S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, pp. 433--481, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that take into account the possible existence of modelling errors. In this context, the recent prominence of H # and set membership robust filtering techniques [1, 2] appears to have overshadowed another fruitful approach proposed approximately 20 years ago by Kassam, Poor and their collaborators [3, 4, 5, 6]. In this approach, which was inspired by Huber s pioneering work [7] in robust statistics, the actual joint spectral density of the signal and observations is assumed to belong to a neighborhood of the nominal model. This neighborhood or uncertainty class can be specified in a variety of ways. It ....

....observations is assumed to belong to a neighborhood of the nominal model. This neighborhood or uncertainty class can be specified in a variety of ways. It can be based on an # contamination model of the type originally considered by Huber, a total variation model [4, 5] or a spectral band model [3, 6] wherein the power spectral densities (PSDs) specifying the signal and observations are required to stay within a band centered on the nominal PSD. Other classes that have been considered in the literature [6] include p point models which allocate fixed amounts of power to certain spectral bands ....

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S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey," Proc. IEEE, vol. 73, pp. 433--481, Mar. 1985.

.... spread spectrum multiple access (DS SSMA) systems using TCM has been studied in [2] Most of the work on TCM has been accomplished under the additive white Gaussian noise (AWGN) assumption [2,9,10] It is well known that sometimes the Gaussian noise assumption cannot be entirely justified [5]. The non Gaussian nature of atmospheric noise, 0165 1684 98 19.00 # 1998 Published by Elsevier Science B.V. All rights reserved. PII S0165 1684(97)00228 4 impulses caused by turning on some electrical devices, etc. are among the typical examples. Therefore, it is worthwhile to study the e#ects ....

S.A. Kassam, H.V. Poor, Robust techniques for signal processing: a survey, Proc. IEEE 73 (March 1985) 433---481.

....in communications, and performance analyses of algorithms. 9.2 Infinite variance models Heavy tailed non Gaussian processes, particularly the Gaussian mixture model and the Cauchy r.v. have long been used to develop and test signal processing techniques which are robust to impulsive noise [194]. The Cauchy r.v. has infinite variance, and is a special case of an alpha stable r.v. It is easy to create a Cauchy r.v. as the ratio of two (possibly correlated) Gaussian random variables and Feller [108] shows how the Cauchy r.v. arises in an example with rotating mirrors. Stable r.v. s result ....

S. Kassam and H. Poor, "Robust techniques for signal processing: a survey," Proc IEEE, vol. 73, pp. 433--481, 1985.

....under nominal conditions and acceptable performance for signal and noise processes that deviate from the nominal within classes of possible characteristics. The theory of robust statistics can be found in [6] a survey on applications to signal processing and communications is presented in [8]. Regarding decentralized detection, in [2] robust data fusion rules are derived for general uncertainty classes described by 2 alternating Choquet capacities [7] using error probability as the performance criterion. In this paper, we consider a decentralized detection scheme with asymptotically ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: A survey", Proceedings of the IEEE, Vol. 73, No. 3, pp. 433-481, March 1985.

.... He is now with the Department of Electrical and Electronics Engineering, Eastern Mediterranean University, Gazimagusa, TRNC (e mail: aykut.hocanin emu.edu.tr) e.g. Gaussian channel) and yet do not face catastrophy when the noise distribution is not nominal (e.g. unlike linear schemes) [9]. Note that suboptimality here refers to very good performance that is slightly worse than that of the nominal optimal detector estimator. In this paper, we consider the performance advantages offered by single user robust detectors in a channel corrupted by multiple access interference and ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey", Proceedings of the IEEE, Vol. 73, No. 3, pp. 433-481, March 1985.

....is paid to the design of the diversity combiner. In general, FFH is inherently near far resistant. Under the described channel conditions, receivers that are optimized for specific channel noise can suffer significant performance degradation for even small deviations from the nominal assumptions [1]. To overcome these difficulties, nonparametric methods have been investigated [2] Such methods are robust against changing noise statistics because they use only coarse information about the received signal. For example, at the output of the Fourier transformer in a non coherent FH receiver, ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, pp. 433--481, Mar. 1985.

....in the presence of impulsive noise is of increasing interest. In general, the Gaussian noise assumption does not provide a valid model for channels corrupted by impulsive noise [1] and linear, matched filter receivers often perform sub optimally in the presence of non Gaussian impulsive noise [2,3]. A broad class of impulsive processes found in practice is modeled accurately [1] by symmetric ff stable (SffS) random variables. In addition, under certain well defined conditions, multiple access noise also is SffS [4,5] The characteristic function of SffS noise is [1] OE( 1 ; 2 ) ....

