximum-likelihood estimate, parameter estimation performance bound.
I. INTRODUCTION
I
N the framework of signal parameter estimation, the embedded parameters can be divided into two categories: those being estimated and those assumed known. We call the latter the background parameter set. For example, in a bearing estimation problem, the signal bearing is the parameter to be estimated and the background parameter set may include the position of each sensor in the receiver array. Other example background sets include the channel model parameters in a communication problem or waveguide parameters in underwater passive source localization. For nonlinear parameter estimation, the typical performance demonstrates a so-called threshold phenomenon. At high SNR, some asymptotic performance bound (say, the Cramér–Rao lower bound (CRB) [1]) can be achieved by a maximum-like-
Manuscript received June 26, 2002; revised October 30, 2003. This work was supported by the Office of Naval Research under Grants N00014-01-1-0817 and N00014-01-1-0257 and by a Woods Hole Oceanographic Institution Education Fellowship. W. Xu was with the Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 USA. He is now with RD Instruments, San Diego, CA 92131 USA (e-mail: [email protected]). A. B. Baggeroer is with the Departments of Ocean and Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: [email protected]). K. L. Bell is with the Department of Applied and Engineering Statistics, George Mason University, Fairfax, VA 22030 USA (e-mail: [email protected]). Communicated by V. V. Veeravalli, Associate Editor for Detection and Estimation. Digital Object Identifier 10.1109/TIT.2004.825023
lihood estimator (MLE); below some signal-to-noise ratio (SNR), the MLE performance departs from the asymptotic behavior due to the ambiguity caused by the nonlinearity. However, when the assumed background is mismatched to the true one, the detected signal observation differs from the modeled one even at the true values of the parameters being estimated. Accordingly, the performance is degraded even at high SNR and the CRB is no longer accurate. A typical way to avoid this problem is to include the background parameters in the parameter set being estimated. This significantly increases the computational and analytical complexity, particularly when the dimensionality of the parameter space is high. Instead, an environmental/system model is often assumed and some of the parameter values are assigned per one’s best knowledge. Hence, it is very common that the assumed environmental/system data differ from the actual physical conditions of an application, and sensitivity to mismatch becomes an important factor in evaluating a parameter estimation algorithm. Environmental mismatch refers to uncertainty in the environmental model, e.g., the communication channel or wave-propagation channel; system mismatch refers to errors in the receiving system like array shape and Doppler. Earlier research [2]–[4] has developed some interesting results in the context of array processing, where accurate sensor information (position and pattern) is hardly available. The desired parameters (bearings of sources) are nonrandom, while the sensor uncertainties are modeled as random perturbations. A hybrid CRB is exploited for the attainable accuracy in the estimation of both the desired parameters and the sensor parameters. In order to make the analysis tractable, some of the terms are evaluated at the nominal parameter values, which otherwise should be averaged over the a priori distribution of the random sensor parameters. The validity of this approximation requires that the variance of each parameter is small, and the bound becomes a pseudo-CRB, losing its theoretical properties. Nevertheless, the mismatch between the nominal sensor parameters and the modeled uncertainties is included, and the bound appears to give realistic results. Mismatch analysis has also long been an important topic in matched-field processing for passive source localization [5], which requires accurate environmental/system information. So far, few results have been reported to connect the performance measure (e.g., bias or mean-square error (MSE)) with the size of mismatch. This paper presents a lower bound on MSE under the given background mismatch. It is a Ziv–Zakai-type bound derived from a mismatched likelihood ratio test (MLRT). Such mismatched test has been addressed in the communication litera-
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ture. For example, an approximate error analysis is developed for a maximum-likelihood receiver, which uses a mismatched channel model to decode the information sequence [6]. Compared to the hybrid CRB, the bound developed here does not impose any theoretical limitation on the size of mismatch; and it includes the threshold behavior as well. The major developments are done in the context of deterministic mismatch, however, it is straightforward to extend to the case of stochastic mismatch but with increased computational complexity. The concept of mismatch analysis works well with the Ziv–Zakai bound because the likelihood ratio test (LRT) depends on both the observation and the assumed statistical information about the observation. This enables us to separate the observation and the assumed statistics in evaluating the error probability. In contrast, in deriving the Bayesian Cramér–Rao bound [7] or Weiss–Weinstein bound [8], the observation vector is integrated out and the bound can only be expressed in terms of the assumed statistics. Evaluation of the error probability of the MLRT determines the validity and tightness of the bound. For the mismatch-free case, there have been considerable efforts analyzing the detection performances under different data models [7], [9], [10]. However, except for some simple cases, direct calculation of the error probability is difficult and some approximations or bounds have been developed. Those small-error approximations based on the distribution tail are no longer applicable to the case with mismatch since now the error probability can be close to one. In a recent development [11], some closed-form expressions for error analysis of the regular LRT are derived for a random signal model typical of many detection and estimation problems. This analysis will be exploited to develop the exact probability of error for the MLRT addressed in this paper. This not only benefits the bound’s evaluation but also facilitates mismatch analysis in terms of the ambiguity function. In general, the effect of mismatch includes two aspects. First, it causes degradation in the main-lobe peak. Second, it shifts the peak away from the correct position, leading to biased estimates. We shall see how it determines the behavior of the MLE. The rest of the paper is organized as follows. Section II introduces a probabilistic framework for parameter estimation under mismatch and derives a lower bound following the Ziv–Zakai approach [12], [13]. In Section III, the mismatch analysis is developed using a random signal model typical of the bearing estimation/passive source localization problem. Section IV then presents some demonstration examples, in which the array tilt angle is mismatched. Section V concludes the paper.
where any random observation vector which depends on a set of parameters and ; the parameter vector to be estimated; any other parameter vector, deterministic or random, not included in but necessary for specifying the pdf of ; and a set of constant scalars, vectors, or matrices, directly defining the pdf. It is a function of the embedded parameters, and , and can be constructed from the moments, for example, the mean and covariance matrix. For example, given is a real Gaussian random variable with but known variance , we have unknown mean
(2) Then , and . To describe a mismatched probability model, we introduce a so-called mismatched probability density function (3) The mismatched pdf has the same form as the regular pdf but is a function of the assumed background parameter now set , while the observation vector is, as always, a function of includes all the true background parameter set . Clearly, the parameters assumed for the background model except those could have being estimated, and some of the parameters in the same values as in . Following the example in (2), we now assume the variance is mismatched, given by . The mismatched pdf is then defined by
(4) Thus, , , , and . Equation (3) actually summarizes the data probability model often used in developing a parameter estimation algorithm, although it is often not explicitly specified that the practical observation vector behaves according to a different background parameter set. For example, the MLE can be expressed as
II. MEAN-SQUARE ERROR BOUND UNDER BACKGROUND MISMATCH
(5)
A. Probabilistic Model for Background Parameter Mismatch From the view of parameter estimation, the environmental/ system mismatch brings mismatch to the underlying data probability model. Let us first state the regular probability density function (pdf) by (1)
B. Derivation of a Ziv–Zakai-Type Lower Bound Consider first the estimation of a scalar parameter based upon noisy observations . The unknown parameter is assumed . A uniform distribution random with known prior pdf for is used throughout this paper except in over Section II-C.3, where the bound is generalized to arbitrarily
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Fig. 1.
