Asymptotic properties of the algebraic constant modulus algorithm

Abstract

Abstract-The algebraic constant modulus algorithm (ACMA) is a noniterative blind source separation algorithm. It computes jointly beamforming vectors for all constant modulus sources as the solution of a joint diagonalization problem. In this paper, we analyze its asymptotic properties and show that (unlike CMA) it converges to the Wiener beamformer when the number of samples or the signal-to-noise ratio (SNR) goes to infinity. We also sketch its connection to the related JADE algorithm and derive a version of ACMA that converges to a zero-forcing beamformer. This gives improved performance in applications that use the estimated mixing matrix, such as in direction finding.Index Terms-Array signal processing, blind beamforming, constant modulus algorithm, simultaneous diagonalization.

The Gaussian Assumption in Second-Order Estimation Problems in Digital Communications

“…The study of in Appendix II comes to the conclusion that the Gaussian assumption is optimal at high SNR unless some eigenvalues of the kurtosis matrix are equal to 1. This condition is closely related to the constant modulus property of the nuisance parameters [13]. To validate this result, let us obtain the asymptotic expression of (10) as the noise variance goes to zero .…”

“…The kurtosis matrix is null in the Gaussian case and provides the complete non-Gaussian information on the nuisance parameters that second-order NDA estimators are able to exploit [8]. In case of circular complex nuisance parameters, is given by the following diagonal matrix: (6) where is the fourth-order moment of the nuisance parameters [8], [13]. It can be shown that for PSK constellations [13], for QAM and APSK constellations and in the Gaussian case.…”

“…In case of circular complex nuisance parameters, is given by the following diagonal matrix: (6) where is the fourth-order moment of the nuisance parameters [8], [13]. It can be shown that for PSK constellations [13], for QAM and APSK constellations and in the Gaussian case. If the nuisance parameters are noncircular, e.g., in case of binary-phase-shift keying (BPSK) or CPM modulations, the exact can be obtained numerically.…”

“…Regarding the CM property, the constant modulus algorithm (CMA) was introduced by Treichler et al [11] in the area of adaptive equalization, and it has been widely studied in the last decade (see [12], [13], and references therein). The CMA has been later applied to blind parametric estimation problems in which the mixing matrix is structured (parameterized), and the aim is to determine the value of these parameters [14].…”

“…In particular, we have that (4) where is the covariance matrix of the innovation vector when the nuisance parameters are not Gaussian distributed. The closed-form expression of was derived in [8], [13], obtaining that (5) where and matrix contains all the fourth-order cumulants (kurtosis) of the vector of nuisance parameters [8]. The kurtosis matrix is null in the Gaussian case and provides the complete non-Gaussian information on the nuisance parameters that second-order NDA estimators are able to exploit [8].…”