1. Introduction
Independent component analysis (ICA) is a technique of blind source separation (BSS) and extensively has been used in several systems due to its simplicity and applicability in practical scenarios [1-3] from the last two decades. Various applications of ICA have been reported in the literature e.g., vibration analysis [4, 5], robotics [6], fault diagnosis [7], biomedical signal processing [8], speech processing [9], and wireless communication systems [10, 11]. In wireless communication, ICA is mainly used in multiple input multiple output (MIMO) systems. The MIMO system consists of multiple transmit and receive antennas that make it an ideal candidate for the ICA algorithms.
In general, the ICA algorithms assumed static or quasi static wireless channels [1]. Several ICA based communication systems are presented in the literature. We summarized a few important research articles as follows. A semi-blind receiver structure is presented in [12], that overcomes the phase and permutation indeterminacies in the ICA post processed data by using a minimum number of pilot bits attached with each transmit data block in a quasi-static scenario. Although, the authors in [12] have added a small number of bits with low computational cost but achieved good bit-error-rate (BER) performance. Another MIMO based communication system is presented in [13], that have improved bandwidth efficiency without considering the spatial coding. An ICA based low complexity orthogonal frequency division multiplexing (OFDM) based receiver structure for MIMO system is proposed in [14] with enhanced performance for large sub-carriers.
1.1 Motivation
Fast and accurate ICA algorithms are presented in the literature [15] for static and quasi-static wireless channels. Performance of these algorithms degrades if variations in the wireless channels are such that the quasi stationarity condition does not hold. However, the practical wireless channels are time varying and un-mixing of the mixed received signals becomes a challenging work for ICA to blindly un-mix them in a time varying scenario [16]. If we reduce the lengths of the processing data blocks to get the quasi stationarity, then the performance of the batch ICA algorithms further degrades. Performance degradation occurs due to the statistical parameters used in the update rules of the ICA algorithms. To the best of our knowledge, we are the first to dicuss the applications of the batch ICA algorithms in time varying wireless channels.
1.2 Contribution
In this paper, we propose an ICA based MIMO transceiver for time varying wireless channels. The proposed transceiver system also performs well if variations in the channels are such that the quasi stationarity condition becomes invalid even for smaller data blocks. To the best of our knowledge, no such a transceiver is presented in the literature that efficiently utilized the existing ICA algorithms in a time varying scenario. The transmitter side of this transceiver system transmits multiple copies of each sample of the source data. The receiver receives mixtures of the transmitted source signals. The multiple mixed copies are averaged and then up-sampled. The up-sampled data is un-mixed through the ICA algorithms and then down sampled to get back the original data blocks lengths.
Due to the use of samples copier (SC), samples averaging (SA) and up/down sampling operations used in the proposed transceiver system, we name it the MIMO SCSA-UD transceiver system. The MIMO SCSA-UD transceiver system improves the quality of the received signal in highly dynamic communication environment at the cost of reduced bandwidth efficiency due to the transmission of multiple copies of each source data sample. By highly dynamic or highly time varying we mean that the channels characteristics are changing for each transmitted sample. Moreover, we also propose the automatic values adjustment (AVA) technique in this paper. This technique controls the functions of the SC and the UD sampling operations blocks in such a way if there are no variations in the channel or the channel is quasi static then the SC block will copy each sample once. By this way the bandwidth efficiency will not be affected in quasi static channels condition. In short we can say that the bandwidth efficiency of the proposed transceiver becomes equal to the bandwidth efficiency of the existing MIMO system in quasi static channels conditions.
Performance of the proposed transceiver is evaluated over quadrature amplitude modulated (QAM) signals. The performance evaluation criterion is signal to interference ratio (SIR).
1.3 Notations and Organization
The notations used in this paper are such that the capital size bold letters are used for matrices and the small size bold letters represent vectors. Symbols and abbreviations used in this paper are given in Table 1 and 2 respectively.