Valley filling operation in (13). f
(1) =
P-
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(; + 1)d.
distributed vector parameter estimation. For any estimator we are interested in lower-bounding the MSE
,
(6)
parameter distribution (then an equally likely hypotheses test), we have -
where is the expectation with respect to the joint pdf of and , and can be an arbitrary bound. Some lower bounds have been developed in such a Bayesian framework [7], [14], [12], [8]. A typical one of interest is the Ziv–Zakai bound [14], [12], which is a detection theory bound derived from the error analysis of the binary hypotheses test [7]. In the context of parameter estimation, the binary hypotheses , is stated by test for two possible parameter values, and
-
(10)
is the probability that, given the true where , and parameter at , the test decision is is the error probability given that the true parameter is . In both cases, a mismatched background is used. To connect the error analysis in the above detection problem , the same to the mean-square estimation error, denoted by technique used by Chazan, Ziv, and Zakai [12], [13] can be followed. It is straightforward to show that
(7) where can be considered as the parameter perturbation. Under and mismatched background, the pdfs under the hypotheses are given by
(8) respectively. Note that one does not know the true background parameter set and uses an assumed background parameter set. Therefore, either the Bayes or Neyman–Pearson criterion [7] again leads us to an optimum test based on the likelihood ratio. To minimize the total probability of error, this LRT is stated by
(9)
We call (9) the mismatched LRT and denote as the associated error probability. For a uniform
-
(11)
which is the same as the Chazan–Ziv–Zakai bound but with the error probability associated with the regular LRT replaced by the one associated with the mismatched LRT. Applying the Bellini–Tartara valley filling function [15] (see also Fig. 1) (12) to (11) yields an improved bound
-
(13)
As shown in Fig. 1, if the bracketed function has a significant oscillatory structure (e.g., in an ambiguity-prone problem), this bound could be much tighter than the original one.
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In the cases with is further simplified
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 4, APRIL 2004
-
-
-
, the bound
be modified to include the background mismatch following -dimensional vector the derivation leading to (13). For any , the extended bound parameter with prior distribution of states
(14)
C. Some Generalizations
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1) Upper Bound on Bias: It is well known that the MSE is a sum of the local variance plus the bias squared both averaged over the prior parameter space
(15) where denotes some arbitrary estimates of a particular and is the mean estimate, i.e., . For the second term we have (16) where is the averaged bias, and the equality holds if and only if is independent of . Note that the averaged variance is always greater than zero. When the bound in (13) is achieved (which has been observed to be the case at high SNR in many problems), we immediately obtain an upper bound for the averaged bias MZZB
(17)
where MZZB is the MSE bound specified in (13). In fact, the bound in (17) is often tight since the variance is close to zero at high SNR. 2) Stochastic Mismatch: So far, the mismatch has been assumed to be deterministic or local. That means, given an as, we look at the mismatch effect for a sumed background given true background . The stochastic mismatch can be inas a random cluded in a straightforward way by modeling over gives us set. Integrating (13) with respect to
-
(18)
Obviously, this is much more complicated in analysis and comis high. putation particularly when the dimensionality of Some approximation or simplification such as the one in [2] is possible but beyond the scope of this paper. Usually, given the environmental/system uncertainty, one may evaluate (13) for a . This is often judicially chosen set of possible mismatch enough to characterize the mismatch effect as long as the size of the uncertainty is small. 3) Vector Parameter Estimation and Nonuniform Prior: Bell et al. extended the Bellini–Tartara bound to arbitrarily distributed vector parameter estimation [16]. This can again
(19)
-dimensional weighting vector. One can obwhere is any tain the bound for each individual parameter by choosing apgives the bound for propriately. For example, the first parameter. , the develIn summary, given the size of mismatch, oped Ziv–Zakai-type bound can be used to evaluate the MSE in parameter estimation. It is obvious that forms the kernel in the bound’s evaluation. In general, when a closed-form error probability for the regular LRT is available, the mismatched error probability is solved in a similar way, particularly when the likelihood ratio can be expressed in terms of the correlation between the observation vector and the computed replica (matched filter). In that case, replacing the matched replica with the mismatched replica does not change the form of the statistics of the likelihood ratio. Such an example is addressed in Section III-A. III. ERROR ANALYSIS UNDER BACKGROUND MISMATCH In this section, we develop mismatch analysis for a data model with random signal embedded in random noise. Specifically, the mismatched LRT and its associated error probability are derived and the MLE error under mismatch is analyzed. It is worth noting that the studied data model exists in many detection and estimation problems including bearing estimation/passive source localization as well as various communication problems, thus, the approach developed here can be readily applied to those cases. In bearing estimation/passive source localization [16], [5], a source (electromagnetic or acoustic) radiates narrowor broad-band signals and the signal field is sampled by an array of receiving sensors. The source is usually assumed to be a stationary random process. Consequently, when the observation time is long enough (so that the window effect in spectral estimation can be ignored), individual discrete Fourier transform (DFT) bins of the received source signal are uncorrelated with each other [7]. In general, a number of snapshots are available for each frequency bin, and the number of snapshots is determined by the observation time and the correlation time of the source process. Suppose we have independent measurements for each of frequencies, , . The complex envelope vector of the received signal can be expressed as
(20) where
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includes the source bearing/location parameters to be estimated; the background parameter set required to define the signal propagation channel and the receiving system; an vector representing the th snapshot of the th frequency bin and is the number of receiving sensors; a random process incorporating amplitude and phase variability with a power spectral density equal of the source; a vector channel transfer function (unit impulse response function) for the propagation between the source and the receiving sensors; and the th snapshot of the stationary white noise vector in the th frequency bin. Given the data model in (20), dependence on channel/system parameters is embedded in the channel transfer function term. Traditional bearing estimation assumes a plane wave model is defined by some harmonic terms at [16], so the reference sensor and their delayed versions at other sensor positions. Recent developments in underwater passive source localization [5] exploit a full wave modeling of signal channel propagation. They are particularly useful for the case of a large vertical/horizontal aperture where nonplane wave signals are encountered. Both the signal and noise terms are assumed to follow a zero-mean complex Gaussian distribution. The covariance is given by matrix for
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Now we assume some mismatch in background parameters, is different from . The mismatched pdf is available thus, by modifying the parameter dependence in (23), which is
(24) The maximum-likelihood estimate in (5) can then be derived under mismatch. Using the eigendecomposition of (21) and the Woodbury identity [1], we have
(25)
(26) is the norm of the complex vector and where is the normalized channel transfer function defined by (27)
(21) denotes the complex conjugate transpose operation, where is an identity matrix, and and denote the variances of the signal process and the noise process, respectively. Denote
(22) represents the matrix transpose operation. Obviis an column vector. Since , , , are uncorrelated across frequencies and snapshots, the covariance matrix of is a block-diagonal as the diagonal component. Then the matrix with conditional pdf of is
where ously,
We assume known and and thus maximize (24) directly. Substituting (25) and (26) into (24), assuming a -independence of the norm of the transfer function,1 and rejecting constant terms, the maximum-likelihood estimate of is given by (28) and . Note that (28) is actually independent of Application of the MLE to random parameter estimation is justified by assuming a uniform a priori parameter distribution. In this case, the MLE is equivalent to the maximum a posteriori estimate (MAP) [1]. A. Error Probability Associated With the Mismatched LRT Having obtained the MLE in (28), the mismatched likelihood ratio in (9) immediately follows
(29) (23) where
denotes the matrix determinant.
1This is the case in bearing estimation and many other practical problems. However, this assumption can be removed based on a recent development on MLE error analysis under spatially correlated noise field [17].
