Cancellation of Polarized Impulsive Noise Using ... - Semantic Scholar

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 12, DECEMBER 1998

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Cancellation of Polarized Impulsive Noise Using an Azimuth-Dependent Conditional Mean Estimator Umberto Spagnolini

Abstract—The separation of signals from noisy vector measurements is obtained by taking advantage of the Middleton Class A model of noise amplitude and the correlation of the components of the noise process due to their polarization. The signal is assumed non-Gaussian to be white Gaussian. Noise is a superposition of processes, each with a fixed azimuth of polarization. Neither the number of processes ( ) nor their azimuths are known. The separation of signal from noise is based on the conditional mean estimators. In addition to the optimum estimator, which can be derived from a knowledge of the bivariate density functions, two suboptimum solutions for polarized noise are discussed: the circularly symmetric estimator and the azimuth-dependent one. Circular symmetry is suitable for the nonpolarized noise vector, whereas the azimuth-dependent estimator is tailored to polarized noise. The azimuth-dependent approach consists of two steps: First, the data vector process is discretized into azimuth sectors, and then, in those classified as noisy, the signal is separated from the noise. Statistical model parameters of random processes are estimated by using the optimum classification, based on the likelihood ratio test (decision-directed method). Iterative whitening methods are also discussed for correlated vector signals. Numerical examples show the effectiveness of the above technique in canceling polarized noise.

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Index Terms— Adaptive estimation, electromagnetic interference, estimation, geophysical signal processing, impulse noise, interference suppression, least mean square methods, signal restoration.

I. INTRODUCTION

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AN-MADE electromagnetic interference (EMI) limits KHz) electhe reliability of some low-frequency tromagnetic methods in geophysical investigations. EMI is basically an impulsive noise as it tends to occur randomly and be of high amplitude and short duration. Low-frequency EMI is mainly associated with power lines, electrified railways, grounding devices, and storm lightning [20]. The impulsive nature of noise is the dominant feature, particularly close to noise sources, as lowpass filtering of the EMI occurs with distance. Noise sources are fixed in space, and the two (or three) components of the noise vector process are strongly correlated; as a consequence, for each noise source, the unit vector of the process (or the vector’s azimuth) is fixed. Azimuths for each noise process depend on the medium and the location of the (uni or bipolar) source [5]. This Manuscript received October 28, 1996; revised April 29, 1998. The associate editor coordinating the review of this paper and approving it for publication was Prof. Antonio Cantoni. The author is with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, Milano, Italy. Publisher Item Identifier S 1053-587X(98)08689-9.

EMI is referred to as polarized impulsive noise. This paper focuses on the cancellation of the polarized impulsive noise by taking advantage of both the statistical properties of noise vector processes and the polarization of each of the processes. Man-made electromagnetic noise is modeled by heavy-tailed non-Gaussian amplitude distribution [18]. Exploitation of the polarization in the restoration of the vector signal was recently proposed [17]; however, the model was highly simplified. In this paper, the optimum and suboptimum cancellation for white and correlated processes is proposed in accordance with the polarized noise model. In addition to the geophysical applications that first motivated this research, the detection and/or the cancellation of impulsive noise has been addressed in the literature by several authors [4], [6], [8]. In many cases, the detection and/or cancellation of impulsive noise leads to the conventional signal processing algorithms (i.e., those optimized with respect to Gaussian noise) becoming more robust [7], [9]. of noisy data is The vector sequence considered to be a superposition of Gaussian vector sequence (which is also indicated as a signal) and impulsive noise vector (1) can be assumed to be independent of each Signal samples other and independent and identically distributed (i.i.d.) with Gaussian probability density function (pdf). Polarized noise is modeled as a superposition of random sequence processes (2) where each process is characterized by a deterministic (but Noise processes are not known) azimuth of polarization independent of each other and of the signal. For each noise is i.i.d. and is characterized source, the random process by heavy-tailed pdf (non-Gaussian process); the noise samples (when present) have Poisson distributed time positions. Thus, and are correlated. processes in (1) is i.i.d., its separation from any If the process non-Gaussian process is accomplished by a zero-memory nonlinearity (ZNL). The choice discussed in Section II for ZNL is the minimum mean square error (MMSE) estimator, which is The shape of the given by the conditional expectation ZNL estimator for the separation (and cancellation) of noise in is dominated by the statistical vector sequence

