Adaptive Step Size Gradient Ascent ICA Algorithm for Wireless MIMO Systems

Independent component analysis (ICA) is a technique of blind source separation (BSS) used for separation of the mixed received signals. ICA algorithms are classified into adaptive and batch algorithms. Adaptive algorithms perform well in time-varying scenario with high-computational complexity, while batch algorithms have better separation performance in quasistatic channels with low-computational complexity. Amongst batch algorithms, the gradient-based ICA algorithms perform well, but step size selection is critical in these algorithms. In this paper, an adaptive step size gradient ascent ICA (ASS-GAICA) algorithm is presented. +e proposed algorithm is free from selection of the step size parameter with improved convergence and separation performance. Different performance evaluation criteria are used to verify the effectiveness of the proposed algorithm. Performance of the proposed algorithm is compared with the FastICA and optimum block adaptive ICA (OBAICA) algorithms for quasistatic and time-varying wireless channels. Simulation is performed over quadrature amplitude modulation (QAM) and binary phase shift keying (BPSK) signals. Results show that the proposed algorithm outperforms the FastICA and OBAICA algorithms for a wide range of signal-to-noise ratio (SNR) and input data block lengths.


Introduction
Independent component analysis (ICA) [1][2][3] is a signal processing technique used for estimation [4,5] of the source signals from the mixed received signals without a priori knowledge of the source signals and the channel condition [3,[6][7][8].Before ICA principal component analysis (PCA) [9,10] was used as a blind source separation (BSS) technique.PCA is based on second-order statistics and makes the data uncorrelated.ICA is based on high-order statistics and makes the data independent.Independence is achieved through maximization of non-Gaussianity.Moreover, ICA has attracted attention from the last two decades due to its effective performance and applicability in practical scenarios [11].It has various applications in different fields of engineering, that is, vibration analysis [12,13], robotics [14], machinery fault diagnosis [15,16], biomedical signal processing [17,18], speech processing [19], and wireless communication [20,21].Its applications in wireless communication systems mainly include multiple input multiple output (MIMO) systems.
e ICA algorithms can be classified into adaptive and batch ICA algorithms [22].e adaptive algorithms perform well in time-varying mixing scenario with high-computational cost and slow convergence.e batch ICA algorithms have less computational complexity and better separation performance in case of quasistatic mixing scenario.Moreover, amongst the batch algorithms, the gradient ascent ICA (GAICA) algorithm performs well, but step size selection is critical in this algorithm.A smaller value of the step size will produce better results with slow convergence, while a larger step size will result in fast convergence with worse performance of the algorithm.An improper choice of the step size will produce worse performance or demand for large iterations [23].
In the literature, various step size selection techniques are presented [24][25][26][27], where all these techniques are based on the fact that a smaller step size produces better results with slow convergence and larger step size provides fast convergence with worse performance.In [24], a so-called modified gradient-based ICA algorithm is developed.is algorithm converges for various values of the step size parameter if the objective function is properly selected.e authors in [25] presented a gradient ascent ICA algorithm, where the algorithm is terminated when the relative change in the unmixing matrices becomes less than a certain defined value for various step sizes.Furthermore, a diminishing step size gradient-based ICA technique is proposed in [26,27], where the step size is adapted in such a way that its initial value is high and which is repeatedly reduced with increasing iterations in order to converge to the optimal value.
In this paper, an adaptive step size gradient ascent ICA (ASS-GAICA) algorithm is developed.e proposed algorithm is free from the selection of the step size parameter.
e objective function used for maximization of the non-Gaussianity in our algorithm is an absolute value of entropy [6].e basic principle used in the derivation of the algorithm is that entropy of the recorded mixed signals is always less than the total entropy of the independent source signals and maximization of entropy produces the separated source signals.In addition to this, different performance evaluation criteria are used to observe the effectiveness of the proposed algorithm.ese criteria include signal-to-interference ratio (SIR), symbol error rate (SER), error standard deviation (ESTD) of the signal, and norm of the product of mixing and unmixing matrices (Norm-WA).It is shown in the simulation that performance of the proposed algorithm is better than the fixed step size GAICA algorithm.Performance of the proposed ASS-GAICA algorithm is also compared with the well-known ICA algorithms: the FastICA [19] and the OBAICA algorithms [28].
Notations used in this paper are (.)T , (.) ′ , and (.) ″ for transpose, derivative, and second-order derivative of a vector or a matrix, respectively.Capital bold face letters represent matrices, and the small bold face letters are used for vectors.Symbols and abbreviations used in this paper are listed in Tables 1 and 2, respectively.
Rest of the paper is organized as follows: Section 2 presents the ICA signals' model and assumptions.e ICA model is purely based on multidimensional data. is section presents the basic idea that how to model the MIMO signals for ICA algorithm.e mathematical formulation of the proposed ASS-GAICA algorithm is demonstrated in Section 3. Section 4 illustrates the simulation results.e results are demonstrated for quasistatic and time-varying mixing scenario utilizing various performance evaluation criteria.Finally, the conclusion is drawn in Section 5.