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The minimum error probability associated with the misand . matched LRT involves , Due to the symmetry, we only need to solve which is
For the case with multiple frequency components, an exact error probability is also available
-
(39)
where
(40) (30) and are then obtained by replacing and accordingly. Substituting the resulting error probabilities into (13) or (19), we are able to evaluate the modified Ziv–Zakai bound under the given background mismatch . The error probability in (30) is solved following the approach developed for the regular LRT [11]. The detailed derivations are given in the Appendix . To simplify the notations, three related correlation terms are first defined (31) (32) (33) These are also called signal ambiguity functions since they are often characterized by a multimodal structure showing a main lobe and many sidelobes, due to highly nonlinear dependences on the embedded parameters. We also find it is useful to define a SNR term SNR For the case with a single frequency component at error probability is (see the Appendix )
(34) , the
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(41) and and , , are solved using (36)– is the partial fraction expansion (38) at each . Note that . For large coefficient for the moment generating function, and , computing can be quite time consuming. B. Mismatch Analysis To implement the error analysis for the mismatched MLE, we and scan within assume a fixed true parameter position at the specified parameter interval; we then look at the error prob. It is shown ability of the corresponding MLRT, to be a good first-order approximation to the exact MLE error probability at [11], provided that the exact one is not available. Calculation of the MLRT error probability involves three correlation terms of the received signal in (31)–(33) with replaced by . The first one, , is for the assumed and the true background parameter sets, with the parameter being estimated at the true position. This measures the peak degradation due to mismatch. The second one, , is for the true, matched parameter set and the scanning, mismatched parameter set. As shown in Section IV, this could introduce peak shift. The third one, , corresponds to the true and the scanning parameter set both under mismatched background. This is replaced similar to the regular ambiguity function with by . Consider the single-frequency and scalar parameter case. We can rewrite the error probability by
(35)
(42)
where
where (36)
and
and
are defined by
(43) For perfectly known background, and for shown that
(37) -
and
(38) respectively.
, it can be easily (44)
Thus, the error probability is inversely related to , which itself is a function of the SNR and the field correlation. For a fixed SNR, the maximum error probability is achieved at the minimum , which corresponds to the maximum field correlation obtained by the closest main-lobe point. Under mismatched background, the situation is different.
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Fig. 2. Uniform linear array. Actual and assumed array positions are also specified.
When is small, although is smaller than one (the peak value without mismatch), it is still the highest correlation value. In this case, the bias in parameter estimation can still be ignored but the MSE would increase. As increases, decreases further such that it is no longer the maximum correlation, i.e., at some , we have In this case, the mismatched error probability is characterized by two operation regions. The first region has and thus, (cf. (37)) and mismatch-free case. Note that
. This is similar to the (45)
where
(46) Obviously, the minimum imum error probability of when
is one, which yields the max(cf. (42)). is obtained
This corresponds to another parameter point which cannot be resolved from the true parameter point by considering the correlation between the mismatched field and the true, mismatch-free field. The second region is specified by In this region,
and
.
the mismatched field at the true parameter position ). From (42), is one ( is zero. Thus, the error probability falls between when and . , the peak correlation point For a reasonable size of is still within the main lobe around in the true parameter. Thus, the second operation region can be on either side of the true parameter point but not on both. As we later see in the examples, errors around the incorrect correlation peak dominate the performance in the high-SNR region. This introduces a mismatch-dependent bias in parameter estimation. At low SNR, the sidelobe contributions through the terms of and become important. IV. EXAMPLES Consider the problem where a plane-wave signal impinges on a uniform linear array of sensors as shown in Fig. 2. The variable denotes the angle (from the horizontal line) of a plane wave arrival. To study the system mismatch, which is typical for as traditional array problems, we choose the array tilt angle the background parameter. The true and assumed tilt angles are and , respectively. Under this configuration, denoted by , where the parameter to be estimated is is assumed known, although might be mismatched. We assume has a uniform distribution on : (48) We further assume a narrow-band source signal centered about a known frequency . The channel transfer function is obtained by choosing the first sensor as the reference sensor (49) where is the wave propagation speed in the medium and is the sensor spacing. For this plane-wave signal model, the ambiguity function in (33) has a nice form
is then given by (47)
which is now smaller than one. The minimum value of is actually zero, which is achieved at high SNR by some close main-lobe points whose mismatched fields are more correlated with the true, mismatch-free field ( ) and also highly correlated with
(50) where . To avoid grating lobes (poles of . (50)), one often choose less than half-wavelength , , For the examples in this section, we choose . We first evaluate the ZZB for the mismatch-free and
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Fig. 3. Signal ambiguity function in bearing estimation. The dotted line corresponds to the true bearing.
Fig. 5. Mismatched ambiguity function in bearing estimation. The bottom panel shows the details around the true bearing. The dotted line corresponds 30 ); the dashdot line corresponds to 40 , at to the true bearing ( which the ambiguity level equals that at the true bearing. The array tilt angle is mismatched by 5 .
Fig. 4. Bearing estimation MSE given by the ZZB (solid line), and the MLE simulations (*). No mismatch is assumed. Three operation regions are also specified.
case [18], i.e., the actual array position is the assumed vertical , cf. Fig. 2). Fig. 3 displays an example position ( 30 (thus, ambiguity function given the true bearing at ). Some periodic ambiguity structure is observed with , where is an integer. sidelobes occurring at Note that varies from 90 to 90 . The error probability associated with the regular LRT, , can be solved from (42) with ( ). As seen from (50) and (34), the ambiguity property as well as the SNR (note that is a constant) is independent of . Accordingly, is only a function of . This greatly simplifies the evaluation of the Ziv–Zakai bound. Fig. 4 shows the evaluation results, in which three operation regions are observed. In the asymptotic region, where the SNR is high, estimation errors are small and linear with the SNR on 3 dB, estimation errors a logarithmic scale. Below SNR
=
increase significantly demonstrating a threshold phenomenon. This is attributed to the sidelobes of the field correlation and defines the transition region. At very low SNR, estimation errors are spread over the a priori parameter interval and the bound approaches the a priori variance. For comparison, the MLE simulation results are also presented on the basis of 4000 Monte Carlo trials at each SNR, where the true parameter for synthesizing the observation is uniformly chosen across the given interval. Clearly, the ZZB closely follows the behavior of the MLE simulations. Below the threshold SNR, MLE MSE are always (up to 7 dB) greater than the predictions. This discrepancy can be explained by noting that in the low-SNR region the optimal estimate is the conditional mean estimate rather than the MLE, which is then no longer achievable by any fundamental performance bound. We then assume there is some mismatch between the assumed . The ambiguity and the actual array positions, i.e., function in (32) is given by
(51) . An example is plotted in Fig. 5 where , 5 , and 30 . A peak degradation for of about 1 dB at the true bearing position (cf. (31)) is observed. we have Besides, at and the highest ambiguity level is now at , which cor. responds to , is calcuThe mismatched error probability, of (same as in Fig. 5). The lated for each and a fixed SNR is 0 dB which is defined on the single-sensor basis. Fig. 6 and , is shows the results. Between greater than one half and actually very close to one between . Outside this region, the highest ambiguity point and
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Fig. 6. Mismatched error probability in bearing estimation. The bottom panel shows the details around the true bearing. The dotted line corresponds to the 30 ); the dashdot line corresponds to 40 , at which true bearing ( the error probability equals that at the true bearing. The single sensor-averaged SNR is 0 dB. The array tilt angle is mismatched by 5 .
=
goes quickly down to zero, particularly on the left side of . Those are actually two operation regions discussed in Section III-B. Although presenting two almost opposite behaviors, they are subject to the same physical rule. That is, when the signal field at one parameter point is closer to the true signal field, the MLE outputs at other parameter points can hardly be beyond that at , particularly with low-level and , the signal field is alway noise. Between more correlated with the true field compared to that at the true bearing, and the error probability increases as SNR increases; outside this region, the mismatched field at the true bearing is always more correlated with the true field compared to those at all other points, and the error probability increases as SNR decreases. This explains why the transition between two regions is so abrupt at high SNR; at low SNR, the transition is much smoother. It is interesting to note that to reduce the estimation error in the high-SNR region, one may want to add some noise. Now it is clear how a bias is introduced by errors around the incorrect ambiguity peak. This bias term can be predicted using the developed bound in (13). Figs. 7 and 8 give the results for 1 and 5 , respectively. We see that as goes ( ), the bound is nearly independent of the away from SNR at high SNR, and is actually dominated by the square of the mismatch-introduced bias. The size of the bias can be approximately computed using the bound in (17). The sign of the bias can be determined from the position of the mismatched 1 , the premain-lobe peak. Using this approach, for ; for 5 , it is . Note that dicted bias is about is a dimensionless quantity. The above predictions can be verified in this simple problem. ; thus, at each the bias in the estimation of Note that is given by (52)
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Fig. 7. Bearing estimation MSE given by the modified ZZB (solid line), and the MLE simulations (*). The array tilt angle is mismatched by 1 . The regular ZZB in Fig. 4 is also shown (dotted line).