1053–587X/98$10.00  1998 IEEE

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model of non-Gaussian process. The non-Gaussian pdf of the for each of the noise processes is based on the amplitude mixture approximation of the Middleton Class A noise model is thus multimodal, and the ZNL [6], [11], [13]; the pdf of estimator needs to be obtained accordingly. The analysis in Section III of the ZNL for nonpolarized noise is preliminary, but necessary, to the one for polarized is assumed as circularly symnoise patterns. In this case, metric processes and can be conveniently characterized with the envelope. To characterize the parameters of the envelope , the decision-directed (DD) method proposed process by Zabin and Poor [27] for the estimation of parameters of Middleton Class A noise has been adopted. In the presence of nonsymmetric densities, it is rather complex to characterize the process and the ZNL estimator. In coherent communication systems, the infase and quadrature components play the role of vector signal components; the circular symmetric noise model has been proposed in order to obtain the locally optimum (LO) detector [6], [14]. To model underwater interference and to derive the LO detector, bivariate pdfs have also been simplified [23]. However, when the pdf is complex, a combination of simpler pdf’s represents a useful approximation (e.g., Gaussian pdf in [1]); this is the solution proposed here to derive a suboptimum ZNL. has In this paper, the bivariate pdf of the process been approximated by the sum of pdf’s along the azimuths, the pdf within each azimuth sector being simplified as cirand the azimuths cularly symmetric. Since the number of the noise processes, as well as the statistical model parameters of each noise process, are not known , the a-priori but should be estimated from observations ZNL estimator has been approximated by a suboptimum solution that is referred to as the azimuth-dependent ZNL (Ais decomposed ZNL) (Section IV). The noise sequence into a set of azimuth-independent non-Gaussian processes (azimuth decomposition). Within the th azimuthal sector, the are approximated as i.i.d. nonenvelope samples Gaussian, independent of the azimuth (as the azimuth is taken into account in azimuth discretization); the parameters of each non-Gaussian process are assumed to be independent of the other sectors and are estimated accordingly. In this way, the A-ZNL follows as the combination of the ZNL for all the azimuth sectors. The second part of the paper deals with correlated signal. be the observations where samples of Let are now correlated. The preprocessing consists of signal by still employing the i.i.d. model (1) for the filtering (residual) process after filtering. Two approaches are discussed in Section V for the correlated signal sequences. The first method takes advantage of the availability of a reference signal signal is assumed to be given by filtering with an unknown multiple-input filter. Separation of signal from noise is performed iteratively with the estimation of the filter response. Noise cancellation is more difficult without any reference is a finitesignal. In robust filtering, it is assumed that order autoregressive (AR) or autoregressive-moving average (ARMA) model, and filter updating is obtained by employ-

ing a nonlinear gain on the residual process [7]–[10], [16], [24]. The low-order prediction filter proposed in Section V estimates, iteratively, the process from Separation of is employed by the A-ZNL the innovation from the noise estimator. Numerical tests carried out with simulated signals show the efficacy of the azimuth decomposition (Section VI). Experimental results concern the separation of man-made polarized impulsive noise from the natural (telluric) EM field. The use of the natural field as signal for subsurface investigations is known as the magnetotelluric (MT) method [15]. The consistency and effectiveness of noise reduction is proved by MT data highly contaminated by impulsive polarized noise. Notation: Bold characters denote vectors; indicates the and convolution operator; are the sequences that represent the two indicomponents along the orthogonal axes; superscript is cates the matrix transposition; the amplitude or the envelope of the vector, and is the corresponding angle or azimuth; denotes the ordered union of disjoint extracted from accordsubsequences is the Dirac function ing to a rule specified in the text; and

II. ZNL

AND

NOISE MODEL

The MMSE estimate of noise mean

from

is the conditional (3)

The conditional mean estimate of a signal is simply the comThe real-valued functions plementary and depend, in general, on both the components of The ZNL (3) transforms each value into the corresponding value Its computation requires some simplifying hypotheses in the definition of realistic probability density functions (pdf’s) that are compatible with the specific application. For the , the choice was description of observed data sequence the mixture model approximation of the Class A Middleton model [11]. Let (4) . The components be the joint pdf of the samples in process are independent and circularly Gaussian ( denotes is the zero-mean Gaussian pdf and the amplitude variance). For a circularly symmetric Gaussian process, it can be shown that the conditional mean estimator according to depends on the pdf of observations (5) is the gradient , and is the pdf independent processes. of the non-Gaussian process for Equation (5) is useful for any analytical derivation of the ZNL

where of the bivariate pdf

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estimator once the pdf is known. However, the shape of the ZNL estimator is basically dominated by the pdf of the non-Gaussian process. is polarized, and the polarization Impulsive noise for th noise process is deterministic. The azimuth is modeled as the mixture pdf of process (6)

