A class of multi-modulus blind deconvolution algorithms using hyperbolic and Givens rotations for MIMO systems
Introduction
Blind methods are important tools that are used in the case where the training phase is unavailable (e.g. speech or biomedical applications), or become unrealistic (e.g. asynchronous wireless networks). Moreover, blind methods can be used to increase the informational throughput and reduce the size of control sequences (overhead) [1], [2]. In such schemes, the available statistics and/or the structural information about the source and/or the channel are usefully exploited [5], [6], [7]. The early attempts [8], [9] were followed by a number of successful methods.
With the ever increasing demand for high data rates and the exceptional behavior of multiple-input multiple-output (MIMO) modern communication systems which exploits the bandwidth at most, the need for developing more efficient blind equalizers or deconvolution methods becomes a necessity. In particular, it has been shown in Ladaycia et al. [1], 2], Rekik et al. [3], Maruta and Ahn [4] that, thanks to the semi-blind solution, the pilot sequence size can be reduced by a factor of 80 to 90% without affecting the channel estimation quality.2
Wireless communications suffer from the multipath fading effect which results into the intersymbol interference (ISI). Moreover, MIMO systems experience an additional interuser interference (IUI) problem. Hence, the blind deconvolution for MIMO systems is a twofold problem: a blind equalization to get rid of the ISI and a blind source separation for the IUI removal.
Recently, efficient BSS methods used to separate QAM signals based on the optimization of the Multi-Modulus Critera using the elementary Givens and Shear3 rotations have been introduced in Shah et al. [10], 11]. These methods, namely Givens Constant Modulus Algorithm (G-CMA) and Hyperbolc G-CMA Algorithm (HD-CMA), are designed for the special case when the signal undergoes a flat fading, where the channels have an instantaneous mixing effect. However, in most practical cases, the channels are convolutive and the aforementioned methods [10], [11], [13] do not apply directly. The approach is also linked to the aspect of pdf-matching or alphabet matching [12] since the consideration of the multimodulus tries to capture more (statistical) information from the other aspects of the source distributions than the power (second-order) of constant modulus. Also, the Multi Modulus (MM) criterion does not take into account all statistics, it is only suitable for “small size” constellations. Hence, for large size QAM constellations (64 and over), pdf fitting or alphabet matching methods would be more appropriate in this case. Machine learning techniques, such as support vector regression (SVR), are also among the proposed solutions for high order QAM signal’s recovery [37].
While modern communications systems use OFDM modulation to transmit over frequency selective channels. OFDM allows to convert the transmission over a convolutive channel into several parallel memoryless channels in which case BSS approaches for instantaneous mixtures can be applied directly. However, the convolutive time domain scenario exists in practice and not all communications standard are based on OFDM (see e.g., [17], [37]). Indeed OFDM has many advantages but possesses many disadvantages as well (e.g., high sensitivity to Carrier Frequency Offset (CFO), Strict Synchronization Requirement (time and frequency), Peak-to-Average Power Ratio (PAPR), Sensitive to Doppler shift, Co-channel Interference in Cellular OFDM, just to name a few) which led to considering other communications standards as, for example, IEEE 802.16 Wireless MAN_ SCa which stands for Wireless Metropolitan Area Network Single carrier.
Our ultimate objective is to extend the previous approach to the more general convolutive mixture case. This extension can be done in several different ways that we have developed and compared in this paper. More precisely, we consider four different approaches. The first one is referred to as Full-BSS deconvolution, whereby we form a spatio-temporal array from the collected data followed by a source de-mixing using BSS techniques, ended by paring and selection using a correlation-based method. The second one is a cascade of linear Second Order Statistics (SOS)-based equalization followed by a BSS demixing approach. The equalization is needed to remove the channel ISI effect, so that the problem is reduced from being of convolutive nature into an instantaneous mixture of signals, whereas the BSS is needed to get rid of IUI effect. Such a cascaded approach is referred to as a two-stage deconvolution approach, and it has been reported in several earlier works, such as [14], [15], [16], [17].