Fig. 8. Bearing estimation MSE given by the modified ZZB (solid line), and the MLE simulations (*). The array tilt angle is mismatched by 5 . The regular ZZB in Fig. 4 is also shown (dotted line).
for small . Ignoring the effects around the endfire region, the averaged bias is solved by (53) for 1 , or In this example, the estimated bias is for 5 , both of which differ from the bound by less than 4%. V. CONCLUSION A Ziv–Zakai-type bound has been developed to bound the MSE of parameter estimation when using incorrect background environmental/system parameters. The basis of this development is the MLRT identifying the true and the assumed background parameter sets embedded in the observation and the signal model respectively. It incorporates the increase in MSE due to estimation bias at high SNR; at low SNR, it
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includes effects of the sidelobe ambiguity as well. It is the first large-error bound capable of incorporating background mismatch. To demonstrate the application of the bound, error analysis of the MLRT is implemented using a random signal model typical of bearing estimation/passive source localization. Error under mismatch is determined by the correlation level degradation at the true parameter position and the location shift of the ambiguity peak, thus explaining the introduced bias at high SNR. Examples of bearing estimation show that the developed bound closely predicts the performance of the maximum-likelihood estimate in the presence of system mismatch, including the MSE in all SNR regions as well as the bias at high SNR. The bound has also been successfully applied to more complicated problems in passive source localization [11]. Evaluation of the scalar-parameter bound with deterministic mismatch can often be accomplished in a computationally efficient manner; generalization to include stochastic mismatch and vector parameter estimation is straightforward in theory but can be challenging to implement. Background mismatch is very common in practical detection/estimation applications. The development in this paper provides a useful tool, as well as opening some new insight, for performance analysis in such problems. For example, it can be seen from (13) that large increase in the MSE matches large increase in the probability error, which starts from an SNR lowerbounded by the SNR at which the channel capacity equals one [19]. By examining the channel capacity change due to background mismatch, one is able to examine how the mismatch influences the position of the threshold SNR.
We express each as (recall that we are assuming the true parameter point is ) (57) This is actually a whitening operation such that each new is complex Gaussian with identity covariance snapshot can be expressed in terms of matrix. Hence, -
(58) , i.e., Note that the above expression is available even for the number of snapshots is smaller than the number of sensors. Denote by matrix eigendecomposition (59) where , and as later shown, only two nonzero . Since is a eigenvalues exist, i.e., unitary matrix, multiplying by does not alter the covariance and then the distribution for Gaussian model [20]. It follows that (60) (61) Therefore, the error probability can be further derived as
APPENDIX DERIVATIONS OF THE ERROR PROBABILITY IN MLRT
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A. Case 1: Single-Frequency Component For a single-frequency component with multiple snapshots, the desired error probability is
(62) The term in parentheses is a weighted sum of two independent degree- complex Chi-squared variables. Thus, we have
-
(63)
(54) We denote data matrix of snapshots by
Note that a positive (55)
Recall that snapshots are independent and identically distributed according to a complex Gaussian distribution. Then the error probability in (54) can be written as
is required in deriving (63).
The statistics of the complex -distribution, , is documented in [21, Appendix A]. Its cumulative distribution function is given by
(64) Applying (64) to (63), we finally obtain (35). To solve the eigenvalues of (59), we first notice that (56) where
represents the trace operation.
(65)
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and
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Note that is a real variable, and the region of convergence [22] is defined by for (66)
share the same eigenvalues. Using (21), we have (73) has a partial-fraction expansion of the form [23] (74) where (75)
(67) In general, for , and are not colinear (otherwise, they are the same vector). It can be easily shown that (67) expands a two-dimensional space, and the corresponding eigenvectors are linear combinations and . Thus, multiplying (67) by of ( is a complex number) gives . Setting the coefficients and to zero yields a quadratic of equation of
(76) Note that for
and
(77)
(68) and for
and
where and are defined in (37) and (38), respectively. The solution for eigenvalues is then given by (36). B. Case 2: Multiple-Frequency Component Using a similar derivation for single-frequency component, the error probability for the multiple-frequency case is given by
(78) Therefore, the pdf of
is given by
(69)
. (79)
Now we define
Finally, the error probability is (70)
-
. To find the pdf of , we so that first observe that each Chi-squared variable is independent of the others, so the moment generating function is easily solved
(71) , , are computed using (37), (38) and (36). Clearly (see also comments below (63))
(72)
(80)
ACKNOWLEDGMENT The authors wish to thank Dr. C. Richmond for discussions on error analysis of the likelihood ratio test. The derivation of a closed-form LRT error probability directly came from those discussions. They are also deeply grateful to the anonymous reviewers for valuable suggestions which have been incorporated into the manuscript.
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REFERENCES [1] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: PTR Prentice-Hall, 1993. [2] Y. Rockah and P. M. Schultheiss, “Array shape calibration using sources in unknown locations—Part I: Far-field sources,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 286–299, Mar. 1987. [3] J. X. Zhu and H. Wang, “Effects of sensor position and pattern perturbations on CRLB for direction finding of multiple narrow-band sources,” in Proc. IEEE 4th Annu. ASSP Workshop on Spectrum Estimation and Modeling, 1988, pp. 98–102. [4] B. Wahlberg, B. Ottersten, and M. Viberg, “Robust signal parameter estimation in the presence of array perturbations,” in Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing (ICASSP’91), 1991, pp. 3277–3280. [5] A. B. Baggeroer, W. A. Kuperman, and P. N. Mikhalevsky, “An overview of matched field methods in ocean acoustics,” IEEE J. Oceanic Eng., vol. 18, pp. 401–424, Oct. 1993. [6] K. Balachandran and J. B. Anderson, “Error performance of mismatched receivers for linear-coded modulation,” IEEE Trans. Commun., vol. 48, pp. 1061–1065, July 2000. [7] H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: Wiley, 1968. [8] E. Weinstein and A. J. Weiss, “A general class of lower bounds in parameter estimation,” IEEE Trans. Inform. Theory, vol. 34, pp. 338–342, Mar. 1988. [9] H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part III. New York: Wiley, 1971. [10] K. L. Bell, “Performance bounds in parameter estimation with application to bearing estimation,” Ph.D. dissertation, George Mason Univ., Fairfax, VA, 1995. [11] W. Xu, “Performance bounds on matched-field methods for source localization and estimation of ocean environmental parameters,” Ph.D. dissertation, MIT, Cambridge, MA, 2001.