ICA Signals' Model and Assumptions
A multiuser multiple input multiple output (MIMO) [29] wireless communication system with M T transmitting antennas and M R receiving antennas is considered.For the sake of simplicity, we have M T � M R � M. e transmitted source signal vectors are s 1 , s 2 , . . ., s M , where s M � [s M1 , s M2 , . . ., s ML ] and the received mixed signals are x 1 , x 2 , . . ., x M .e block diagram of the system model is shown in Figure 1. e ICA postprocessed estimated signals are y 1 , y 2 , . . ., y M .e role of the ICA algorithms is to unmix the received mixed signals in order to find the actual transmitted signals [3]. e received mixed signals can be modeled in matrix form as follows: where X is M × L mixed data matrix, A is M × M mixing matrix, and S is M × L source data matrix.Equation ( 1) represents the noise free model of ICA. e noisy ICA model can be represented as  2 Mobile Information Systems where N represents the additive white Gaussian noise with dimension M × L. e addition of noise further degrades the separation performance of the ICA algorithm.Moreover, it is known that inverse of the mixing matrix exist which is known as the unmixing matrix and denoted by W. e role of the ICA algorithms is to estimate the unmixing matrix.
After estimating the unmixing matrix, the resultant unmixed signals can be modeled as follows: After estimating the unmixing matrix W, the mixing source signals can be separated from the mixed signals by using (3).Accuracy of the ICA algorithms depends on proper estimation of the unmixing matrix.In case, if the mixing matrix changes dynamically within the processing data blocks, then unmixing becomes difficult.
e basic assumption of ICA used in this paper is given below: is assumption is also true because we consider the communication signals which are non-Gaussian in nature.
(iii) e mixing matrix A is assumed to be square.e validity of this assumption can be observed from the equal number of the transmitting and receiving antennas.