A. The Optimum ZNL A process with the independent components and models the nonpolarized noise and can describe a situation of randomly varying polarization and/or of complex patterns of noise sources distributed uniformly along the azimuths (isotropic noise). The corresponding Gaussian mixture pdf is circularly symmetric (10)

indicates the impulse occurrence probability (i.e., where impulses are equally likely to occur at any point in the is the amplitude variance, and Dirac sequence), simplifies a smaller variance term. Model (6) function corrupts data with outliers about implies that the process % of the time, whereas the remaining % is zero. The ZNL estimator can be derived according to the (known) statistical parameters of signal and polarized noise. In the following, the ZNL is derived for one polarized noise ; the nonpolarized noise model for is discussed in Section III. A. ZNL for One Polarized Non-Gaussian Process In the case of one polarized non-Gaussian process with a known polarization azimuth, the ZNL estimator is derived analytically. Without any loss of generality, the noise In this case, the bivariate pdf of polarization is and are (7) (8) and (For convenience, variances refer to the background process and the impulsive process, respectively). From (5), we have the analytical solution of the ZNL estimator

indicates the Like model (6), here, noise variance, and is the impulse occurrence probability. is The bivariate pdf of (11) , and where be derived from (5) as

The ZNL estimator can (12)

and the kernel function is

(13) is also useful for an understanding Kernel of the shape of the ZNL estimators. Since , the kernel is expected to be if the observed sequence has low values (for background process), whereas it is approximately zero (blanking) for observations with large amplitudes. depends on amplitude only and Kernel is circularly symmetric. The ZNL estimator for nonpolarized noise is azimuth-independent and preserves the azimuths as B. Decision-Directed (DD) Estimation of Gaussian Mixture Model Parameters The statistical model for a circularly symmetric process (11) can be described using the corresponding mixture approximation of a canonical Class A model for the envelope (i.e., polar coordinate system)

(9) only and is decoupled As expected, ZNL (9) depends on on the two components. Along the azimuth that corresponds ), the ZNL estimator to noise polarization (i.e., is azimuth invariant as the ZNL for scalar signals. The ZNL estimator for any polarization is derived from (9) by rotating the reference axes. III. THE NONPOLARIZED NOISE In Appendix A, it is shown that polarized noise process approaches the nonpolarized model asymptotically (for Since the nonpolarized model is used as an approximation in the suboptimal ZNL (Section IV), it is of the pdf discussed first.

(14) The envelope pdf consists of a superposition of Rayleigh pdf’s (note that ). Only the knowledge of is needed to of the mixture model (11). estimate the parameters and their relationship The evaluation of the moments of with the statistical model (14) allows an estimation of the noise model parameters by using the method of moments (MoM) [12], [26]. The main advantages of the MoM method are computational efficiency and consistency. However, convergence can be considered slow, and invalid estimates (e.g., and/or ) can be easily obtained for a small data set. The decision-directed (DD) method for estimation of model parameters is based on the optimum classification of the as background or impulsive samples in sequence samples [27]. The optimality criterion is the minimization

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of the probability of incorrect classification. The likelihood ratio test (15) provides the optimum decision criterion to discriminate beor backtween samples belonging to the impulsive processes. Since none of the parameters in ground (15) are known, the inequality (15) is solved iteratively for ; iterations converge to a consistent the whole sequence solution according to [27]. The iterative approach for the estimation of model parameters for the nonpolarized Gaussian mixture is described. Let be the statistical parameters estimated at th ; the likelihood ratio iteration from the overall sequence to belong test (15) allows the classification of the samples or background processes to impulsive

(a)

(16) Once samples of sequence where have been classified, we can use, for the next iteration, the second moments of these samples, which are shown to belong to the background or impulsive process

(17)

and are the lengths of sequences and , respectively. The frequency of impulsive samples represents an estimate of probability at the next iteration (18) This recursive algorithm converges to the estimates of model (or similarly parameters when ). We have always experienced the convergence of the algorithm, even for small data samples (up to 1000 samples), equal to the second by choosing, for initialization, and moment of the overall sequence (with or 0.1). The estimates of the model parameters are biased mainly for small interference to background ratio (I/B) Fig. 1 shows both the estimated I/B at convergence (i.e., ) as well as the probability with respect and It can be concluded that the estimated to parameters for nonpolarized noise converge to the true model for I/B larger than 12–14 dB almost independently of probability.

