Blind constant modulus equalization via convex optimization

Abstract

In this paper, we formulate the problem of blind equalization of constant modulus (CM) signals as a convex optimization problem. The convex formulation is obtained by performing an algebraic transformation on the direct formulation of the CM equalization problem. Using this transformation, the original nonconvex CM equalization formulation is turned into a convex semidefinite program (SDP) that can be efficiently solved using interior point methods. Our SDP formulation is applicable to baud spaced equalization as well as fractionally spaced equalization. Performance analysis shows that the expected distance between the equalizer obtained by the SDP approach and the optimal equalizer in the noise-free case converges to zero exponentially as the signal-to-noise ratio (SNR) increases. In addition, simulations suggest that our method performs better than standard methods while requiring significantly fewer data samples.

A Convex Relaxation Approach to Higher-Order Statistical Approaches to Signal Recovery

“…More specifically, the new implementation leverages a convex optimization relaxation that can be applied to CMA, SWA, and MED algorithms. We show that our convex formulation requires less resource and improves the efficiency of the convex formulation in [21]. We further generalize the formulation to accommodate the semiblind algorithms when a small number of training symbols are available to assist the receiver signal recovery and separation.…”

“…In this section, we provide a real-valued representation of the conventional batch CMA cost. This representation is one of the keys to reduce parameter space compared to the work of [21] in which the formulation is based on complex values. This representation is also applied to other blind channel equalization and blind source separation costs and will be shown in the latter sections.…”

“…Unfortunately, local convergence remains a major obstacle, that requires good initialization strategies. Another approach to mitigate the local convergence problem is through relaxation by "lifting" the receiver parameters to a new parameterization that covers a larger parameter space over which the cost function is convex [20], [21]. Once the convex problem is uniquely solved, the solution is mapped back to the original restrictive parameter space.…”

“…In this work, we study means to improve the convergence of general blind and semiblind equalization algorithms by generalizing the cost modification principle presented in [21]. We can therefore develop a more general batch implementation of well-known blind equalization and source separation algorithms that minimize cost functions involving the fourth order statistics of system output signals.…”

Approximation methods for complex polynomial optimization

“…If the degree of complex polynomial is beyond quadratic, say quartic, several applications in signal processing can be found in the literature. Maricic et al [20] proposed a quartic polynomial model for blind channel equalization in digital communication. Aittomäki and Koivunen [1] discussed the problem of beam-pattern synthesis in array signal processing problem and formulated it to be a complex quartic minimization problem.…”