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This paper proposes a two-stage fast convergence adaptive complex-valued independent component analysis based on second-order statistics of complex-valued source signals. The first stage constructs a cost function by extending the real-valued whiten cost function to a complex-valued domain and optimizes the cost function using a complex-valued gradient. The second stage uses the restriction that the pseudocovariance matrix of the separated signal is a diagonal matrix to construct the cost function and the geodesic method is used to optimize the cost function. Compared with other adaptive complex-valued independent component analysis, the proposed method shows a faster convergence rate and smaller error. Computer simulations were performed on synthesized signals and communications signals. The simulation results demonstrate the validity of the proposed algorithm.

Blind source separation (BSS) is the separating of a set of source signals from a set of mixed signals without the aid of information (or with very little information) about either the source signals or the mixing process. Independent component analysis (ICA) is an attractive approach for solving blind source separation problems. ICA can be divided into real-valued ICA and complex-valued ICA according to the mixed signals. Complex-valued ICA is widely used to estimate the mixing matrix or to separate complex-valued mixed signals, such as frequency domain signals [

Studies of complex-valued ICA can be divided into three categories. The first category includes methods based on a nonlinear function, such as complex-valued fastICA (C-fastICA) [

The major advantage of SUT is that “

To increase the rate of convergence, a fast complex-valued ICA method is proposed in this work. The proposed method first extends the real-valued whitening process to a complex-valued domain to provide unit variance for the processed signal. Second, this work uses the restriction that the pseudocovariance matrix of the separated signals is a diagonal matrix to construct cost function and optimize the cost function using the geodesic method. This avoids computing the square root and inverse of the separating matrix and also keeps the separating matrix to be an orthogonal matrix, without any forcing operation. This improves the convergence speed of the proposed method compared to the other adaptive methods.

Generally, a linear complex-valued ICA model that is noise-free can be expressed as follows:

Assume a complex-valued random column vector

For any complex random vector

(1) Whitening the complex-valued observed signals

(2) Determining the separating matrix of the whitened signal by use of Takagi’s factorization: this is done according to

In this section, we describe an adaptive fast convergence complex-valued ICA algorithm based on second-order statistics, used in the SUT method. This is unlike other adaptive complex-valued ICA methods that simultaneously force separated signals to comply with second-order statistics. Instead, this method uses an adaptive serial updating method to realize the SUT. First, we use an adaptive method to whiten the observed signals. The cost function used in real-value whitening is directly extended to the complex-valued signal. The cost function is given as follows:

To overcome this problem, we use a geodesic method to search the optimized separating matrix

Using the geodesic method with self-tuning [

Initialize the whitening matrix and separating matrix using unit matrix, learning rate

Use (

Compute the gradient of the cost function in Riemannian space, which can be expressed as follows:

where

Compute the rotation matrix

If

where

If

Update the separating matrix

If

In order to test the algorithm, we used five synthesized signals with different spectral coefficients, three digital communication signals with different spectral coefficients, and three synthesized signals of which two signals have same spectral coefficients as the source signals. For simplicity, we directly used the expectation of the signal instead of the instantaneous value. Quality of separation was assessed using the performance index (PI), a widely used index in ICA. PI can be expressed as [^{−2} gives quite a good performance” [

In the first experiment, five complex-valued synthesized source signals with 10000 samples were used, constructed as follows:

In contrast, convergence curves are shown in Figure

Convergence curves of four methods with synthesized signals.

Average convergence curves for the four methods are shown in Figure

Average convergence curves of four methods with synthesized signals.

In the second experiment, we supposed that three digital communication signals (8QAM, 4QAM, and BPSK) impinge on a uniform linear antenna array with three elements from directions of 10°, 25°, and 70°. In Figure

Original signals, mixed signals, and separated signals in the digital communication system.

The average convergence curves for the four methods are shown in Figure

Average convergence curves of the four methods for digital communication signals.

In the third experiment, three random complex-valued signals were used as source signals, with spectral coefficients of 0, 0.6, and 0.6. Their imaginary and real parts were generated by a random uniform distribution function. Average convergence curves from an average of 100 different simulation runs with a learning rate of 0.01 are shown for the four methods in Figure

Average convergence curves of all methods for source signals; two of these signals have the same spectral coefficients.

The proposed method has two stages. The convergence curves shown in all figures are the convergence curves only for the second stage. For the first stage, the whitening signal converges to the unit matrix in first experiment after about 600 iterations and after about 100 iterations in the second and third experiments. Compared with other methods, the total iterations required for the proposed method are far less than other methods.

This paper proposes an adaptive complex-valued ICA method for noncircular signals based on second-order statistics and the geodesic method. The proposed method has faster convergence and smaller error than the other adaptive methods. For different mixing source signals, the proposed method has better performance and faster convergence than the Scott method. For source signals with different spectral coefficients, the proposed method and the SUT method have almost the same error. However, the SUT method is not suitable for source signals that some of source signals have the same spectral coefficients.

The authors declare that they have no competing interests.

This work was supported by the National Natural Science Foundation of China (61271115) and the Foundation of Jilin Educational Committee (2015235).