(b)

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Fig. 1. Bias of DD method. (a) Estimated interference variance normalized ( )2 ( )2 =B versus interference to backto estimated background variance I ground ratio (I/B) evaluated for p = f0:01; 0:05; 0:1; 0:25g: (b) Estimated probability p( ) versus I/B and p: True values are indicated with dashed lines.

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IV. THE A-ZNL

FOR

POLARIZED NOISE

In principle, given the bivariate pdf of the process , the ZNL estimator of the noise (or signal) can be evaluated for any situation (see Section II). However, if the parameters need to be estimated from a limited data of the pdf set, an efficient but suboptimal method for model parameter estimation, as well as for the ZNL, should be defined for any practical use. Let us evaluate the gradient in (5) by trasforming the variables into the corresponding polar coordinate system

(19) is independent of the azimuth (i.e., If the bivariate pdf ), the ZNL estimator depends on the envelope

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only, as for the circularly symmetric pdf. In the presence depends on the of polarized noise, the bivariate pdf azimuth (i.e., the pdf along the azimuths is multimodal), and the circularly symmetric ZNL is not at all applicable. However, is divided into azimuth sectors (azimuth if the process within each azimuth sector is discretization), the pdf approximately independent of the azimuth. The rationale of the A-ZNL is to consider one azimuth sector at a time and to approximate the ZNL within each sector as being azimuth independent. In other words, within the th azimuth sector, is assumed, and therefore, the process is locally nonpolarized. The suboptimal solution corresponds to a decomposition of into multiple azimuthal-independent mixture the sequence processes. For each azimuth, or azimuth interval, the nonis assumed independent of the neighGaussian process boring azimuths, and the corresponding envelope evaluated within the th azimuth interval is circularly symmetrical. These approximations simplify the problem of estimating the number of non-Gaussian processes and their polarization by assuming up to one non-Gaussian process for each azimuth sector. The main advantage of this approach arises from the possibility of evaluating, in principle, the ZNL estimator of noise (or signal) for any unknown number of polarized noise processes with any azimuth distribution.

and are azimuth-dependent parameters. The AZNL estimator for the whole sequence is the combination of the suboptimum ZNL’s for all the azimuths: Therefore, the A-ZNL is a combination of ZNL for nonpolarized noise applied in parallel to the sectors with the coupling through the background model Because of the way the A-ZNL estimator has been defined, the azimuth is preserved as for the isotropic noise model. This effect needs to be evaluated in comparison with derived from (5). For small values of the optimum ZNL , the optimum ZNL is azimuth invariant as, for , the is mostly dominated by the circularly symmetric pdf background term. Furthermore, if the th azimuth interval is not affected by impulsive noise (e.g., the azimuth interval is orthogonal to the azimuth of noise) or it includes one domand are azimuth inant polarized noise source, both invariant. Distortions that arise from the azimuth invariance of the A-ZNL estimator are negligible in many cases. Based on the statistical model of the A-ZNL, azimuth discretization has to be chosen at least on a basis of one polarized noise source per sector. However, azimuth discretization is basically an implicit (polar) sampling of the pdf ; therefore, a careful sampling analysis shows that the should be lower than or, azimuth interval equivalently,

A. The Suboptimum A-ZNL Estimator

B. Decision-Directed (DD) Estimation of Model Parameters

Let us consider the azimuthal discretization of the sequence into disjoint subsequences

(20) Within the th azimuth inwhere is modeled as terval, the envelope of the process the mixture (14): Variance of the background process is independent of the azimuth as it is circularly Gaussian; probability and variance of the non-Gaussian process that model the impulsive interference within the th azimuth are and , respectively. Since the way chosen here to take into account the polarization is by azimuth discretization, model parameters of impulsive interference are estimated independently for each azimuth interval. Within each azimuth, the conditional mean estimator is the one for nonpolarized noise discussed in the previous section. Therefore, the suboptimum ZNL is (the hat indicates that the parameter or function is referred to as the suboptimum ZNL) (21) where the kernel for the Rayleigh mixture is similar to (13), as in (22), shown at the bottom of the page, where

The DD method for iterative estimation of impulsive noise and is performed independently on each parameters sector. The classification between background and impulsive samples has to be carried out by constraining the likelihood in every ratio test to have the same background variance sector. The th iteration of the DD method is based, for , on the classification rule between each subsequence and background samples. The impulsive likelihood ratio test adapted from (16) is