Table 1.List of Symbols
Table 2.List of Abbreviations
Remaining paper is organized as follows. The signal model and overview of ICA is given in Section 2, Section 3 presents the proposed MIMO SCSA-UD transceiver system. Section 4 illustrates the simulation results. Finally, conclusion is given in Section 5.
2. The Signal Model and Overview of ICA
We consider a multi-user MIMO wireless communication system with MT transmit and MR receive antennas. For the sake of simplicity, it is considered that MT = MR = M. The transmitted signals vectors are s1, s2,…, sM, where sm = [sm1, sm2,…, smL] and the received mixed signals are x1, x2,…,xM. The system model for ICA signals processing is shown in Fig. 1. The ICA post process reconstructed signals are represented by y1, y2,…, yM. Where the post processed signals are the resultant un-mixed signals of the ICA algorithms. The un-mixed signals are the estimated resultant source signals transmitted by different antennas. The ICA algorithms un-mix the received mixed signals in order to find the actual transmitted source signals. The received mixed signals can be modeled as follows
where, X is M × L mixed data matrix, H is M × M complex valued channel matrix and S is M × L matrix that contains the independent and non-Gaussian source signals. Though the ICA research can be divided into the following categories
Fig. 1.The existing MIMO transceiver system for ICA signals processing.
We use the square channel or mixing matrix in our analysis. As a future work, we would like to extend it for non-square case utilizing the techniques of non-square mixing scenario.
Equation (1) represents the noise free model of ICA. The noisy ICA model can be represented as
where, N is M×L matrix, which represents the complex valued additive white Gaussian noise with variance one and assumed independent to the source signals. The addition of noise also degrades the performance of the ICA algorithms. The role of the ICA algorithms is to estimate the un-mixing matrix W, where W=H-1. After estimating the un-mixing matrix the desired un-mixed signals can be represented as follows
Once the un-mixing matrix W is determined, then, the individual component signals can be separated from the mixed signals by using equation (3). Accuracy of the ICA algorithms depends on the criterion used for estimating the un-mixing matrix.
In case of highly time varying channels, the coefficients of the channel matrix H changes for each incoming data sample and the un-mixing becomes a challenging work for the ICA algorithms.
Assumptions used in this paper are given below
There are some indeterminacies in the ICA post processed data, as given below
3. The Proposed MIMO SCSA-UD transceiver system
In this section, the proposed transceiver system called the MIMO SCSA-UD transceiver is presented. The MIMO SCSA-UD transceiver improves the separation performance of the batch ICA algorithms in highly time varying channels. In case of highly time varying channels the quasi stationary condition becomes invalid even for smaller data blocks. The proposed transceiver system is given in Fig. 2.
Fig. 2.The proposed MIMO SCSA-UD transceiver system.
The proposed transceiver system first processes the source signals s1, s2 ,…, sM, through samples copier (SC) block that copies each sample Q times before transmission. The source signals are the information signals which we want to transmit. These signals may be speech or any other information signals, which we want to transmit. In order to generate these signals, we first generate random data in matlab using the rand (random) function and then modulate it using the qammod (quardrature amplitude modulation) function of matlab for further processing through the SC block. The transmitted source signals after the SC block are s'1, s'2 ,…, s'M, where s'm = [s'm1, s'm2 ,…, s'mLQ]. The SC process for each source data vector is demonstrated as follows
At the receiver side linearly mixed data vectors x1, x2,…,xM are received and passed through the samples separator (SS) and samples averaging (SA) blocks. The SS block separates the mixed received samples inserted by the SC block and the SA block calculates their average values. The signal processing of the SS and the SA blocks is shown in Fig. 3, where 1D represents shift right and -1D represents shift left by one sample. This figure shows that first the mixed data vector x(n) is decimated by integer value Q to discard the extra inserted samples, then delayed and down sampled x(n) Q times and then shift back all the decimated vectors to remove the delays, finally add them up and divide by Q to get the average values. This process is demonstrated mathematically as follows
where x1m1, x2m1, …, xQm1 represent the mixed received copies of samples that are transmitted Q times, where the transmitted source symbols are s1m1, s2m1, …, sQm1.