S.A. Kassam and H.V. Poor, "Robust Techniques for Signal Processing: A Survey," Proc. IEEE, vol. 73, No. 3, Mar. 1995, pp 433-481.

....all correlations within prescribed uncertainty classes and is thus insensitive to limited deviations from the nominal operating conditions. Our results extend a previously proposed minimax robust time invariant Wiener lter based on the so called p point uncertainty model for stationary processes [3, 4]. The paper is organized as follows. Section 2 introduces a p point uncertainty model for nonstationary processes, and Section 3 derives the corresponding minimax robust timevarying Wiener lter. Intuitively appealing time frequency Funding by FWF grant P11904 TEC. 1 The correlation operator ....

....) E fx(t) x (t 0 )g. In a discrete time setting, Rx would be a matrix. formulations are presented in Section 4. Finally, numerical simulations are provided in Section 5. 2 NONSTATIONARY p POINT UNCERTAINTY MODEL Generalizing the p point uncertainty model for stationary random processes [3, 4], we propose an uncertainty model for nonstationary random processes which describes the designer s uncertainty about the actual correlations. Let the orthogonal subspaces X i , i = 1; 2; N be a partition of the space L2 (R) of square integrable functions, i.e. L N i=1 X i = L2 (R) and ....

S. A. Kassam and H. V. Poor, \Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, pp. 433-481, March 1985.

....expressed as emin(Rs ; Rn) e(HW ; Rs ; Rn) tr Rs (Rs Rn ) 1 Rn : 3) The Wiener lter s sensitivity to deviations of the actual correlations from the nominal correlations motivates the use of minimax robust Wiener lters. This paper extends the robust Wiener lters proposed in [2] [5] for stationary processes to the nonstationary case (see also [6, 7] Complementing the introduction of robust time varying Wiener lters in [8] Section 2 provides a fundamental result that facilitates the calculation of such lters. A further simpli cation is achieved in Section 3 by a ....

....of TF uncertainty classes e S, e N and we provide closed form expressions for the respective robust TF Wiener lters e HR . p Point Model. Let fR i g i=1;2; N be a partition of the TF plane, i.e. S N i=1 R i = R 2 and R i R j = for i 6= j. Extending the stationary case de nition in [3, 5], so called p point uncertainty classes can be de ned for WVS as [8] e S = n W s (t; f) ZZ R i W s (t; f) dtdf = s i ; i = 1; 2; N o e N = n W n (t; f) ZZ R i W n (t; f) dtdf = n i ; i = 1; 2; N o ; i.e. as the sets that contain all pseudo WVS having prescribed ....

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S. A. Kassam and H. V. Poor, \Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, pp. 433-481, March 1985.

....P11904 TEC. 1 Second, it is less sensitive to deviations from the nominal operating conditions in fact, it maintains constant performance for all second order statistics within prescribed uncertainty classes. Our results extend minimax robust time invariant Wiener lters for stationary processes [1, 2] to the nonstationary case. For simplicity, we restrict ourselves to a nonstationary version of the so called p point uncertainty model, although other uncertainty models can be used as well (see [3] for a generalized formulation) The paper is organized as follows. Based on a novel p point ....

.... Point Uncertainty Model In order to model the designer s uncertainty about the actual correlations, we next propose an uncertainty model for nonstationary random processes [9, 10] This uncertainty model generalizes the so called p point uncertainty model for stationary processes considered in [1, 2] to the nonstationary case. An approximate time frequency reformulation of this model will be presented in Section 4. The novel nonstationary p point uncertainty model is based on a partition of the space L 2 (R) of square integrable functions into N mutually orthogonal subspaces X i with i = 1; ....

S. A. Kassam and H. V. Poor, \Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, pp. 433-481, March 1985.

....pseudo Wiener filter) to variations of the secondorder statistics motivates the use of minimax robust Wiener filters that optimize the worst case performance within specified uncertainty classes of operating conditions. Recently, the minimax robust time invariant Wiener filters introduced in [50 53] for stationary processes were extended to the nonstationary case, and time frequency designs of robust time varying Wiener filters were proposed [54,55] Particularly simple and intuitively appealing results were obtained for so called p point uncertainty classes. Let fR i g i=1;2; K be a ....