(23)

is The background process is circularly symmetric ( independent of the azimuth); samples classified as background The recursive formulae for DD are model parameter estimation are not reported here but are a simple extension of the previously discussed relationships in (17) and (18), and the same background variance has been chosen. are The estimated parameters at convergence indicated herein with the explicit dependence on the azimuth

(22)

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The DD model estimate might converge, for azimuths ; in this less influenced by impulsive noise, to is assigned. case, In spite of the A-ZNL requirements based on the sam, the accuracy of with limited pling of be observations calls for large azimuth intervals. Let the minimum number of the overall observations and samples needed for reliable estimates from the DD estimator; the average number of samples within the azimuth sector should verify the inequality (for noise process with the smaller probability) (24) where the term 1/4 approximately takes into account for the Reasonazimuth sectoring and polar sampling of pdf able values of DD convergence for nonpolarized noise model seems to be

process. The use of restored signal and cleaning steps for outlier blanking are reproposed here as being extensively discussed in the context of robust signal processing [7]. The separation of the non-Gaussian pdf of the transmitted data sequence in the communication system is a property exploited in blind deconvolution [2]. Methods discussed here can be considered to be an extension of those for scalar signals [3]. (For the sake of simplicity, the sample index is understood in the notation; all statistical processes are stationary.) A. Filter Identification Samples of the components in the reference sequence are zero mean i.i.d. Gaussian process with , and The reference variance Signal components are given is independent of noise by the convolution with the (unknown) dual-input/dual-output filter (28)

C. A-ZNL for One Polarized Non-Gaussian Process The corresponding A-ZNL estimator in closed form is derived by considering the polar coordinate system for the pdf (8). For a given azimuth , the circularly equivalent pdf is

(25) where it has been defined, for convenience, The A-ZNL is where kernel is now azimuth-dependent

(26) For those azimuths orthogonal to polarized noise, it follows , and that Since, for , A-ZNL is different from (9) (i.e., ), we verified the accuracy of the proposed solution by simulations. V. FILTERING

OF

CORRELATED SIGNALS

The model (1) of the correlated signal noise is reproposed here

and the polarized

In the following, we or, more compactly, from present an iterative method to restore the signal by estimating iteratively the filter Let denote the filter response at the th iteration; it differs from the “true” filter mainly by the residual impulse response Each iteration of the restoration algorithm is based on three steps: ; 1) estimation of the signal vector 2) separation of the identification error and the non-Gaussian process from the residual (see Section II); process 3) update of the filter response with an estimate of the identification error. is independent of The identification error process ; if the sequence is long enough, the process is Gaussian (with the Central Limit Theorem). Since is dependent on the residual impulse response of the , has to be iteratively minimized to obtain filter a better filter estimation. Therefore, there is the need to from identification error in the residual separate Let the noise estimate from be ; the restored signal at the th iteration is thus obtained by noise cancellation as in (29)

(27) The different notation is to distinguish the correlated from the white signal model. The approaches for correlated signals involve the iterative use of a filter to predict signal samples from a reference signal (filter identification) or from a cleaned data sequence (inverse filter or iterative prewhitening) [25], which is similar to the non-Gaussian Kalman filtering for scalar [10] and multidimensional signals [24]. In the iterative that can be filtering approach, it is the residual process , where described by two independent terms the first term is the white Gaussian process of the identification is the non-Gaussian error (residual impulse response), and

is a reasonable estimation of the noise In other words, if (at the first iteration, , or equivasequence is implicitly assumed), represents lently, an estimate of the restored signal. Successive refinements of the filter and ZNL improve signal estimation. is obtained The updated filter at the next iteration from the minimization of the mean square error (MSE) between a cleaned version of the signal estimated at a previous and the signal estimated from at the current iteration iteration (30)

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In the adaptive approach (i.e., one iteration per time sample), the filter adjustment is derived from the stochastic gradient method [22] (31) where is the step size. Similarly, the filter may be obtained normal equations ( as the solution of the system of is the length of the filter for each component)

