- Research Article
- Open Access
Recursive and Fast Recursive Capon Spectral Estimators
EURASIP Journal on Advances in Signal Processing volume 2007, Article number: 045194 (2007)
The Capon algorithm, which was originally proposed for wavenumber estimation in array signal processing, has become a powerful tool for spectral analysis. Over several decades, a significant amount of research attention has been devoted to the estimation of the Capon spectrum. Most of the developed algorithms thus far, however, rely on the direct computation of the inverse of the input correlation (or covariance) matrix, which can be computationally very expensive particularly when the dimension of the matrix is large. This paper deals with fast and efficient algorithms in computing the Capon spectrum. Inspired from the recursive idea established in adaptive signal processing theory, we first derive a recursive Capon algorithm. This new algorithm does not require an explicit matrix inversion, and hence it is more efficient to implement than the direct-inverse approach. We then develop a fast version of the recursive algorithm based on techniques used in fast recursive least-squares adaptive algorithms. This new fast algorithm can further reduce the complexity of the recursive Capon algorithm by an order of magnitude. Although our focus is on the Capon spectral estimation, the ideas shown in this paper can also be generalized and applied to other applications. To illustrate this, we will show how to apply the recursive idea to the estimation of the magnitude squared coherence function, which plays an important role for problems like time-delay estimation, signal-to-noise ratio estimation, and doubletalk detection in echo cancellation.
Marple SL Jr.: Digital Spectral Analysis with Applications. Prentice-Hall, Englewood Cliffs, NJ, USA; 1987.
Kay SM: Modern Spectral Estimation: Theory and Application. Prentice-Hall, Englewood Cliffs, NJ, USA; 1988.
Stoica P, Moses RL: Introduction to Spectral Analysis. Prentice-Hall, Upper Saddle River, NJ, USA; 1997.
Capon J: High-resolution frequency-wavenumber spectrum analysis. Proceedings of the IEEE 1969,57(8):1408-1418.
Capon J: Maximum-likelihood spectral estimation. In Nonlinear Methods of Spectral Analysis. Edited by: Haykin S. Springer, New York, NY, USA; 1979:155-179.
Lacoss RT: Data adaptive spectral analysis methods. Geophysics 1971,36(4):661-675. 10.1190/1.1440203
Stoica P, Jakobsson A, Li J: Matched-filter bank interpretation of some spectral estimators. Signal Processing 1998,66(1):45-59. 10.1016/S0165-1684(97)00239-9
Li H, Li J, Stoica P: Performance analysis of forward-backward matched-filterbank spectral estimators. IEEE Transactions on Signal Processing 1998,46(7):1954-1966. 10.1109/78.700967
Wang Y, Li J, Stoica P: Rank-deficient robust Capon filter bank approach to complex spectral estimation. IEEE Transactions on Signal Processing 2005,53(8, part 1):2713-2726.
Frazho AE, Sherman PJ: On the convergence of the multichannel maximum likelihood point spectrum estimator. IEEE Transactions on Signal Processing 1991,39(5):1210-1212. 10.1109/78.80956
Austin RT, England AW, Wakefield GH: Special problems in the estimation of power-law spectra as applied to topographical modeling. IEEE Transactions on Geoscience and Remote Sensing 1994,32(4):928-939. 10.1109/36.298021
Li J, Stoica P: An adaptive filtering approach to spectral estimation and SAR imaging. IEEE Transactions on Signal Processing 1996,44(6):1469-1484. 10.1109/78.506612
Frost OL: An algorithm for linearly constrained adaptive array processing. Proceedings of the IEEE 1972,60(8):926-935.
Applebaum SP, Chapman DJ: Adaptive arrays with main beam constraints. IEEE Transactions on Antennas and Propagation 1976,24(5):650-662. 10.1109/TAP.1976.1141416
Cox H, Zeskind RM, Owen MM: Robust adaptive beamforming. IEEE Transactions on Acoustics, Speech, and Signal Processing 1987,35(10):1365-1376. 10.1109/TASSP.1987.1165054
Steele AK, Byrne CL, Riley JL, Swift M: Performance comparison of high resolution bearing estimation algorithms using simulated and sea test data. IEEE Journal of Oceanic Engineering 1993,18(4):438-446. 10.1109/48.262294
Stoica P, Wang Z, Li J: Robust Capon beamforming. IEEE Signal Processing Letters 2003,10(6):172-175. 10.1109/LSP.2003.811637
Lee M-S, Katkovnik V, Kim Y-H: System modeling and signal processing for a switch antenna array radar. IEEE Transactions on Signal Processing 2004,52(6):1513-1523. 10.1109/TSP.2004.827204
Li J, Stoica P, Wang Z: On robust Capon beamforming and diagonal loading. IEEE Transactions on Signal Processing 2003,51(7):1702-1715. 10.1109/TSP.2003.812831
Hawkes M, Nehorai A: Acoustic vector-sensor beamforming and Capon direction estimation. IEEE Transactions on Signal Processing 1998,46(9):2291-2304. 10.1109/78.709509
Martin N, Mars J, Martin J, Chorier C: A Capon's time-octave representation application in room acoustics. IEEE Transactions on Signal Processing 1995,43(8):1842-1854. 10.1109/78.403343
Ma W-K, Ching P-C, Vo B-N: Crosstalk resilient interface cancellation in microphone arrays using Capon beamforming. IEEE Transactions on Speech and Audio Processing 2004,12(5):468-476. 10.1109/TSA.2004.833011
Murthi MN, Rao DB: Minimum Variance Distortionless Response (MVDR) modeling of voiced speech. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '97), April 1997, Munich, Germany 3: 1687–1690.
