An analytical constant modulus algorithm

Abstract

Abstract-Iterative constant modulus algorithms such as Godard and CMA have been used to blindly separate a superposition of co-channel constant modulus (CM) signals impinging on an antenna array. These algorithms have certain deficiencies in the context of convergence to local minima and the retrieval of all individual CM signals that are present in the channel. In this paper, we show that the underlying constant modulus factorization problem is, in fact, a generalized eigenvalue problem, and may be solved via a simultaneous diagonalization of a set of matrices. With this new, analytical approach, it is possible to detect the number of CM signals present in the channel, and to retrieve all of them exactly, rejecting other, non-CM signals. Only a modest amount of samples are required. The algorithm is robust in the presence of noise, and is tested on measured data, collected from an experimental set-up.

Blind separation of BPSK sources with residual carriers

“…A convenient way to find the basis goes via a QR factorization of 1 P : An important result is that, for sufficiently independent conditions, δ [4]. The remaining problem is to find out which linear combinations of the {y k } lead to a vector y that can be written as y ¦ ¯t ⊗ t. The latter problem is conveniently rephrased by working with a matrix Y ¦ tt * .…”

“…This is essentially a generalized eigenvalue problem. Several algorithms are available, e.g., based on Jacobi iterations [1,3,4,[9][10][11][12]. Since we usually have a good starting point from the eigenvalue problem of a pair of matrices, 3 such iterations usually converge extremely fast, in two or three iterations, be it to a local optimum.…”

“…For constant-modulus signals, a successful algorithm is the Analytic Constant Modulus Algorithm (ACMA) [4], which solves an overdetermined set of quadratic equations. The algorithm has been specialized to separate superpositions of binary {±1} or {0 1} signals [5].…”

Joint angle and delay estimation using shift-invariance techniques

“…Method " ": A third method is similar to the one proposed in [20] and can be called a "super"-generalized Schur method as it tries to compute a (Schur) decomposition for more than two matrix pencils. It is an attempt to find unitary , , and to make all four matrices in (18) as much upper triangular as possible by a straightforward extension of the usual iteration [18].…”

An analysis of constant modulus algorithm for array signal processing

“…The second class of approaches obtain the signal estimate in a single step. One example is the algebraic CMA (ACMA) proposed by Van der Veen [12]. Unlike multistage CMA, these approaches are not subject to estimation error propagation.…”