[12] D. Chazan, M. Zakai, and J. Ziv, “Improved lower bounds on signal parameter estimation,” IEEE Trans. Inform. Theory, vol. IT-21, pp. 90–93, Jan. 1975. [13] A. J. Weiss and E. Weinstein, “Fundamental limitations in passive time delay estimation—Part I: Narrow-band systems,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, pp. 472–486, Feb. 1983. [14] J. Ziv and M. Zakai, “Some lower bounds on signal parameter estimation,” IEEE Trans. Inform. Theory, vol. IT-15, pp. 386–391, May 1969. [15] S. Bellini and G. Tartara, “Bounds on error in signal parameter estimation,” IEEE Trans. Commun., vol. COM-22, pp. 340–342, Mar. 1974. [16] K. L. Bell, Y. Steinberg, Y. Ephraim, and H. L. Van Trees, “Extended Ziv–Zakai lower bound for vector parameter estimation,” IEEE Trans. Inform. Theory, vol. 43, pp. 624–637, Mar. 1997. [17] W. Xu and C. D. Richmond, “Quantitative ambiguity analysis for matched-field source localization under spatially-correlated noise field,” in Proc. MTS/IEEE OCEANS’03, 2003, pp. 922–927. [18] K. L. Bell, Y. Ephraim, and H. L. Van Trees, “Explicit Ziv-Zakai lower bound for bearing estimation,” IEEE Trans. Signal Processing, vol. 44, pp. 2810–2824, Nov. 1996. [19] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. [20] A. Steinhardt, “Adaptive multisensor detection,” in Adaptive Radar Detection and Estimation, S. Haykin and A. Steinhardt, Eds. New York: Wiley, 1992, ch. 3, pp. 126–137. [21] C. D. Richmond, “Performance of the adaptive sidelobe blanker detection algorithms in homogeneous environments,” IEEE Trans. Signal Processing, vol. 48, pp. 1235–1247, May 2000. [22] A. V. Oppenheim and A. S. Willsky, Signals and Systems. Upper Saddle River, NJ: Prentice-Hall, 1997. [23] C. D. Richmond, “Performance of a class of adaptive detection algorithms in nonhomogeneous environments,” IEEE Trans. Signal Processing, vol. 48, pp. 1248–1262, May 2000.
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A Bound on Mean-Square Estimation Error With Background Parameter Mismatch Wen Xu, Member, IEEE, Arthur B. Baggeroer, Fellow, IEEE, and Kristine L. Bell, Senior Member, IEEE
Abstract—In typical parameter estimation problems, the signal observation is a function of the parameter set to be estimated as well as some background (environmental/system) parameters assumed known. The assumed background could differ from the true one, leading to biased estimates even at high signal-to-noise ratio (SNR). To analyze this background mismatch problem, a Ziv–Zakai-type lower bound on the mean-square error (MSE) is developed based on the mismatched likelihood ratio test (MLRT). At high SNR, the bound incorporates the increase in MSE due to estimation bias; at low SNR, it includes the threshold effect due to estimation ambiguity. The kernel of the bound’s evaluation is the error probability associated with the MLRT. A closed-form expression for this error probability is derived under a random signal model typical of the bearing estimation/passive source localization problem. The mismatch is then analyzed in terms of the related ambiguity functions. Examples of bearing estimation with system (array shape) mismatch demonstrate that the developed bound describes the simulations of the maximum-likelihood estimate well, including the sidelobe-introduced threshold behavior and the bias at high SNR. Index Terms—Background mismatch, maximum-likelihood estimate, parameter estimation performance bound.
I. INTRODUCTION
I
N the framework of signal parameter estimation, the embedded parameters can be divided into two categories: those being estimated and those assumed known. We call the latter the background parameter set. For example, in a bearing estimation problem, the signal bearing is the parameter to be estimated and the background parameter set may include the position of each sensor in the receiver array. Other example background sets include the channel model parameters in a communication problem or waveguide parameters in underwater passive source localization. For nonlinear parameter estimation, the typical performance demonstrates a so-called threshold phenomenon. At high SNR, some asymptotic performance bound (say, the Cramér–Rao lower bound (CRB) [1]) can be achieved by a maximum-like-
Manuscript received June 26, 2002; revised October 30, 2003. This work was supported by the Office of Naval Research under Grants N00014-01-1-0817 and N00014-01-1-0257 and by a Woods Hole Oceanographic Institution Education Fellowship. W. Xu was with the Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 USA. He is now with RD Instruments, San Diego, CA 92131 USA (e-mail: [email protected]). A. B. Baggeroer is with the Departments of Ocean and Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: [email protected]). K. L. Bell is with the Department of Applied and Engineering Statistics, George Mason University, Fairfax, VA 22030 USA (e-mail: [email protected]). Communicated by V. V. Veeravalli, Associate Editor for Detection and Estimation. Digital Object Identifier 10.1109/TIT.2004.825023
lihood estimator (MLE); below some signal-to-noise ratio (SNR), the MLE performance departs from the asymptotic behavior due to the ambiguity caused by the nonlinearity. However, when the assumed background is mismatched to the true one, the detected signal observation differs from the modeled one even at the true values of the parameters being estimated. Accordingly, the performance is degraded even at high SNR and the CRB is no longer accurate. A typical way to avoid this problem is to include the background parameters in the parameter set being estimated. This significantly increases the computational and analytical complexity, particularly when the dimensionality of the parameter space is high. Instead, an environmental/system model is often assumed and some of the parameter values are assigned per one’s best knowledge. Hence, it is very common that the assumed environmental/system data differ from the actual physical conditions of an application, and sensitivity to mismatch becomes an important factor in evaluating a parameter estimation algorithm. Environmental mismatch refers to uncertainty in the environmental model, e.g., the communication channel or wave-propagation channel; system mismatch refers to errors in the receiving system like array shape and Doppler. Earlier research [2]–[4] has developed some interesting results in the context of array processing, where accurate sensor information (position and pattern) is hardly available. The desired parameters (bearings of sources) are nonrandom, while the sensor uncertainties are modeled as random perturbations. A hybrid CRB is exploited for the attainable accuracy in the estimation of both the desired parameters and the sensor parameters. In order to make the analysis tractable, some of the terms are evaluated at the nominal parameter values, which otherwise should be averaged over the a priori distribution of the random sensor parameters. The validity of this approximation requires that the variance of each parameter is small, and the bound becomes a pseudo-CRB, losing its theoretical properties. Nevertheless, the mismatch between the nominal sensor parameters and the modeled uncertainties is included, and the bound appears to give realistic results. Mismatch analysis has also long been an important topic in matched-field processing for passive source localization [5], which requires accurate environmental/system information. So far, few results have been reported to connect the performance measure (e.g., bias or mean-square error (MSE)) with the size of mismatch. This paper presents a lower bound on MSE under the given background mismatch. It is a Ziv–Zakai-type bound derived from a mismatched likelihood ratio test (MLRT). Such mismatched test has been addressed in the communication litera-
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ture. For example, an approximate error analysis is developed for a maximum-likelihood receiver, which uses a mismatched channel model to decode the information sequence [6]. Compared to the hybrid CRB, the bound developed here does not impose any theoretical limitation on the size of mismatch; and it includes the threshold behavior as well. The major developments are done in the context of deterministic mismatch, however, it is straightforward to extend to the case of stochastic mismatch but with increased computational complexity. The concept of mismatch analysis works well with the Ziv–Zakai bound because the likelihood ratio test (LRT) depends on both the observation and the assumed statistical information about the observation. This enables us to separate the observation and the assumed statistics in evaluating the error probability. In contrast, in deriving the Bayesian Cramér–Rao bound [7] or Weiss–Weinstein bound [8], the observation vector is integrated out and the bound can only be expressed in terms of the assumed statistics. Evaluation of the error probability of the MLRT determines the validity and tightness of the bound. For the mismatch-free case, there have been considerable efforts analyzing the detection performances under different data models [7], [9], [10]. However, except for some simple cases, direct calculation of the error probability is difficult and some approximations or bounds have been developed. Those small-error approximations based on the distribution tail are no longer applicable to the case with mismatch since now the error probability can be close to one. In a recent development [11], some closed-form expressions for error analysis of the regular LRT are derived for a random signal model typical of many detection and estimation problems. This analysis will be exploited to develop the exact probability of error for the MLRT addressed in this paper. This not only benefits the bound’s evaluation but also facilitates mismatch analysis in terms of the ambiguity function. In general, the effect of mismatch includes two aspects. First, it causes degradation in the main-lobe peak. Second, it shifts the peak away from the correct position, leading to biased estimates. We shall see how it determines the behavior of the MLE. The rest of the paper is organized as follows. Section II introduces a probabilistic framework for parameter estimation under mismatch and derives a lower bound following the Ziv–Zakai approach [12], [13]. In Section III, the mismatch analysis is developed using a random signal model typical of the bearing estimation/passive source localization problem. Section IV then presents some demonstration examples, in which the array tilt angle is mismatched. Section V concludes the paper.