In the third method, we perform Blind Deconvolution (BD) through considering one composite cost function which penalizes the Multi-Modulus (MM) criterion and the cross-correlation between the different source signals. The optimization of the cost function is done using elementary Givens and Shear rotations resulting in two different algorithms, namely, Givens and Hyperbolic Givens MM Deconvolution algorithms G-MMDA and HG-MMDA, respectively. The last approach is a deflation-based method, such that each time one single source is extracted, then it is waived from the convolutive mixture using an appropriate subspace projection. This is repeated until all desired sources are extracted. This paper has mainly three contributions:
- C1.
It proposes, for the first time, several deconvolution algorithms based on the alphabet nature of the constellation signals (MM criterion) using the elementary Givens and Shear rotations.
- C2.
It reviews and compares the state-of-the-art deconvolution approaches (i.e. the four previously mentioned solutions) that exploit the constant or multi-modulus property to perform blind deconvolution.
- C3.
Unlike that work of Shah [11], which is done for the case of instantaneous (flat channel) mixtures, ours is a generalization which covers the case of convolutive MIMO channels.
The paper is organized as follows: Section 2 presents the data model, briefs the BD principle, and defines the used cost function as well as Givens and Shear rotations. Sections 3 and 4 outline the four different approaches and the corresponding derived algorithms. Section 5 compares the four different approaches in terms of computational cost and source signal restoration quality. Finally, Section 6 concludes the paper.
The list of used notations along with their description are given in Table 1.
Section snippets
System model and objectives
Consider a MIMO system with transmitters, all transmitting at the same time over the same frequency band, and receiving antennas. The received signal is assumed to be sampled at the symbol rate; however, fractional sampling can be easily incorporated by increasing the dimension of the received data vector. The transmitted data is assumed to pass through a frequency selective channel, i.e., ISI is present. The collected samples at time instant is modeled as:
Two-stage deconvolution algorithms
By imitating the Jacobi-like algorithms [30], [32], one can write the unitary matrix as a product of elementary Givens rotations, according to:where stands for the number of needed sweeps (iterations) until convergence. The optimal Givens rotations’ angles ( and ) are obtained by optimization of the MM cost function given in (8).
Unfortunately, it can be shown that closed form or simple solutions for this optimization problem are not possible due to the
One-stage deconvolution algorithms
In contrary to the proposed methods in Section 3, this section proposes two BD methods which perform the deconvolution in one stage. The first method is developed based on optimizing a hybrid criterion while the second one is a deflation-based approach.
Computational cost
Equalization-BSS approach is of two steps: in the first one, we use the MNS method which costs approximately flops, whereas the second step is the BSS which costs for the whitening plus the cost of the chosen separation method. For example, the latter costs flops/sweep if G-MMA is chosen. Table 3 provides the numerical cost per one sweep for the proposed method and considers the Givens rotations, hence to find the total cost one need to multiply the
Conclusion
This paper tackled one of the fundamental problems in wireless communications, more precisely, performing blind deconvolution and channel estimation for convolutive MIMO systems without the help of pilot data. In this paper, four different batch BD algorithms are presented; the first two, namely Full-BSS and Equalization-BSS, are classified as a two-step methods, while the last two, namely, Hybrid and deflation-based, are one step iterative approaches. The proposed algorithms are designed using
Credit author statement
The authors of this work share equal responsibility in the production of this work.
Funding
This work was supported by the Deanship of Scientific Research at KFUPM under Research Grant SB181001.
Declaration of Competing Interest
The authors declare that they have no competing interests.
Acknowledgments
The authors acknowledge the support provided by the Deanship of Scientific Research at KFUPM under Research Grant SB181001. Also, the authors would like to thank the anonymous reviewers and the Handling Editor for their constructive suggestions that have helped improve the paper.
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