Murthi MN, Rao BD: All-pole modeling of speech based on the minimum variance distortionless response spectrum. IEEE Transactions on Speech and Audio Processing 2000,8(3):221-239. 10.1109/89.841206
Musicus BR: Fast MLM power spectrum estimation from uniformly spaced correlations. IEEE Transactions on Acoustics, Speech, and Signal Processing 1985,33(5):1333-1335. 10.1109/TASSP.1985.1164696
Ekman T, Jakobsson A, Stoica P: On the efficient implementation of the Capon spectral estimator. Proceedings of the 10th European Signal Processing Conference (EUSIPCO '00), September 2000, Tampere, Finland 1221–1224.
Alty SR, Jakobsson A, Larsson EG: Efficient time-recursive implementation of matched filterbank spectral estimators. IEEE Transactions on Circuits and Systems I: Regular Papers 2005,52(3):516-521.
Wei L, Marple SL Jr.: A new least-squares-based minimum variance spectral estimator fast algorithm. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '05), March 2005, Philadelphia, Pa, USA 4: 405–408.
Tian Z, Bell KL, Van Trees HL: A recursive least squares implementation for LCMP beamforming under quadratic constraint. IEEE Transactions on Signal Processing 2001,49(6):1138-1145. 10.1109/78.923296
Pados DA, Karystinos GN: An iterative algorithm for the computation of the MVDR filter. IEEE Transactions on Signal Processing 2001,49(2):290-300. 10.1109/78.902111
Jakobsson A, Marple SL Jr., Stoica P: Computationally efficient two-dimensional Capon spectrum analysis. IEEE Transactions on Signal Processing 2000,48(9):2651-2661. 10.1109/78.863072
Liu Z-S, Li H, Li J: Efficient implementation of Capon and APES for spectral estimation. IEEE Transactions on Aerospace and Electronic Systems 1998,34(4):1314-1319. 10.1109/7.722716
Haykin S: Adaptive Filter Theory. 4th edition. Prentice-Hall, Englewood Cliffs, NJ, USA; 2002.
Xu Z, Tsatsanis MK: Blind adaptive algorithms for minimum variance CDMA receivers. IEEE Transactions on Communications 2001,49(1):180-194. 10.1109/26.898261
Carter GC: Coherence and time delay estimation. In Signal Processing Handbook. Edited by: Chen CH. Marcel Dekker, New York, NY, USA; 1988:443-482.
Benesty J, Gänsler T, Morgan DR, Sondhi MM, Gay SL: Advances in Network and Acoustic Echo Cancellation. Springer, Berlin, Germany; 2001.
Welch PD: The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Transactions on Audio and Electroacoustics 1967,15(2):70-73. 10.1109/TAU.1967.1161901
Lagunas MA, Santamaria ME, Gasull A, Moreno A: Maximum likelihood filters in spectral estimation problems. Signal Processing 1986,10(1):19-34. 10.1016/0165-1684(86)90062-9
Lagunas MA, Gasull A: Measuring true spectral density from ML filters (NMLM and q-NMLM spectral estimates). Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '84), March 1984, San Digo, Calif, USA 1: 608–611.
Gray RM: Toeplitz and circulant matrices: a review. In Internal Report. Stanford University, Stanford, Calif, USA; 2002.
Bellanger MG: Adaptive Digital Filters and Signal Analysis. Marcel Dekker, New York, NY, USA; 1987.
Fox L: An Introduction to Numerical Linear Algebra. Clarendon Press, Oxford, UK; 1964.
Lagunas MA, Santamaria ME, Gasull A, Moreno A: Cross spectrum ML estimate. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '85), April 1985, Tampo, Fla, USA 77–80.
Benesty J, Chen J, Huang Y: A generalized MVDR spectrum. IEEE Signal Processing Letters 2005,12(12):827-830.