....proposed [54,55] Particularly simple and intuitively appealing results were obtained for so called p point uncertainty classes. Let fR i g i=1;2; K be a partition of the time frequency plane, i.e. S K i=1 R i =R 2 and R i R j = for i 6= j. Extending the stationary case definition in [51,53], p point uncertainty classes can be defined for Wigner Ville spectra as [54] S = n W (0) s (t; f) ZZ R i W (0) s (t; f) dtdf = s i ; i = 1; K o N = n W (0) n (t; f) ZZ R i W (0) n (t; f) dtdf = n i ; i = 1; K o ; i.e. as the sets which contain all ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, pp. 433--481, March 1985.

....accordingly, the prediction schemes obtained are distribution dependent. An exception to this is the line of work related to robust filtering which was carried out by researchers from the signal processing and information theoretic communities in the seventies and eighties (cf. 9] 11] 19] [25], 26] 36] and references therein) The setting considered in robust filtering, however, is completely different than the one we consider in this paper. The typical problem considered in robust filtering is that where the power spectral densities of the signal and the noise (assumed jointly ....

S. A. Kassam and H. V. Poor, "Robust Techniques for Signal Processing: A Survey," Proceedings of the IEEE, vol. 73, no. 3, March 1985.

....e y 1 A 1 oe 2 l e Gamma y 1 A 1 oe 2 l (9) The estimator in (9) is a robust estimator. Fig. 2 gives an example of the soft estimate of b1 using (9) in 2 term Gaussian mixture noise. The re descending behavior of the estimator for large observations is typical of a robust estimator [11]. When no prior information is available, P r(b1 = 1) P r(b1 = Gamma1) 1 2 , and equation (9) becomes Efb1 jy1 g = P L l=1 l oe l e Gamma y 2 1 A 2 1 2oe 2 l sinh( y 1 A 1 oe 2 l ) P L l=1 l oe l e Gamma y 2 1 A 2 1 2oe 2 l cosh( y 1 A 1 oe 2 l ) ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey," Proceedings of IEEE, vol. 73, pp.433-481, March 1985.

....slightly smaller probability of bit error, when decisions are generated using a fixed, relatively small delay. We note that in the approach taken here we use a parametric model for the noise and attempt to estimate the parameters. A different approach is to use the theory of robust detection [19], 20] which is based on much less assumptions on the form of the noise pdf. In fact, we discuss in [11] how our approach does perform robust processing, which is an interesting connection to the work in [19] 20] 7 ....

....to estimate the parameters. A different approach is to use the theory of robust detection [19] 20] which is based on much less assumptions on the form of the noise pdf. In fact, we discuss in [11] how our approach does perform robust processing, which is an interesting connection to the work in [19], 20] 7 ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey," Proc. IEEE, Vol. 73, No. 3, pp. 433 - 481, March 1985.

....y 1 A 1 oe 2 l e Gamma y 1 A 1 oe 2 l # (9) 8 The estimator in (9) is a robust estimator. Fig. 2 gives an example of the soft estimate of b 1 using (9) in 2 term Gaussian mixture noise. The re descending behavior of the estimator for large observations is typical of a robust estimator [20]. When no prior information is available, P r(b 1 = 1) P r(b 1 = Gamma1) 1 2 , and equation (9) becomes Efb 1 jy 1 g = P L l=1 l oe l e Gamma y 2 1 A 2 1 2oe 2 l sinh( y1 A1 oe 2 l ) P L l=1 l oe l e Gamma y 2 1 A 2 1 2oe 2 l cosh( y1 A1 oe 2 l ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey," Proceedings of IEEE, vol. 73, pp.433-481, March 1985.

....interval of the Doppler frequency and the low end of the SIR range can indeed be used over the whole parameter range. If the operating area is bounded by SIR= 15,25]dB and fD = 0; 160]Hz, then this lter does in fact constitute a minimax robust design, since the so called saddle point condition [5] is ful lled: The resulting performance attains its worst value at the nominal (worst case) design point. In the most critical regions, with low SIR and or high Doppler frequency, the performance for an AR 2 I based design is about the same as for an AR 2 based design. VI.C BER Performance for ....

S. A. Kassam and V. Poor, \Robust techniques for signal processing: a survey," Proc. of the IEEE, vol. 73, pp. 433-481, 1985.

....and the averaged covariance function (4.4) with D 2 U[0:01 0:03] dotted) and D 2 U[0:015 0:025] dashed) C. Robust Design of Adaptation Laws If the Doppler frequency D is uncertain, one could minimize the worst case e ect of this uncertainty by performing a minimax robust lter design [7, 15]. A less conservative, and often much less computationally demanding, alternative is to regard D as a random variable. The MSE tracking performance resulting from the outcomes of D is averaged and we minimize this average [22, 20] As is shown in Section 3.5 of [9] this problem can be solved by ....