where any random observation vector which depends on a set of parameters and ; the parameter vector to be estimated; any other parameter vector, deterministic or random, not included in but necessary for specifying the pdf of ; and a set of constant scalars, vectors, or matrices, directly defining the pdf. It is a function of the embedded parameters, and , and can be constructed from the moments, for example, the mean and covariance matrix. For example, given is a real Gaussian random variable with but known variance , we have unknown mean
(2) Then , and . To describe a mismatched probability model, we introduce a so-called mismatched probability density function (3) The mismatched pdf has the same form as the regular pdf but is a function of the assumed background parameter now set , while the observation vector is, as always, a function of includes all the true background parameter set . Clearly, the parameters assumed for the background model except those could have being estimated, and some of the parameters in the same values as in . Following the example in (2), we now assume the variance is mismatched, given by . The mismatched pdf is then defined by
(4) Thus, , , , and . Equation (3) actually summarizes the data probability model often used in developing a parameter estimation algorithm, although it is often not explicitly specified that the practical observation vector behaves according to a different background parameter set. For example, the MLE can be expressed as
II. MEAN-SQUARE ERROR BOUND UNDER BACKGROUND MISMATCH
(5)
A. Probabilistic Model for Background Parameter Mismatch From the view of parameter estimation, the environmental/ system mismatch brings mismatch to the underlying data probability model. Let us first state the regular probability density function (pdf) by (1)
B. Derivation of a Ziv–Zakai-Type Lower Bound Consider first the estimation of a scalar parameter based upon noisy observations . The unknown parameter is assumed . A uniform distribution random with known prior pdf for is used throughout this paper except in over Section II-C.3, where the bound is generalized to arbitrarily
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Fig. 1.
Valley filling operation in (13). f
(1) =
P-
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(; + 1)d.
distributed vector parameter estimation. For any estimator we are interested in lower-bounding the MSE
,
(6)
parameter distribution (then an equally likely hypotheses test), we have -
where is the expectation with respect to the joint pdf of and , and can be an arbitrary bound. Some lower bounds have been developed in such a Bayesian framework [7], [14], [12], [8]. A typical one of interest is the Ziv–Zakai bound [14], [12], which is a detection theory bound derived from the error analysis of the binary hypotheses test [7]. In the context of parameter estimation, the binary hypotheses , is stated by test for two possible parameter values, and
-
(10)
is the probability that, given the true where , and parameter at , the test decision is is the error probability given that the true parameter is . In both cases, a mismatched background is used. To connect the error analysis in the above detection problem , the same to the mean-square estimation error, denoted by technique used by Chazan, Ziv, and Zakai [12], [13] can be followed. It is straightforward to show that
(7) where can be considered as the parameter perturbation. Under and mismatched background, the pdfs under the hypotheses are given by
(8) respectively. Note that one does not know the true background parameter set and uses an assumed background parameter set. Therefore, either the Bayes or Neyman–Pearson criterion [7] again leads us to an optimum test based on the likelihood ratio. To minimize the total probability of error, this LRT is stated by
(9)
We call (9) the mismatched LRT and denote as the associated error probability. For a uniform
-
(11)
which is the same as the Chazan–Ziv–Zakai bound but with the error probability associated with the regular LRT replaced by the one associated with the mismatched LRT. Applying the Bellini–Tartara valley filling function [15] (see also Fig. 1) (12) to (11) yields an improved bound
-
(13)
As shown in Fig. 1, if the bracketed function has a significant oscillatory structure (e.g., in an ambiguity-prone problem), this bound could be much tighter than the original one.
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-
-
-
, the bound
be modified to include the background mismatch following -dimensional vector the derivation leading to (13). For any , the extended bound parameter with prior distribution of states
(14)
C. Some Generalizations
-
1) Upper Bound on Bias: It is well known that the MSE is a sum of the local variance plus the bias squared both averaged over the prior parameter space
(15) where denotes some arbitrary estimates of a particular and is the mean estimate, i.e., . For the second term we have (16) where is the averaged bias, and the equality holds if and only if is independent of . Note that the averaged variance is always greater than zero. When the bound in (13) is achieved (which has been observed to be the case at high SNR in many problems), we immediately obtain an upper bound for the averaged bias MZZB
(17)
where MZZB is the MSE bound specified in (13). In fact, the bound in (17) is often tight since the variance is close to zero at high SNR. 2) Stochastic Mismatch: So far, the mismatch has been assumed to be deterministic or local. That means, given an as, we look at the mismatch effect for a sumed background given true background . The stochastic mismatch can be inas a random cluded in a straightforward way by modeling over gives us set. Integrating (13) with respect to
-
(18)
Obviously, this is much more complicated in analysis and comis high. putation particularly when the dimensionality of Some approximation or simplification such as the one in [2] is possible but beyond the scope of this paper. Usually, given the environmental/system uncertainty, one may evaluate (13) for a . This is often judicially chosen set of possible mismatch enough to characterize the mismatch effect as long as the size of the uncertainty is small. 3) Vector Parameter Estimation and Nonuniform Prior: Bell et al. extended the Bellini–Tartara bound to arbitrarily distributed vector parameter estimation [16]. This can again
(19)
-dimensional weighting vector. One can obwhere is any tain the bound for each individual parameter by choosing apgives the bound for propriately. For example, the first parameter. , the develIn summary, given the size of mismatch, oped Ziv–Zakai-type bound can be used to evaluate the MSE in parameter estimation. It is obvious that forms the kernel in the bound’s evaluation. In general, when a closed-form error probability for the regular LRT is available, the mismatched error probability is solved in a similar way, particularly when the likelihood ratio can be expressed in terms of the correlation between the observation vector and the computed replica (matched filter). In that case, replacing the matched replica with the mismatched replica does not change the form of the statistics of the likelihood ratio. Such an example is addressed in Section III-A. III. ERROR ANALYSIS UNDER BACKGROUND MISMATCH In this section, we develop mismatch analysis for a data model with random signal embedded in random noise. Specifically, the mismatched LRT and its associated error probability are derived and the MLE error under mismatch is analyzed. It is worth noting that the studied data model exists in many detection and estimation problems including bearing estimation/passive source localization as well as various communication problems, thus, the approach developed here can be readily applied to those cases. In bearing estimation/passive source localization [16], [5], a source (electromagnetic or acoustic) radiates narrowor broad-band signals and the signal field is sampled by an array of receiving sensors. The source is usually assumed to be a stationary random process. Consequently, when the observation time is long enough (so that the window effect in spectral estimation can be ignored), individual discrete Fourier transform (DFT) bins of the received source signal are uncorrelated with each other [7]. In general, a number of snapshots are available for each frequency bin, and the number of snapshots is determined by the observation time and the correlation time of the source process. Suppose we have independent measurements for each of frequencies, , . The complex envelope vector of the received signal can be expressed as
(20) where
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includes the source bearing/location parameters to be estimated; the background parameter set required to define the signal propagation channel and the receiving system; an vector representing the th snapshot of the th frequency bin and is the number of receiving sensors; a random process incorporating amplitude and phase variability with a power spectral density equal of the source; a vector channel transfer function (unit impulse response function) for the propagation between the source and the receiving sensors; and the th snapshot of the stationary white noise vector in the th frequency bin. Given the data model in (20), dependence on channel/system parameters is embedded in the channel transfer function term. Traditional bearing estimation assumes a plane wave model is defined by some harmonic terms at [16], so the reference sensor and their delayed versions at other sensor positions. Recent developments in underwater passive source localization [5] exploit a full wave modeling of signal channel propagation. They are particularly useful for the case of a large vertical/horizontal aperture where nonplane wave signals are encountered. Both the signal and noise terms are assumed to follow a zero-mean complex Gaussian distribution. The covariance is given by matrix for
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Now we assume some mismatch in background parameters, is different from . The mismatched pdf is available thus, by modifying the parameter dependence in (23), which is
(24) The maximum-likelihood estimate in (5) can then be derived under mismatch. Using the eigendecomposition of (21) and the Woodbury identity [1], we have
(25)
(26) is the norm of the complex vector and where is the normalized channel transfer function defined by (27)
(21) denotes the complex conjugate transpose operation, where is an identity matrix, and and denote the variances of the signal process and the noise process, respectively. Denote
(22) represents the matrix transpose operation. Obviis an column vector. Since , , , are uncorrelated across frequencies and snapshots, the covariance matrix of is a block-diagonal as the diagonal component. Then the matrix with conditional pdf of is
where ously,
We assume known and and thus maximize (24) directly. Substituting (25) and (26) into (24), assuming a -independence of the norm of the transfer function,1 and rejecting constant terms, the maximum-likelihood estimate of is given by (28) and . Note that (28) is actually independent of Application of the MLE to random parameter estimation is justified by assuming a uniform a priori parameter distribution. In this case, the MLE is equivalent to the maximum a posteriori estimate (MAP) [1]. A. Error Probability Associated With the Mismatched LRT Having obtained the MLE in (28), the mismatched likelihood ratio in (9) immediately follows
(29) (23) where
denotes the matrix determinant.