Proposed ASS-GAICA Algorithm
e absolute value of entropy is used as an objective function in the derivation of the algorithm.e symbols used in the formulation of the algorithm are listed in Table 1.Furthermore, it is known from the ICA model that the source signals have non-Gaussian distributions.e CDF and PDF of the transmitted source signals are represented by g and g ′ , respectively.It is considered in [22] that the PDF of the supper-Gaussian signal can be represented by 2 tanh(Y) and sub-Gaussian by Y 3 .Received signals are the linear mixtures of the transmitted source signals, and the source signals can be estimated from the mixed received signals through the ICA algorithm.e ICA algorithm basically estimates the unmixing matrix W [3]. General form of the gradient ascent ICA algorithm is given in Algorithm 1.Hence, entropy of the estimated signals y 1 , y 2 , . . ., y M for jth iteration can be written in vector matrix form as Maximization of entropy as given in (4) tends toward maximization of the non-Gaussianity.In order to achieve this objective, the gradient of h(j) with respect to each element of W can be written as where Now, general form of the update equation for W can be written as After substituting ( 5) in ( 7), we will get the following updated formula for W: Equation ( 8) represents the fixed step size GAICA algorithm where the positive sign used in (8) indicates the gradient ascent nature of the algorithm.is equation updates the unmixing matrix W in such a way to further maximize the entropy while utilizing a constant step size.e optimal step size η selection is difficult in this equation.e improper value of η will result in worse performance or slow convergence of the algorithm.Moreover, in order to achieve the adaptive step size mechanism, first determine h(j + 1) at (j + 1)th iteration.en, approximate h(j + 1) up to the second-order terms by using Taylor's series (1) Initialization Whiten the mixed received data Choose a random initial unmixing matrix W Select the number of maximum iterations (J) (2) for j � 1 to J do (3) Y � WX (4) Update the unmixing matrix using the following equation: W(j + 1) � W(j) + η∇ h (j) (5) Terminate the loop if ||h(j) − h(j − 1)|| < δ, where δ is very small (6) end for ALGORITHM 1: Gradient ascent ICA algorithm.
Mobile Information Systems expansion.Finally, adjust the value of η in such a way to further maximize the entropy while processing from jth iteration to (j + 1)th iteration.In order to achieve these objectives first, we determine the entropy for (j + 1)th iteration as After applying Taylor's series expansion over (9), the resultant equation can be written as where ΔW � W(j + 1) − W(j).
For smaller values of ΔW, h(j + 1) can be approximated up to second-order terms in (10), and this is indeed the case in our algorithm, which is shown in the simulation section.Using this fact in combination with (5), we can write (10) as From ( 8), we have After combining ( 12) and ( 11), we get In order to achieve the adaptive step size selection criterion, adjust η in such a way to further maximize h(j + 1).To achieve this objective, evaluate the gradient of h(j + 1) with respect to η as z zη h T (j + 1)h(j + 1)   � 0.
Now after combining ( 13) and ( 14), we get the following resultant equation: Now substituting ( 15) in ( 8), we have Let then ( 16) can be written as e updated formula (18) involves inversion of the denominator part of Q, that is, W(j)C(j)X(j) + I, having same dimension as matrix W. It means that the proposed algorithm is more suitable for a smaller number of the source signals because the computational complexity of the matrix inversion is O(M 3 ).In case of large M, an estimate of Q can be developed.Another factor effecting the computational complexity is the data block length L, and hence computational complexity of the proposed algorithm is O(LM 3 ) that is approximately similar to the computational complexity of the fixed step size GAICA algorithm.e proposed algorithm is summarized in Algorithm 2. We use ||h(j) − h(j − 1)|| < δ as a stopping criterion in our algorithm.

Results and Discussion
Performance of the proposed ASS-GAICA algorithm is demonstrated in this section.A dual antenna MIMO transceiver system is considered for simulations.
e receiver receives linear mixtures of the transmitted signals.e channel is assumed AWGN with frequency flat fading.Simulation is performed over BPSK and 16-QAM signals with SNR ranging from −2 dB to 10 dB and input data block lengths ranging from 50 to 1000 samples using Monte-Carlo C(j) � E[g″(W T (j)X(j))/g′(W T (j)X(j))] (6) Calculate matrix Q as Q � W(j)/[W(j)C(j)X(j) + I] (7) Update the unmixing matrix using the following equation: where δ is very small (9) end for ALGORITHM 2: e ASS-GAICA algorithm.4 Mobile Information Systems simulation.We consider the quasistatic and the timevarying wireless channels.e simulation parameters are shown in Table 3.
e order, phase, and amplitude indeterminacies are resolved by using a technique presented in [30], where a few extra samples are transmitted which are known to the receiver in advance. is technique is insensitive to the data block lengths.
Different performance evaluation criteria are used to measure the effectiveness of the proposed algorithm which is given below.
(1) Signal-to-interference ratio (SIR): e SIR represents the average ratio of the desired signal power to the power of the estimation error.e SIR in dB can be written as (2) Error standard deviation (ESTD) of the signal: Error signal is the difference of the source signal and the algorithm postprocessed signal.e ESTD can be defined as where e(n) � s(n) − y(n).
(4) Norm of the product of mixing and unmixing matrices (Norm-WA): For ideal case the norm of the product of W and A will be one because A � W −1 .A value of Norm-WA near to one represents efficient separation.Mathematically it can be defined as follows: Simulation results for quasistatic wireless channels are demonstrated as follows.
First, we demonstrate the convergence behavior of the proposed ASS-GAICA algorithm and the fixed step size GAICA algorithm.e GAICA is an algorithm where the step size can be adjusted experimentally.e smaller value of the step size improves the separation performance with slower convergence, while the larger value increases the convergence speed with less accurate results.In order to compare the results of the GAICA algorithm with our proposed algorithm, we consider two values of the step size parameter, that is, η � 0.1 and 0.2.e results are shown in Figure 2 for 16-QAM signals with L � 100 and SNR of 10 dB.We consider the maximum number of iterations as 50 for both the algorithms.e step size values considered for the GAICA algorithm are η � 0.1 and 0.2.It can be observed from the results that further decrease in the step size parameter will further reduce the convergence speed.Similarly, increase in the step size will produce worse performance of the algorithm.Hence, we restrict ourselves to these two values of the step size parameter.e proposed algorithm properly converges in 22 iterations, while the GAICA algorithm converges in 40 iterations for η � 0.1.If we increase the value of η to 0.2, then the GAICA algorithm converges faster with less accurate steady state results in 50 iterations.In summary, we can say that the GAICA algorithm converges faster for larger step size with less accurate steady state results and vice versa, while the proposed algorithm improves the convergence speed as well as quality of the results.
Secondly, the performance of the proposed algorithm is compared with the well-known algorithm of ICA, FastICA algorithm and OBAICA algorithm using different performance measures as a cross-check.Figures 3 and 4 show the effectiveness of the proposed algorithm in terms of the SIR for different block lengths and different SNRs using 16-QAM signals.In Figure 3, we consider the fixed SNR of 10 dB with different block lengths, while in Figure 4, the block length is kept 1000 samples with different values of SNR.
e performance of the proposed algorithm is again evaluated utilizing the error standard deviation (ESTD) of the signal.
e results are shown in Figures 5 and 6 for different blocks lengths and SNR using 16-QAM signals.e