(32) , where the or, equivalently, represents the correlation between the sequences matrix indicated by subscripts. The block Toeplitz matrix for i.i.d. With respect to the adaptive approach reference is (31), each iteration (32) now corresponds to a matrix inversion performed by taking into account the overall sequences. The adaptive approach (31), the batch estimate (32), and the recursive least squares (RLS) method are equally valid, and the estimation of the filter response is in no way critical. Adaptive and RLS approaches have the advantage of realdenote the time implementation. Let and are negligible at residual; we require that convergence. From (31) (33) [or holds true at convergence. For small values ], the ZNL can be approximated basically with first-order series [in the neighborhood of the origin, is circularly symmetric] Close to convergence, the mixture (or ), which leads to parameters are , and (33) becomes the usual orthogonality On the other hand, the ZNL for becomes , and thus, (33) large is satisfied. In any case, we experienced the convergence by even if, with initializing the procedure with low SNR, the convergence to the global minimum could be critical (see, e.g., [9]). Since the statistical parameters change with iterations, ZNL should be estimated from changed accordingly. Convergence to “true” filter (if needed) depends on the ZNL estimator that should be congruent with and For non-i.i.d. or the statistical model of , the signal can correlated reference sequences with [19]. be equally restored even if are different from Note that the iterations to estimate the iterations of the DD method discussed in Sections III and IV; as in DD iterations, the statistical model parameters (not the restored signal or the prediction filter) have to be The DD algorithm estimated from the overall sequence for estimation of mixture parameters can also be adapted for real time; DD iterations are included as an inner loop in iterative data cleaning.

B. Inverse or Prewhitening Filter Without any reference signal, the signal is usually modeled as a lowpass or finite-order AR process. Signal restoration involves the estimation of the low-order AR approximation to based on a prediction filter (note that the process ). As in a previous section, the filter response is updated iteratively after having separated the prediction error from the residual process. The signal is a filtered i.i.d. Gaussian process , where diag The prediction is the one that allows the whitening of the difference filter Two whitening filters are decoupled and act separately as scalar filters on both components ; filters can be either causal (prediction filter) or symmetric (smoothing-type filter) [9]. Still using an iterative approach, let be the prediction filter at the th iteration. If is an , then estimate of the process represents an estimate of the signal from The residual is thus described by a Gaussian process and the noise (34) plays the role of the reference signal. The restored signal The Gaussian process in (34) is (35) , it includes the process due to an incomplete In addition to and the filtered residual estimate of the whitening filter that depends on incomplete noise estimation noise (the Gaussian model for this term becomes more realistic after is long enough to let few iterations or if the filter the Central Limit Theorem hold true). The ZNL estimator allows us to separate, in the residual (34), the non-Gaussian from the other terms. The prediction filter noise sequence at the next iteration is the one that minimizes the MSE for the restored sequence (36) For the solution of (36), we may follow an adaptive approach or iterative matrix inversion similar to (31) or (32). In addition, convergence occurred in only a few iterations according to [9]. , or even one sample The low-order prediction filter predictor as shown in [21], represents an approximation of the AR model that is adequate for the separation of non-Gaussian from Gaussian processes, provided that the ZNL estimator has been chosen accordingly. VI. NUMERICAL EXAMPLES A. Simulations The performance analysis for a Gaussian background process superimposed on one polarized process has been consists of 105 achieved by simulation. The sequence random samples with one polarization azimuth in and varying characteristics; and

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(a)

(b) Fig. 2. Simulation of suboptimum model for one source of noise. (a) Estimated variance in the direction of noise source normalized to background 2 versus interference to background ratio (I/B) ^I2 ( = 1 )= ^B variance  evaluated for p1 = f0:01; 0:05; 0:1; 0:25g: (b) Probability p^( = 1 ) estimated from DD method versus I/B and p1 :

I/B

ranges from 3 dB up to 40 dB All the simulations have been scaled to I/B and the background by choosing have been estimated with the DD method and probability sectors). Fig. 2 shows an azimuth interval of 10 ( (noise source the model parameters estimated at seems to be not very sensitive to the azimuth). (at least for the practical values used here); probability for small values of it is overestimated and sensitive to I/B, according to the nonpolarized noise model in Fig. 1. is underestimated (or overestimated) in Probability is overestimated the interval of values where for large (or underestimated). The 3 dB of in is better approximated I/B is because by the Gaussian mixture [17]. Since the threshold of the and , the bias A-ZNL estimator depends on

Fig. 3. Root mean square error (RMSE) in the estimation of polarized noise (one source of polarized noise). Dashed line: optimum ZNL with given model parameters; solid line: suboptimum A-ZNL with model parameters estimated with DD method (azimuth quantization: 10 ); dash-dot line: optimum ZNL for nonpolarized noise with given model parameters.