....the Doppler frequency and the low end of the SIR range can indeed be used over the whole parameter range. If the operating area is considered to be bounded by SIR= 15,25]dB and fD = 0; 160]Hz, this lter does in fact constitute a minimax robust design, since the so called saddle point condition [7] is ful lled: The resulting performance attains its worst value at the nominal (worst case) design point. In the most critical regions, with low SIR and or high Doppler frequency, the performance for an AR 2 I based design is about the same as for an AR 2 based design. In the at fading case, ....

S. A. Kassam and V. Poor, \Robust techniques for signal processing: a survey," Proc. of the IEEE, vol. 73, pp. 433-481, 1985.

....are available. These standard antenna diversity combining techniques are designed for Gaussian noise, so they perform linear processing of the antenna measurements. In general, reception schemes designed for Gaussian noise environments perform very poorly when impulsive noise is present [8]. In particular, it has been demonstrated in [9] that maximal ratio combining, equal gain combining, and selection diversity are not effective in impulsive noise environments. The poor performance is explained intuitively by noting that schemes designed for Gaussian noise cases expect noise ....

....not known to be weak, but an explicit receiver structure is assumed which is optimum only for Gaussian noise cases. This structure does not allow the limiting needed to achieve good performance for heavy tailed noise cases, and its performance will be poor for such cases based on previous studies [8, 9]. Instead of designing a receiver for a particular noise distribution, another approach is to develop a receiver that adapts to its environment. Indeed, even a very simple adaptive receiver that uses a weighted combination of linear and hard limiting characteristics provides significant ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey," Proceedings IEEE, vol. 73, pp. 433-481, March 1985.

.... vision problems are ill posed, they are sensitive to noise, and their solutions might be incorrect if the actual observations do not fit the model assumed [1] To implement a robust computer vision algorithm, a stable system effectively coping with noise and distortions in the input is required [2], 3] In spite of the usefulness of the motion field, conventional motion estimation methods suffer from such problems [4] 5] This paper proposes a motion estimation algorithm robust to motion discontinuity and noise. Because of ill posedness, it is difficult to find a solution in all cases ....

# S.A. Kassam and H.V. Poor, "Robust Techniques for Signal Processing: A Survey," Proc. IEEE, vol. 73, no. 3, pp. 433-481, Mar. 1985.

....for the whole class of possible models, is obtained by minimizing the squared estimation error, averaged both with respect to model errors and the noise. Most previous suggestions for robust filter design have been based on the minimax approach. See e.g. D Appolito and Hutchinson (1972) and Kassam and Poor (1985). Apart from leading to a much simpler design methodology, the approach proposed here avoids two drawbacks of robust minimax design. First, the descriptions of model uncertainties may have soft bounds. These are more readily obtainable in a noisy environment than the hard bounds required for ....

Kassam, S A and H V Poor (1985). Robust techniques for signal processing: a survey. Proc. IEEE, vol 73, pp 433-- 481.

....a drastic degradation in performance even when deviations from nominal assumptions are small. This is the reason for constructing robust signal processing algorithms, i.e. methods that offer good performance under nominal conditions and acceptable performance for a wide range of other conditions [79], 80] Even though robustness was not emphasized in this thesis, it motivated us to use limiting nonlinearities. Prediction with Neural Networks The Volterra predictors used in noise estimation cancellation detectors fall considerably short of our expectations based on the performance of longer ....

S. Kassam and H. Poor, "Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, No.3, pp. 433--481, March 1985.

....there exists a saddle point solution. One may then search for a least favourable pair of signal and noise spectra, in prespecified uncertainty classes. The optimal estimator is a filter designed for that pair. See [6] 12] 17] 19] 20] 22] 27] and the survey paper by Kassam and Poor [13]. Uncertainties can be described in a state space framework. See e.g. 10] 18] and [28] The computational effort involved in minimax design is considerable. Closed form solutions do mostly not exist. Apart from leading to a much simpler design methodology, the approach proposed here avoids ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey," Proceedings of the IEEE, vol 73, (1985) 433-- 481.

....In this note we consider the following detection problem. The nominal signals are known, but the actual signals present in the observed data can be corrupted slightly. This situation may arise because of imperfect prepagation characteristic,element position uncertainties, or element gain variation [10,11]. The probability distribution of the noise is known precisely, or some small deviation exists from the nominal model. This situation occurs frequently in applications. 6] To derive the robust detector which is insensitive to the corruption of signals and deviation from nominal noise model, we ....