Fig. 3.The signal processing of the samples separator and samples averaging blocks.
In the next step, the mixed data vectors x'm are up-sampled before un-mixing. The up-sampling (US) operation increases the data blocks lengths by using an integer factor Z. The US block generates (Z - 1) samples between two adjacent samples. The up-sampling is a two steps process, (1) zero insertion, and (2) the samples generation. The zero insertion process for a mixed received data block x'(n) can be represented as follows
where, δ is the unit impulse function.
After zero insertion the samples generation process is performed using the linear interpolator of [24]. The impulse response of the linear interpolator represented by h(n) and can be written as follows
The up-sampled mixed data vectors are x↑m = [x↑m1, x↑m2, …., x↑mZL], where m=1,2,…,M. The up-sampled mixed data is un-mixed by using the complex valued FastICA algorithm [25]. This algorithm is used as a bench mark to evaluate the effectiveness of the proposed transceiver system. The resultant un-mixed data vectors are y↑1, y↑2, …., y↑M. The un-mixed data is then down sampled using the decimation parameter Z to get back the original data blocks lengths. The DS or decimator discards (Z -1) number of samples and keeps the remaining samples. The decimated version of the data block y↑(n) can be represented as follows
The decimated data vectors are y1, y2, …, yM, where ym = [ym1, ym2,…, ymL].
The indeterminacies in the resultant reconstructed data can be resolved by transmitting few known symbols with each data block as performed in [12].
Now, the important thing is how to select the optimal values of the key parameters Z and Q. For this purpose, we propose the automatic values adjustment (AVA) technique. This technique is responsible for selection of the desired values of Z and Q according to the channels conditions. The AVA technique is presented in Fig. 4. The AVA technique initially selects the SC parameter Q and the UD sampling parameter Z equal 1 and computes the signals to interference ratios (SIRs) of the resultant un-mixed signals y1, y2, …, yM. The SIR represents the average ratio of the desired signal power to the power of the estimation error and is defined as follows
where i=1,2,...,M.
Fig. 4.The automatic values adjustment (AVA) technique for the key parameters Q and Z.
SIR is used as a criteria to evaluate the results of the proposed and existing MIMO systems using the FastICA and OBAICA algorithms in both the transceivers. Basically, we are transmitting M number of signals utilizing M number of antennas. Equation (9) calculates the SIRs of all the transmitted data blocks. After calculating the SIRs we compute the average SIR (SIRavg) as follows
Now compare the value of the SIRavg with the predefined threshold μ, where this constant defines the quality of the resultant un-mixed signals. If SIRavg > μ, then keep the same values of Q and Z. In this scenario, the signal processing of the proposed transceiver system becomes equal to the signal processing of the existing MIMO system shown in Fig. 1. Moreover, if SIRavg ≤ μ, due to variations in the wireless channels, then the AVA technique will increase the values of Q and Z iteratively up to the desired limit. Furthermore, the signal processing steps of the proposed transceiver system with the AVA technique are summarized as follows
Computational complexity of the FastICA algorithm utilizing the existing MIMO system shown in Fig 1 is O(ML). The proposed transceiver system has the complexity of O(MLZQ). Though, the proposed transceiver system will increase the computational complexity as compared to the complexity of the existing MIMO system in time varying channels. However, the proposed system gives increased accuracy of the results in time varying scenario, which is shown in Section 4. Moreover, in case if the channel is quasi static then the complexity of the proposed transceiver becomes equal to the computational complexity of the existing MIMO system, which we shown in detail in the automatic values adjustment (AVA) technique in Fig. 4. In short, we can say that the computational complexity and bandwidth utilization of the proposed system increases if the channels become highly time varying, where the channels are varying for each transmitted source symbol.
4. Results and Discussion
In this section, the performance of the proposed transceiver system is simulated and discussed. A dual antenna MIMO transceiver system is considered for simulation purpose. The transmitter transmits quadrature amplitude modulated (QAM) signals with equal carrier frequencies and power. Signal to noise ratio (SNR) ranges from -3dB to 10dB, throughout the simulations. Additive white Gaussian noise (AWGN) channels with frequency flat fading are used. The receiver receives linear mixtures of the transmitted source signals.
