The Gaussian assumption in second-order estimation problems in digital communications

This paper deals with the goodness of the Gaussian assumption when designing second-order blind estimation methods in the context of digital communications. The low- and high-signal-to-noise ratio (SNR) asymptotic performance of the maximum likelihood estimator - derived assuming Gaussian transmitted symbols - is compared with the performance of the optimal second-order estimator, which exploits the actual distribution of the discrete constellation. The asymptotic study concludes that the Gaussian assumption leads to the optimal second-order solution if the SNR is very low or if the symbols belong to a multilevel constellation such as quadrature-amplitude modulation (QAM) or amplitude-phase-shift keying (APSK). On the other hand, the Gaussian assumption can yield important losses at high SNR if the transmitted symbols are drawn from a constant modulus constellation such as phase-shift keying (PSK) or continuous-phase modulations (CPM). These conclusions are illustrated for the problem of direction-of-arrival (DOA) estimation of multiple digitally-modulated signals.Peer ReviewedPostprint (published version

Maximum Likelihood-based Direction-of-arrival Estimator For Discrete Sources

This paper addresses the problem of direction-of-arrival (DOA) parameter estimation in array processing when the signals are inherently discrete, which is the case mainly in the digital communication context. Based on the particular structure of the signal space in the data model, a maximum likelihood-based approach is introduced. The strategy consists in transforming the parameter estimation problem into a decision task. It is shown through numerical simulations that the proposed solution closely follows the performance limit given by the Cramér-Rao bound. Some important features of the technique are as follows: (i) it is capable of handling any number of sources, provided that the number of sensors is greater than or equal to two and the number of snapshots is sufficiently greater than the cardinality of the signal space; (ii) the estimation quality is not affected by the angle and phase separation; and (iii) it offers the possibility to deal with uncalibrated arrays. © 2013 Springer Science+Business Media New York.32524232443Abeida, H., Delmas, J.P., MUSIC-like estimation of direction of arrival for noncircular sources (2006) IEEE Trans. Signal Process., 54 (7), pp. 2678-2690. , 10.1109/TSP.2006.873505 10.1109/TSP.2006.873505Anand, K., Mathew, G., Reddy, V., Blind separation of multiple co-channel BPSK signals arriving at an antenna array (1995) IEEE Signal Process. Lett., 2 (9), pp. 176-178. , 10.1109/97.410546 10.1109/97.410546Araki, S., Sawada, H., Mukai, R., Makino, S., DOA estimation for multiple sparse sources with normalized observation vector clustering (2006) IEEE International Conference on Acoustics, Speech and Signal Processing, , ICASSP 2006 Proceedings 5 10.1109/ICASSP.2006.1661205Attux, R., Suyama, R., Ferrari, R., Junqueira, C., Krummenauer, R., Larzabal, P., Lopes, A., A clustering-based method for DOA estimation in wireless communications (2007) 15th European Signal Processing Conference, pp. 262-266. , EURASIP PoznańBresler, Y., Macovski, A., Exact maximum likelihood parameter estimation of superimposed exponential signals in noise (1986) IEEE Trans. Acoust. Speech Signal Process., 34 (5), pp. 1081-1089. , 10.1109/TASSP.1986.1164949De Castro, L.N., (2006) Fundamentals of Natural Computing: Basic Concepts, Algorithms and Applications (Chapman & Hall/CRC Computer and Information Sciences), , Chapman & Hall LondonDe Castro, L.N., Timmis, J., An artificial immune network for multimodal function optimization (2002) Proceedings of the 2002 Congress on Evolutionary Computation, pp. 699-704. , Honolulu, HI, USA CEC'02 1 10.1109/CEC.2002.1007011Chargé, P., Wang, Y., A root-MUSIC-like direction