1This is the case in bearing estimation and many other practical problems. However, this assumption can be removed based on a recent development on MLE error analysis under spatially correlated noise field [17].
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The minimum error probability associated with the misand . matched LRT involves , Due to the symmetry, we only need to solve which is
For the case with multiple frequency components, an exact error probability is also available
-
(39)
where
(40) (30) and are then obtained by replacing and accordingly. Substituting the resulting error probabilities into (13) or (19), we are able to evaluate the modified Ziv–Zakai bound under the given background mismatch . The error probability in (30) is solved following the approach developed for the regular LRT [11]. The detailed derivations are given in the Appendix . To simplify the notations, three related correlation terms are first defined (31) (32) (33) These are also called signal ambiguity functions since they are often characterized by a multimodal structure showing a main lobe and many sidelobes, due to highly nonlinear dependences on the embedded parameters. We also find it is useful to define a SNR term SNR For the case with a single frequency component at error probability is (see the Appendix )
(34) , the
-
(41) and and , , are solved using (36)– is the partial fraction expansion (38) at each . Note that . For large coefficient for the moment generating function, and , computing can be quite time consuming. B. Mismatch Analysis To implement the error analysis for the mismatched MLE, we and scan within assume a fixed true parameter position at the specified parameter interval; we then look at the error prob. It is shown ability of the corresponding MLRT, to be a good first-order approximation to the exact MLE error probability at [11], provided that the exact one is not available. Calculation of the MLRT error probability involves three correlation terms of the received signal in (31)–(33) with replaced by . The first one, , is for the assumed and the true background parameter sets, with the parameter being estimated at the true position. This measures the peak degradation due to mismatch. The second one, , is for the true, matched parameter set and the scanning, mismatched parameter set. As shown in Section IV, this could introduce peak shift. The third one, , corresponds to the true and the scanning parameter set both under mismatched background. This is replaced similar to the regular ambiguity function with by . Consider the single-frequency and scalar parameter case. We can rewrite the error probability by
(35)
(42)
where
where (36)
and
and
are defined by
(43) For perfectly known background, and for shown that
(37) -
and
(38) respectively.
, it can be easily (44)
Thus, the error probability is inversely related to , which itself is a function of the SNR and the field correlation. For a fixed SNR, the maximum error probability is achieved at the minimum , which corresponds to the maximum field correlation obtained by the closest main-lobe point. Under mismatched background, the situation is different.
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Fig. 2. Uniform linear array. Actual and assumed array positions are also specified.
When is small, although is smaller than one (the peak value without mismatch), it is still the highest correlation value. In this case, the bias in parameter estimation can still be ignored but the MSE would increase. As increases, decreases further such that it is no longer the maximum correlation, i.e., at some , we have In this case, the mismatched error probability is characterized by two operation regions. The first region has and thus, (cf. (37)) and mismatch-free case. Note that
. This is similar to the (45)
where
(46) Obviously, the minimum imum error probability of when
is one, which yields the max(cf. (42)). is obtained
This corresponds to another parameter point which cannot be resolved from the true parameter point by considering the correlation between the mismatched field and the true, mismatch-free field. The second region is specified by In this region,
and
.
the mismatched field at the true parameter position ). From (42), is one ( is zero. Thus, the error probability falls between when and . , the peak correlation point For a reasonable size of is still within the main lobe around in the true parameter. Thus, the second operation region can be on either side of the true parameter point but not on both. As we later see in the examples, errors around the incorrect correlation peak dominate the performance in the high-SNR region. This introduces a mismatch-dependent bias in parameter estimation. At low SNR, the sidelobe contributions through the terms of and become important. IV. EXAMPLES Consider the problem where a plane-wave signal impinges on a uniform linear array of sensors as shown in Fig. 2. The variable denotes the angle (from the horizontal line) of a plane wave arrival. To study the system mismatch, which is typical for as traditional array problems, we choose the array tilt angle the background parameter. The true and assumed tilt angles are and , respectively. Under this configuration, denoted by , where the parameter to be estimated is is assumed known, although might be mismatched. We assume has a uniform distribution on : (48) We further assume a narrow-band source signal centered about a known frequency . The channel transfer function is obtained by choosing the first sensor as the reference sensor (49) where is the wave propagation speed in the medium and is the sensor spacing. For this plane-wave signal model, the ambiguity function in (33) has a nice form
is then given by (47)
which is now smaller than one. The minimum value of is actually zero, which is achieved at high SNR by some close main-lobe points whose mismatched fields are more correlated with the true, mismatch-free field ( ) and also highly correlated with
(50) where . To avoid grating lobes (poles of . (50)), one often choose less than half-wavelength , , For the examples in this section, we choose . We first evaluate the ZZB for the mismatch-free and
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Fig. 3. Signal ambiguity function in bearing estimation. The dotted line corresponds to the true bearing.
Fig. 5. Mismatched ambiguity function in bearing estimation. The bottom panel shows the details around the true bearing. The dotted line corresponds 30 ); the dashdot line corresponds to 40 , at to the true bearing ( which the ambiguity level equals that at the true bearing. The array tilt angle is mismatched by 5 .
