A-CMN

M. Novey and T. Adali, "Adaptable nonlinearity for complex maximization of nongaussianity and a fixed-point algorithm," in Proc. IEEE Workshop on Machine Learning for Signal Processing (MLSP), Maynooth, Ireland, pp. 79-84, Sep. 2006.

Brief introduction:
The adaptive complex maximization of nongaussianity (A-CMN) algorithm provides significant performance improvement by adapting the nonlinearity to the source distribution. In A-CMN, the choice of nonlinearity provides a reasonable approximation to the differential entropy of the source that is being estimated. The sources assumed in A-CMN are the symmetric exponential power family of distributions that are parameterized by a scale parameter and a shape parameter. To adapt the nonlinearity to the source distribution, the shape parameter is estimated from the data using maximum likelihood estimation.

Parameters of the algorithm:
X: mixtures Type: to use a circular version of A-CMN or noncircular A-CMN

T-CMN

M. Novey and T. Adali, "Complex ICA by negentropy maximization," IEEE Trans. Neural Networks, vol. 19, no. 4, pp. 596-609, April 2008.

Brief introduction:
In this algorithm, the cost function J(w)= E{|G(w^Hx)|²} is used for complex ICA where the function G is a complex analytic function such as polynomials or transcendental functions. Gradient update rule and quasi-Newton update rule are derived for this cost function. T-CMN algorithm is closely related to circular complex FastICA algorithm. However, different from circular FastICA, T-CMN can generate an asymmetric class of functions that are able to deal with noncircular sources.

Parameters of the algorithm:
X: mixtures
Type: to choose the function G used in the algorithm, such as 'atanh', 'asinh', 'tanh', 'cosh', 'acosh', '(.)²', 'asin', 'tan', 'polynomial'.

Circular complex FastICA

E. Bingham and A. Hyvarinen, "A fast fixed-point algorithm for independent component analysis of complex valued signals," International Journal of Neural Systems, vol. 10, no. 1, pp. 1-8, 2000.

Brief introduction:
The cost function J(w)= E{G(|w^Hx|²)} is used for the circular complex FastICA algorithm where the function G is a real-valued smooth even function, such as log function. Since the cost function depends only on the magnitude of the separated sources, circular complex FastICA deals with circular sources and cannot separate noncircular sources. A quasi-Newton update rule is implemented in the code and simulations results show its good performance with circular sources.

Parameters of the algorithm:
X: mixtures
Type: to choose the function G used in the code, such as 'log', 'kurtosis', 'square root'.

Noncircular complex FastICA

M. Novey and T. Adali, "On extending the complex FastICA algorithm to noncircular sources, " IEEE Trans. Signal Processing, vol. 56, no. 5, pp. 2148-2154, May 2008.

Brief introduction:
This algorithm is closely related to the circular complex FastICA algorithm since the derivation starts with the same cost function J(w)= E{G(|w^Hx|²)}. Different from circular complex FastICA, this algorithm doesn't assume circular sources. In the derivation of quasi-Newton algorithm, the authors make some approximations on the Hessian matrix and introduce a pseudo-covariance term in the final fixed-point update rule. This modification provides significant improvement in performance when confronted with noncircular sources, specifically with sub-Gaussian noncircular signals such as binary phase-shift keying signals.

Parameters of the algorithm:
X: mixtures
Type: to choose the function G used in the code, such as 'log', 'kurtosis', 'square root'.

C-QAM

M. Novey and T. Adali, " Complex fixed-point ICA algorithm for separation of QAM sources using Gaussian mixture model," in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), Honolulu, Hawaii, April 2007.

Brief introduction:
The complex quadrature amplitude modulated (C-QAM) algorithm is introduced for separation of QAM sources through independent component analysis by maximizing negentropy. The algorithm matches the input QAM distribution through a mixture of Gaussian kernels and uses fixed-point updates. C-QAM provides improved performance over a wide range of operating conditions such as low signal-to-noise ratio, small sample sizes, and large number of sources.

Parameters of the algorithm:
X: mixtures