S.A. Kassam and H.V. Poor, Robust techniques for signal processing: A survey. Proc. IEEE, 73:433-481,1985.

....(MSE) J 1 = sup Delta(k) traceE(z(k) Gamma z(k) z(k) Gamma z(k) T (2) can be minimized by solving two coupled Riccati equations, combined with a one dimensional numerical search. This represents a significant computational simplification, as compared to previous minimax designs [4] [9], 10] 11] In the second approach, a probabilistic description of model errors will be used, as outlined in [16] A set of (true) dynamic systems is assumed to be well described by a set of discrete time, stable, linear and timeinvariant transfer function matrices F = F o DeltaF : 3) We ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey," Proceedings of the IEEE, vol. 73, pp. 433--481, 1985.

....models. Error models are by necessity imprecise; exact modelling of the unmodelled dynamics would be a contradiction in terms. Hard bounds in the frequency domain are one example of specifications, commonly characterized as j DeltaF ( j L( 1. 1) For other models of spectral uncertainty, see Kassam and Poor (1985), p 438, or Goodwin and Salgado (1989) Error models may be obtained from off line experiments or via on line adaptation. Ways to utilize estimates of model uncertainty are of value in adaptive as well as in fixed filter design. We will consider mainly estimation problems, with scalar stationary ....

....as well as in the plant dynamics. Minimax design implies the search for a saddle point solution. One may search for a least favourable pair of signal and noise spectra, in prespecified uncertainty classes. Under certain conditions, the optimal estimator is a filter designed for that pair. See Kassam and Poor (1985) and also Kassam and Lim (1977) Poor (1980) Moustakides and Kassam (1983,1985) and Vastola and Poor (1984) Uncertainties can be described in a state space framework. See e.g. Martin and Minz (1983) 1 For methods, see e.g. Nagpal and Khargonekar (1991) The main known connection of H1 ....

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Kassam, S. A. and H. V. Poor (1985). Robust techniques for signal processing: a survey. Proceedings of the IEEE, vol 73, pp 433-- 481.

....with respect to a set of models, one can search for models whose optimal filter gives the worst (nominal) performance, and use the corresponding filter. This is a much simpler task, but can still be computationally demanding. See [18] 27] 30] 36] and the survey paper by Kassam and Poor [19]. The condition minR max F = max F minR is not fulfilled in numerous problems, which makes them very difficult to solve. See e.g. Example 5 in [35] and the example in Section 4 below. Kalman filter like estimators have recently been developed for systems with structured and possibly time varying ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey," Proceedings of the IEEE, vol. 73, pp. 433--481, 1985.

....There are many aspects of this problem that we have not treated here. Issues such as robustness and nonGaussian noise have been touched upon very briefly; however, these are important issues in applications, and more detailed treatments of these issues can be found, for examples, in [1] 58] [60], and [74] We also mentioned briefly RKHS methods for the detection of non Gaussian signals, and for signal and noise models exhibiting fractal behavior. Further techniques for the detection of non Gaussian signals are reviewed in [30] and methods for exploiting self similarity are described, ....

S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, no. 3, pp. 433--481, Mar. 1985.

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S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, pp. 433--481, 1985.

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S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, pp. 433--481, Mar. 1985.

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S. A. Kassam and H. V. Poor. Robust techniques for signal processing: A survey. Proceedings of the IEEE, 73(3):443-481, March 1985.

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S. A. Kassam and V. Poor, "Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, pp. 433--481, Mar. 1985.

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S. Kassam and H. Poor, "Robust techniques for signal processing: a survey," Proceedings of the IEEE, vol. 73, pp. 433--481, 1985.

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S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey," Proc. of the IEEE, vol. 73, pp. 433481, Mar. 1985.

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S. A. KASSAM AND H. V. POOR. Robust techniques for signal processing: a survey. Proceedings of the IEEE, 73(3):433--481, March 1985.

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S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: A survey," IEEE Proc., vol. 73, pp. 433--481, Mar. 1985.

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S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: A survey," Proc. IEEE, vol. 73, pp. 433--481, Mar. 1985.

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S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey," Proc. IEEE 73, 433--481 #1985#.

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S. A. Kassam and H. V. Poor, "Robust techniques for signal processing: a survey," Proc. IEEE, Vol. 73, No. 3, pp. 433 - 481, March 1985.

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S. A. Kassam and H. V. Poor. Robust techniques for signal processing: A survey. Proc. IEEE, 73:433--481, 1985.

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