in the estimated parameters is consistent with the expected threshold. This is proved by the results in Fig. 3, showing the root mean square error (RMSE) derived from for the A-ZNL estimator (21). In the same figure, the RMSE of the optimum ZNL estimator for polarized noise (9) and the RMSE of the ZNL estimator for nonpolarized noise [(12), (13)] are shown for reference; for both cases, the “true” model are assumed as given. The RMSE parameters of the A-ZNL estimator is lower, is bounded by the optimum ZNL estimator, and behaves similarly to the ZNL estimator for nonpolarized noise for high I/B (mainly because of the azimuth invariance of both ZNL’s). This comparison validates the suboptimal solution proposed here and suggests the use of circularly symmetrical ZNL, even in conditions of polarized noise, since it represents a conservative solution whenever the azimuth of interference is not known. However, the estimation of model parameters for circularly symmetrical ZNL cannot be performed consistently in a polarized noise environment. A simple numerical example with two noise processes allows us to take advantage of the A-ZNL. Let and be the parameters of the two noise processes; variances are normalized so The bivariate ZNL estimators and that represent the transformations that maps each value into the corresponding value The mapping function is rather complicated to visualize; however, according to the circular and the A-ZNL, we have chosen symmetry of the process a polar system. The envelope of the mapping function can be normalized to as For comparison with the kernel of the A-ZNL, the complementary function is shown in Fig. 4. The same figure also shows the mapping function for the azimuths Superimposed, for reference, in both figures are the contour lines of the bivariate pdf The ZNL is sensitive to the azimuth of noise processes and thresholds the observations accordingly. The radial

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along azimuths and are biased according to the values obtained for one polarized source, as shown in Fig. 2. The polarized noise model and the A-ZNL proposed here would be preferable to the circularly symmetric one when the is rather complex, mainly when it is multimodal pdf across the azimuths. According to Appendix A, this occurs when the number of equally distributed azimuths of the noise or when the azimuths process is small are clustered around few values. B. Experimental Results

(a)

(b) Fig. 4. Example of two polarized noises along 1 = =2 and 2 = =4: (a) ZNL estimator normalized to the envelope: 1 g [z ] = z : Arrows indicate the azimuths of the two noise processes; contour lines are the bivariate pdf fz (z ): (b) Azimuth of the ZNL estimator g [z ]:

0j

from impulsive EMI has The restoration of the signal been investigated in magnetotellurics [15] for dual-input/dualoutput filter identification [19] or with an approximate ZNL estimator for polarized noise [17]. The experimental data represents a sequence (approximately 5 104 samples) of lowHz) telluric electric (E) field that shows frequency ( polarized noise (the electric field was measured close to an urbanized area and a railway network). In the magnetotelluric method, the magnetic field represents the reference sequence , processes are approximately stationary [15], and has to be prewhitened to employ the i.i.d. model of Section V [19]. is the one estimated after two iterThe residual sequence ations of batch processing and contains both the identification error (circularly Gaussian) and the polarized noise. Fig. 6(a) shows the polar plot (hodogram) of the residual sequence Polarizations of noise process are along two dominant directions that represent the most likely EM noise sources (e.g., 40 seems to depend on grounding devices of the noise on 80 is mainly associated with trains urbanized area, and moving along the rails as polarization is slowly changing). and the probability The estimated interference estimated with the DD method are in Fig. 6(b) and (c), respectively. The azimuth dependent parameters are consistent with the corresponding hodogram, like the thresholds of the kernel

jjj

distribution of angles in Fig. 4(b) indicates that ZNL preserves ), the azimuths (azimuth preservation implies that along, or perpendicular and this occurs for small values of to, the azimuths of the noise processes. A-ZNL has been derived from the azimuthal decomposition in a simulated of 2 105 samples, and the azimuth interval is sequence , and 4 . Fig. 5(b) shows the estimated parameters for varying azimuths. The kernel of the Fig. 5(d) shows shown in Fig. 5(a) and (c) A-ZNL using different representations has a behavior similar in shape to that in Fig. 4 (azimuth preservation is implicit in the suboptimum ZNL approximation). The thresholds of the A-ZNL approximately evaluated in terms of standard deviation follow the so-called “three-sigma” rule that is frequently used in statistics when these are evaluated along the azimuths and . The threshold is larger (approximately ) for the other azimuths. In addition, the estimated model parameters