We consider both the time varying and the quasi static wireless channels. In case of time varying channels the condition of the quasi stationarity becomes invalid even for smaller lengths of the processing data blocks and the coefficients of the channel matrix H are changing within the processing data blocks. The channel matrix used in our simulation is given below,
where coefficients δ1, δ2, δ3 and δ4 of the matrix H are complex valued random variables. We generate them randomly in our simulation for time varying as well as quasi static channels. In quasi static case, we keep the channel coefficients fixed during the processing data blocks, while in time varying scenario these coefficients are changed randomly for each transmitted data sample.
The performance measure used in our simulation is signal to interference ratio (SIR). Simulation is performed using Monte Carlo simulation for each input data block. The input data blocks lengths ranges from 20 to 200 samples.
Different parameters used in our simulation are summarized in Table 3.
Table 3.Simulation parameters
In our simulations, first, we evaluate the performance of the proposed MIMO SCSA-UD transceiver system utilizing the quasi static as well as the time varying wireless channels, without utilizing the automatic value adjustment (AVA) technique. SNR is kept constant with a value of 10dB. In this simulation setup, we keep the value of Q equal 1 to clearly observe the effect of up/down sampling parameter Z with the values ranges from 1 to 4. We consider the smaller values of Z to protect oversampling and also to reduce the computational complexity of the proposed transceiver system. For detailed discussion regarding the value of Z one can refer to [24]. Results are demonstrated in Fig. 5. Performance improvements can be observed from this figure for different values of Z even in time varying channels.
Fig. 5.Results of the proposed MIMO SCSA-UD transceiver utilizing the SC parameter Q=1, and changing the value of Z from 1 to 4, in quasi static as well as time varying channels conditions.
Secondly, we evaluate the performance of the proposed system using the AVA technique. The AVA technique is used with fixed value of Q equal 1, for variable blocks lengths. Other parameters of the simulation setup are kept same as in the above experiment. Simulation is performed over quasi static and time varying conditions with different values of the predefined threshold. In case of quasi static channels we use μ= 11, 8, 7, 6, and in case of time varying scenario we use μ= 8, 7, 6, 5 respectively. Results are given in Fig. 6. One can observe from this figure that once the value of the predefined threshold is set to any value, then the AVA technique updates the value of Z from 1 to the desired value depending upon the value of μ for different blocks lengths and channel conditions.
Fig. 6.Performance of the proposed transceiver system utilizing the AVA technique with the value of Q equal 1, and Z changes from1 to 4 in both the quasi static and time varying scenarios.
In the third simulation setup, performance of the MIMO SCSA-UD transceiver using the AVA technique with Q=1, and μ= 8, is compared with the FastICA [25] and the OBAICA [16] algorithms for different blocks lengths and fixed SNR of 10dB, in time varying scenario. Results are shown in Fig. 7. Performance improvement of the MIMO SCSA-UD transceiver can be observed from this figure even at larger blocks lengths in time varying scenario. Actually, for smaller data blocks, the expectation operator used in the update rule of the algorithm fails to produce correct results. By increasing the data block length, the accuracy of the results also increases. However, in a time varying scenario, the batch ICA algorithms are unable to track the variations and at larger block length the variations increases. Due to this reason, in our simulation, first the accuracy of the results increases and then decreases. The important point is that the results of the proposed system are better at smaller as well as larger data blocks.
Fig. 7.Performance comparison of the proposed system with the FastICA and OBAICA algorithms in time varying channels. The AVA technique is used in the proposed system with Q=1, and μ=8, where values of Z changes from 1 to 4.
Furthermore, we also compared the results for different SNRs utilizing a constant block length of 30 samples. Results are given in Fig. 8. This figure shows superior performance of the proposed system for different values of the SNRs and smaller data block length of 30 samples.
Fig. 8.Performance of the proposed transceiver system in comparison with the existing MIMO system utilizing the FastICA and OBAICA algorithms in both the transceivers for a block length of 30 samples and SNRs ranges from -2 to 10dB.