Conclusion
e entropy-based gradient ascent ICA (GAICA) algorithm has improved accuracy of separation, but the step size selection is difficult in this algorithm.Improper step size will result in worse performance of the algorithm.In this paper, we proposed an adaptive step size gradient ascent ICA (ASS-GAICA) algorithm that selects the optimal step size adaptively.e proposed algorithm has better convergence behavior with improved accuracy of results than the fixed step size GAICA algorithm.Performance of the proposed algorithm is compared with the well-known ICA algorithms, the FastICA and the OBAICA algorithms using binary phase shift keying (BPSK) and quadrature amplitude modulation (QAM) signals in quasistatic and time-varying scenario.Different performance measures, that is, SIR, ESTD, SER, and Norm-WA, are used to evaluate the effectiveness of the proposed algorithm.
e simulation results show that the proposed algorithm outperforms the FastICA and the OBAICA algorithms for various input data blocks and SNR. e proposed algorithm outperforms even for time-varying scenario.
Due to matrix inversion in the update equation, the proposed algorithm is suitable for a smaller number of the mixing source signals.In case of a large number of the mixing signals, some estimate of the matrix Q is required.As a future work, we will develop an estimator of this matrix for large number of the source signals.

Figure 1 :
Figure 1: MIMO transceiver system for ICA signal processing. is figure represents the basic ICA structure for recording multidimensional data.e transmitted signals are generated by independent sources.e receiving antennas receive linear mixtures of the transmitted independent source signals.
(i) e source signals s M are mutually statistically independent.e transmitting antennas transmit different source signals, and thus the independent assumption becomes true.(ii) e source signals s M are non-Gaussian in nature.

Figure 2 :
Figure 2: Convergence characteristics of the proposed ASS-GAICA and GAICA algorithms.e step size values considered in GAICA algorithm are 0.1 and 0.2.

Figure 8 :
Figure 8: SER performance for different SNRs at a data block length of 100 samples.SNRs considered in these results range from −2 to 10 dB.

Figure 9 :
Figure 9: SIR performance of all the three algorithms for BPSKmodulated signals utilizing different data block lengths and fixed SNR of 10 dB.

Table 1 :
List of symbols.
N Noise matrix of size M × L Δ 1 , Δ 2 , Δ 3 , Δ 4Random variables representing time-varying mixing coefficients Table 2: List of abbreviations.QAM Quadrature amplitude modulation BPSK Binary phase shift keying ICA Independent component analysis MIMO Multiple input multiple output SIR Signal-to-interference ratio SER Symbol error rate ESTD Error standard deviation BSS Blind source separation AWGN Additive white Gaussian noise SNR Signal-to-noise ratio CDF Commutative density function PDF Probability density function OFDM Orthogonal frequency division multiplexing PCA Principal component analysis

Table 5 :
Performance comparison of the OBAICA, FastICA, and the ASS-GAICA algorithms for QAM-modulated signals utilizing time-varying wireless channels.