finding method for cyclostationary signals (2005) EURASIP J. Appl. Signal Process., 2005 (1), pp. 69-73. , 10.1155/ASP.2005.69 1107.94334 10.1155/ASP.2005.69Chargé, P., Wang, Y., Saillard, J., A non-circular sources direction finding method using polynomial rooting (2001) Signal Process., 81 (8), pp. 1765-1770. , 10.1016/S0165-1684(01)00071-8 1076.94510 10.1016/S0165-1684(01)00071-8Chargé, P., Wang, Y., Saillard, J., A root-MUSIC algorithm for non circular sources (2001) IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 2985-2988. , ICASSP'01 Proceedings IEEE Comput. Soc. Washington 10.1109/ICASSP.2001. 940276Chargé, P., Wang, Y., Saillard, J., An extended cyclic MUSIC algorithm (2003) IEEE Trans. Signal Process., 51 (7), pp. 1695-1701. , 10.1109/TSP.2003.812834 1996960 10.1109/TSP.2003.812834Chen, C., Zhang, X., Joint angle and array gain-phase errors estimation using PM-like algorithm for bistatic MIMO radars (2012) Circuits Syst. Signal Process, pp. 1-19. , doi: 10.1007/s00034-012-9502-2Delmas, J.P., Asymptotically minimum variance second-order estimation for noncircular signals with application to DOA estimation (2004) IEEE Trans. Signal Process., 52 (5), pp. 1235-1241. , 10.1109/TSP.2004.826170 2061979 10.1109/TSP.2004.826170Delmas, J.P., Abeida, H., Stochastic Cramér-Rao bound for noncircular signals with application to DOA estimation (2004) IEEE Trans. Signal Process., 52 (11), pp. 3192-3199. , 10.1109/TSP.2004.836462 2095600 10.1109/TSP.2004.836462Delmas, J.P., Abeida, H., Cramér-Rao bounds of DOA estimates for BPSK and QPSK modulated signals (2006) IEEE Trans. Signal Process., 54 (1), pp. 117-126. , 10.1109/TSP.2005.859224 10.1109/TSP.2005.859224Delmas, J.P., Abeida, H., Statistical resolution limits of DOA for discrete sources (2006) IEEE International Conference on Acoustics, Speech and Signal Processing, , ICASSP'2006 Proceedings 4 Toulouse France 10.1109/ICASSP.2006.1661112Duda, R.O., Hart, P.E., Stork, D.G., (2000) Pattern Classification, , 2 Wiley-Interscience New YorkEl Kassis, C., Picheral, J., Fleury, G., Mokbel, C., Direction of arrival estimation using EM-ESPRIT with nonuniform arrays (2012) Circuits Syst. Signal Process., 31, pp. 1787-1807. , 10.1007/s00034-012-9397-y 2971326 10.1007/s00034-012-9397-yGardner, W.A., Simplification of MUSIC and ESPRIT by exploitation of cyclostationarity (1988) Proc. IEEE, 76 (7), pp. 845-847. , 10.1109/5.7152 10.1109/5.7152Gershman, A., Stoica, P., MODE with extra-roots (MODEX): A new DOA estimation algorithm with an improved threshold performance (1999) IEEE International Conference on Acoustics, Speech, and Signal Processing, pp. 2833-2836. , Phoenix, AZ ICASSP'99 Proceedings 5 10.1109/ICASSP.1999.761352Gounon, P., Adnet, C., Galy, J., Localisation angulaire de signaux non circulaires (1998) Trait. Signal, 15 (1), pp. 17-23. , 0995.94506Haardt, M., Romer, F., Enhancements of unitary ESPRIT for non-circular sources (2004) IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 101-104. , ICASSP'04 Proceedings 2 10.1109/ICASSP.2004.1326204Inaba, M., Katoh, N., Imai, H., Applications of weighted Voronoi diagrams and randomization to variance-based k-clustering: (Extended abstract) (1994) Proceedings of the Tenth Annual Symposium on Computational Geometry, pp. 332-339. , SCG'94 ACM New York 10.1145/177424.178042 10.1145/177424.178042Kay, S.M., (1993) Fundamentals of Statistical Signal Processing, , Estimation Theory I Prentice Hall Englewood CliffsKrim, H., Viberg, M., Two decades of array signal processing research: The parametric approach (1996) IEEE Signal Process. Mag., 13 (4), pp. 67-94. , 10.1109/79.526899 10.1109/79.526899Kumaresan, R., Tufts, D., Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise (1982) IEEE Trans. Acoust. Speech Signal Process., 30 (6), pp. 833-840. , 10.1109/TASSP.1982.1163974Lavielle, M., Moulines, E., Cardoso, J.F., A maximum likelihood solution to DOA estimation for discrete