As a cross check we compared the bit error rate (BER) performance of the proposed transceiver with the FastICA and OBAICA algorithms for data block length of 30 samples. Other parameters of the simulation setup are kept same as in the previous experiment. Results are demonstrated in Fig. 9 where the proposed system shows superior performance in terms of BER for a block length of 30 samples and different SNRs. It must be noted that the value of Q=1, and Z varies from 1 to 4.
Fig. 9.Bit error rate (BER) performance of the proposed system in comparison with the FastICA and OBAICA algorithms.
In the next simulation setup, we consider the proposed system with different values of Q and Z, where both the values range from 1 to 4. We manually adjust these parameters to clearly observe their effect over un-mixing. The value of Q also kept smaller due to more bandwidth utilization in time varying scenario. Results are given in Fig. 10. In these results first we keep the value of Q=2 and change the value of Z from 1 to 4. Secondly, we keep Q=4 and change Z from 1 to 4, where more improvements in results observed while increasing the value of Q from 2 to 4 for larger blocks lengths in time varying scenario.
Fig. 10.Results of the proposed MIMO SCSA-UD transceiver system for different values of Q and Z without using the AVA technique.
In Fig. 11, we demonstrate the results of the proposed system with the AVA technique. Different values of the predefined threshold μ are used which are 5, 7, and 9. SNR is kept constant with the value of 10dB. The channels are time varying. The AVA technique continuously checks the value of SIRavg in comparison with the predefined threshold μ i.e, for μ=5 the SIRav is greater than 5 for initial block length of 20 samples, so the values of Q and Z remained 1. In case of μ=7, and 9, the values of Q and Z are increased from 1 to the desired values, because the SIRavg is less than 7 or 9 for block length of 20 samples. In this simulation more performance improvements are observed for μ=7, and 9, which is the main contribution of the proposed system. Moreover, the proposed system is also evaluated for different SNRs, with the data block length of 100 samples. Other parameters are same as in the above experiment. Results are given in Fig. 12, which shows performance improvements for larger value of μ.
Fig. 11.Results of the proposed MIMO SCSA-UD transceiver system utilizing the AVA technique for μ=5, 7, and 9.
Fig. 12.Results of the proposed transceiver system utilizing the AVA technique with μ=5, 7, and 9 for data block length of 100 samples and SNRs ranges from -2 to 10dB.
Finally, we demonstrate the performance of the MIMO SCSA-UD transceiver (with μ=7) in comparison with the existing MIMO system utilizing the FastICA and OBAICA algorithms in both the transceivers for time varying scenario. Results are tabulated in Table 4. Results shown in this table are such that the SIR performance of the FastICA algorithm degrades for increased length of the processing data blocks while performance of the proposed transceiver further improves. This is the main achievement of the proposed transceiver system.
Table 4.The SIR performance comparison of the proposed transceiver system with the FastICA and the OBAICA algorithms in time varying scenario.
The results for different SNRs with block length of 100 are given in Fig. 12. This figure also shows superior performance of the proposed MIMO SCSA-UD transceiver system for time varying wireless scenario.
5. Conclusion
In most of the ICA algorithms implemented in wireless communication systems, the channels are considered static or quasi static. Its use in time varying channels to blindly estimate the original signal from the mixed received signals is limited and inefficient. In this paper, the MIMO SCSA-UD transceiver system is proposed that performs well in time varying scenario. The channels are assumed highly time varying with the channel characteristics changing for each transmitted sample. We use the complex valued FastICA algorithm as a benchmark to evaluate the effectiveness of the proposed system. Simulation is performed over QAM modulated signals for various SNRs and input data block lengths. Performance of the MIMO SCSA-UD transceiver is compared with the OBAICA algorithm and the FastICA algorithm. Results show that the proposed transceiver outperforms the OBAICA and FastICA algorithms in highly time varying wireless channels.
As a future work, we would like to extend the proposed approach for non-square case utilizing the techniques of non-square mixing scenario.
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