VII. CONCLUSIONS The zero memory nonlinearity (ZNL) estimator, which is based on the mixture approximation of the Middleton Class A model, allows us to separate non-Gaussian polarized noise from an independent and identically distributed Gaussian process. The polarized noise is assumed to be given by a independent processes, each having a detercombination of ministic (but not known) azimuth of polarization. The optimum ZNL estimator can be derived only if all the statistical parameters of the involved processes are known; therefore, its use is impractical. The suboptimal azimuth-dependent ZNL (A-ZNL) estimator has been obtained by decomposing the combined process (i.e., the Gaussian and the non-Gaussian processes) into azimuthally independent processes. Statistical parameters are estimated by considering, for each azimuth or azimuthal sector, the corresponding envelope process. Statistical parameters are estimated after the classification of the samples in the sequence as Gaussian and non-Gaussian (decision-directed are implicitly estimated method). The number of sources in azimuth discretization. Numerical examples demonstrate

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(a)

(b)

(c)

(d)

Fig. 5. Simulation of the A-ZNL estimator for two polarized noises along 1 = =2 and 2 = =4 (same as Fig. 4). (a) and (c) Kernel of the A-ZNL 2h ^ [jz j;  ^B ^ I ( ); p ^( )]. (b) Estimated  ^I ( )= ^B at first iteration (solid line filled bright gray), during iterations (dotted line) and at convergence estimator  ^B ;  (solid line filled dark gray). (d) Estimated probability p^( ) [same line/color coding as in (b)].

the feasibility of the proposed method and the limited loss of performance compared with the optimum estimator. In addition to the experimental application, the proposed method can be used in other geophysical methods, in the processing of natural EM fields for earthquake prediction, and in radar polarimetry processing.

Let ; the noise pdf can be approximated as [ terms have been neglected] (37) The convolution can be conveniently rewritten in terms of characteristic functions

APPENDIX A ASYMPTOTIC PROPERTIES OF POLARIZED NOISE The validity of ZNL for nonpolarized noise can be shown is circularly by demonstrating that the observation pdf This follows from the analysis of symmetric for the asymptotic properties of polarized noise. For this purpose, the noise model (2) is simplified into a combination of independent noise processes having uniformly with and distributed polarization and the same statistical parameters The convolution of polarized noise pdf’s is evaluated for under the and We have the following approximations is the noise pdf for notation: . the th polarization azimuth

(38)

For

where , it becomes

(39)

SPAGNOLINI: CANCELLATION OF POLARIZED IMPULSIVE NOISE

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i.e., it is a combination of along angles. The pdf

line functions equally spaced

(42) with is circularly symmetric if the 2-D convolution of smooths the combination of equally spaced , the in azimuths. If the angular spacing is for neighboring polarizations in (42) can smoothing of In be (approximately) guaranteed up to amplitude is the number of polarized the case discussed here, noise processes that makes the circularly symmetric model more appropriate for the ZNL estimator. If the polarization azimuths are more or less uniformly distributed, the circular symmetric model is appropriate for However, resolution of two noise processes and in terms of their polarization depends on their azimuth ); for resolution, their polarization should separation (if be Asymptotic properties can be derived (after a more complicated analysis) even without the strict model assumptions of and ); results polarized noise processes (i.e., are similar, provided that polarizations of noise processes are (approximately) uniformly distributed. In any case, circular and ZNL is of interest in suboptimum symmetry of pdf ZNL; in this case, the asymptotic analysis can be carried out locally.

(a)

(b)

ACKNOWLEDGMENT The author is indebted to F. Rocca for his comments. The suggestions of an anonymous reviewer regarding the organization of the manuscript are strongly acknowledged. REFERENCES

(c) Fig. 6. Experimental results. (a) Hodogram of sequence fz g (dots). (b)  ^I ( )= ^B . (c) p ^( ) estimated with DD method (the same line/color coding as in Fig. 5).

and therefore, the noise pdf and the pdf Gaussian

are circularly

(40) ; the noise pdf can be approximated as (by Let for ) [17] neglecting powers

(41)

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Umberto Spagnolini graduated from Politecnico di Milano cum laude, where received the Dottore in Ingegneria Elettronica degree in 1988. Since 1988, he has been with the Dipartimento di Elettronica e Informazione, where he presently holds the position of Associate Professor of Digital Signal Processing. His primary research (and applications) focuses on array processing and wavefield interpolation (mobile communication and geophysics), inverse problems (ground penetrating radar), and parameter estimation (two-dimensional phase unwrapping for SAR), and non-Gaussian EMI reduction. Dr. Spagnolini is a member of the Society of Exploration Geophysicists and the European Association of Geoscientists and Engineers (EAGE). In 1991, he received the Associazione Elettronica ed Elettronica Italiana Award and the Van Weelden Award of the EAGE.