## Complex ICA Algorithms Based on Non-Gaussianity Maximization

We present the code for four complex-valued ICA algorithms specifically designed for noncircular sources:

- Transcendental complex maximization of non-Gaussianity (T-CMN) [1]
- Adaptive complex maximization of non-Gaussianity (A-CMN) [2]
- Complex quadrature amplitude modulation (C-QAM) [3]
- Noncircular FastICA (nc-FastICA) [4]

### T-CMN

The T-CMN algorithm utilizes a cost function comprised of complex transcendental functions to match the log of the source distribution, i.e., entropy. The fixed-point optimization algorithm uses the pseudocovariance matrix directly in the update providing convergence with noncircular sources.

### A-CMN

The A-CMN algorithm adapts the cost function to the unknown source distribution by assuming the sources are realizations from the complex generalized Gaussian distribution (CGGD). The shape parameter and covariance matrix are estimated during the ICA fixed-point update using an MLE. This algorithm is suited for applications where the source distributions are not known a priori.

### C-QAM

The C-QAM algorithm uses a cost function made up of a mixture of complex Gaussian kernel functions that explicitly match the distribution of quadrature amplitude modulated (QAM) and binary phase shift keying (BPSK) signals.

### nc-FastICA

The nc-FastICA algorithm is a modification to the complex FastICA algorithm of Bingham and Hyvarinan (2000). This modification extends the operation to noncircular sources by using the information in the pseudocovariance matrix in the fixed-point update. This modification provides stability with noncircular sources which may be unstable using complex FastICA.

**References:**

[1] M. Novey and T. Adali, "Complex ICA by negentropy maximization,"

*IEEE Trans. Neural Networks*, vol. 19, no. 4, pp. 596-609, April 2008.

[2] 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.

[3] 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.

[4] 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.