Stationary points of the finite length constant modulus optimization

A simple comparison of constant modulus and Wiener criteria for equalization with complex signals

“…The minima of the CM (2,2) cost function come in pairs, for coefficient vectors H and −H [8]. Thus, in the search for the optimal coefficients, it is necessary to express J CM2 as a cost function which is symmetrical in terms of H and −H .…”

Performance Bounds on the Constant Modulus Error Surface

“…Developing Convex Error Surface property. CMA (which is based on minmization of the ConIn [4], we transformed the CM error surface into a quadratic stant Modulus (CM) criterion), assumes that the input to the surface using the Kronecker product. Some relevant results are channel is a modulated signal that has constant amplitude at reviewed here.…”

“…Any deviation of the received signal Lemma 2.1: The constant modulus performance measure is amplitude from the constant value is considered a distortion quadratic in the Kronecker product of the parameter vector introduced by the channel. and its complex conjugate [4] [5] Proof:-The proof is in [4]a II. CM PROBLEM FORMULATION 2 Define I = n, and define the I x 1 kronecker product Let {s(t)} be an unobservable sequence.…”

“…Therefore the problem of estimating {y(t)} can Lemma 2.2. The matrix R,A,A is Hermitian and positive be solved by determining the coefficients wij, i =0,1, ..., in-i, semi-definite [4] [5] in a manner such that the performance measure 5 is minProof: The proof is in [4] v imized. A popular iterative method to solve the CM miniNext, we take the gradient of the new performance surface.…”

“…A popular iterative method to solve the CM miniNext, we take the gradient of the new performance surface. mization problem is the constant modulus adaptive (CMA) From [4], it is clear that the performance measure is quadratic 0-7803-9390-2/06/$20.00 ©C2006 IEEE with respect to 0. It is also clear that the cost function is III.…”