sources (1994) IEEE Seventh SP Workshop on Statistical Signal and Array Processing, pp. 349-352. , 10.1109/SSAP.1994.572515Lee, H.B., The Cramér-Rao bound on frequency estimates of signals closely spaced in frequency (1992) IEEE Trans. Signal Process., 40 (6), pp. 1507-1517. , 10.1109/78.139253 10.1109/78.139253Leshem, A., Van Der Veen, A., Direction-of-arrival estimation for constant modulus signals (1999) IEEE Trans. Signal Process., 47 (11), pp. 3125-3129. , 1064.94518 10.1109/78.796446Li, J., Stoica, P., Liu, Z., Comparative study of IQML and MODE direction-of-arrival estimators (1998) IEEE Trans. Signal Process., 46 (1), pp. 149-160. , 10.1109/78.651203Merz, P., An iterated local search approach for minimum sum-of-squares clustering (2003) Advances in Intelligent Data Analysis V, 5th International Symposium on Intelligent Data Analysis, IDA 2003, pp. 286-296. , Lecture Notes in Computer Science 2810 Springer BerlinRoy, R., (1987) ESPRIT: Estimation of Signal Parameters Via Rotational Invariance Techniques, , Ph.D. thesis, Stanford University, Stanford, CASchell, S.V., Calabretta, R.A., Gardner, W.A., Agee, B.G., Cyclic MUSIC algorithms for signal-selective direction estimation (1989) International Conference on Acoustics, Speech and Signal Processing, pp. 2278-2281. , Glasgow, UK ICASSP'1989 10.1109/ICASSP.1989.266920 10.1109/ICASSP.1989. 266920Schmidt, R.O., Multiple emitter location and signal parameter estimation (1986) IEEE Trans. Antennas Propag., 34 (3), pp. 276-280. , 10.1109/TAP.1986.1143830Shynk, J., Gooch, R., The constant modulus array for cochannel signal copy and direction finding (1996) IEEE Trans. Signal Process., 44 (3), pp. 652-660. , 10.1109/78.489038 10.1109/78.489038Smith, S.T., Statistical resolution limits and the complexified Cramér-Rao bound (2005) IEEE Trans. Signal Process., 53 (5), pp. 1597-1609. , 10.1109/TSP.2005.845426 2148598 10.1109/TSP.2005.845426Stoica, P., Besson, O., Maximum likelihood DOA estimation for constant-modulus signal (2000) Electron. Lett., 36 (9), pp. 849-851. , 10.1049/el:20000620Stoica, P., Nehorai, A., Performance study of conditional and unconditional direction-of-arrival estimation (1990) IEEE Trans. Acoust. Speech Signal Process., 38 (10), pp. 1783-1795. , 10.1109/29.60109 0732.62098 10.1109/29.60109Stoica, P., Sharman, K., Novel eigenanalysis method for direction estimation (1990) IEE Proc., F, Radar Signal Process., 137 (1), pp. 19-26. , 1037504 10.1049/ip-f-2.1990.0004Van Trees, H.L., (2001) Optimum Array Processing. Part IV of Detection, Estimation and Modulation Theory, , Wiley New York 10.1002/0471221090Van Trees, H.L., (2001) Radar-Sonar Signal Processing and Gaussian Signals in Noise. Part III of Detection, Estimation and Modulation Theory, , Wiley New York 10.1002/0471221090Viberg, M., Ottersten, B., Sensor array processing based on subspace fitting (1991) IEEE Trans. Signal Process., 39 (5), pp. 1110-1121. , 10.1109/78.80966 0758.93067 10.1109/78.80966Vizireanu, D., Halunga, S., Simple, fast and accurate eight points amplitude estimation method of sinusoidal signals for DSP based instrumentation (2012) J. Instrum., 7 (4). , doi: 10.1088/1748-0221/7/04/P04001Vizireanu, D.N., A fast, simple and accurate time-varying frequency estimation method for single-phase electric power systems (2012) Measurement, 45 (5), pp. 1331-1333. , 10.1016/j.measurement.2012.01.038 10.1016/j.measurement.2012.01.038Wang, D., Sensor array calibration in presence of mutual coupling and gain/phase errors by combining the spatial-domain and time-domain waveform information of the calibration sources (2012) Circuits, Systems, and Signal Processing, pp. 1-36. , doi: 10.1007/s00034-012-9499-6Xu, G., Kailath, T., Direction-of-arrival estimation via exploitation of cyclostationary - A combination of temporal and spatial processing (1992) IEEE Trans. Signal Process., 40 (7), pp. 1775-1786. , 10.1109/78.143448 0850.62659 10.1109/78.14344