Fig. 4. Bearing estimation MSE given by the ZZB (solid line), and the MLE simulations (*). No mismatch is assumed. Three operation regions are also specified.
case [18], i.e., the actual array position is the assumed vertical , cf. Fig. 2). Fig. 3 displays an example position ( 30 (thus, ambiguity function given the true bearing at ). Some periodic ambiguity structure is observed with , where is an integer. sidelobes occurring at Note that varies from 90 to 90 . The error probability associated with the regular LRT, , can be solved from (42) with ( ). As seen from (50) and (34), the ambiguity property as well as the SNR (note that is a constant) is independent of . Accordingly, is only a function of . This greatly simplifies the evaluation of the Ziv–Zakai bound. Fig. 4 shows the evaluation results, in which three operation regions are observed. In the asymptotic region, where the SNR is high, estimation errors are small and linear with the SNR on 3 dB, estimation errors a logarithmic scale. Below SNR
=
increase significantly demonstrating a threshold phenomenon. This is attributed to the sidelobes of the field correlation and defines the transition region. At very low SNR, estimation errors are spread over the a priori parameter interval and the bound approaches the a priori variance. For comparison, the MLE simulation results are also presented on the basis of 4000 Monte Carlo trials at each SNR, where the true parameter for synthesizing the observation is uniformly chosen across the given interval. Clearly, the ZZB closely follows the behavior of the MLE simulations. Below the threshold SNR, MLE MSE are always (up to 7 dB) greater than the predictions. This discrepancy can be explained by noting that in the low-SNR region the optimal estimate is the conditional mean estimate rather than the MLE, which is then no longer achievable by any fundamental performance bound. We then assume there is some mismatch between the assumed . The ambiguity and the actual array positions, i.e., function in (32) is given by
(51) . An example is plotted in Fig. 5 where , 5 , and 30 . A peak degradation for of about 1 dB at the true bearing position (cf. (31)) is observed. we have Besides, at and the highest ambiguity level is now at , which cor. responds to , is calcuThe mismatched error probability, of (same as in Fig. 5). The lated for each and a fixed SNR is 0 dB which is defined on the single-sensor basis. Fig. 6 and , is shows the results. Between greater than one half and actually very close to one between . Outside this region, the highest ambiguity point and
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Fig. 6. Mismatched error probability in bearing estimation. The bottom panel shows the details around the true bearing. The dotted line corresponds to the 30 ); the dashdot line corresponds to 40 , at which true bearing ( the error probability equals that at the true bearing. The single sensor-averaged SNR is 0 dB. The array tilt angle is mismatched by 5 .
=
goes quickly down to zero, particularly on the left side of . Those are actually two operation regions discussed in Section III-B. Although presenting two almost opposite behaviors, they are subject to the same physical rule. That is, when the signal field at one parameter point is closer to the true signal field, the MLE outputs at other parameter points can hardly be beyond that at , particularly with low-level and , the signal field is alway noise. Between more correlated with the true field compared to that at the true bearing, and the error probability increases as SNR increases; outside this region, the mismatched field at the true bearing is always more correlated with the true field compared to those at all other points, and the error probability increases as SNR decreases. This explains why the transition between two regions is so abrupt at high SNR; at low SNR, the transition is much smoother. It is interesting to note that to reduce the estimation error in the high-SNR region, one may want to add some noise. Now it is clear how a bias is introduced by errors around the incorrect ambiguity peak. This bias term can be predicted using the developed bound in (13). Figs. 7 and 8 give the results for 1 and 5 , respectively. We see that as goes ( ), the bound is nearly independent of the away from SNR at high SNR, and is actually dominated by the square of the mismatch-introduced bias. The size of the bias can be approximately computed using the bound in (17). The sign of the bias can be determined from the position of the mismatched 1 , the premain-lobe peak. Using this approach, for ; for 5 , it is . Note that dicted bias is about is a dimensionless quantity. The above predictions can be verified in this simple problem. ; thus, at each the bias in the estimation of Note that is given by (52)
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Fig. 7. Bearing estimation MSE given by the modified ZZB (solid line), and the MLE simulations (*). The array tilt angle is mismatched by 1 . The regular ZZB in Fig. 4 is also shown (dotted line).
Fig. 8. Bearing estimation MSE given by the modified ZZB (solid line), and the MLE simulations (*). The array tilt angle is mismatched by 5 . The regular ZZB in Fig. 4 is also shown (dotted line).
for small . Ignoring the effects around the endfire region, the averaged bias is solved by (53) for 1 , or In this example, the estimated bias is for 5 , both of which differ from the bound by less than 4%. V. CONCLUSION A Ziv–Zakai-type bound has been developed to bound the MSE of parameter estimation when using incorrect background environmental/system parameters. The basis of this development is the MLRT identifying the true and the assumed background parameter sets embedded in the observation and the signal model respectively. It incorporates the increase in MSE due to estimation bias at high SNR; at low SNR, it
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includes effects of the sidelobe ambiguity as well. It is the first large-error bound capable of incorporating background mismatch. To demonstrate the application of the bound, error analysis of the MLRT is implemented using a random signal model typical of bearing estimation/passive source localization. Error under mismatch is determined by the correlation level degradation at the true parameter position and the location shift of the ambiguity peak, thus explaining the introduced bias at high SNR. Examples of bearing estimation show that the developed bound closely predicts the performance of the maximum-likelihood estimate in the presence of system mismatch, including the MSE in all SNR regions as well as the bias at high SNR. The bound has also been successfully applied to more complicated problems in passive source localization [11]. Evaluation of the scalar-parameter bound with deterministic mismatch can often be accomplished in a computationally efficient manner; generalization to include stochastic mismatch and vector parameter estimation is straightforward in theory but can be challenging to implement. Background mismatch is very common in practical detection/estimation applications. The development in this paper provides a useful tool, as well as opening some new insight, for performance analysis in such problems. For example, it can be seen from (13) that large increase in the MSE matches large increase in the probability error, which starts from an SNR lowerbounded by the SNR at which the channel capacity equals one [19]. By examining the channel capacity change due to background mismatch, one is able to examine how the mismatch influences the position of the threshold SNR.
We express each as (recall that we are assuming the true parameter point is ) (57) This is actually a whitening operation such that each new is complex Gaussian with identity covariance snapshot can be expressed in terms of matrix. Hence, -
(58) , i.e., Note that the above expression is available even for the number of snapshots is smaller than the number of sensors. Denote by matrix eigendecomposition (59) where , and as later shown, only two nonzero . Since is a eigenvalues exist, i.e., unitary matrix, multiplying by does not alter the covariance and then the distribution for Gaussian model [20]. It follows that (60) (61) Therefore, the error probability can be further derived as
APPENDIX DERIVATIONS OF THE ERROR PROBABILITY IN MLRT
-
A. Case 1: Single-Frequency Component For a single-frequency component with multiple snapshots, the desired error probability is
(62) The term in parentheses is a weighted sum of two independent degree- complex Chi-squared variables. Thus, we have
-
(63)
(54) We denote data matrix of snapshots by
Note that a positive (55)
Recall that snapshots are independent and identically distributed according to a complex Gaussian distribution. Then the error probability in (54) can be written as
is required in deriving (63).
The statistics of the complex -distribution, , is documented in [21, Appendix A]. Its cumulative distribution function is given by
(64) Applying (64) to (63), we finally obtain (35). To solve the eigenvalues of (59), we first notice that (56) where
represents the trace operation.
(65)
XU et al.: A BOUND ON MEAN-SQUARE ESTIMATION ERROR WITH BACKGROUND PARAMETER MISMATCH
and
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Note that is a real variable, and the region of convergence [22] is defined by for (66)
share the same eigenvalues. Using (21), we have (73) has a partial-fraction expansion of the form [23] (74) where (75)
(67) In general, for , and are not colinear (otherwise, they are the same vector). It can be easily shown that (67) expands a two-dimensional space, and the corresponding eigenvectors are linear combinations and . Thus, multiplying (67) by of ( is a complex number) gives . Setting the coefficients and to zero yields a quadratic of equation of
(76) Note that for
and
(77)
(68) and for
and
where and are defined in (37) and (38), respectively. The solution for eigenvalues is then given by (36). B. Case 2: Multiple-Frequency Component Using a similar derivation for single-frequency component, the error probability for the multiple-frequency case is given by
(78) Therefore, the pdf of
is given by
(69)
. (79)
Now we define
Finally, the error probability is (70)
-
. To find the pdf of , we so that first observe that each Chi-squared variable is independent of the others, so the moment generating function is easily solved
(71) , , are computed using (37), (38) and (36). Clearly (see also comments below (63))
(72)
(80)
ACKNOWLEDGMENT The authors wish to thank Dr. C. Richmond for discussions on error analysis of the likelihood ratio test. The derivation of a closed-form LRT error probability directly came from those discussions. They are also deeply grateful to the anonymous reviewers for valuable suggestions which have been incorporated into the manuscript.
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