Probabilitybased pilot allocation for MIMO relay Distributed compressed sensing channel estimation
 Abbas AkbarpourKasgari^{1}Email authorView ORCID ID profile and
 Mehrdad Ardebilipour^{1}
https://doi.org/10.1186/s1363401805397
© The Author(s) 2018
Received: 30 August 2017
Accepted: 14 February 2018
Published: 7 March 2018
Abstract
MultipleInput MultipleOutput (MIMO) relay communication systems are used as an efficient system in spectral efficiency and power allocation view point. In these systems, some of the facilities need channel state information (CSI). Besides, new estimation methods based on compressed sensing (CS) are well known for their spectral efficiency and accuracy. In this paper, we have used a Distributed CSbased channel estimation method to improve the accuracy and spectral efficiency of channel estimation for MIMOOrthogonal Frequency Division Multiplexing relay network. Specifically, using Least Squares (LS) estimation increases the accuracy of wellknown Compressive Sampling Matching Pursuit (CoSaMP) algorithm and proposes Blockverified CoSaMP (BvCoSaMP). To improve the accuracy of estimation, we are encountered with a combinatorial optimization which is dealt with probabilitybased approaches in this paper. More particularly, three probabilitybased optimization methods have been proposed to optimize the mutual coherence of measurement matrix called Sequential CrossEntropy (SCE), Extended Estimating of Distribution Algorithm (EEDA), and Parallel CrossEntropy (PCE). All these methods are based on sampling from a Probability Density Function (PDF) which is updated in each iteration using elite samples of the population. The simulation results represent the accuracy and speed of the proposed methods, and the comparison is expressed as well.
Keywords
1 Introduction
MultipleInput MultipleOutput (MIMO) relay communication systems make it possible to increase the data rate and coverage area of communication. Different characteristics of MIMOOrthogonal Frequency Division Multiplexing (OFDM) relaying make it an essential technology to conflict with fading and longdistance impairments [1, 2]. There are some features which need channel state information (CSI) to be available such as coherent demodulation, beamforming, relay selection, and so on [3, 4]. Increasing demand for high data rate communication could be accomplished by increasing the bandwidth efficiency of channel estimation methods which could be met by using compressed sensing (CS)based procedures. Since there is a small number of significant scatterers in the wireless environments, and the delay spread is normally large, the wireless channel could be modeled by sparse signal especially when operating in wide bandwidth. Consequently, CSbased channel estimation approaches could be used to increase the bandwidth efficiency and accuracy of channel estimation. Further to sparsity, MIMO communication systems benefit from the joint support of all the channel ensembles, since there are common scatterers in the environment between antennas of a transmitreceive pair [5]. Consequently, the support vector of the different channel groups is distributed identically. Thus, all the channels of MIMO communication between a transceiver pair are jointly sparse and could be represented in a blocksparse form [6]. By taking into account the block sparsity of MIMO channels, we could use the Distributed CS (DCS)based channel estimation approaches instead of CSbased channel estimation methods. DCSSimultaneous Orthogonal Matching Pursuit (SOMP) was one of the first DCSbased approaches which have been used in SingleInput SingleOutput (SISO)OFDM channel estimation [7]. Subsequently, in [8], DCSSOMP was used to estimate the jointly sparse channel vectors in MIMO transmission. Recently, massive MIMO CSI matrices were estimated using the jointOrthogonal Matching Pursuit (OMP) algorithm which has been proposed in [9] using the DCSbased approaches.
In DCSbased channel estimation methods for MIMOOFDM systems, the measurement matrix can be constructed using random pilot patterns. However, it can be optimized to boost the estimation accuracy using Restricted Isometry Property (RIP). Since there is no known polynomial time approach to optimize RIP, alternatively, the mutual coherence is minimized which is more practical [10]. It is shown that the mutual coherence is related to the position of pilots. As a consequence, one may minimize the mutual coherence by exhaustively searching the best positions for pilots which are intractable from the computational point of view. Hence, utilizing computational programs to generate suboptimal pilot patterns are entirely favorable. To this end, some of the papers in the literature perform evolutionary algorithms to create pilot patterns. Specifically, in [11], two different methods are developed for optimizing pilot allocation in CSbased channel estimation including Genetic Algorithm (GA) and minimizing the largest element in mutual coherence set. Moreover, the pilot allocation for DCSbased channel estimation is proposed in [12] using GAbased suboptimal approach. Additionally, Estimating the Distribution Algorithm (EDA) is used as a promising approach to optimize the pilot positions in SISOOFDM systems [13]. For MIMOOFDM systems, in [12], the authors try to optimize the pilot allocation using GAbased approach which was the extension of [14]. Other papers including [15–17] focus on the SISOOFDM systems and try to generate pilot sequences to increase the compressed channel estimation accuracy.
In this paper, we have developed a DCSbased approach called Blockverified Compressive Sampling Matching Pursuit (BvCoSaMP). Specifically, we have used the joint sparsity behavior of MIMO channels to produce a block sparse measurement matrix. Fortunately, using a permutation matrix, we have generated the measurement matrix and received pilot subcarriers. Using the measurement matrix and received pilot vectors, we have developed a distributed form of Compressive Sampling Matching Pursuit (CoSaMP) algorithm. To utilize CoSaMP successfully, we have verified it by using Least Squares (LS) estimation on the positions of the channel vectors which are suggested by BlockCoSaMP (BCoSaMP) algorithm. Hence, BCoSaMP is completed by an LS estimation phase, and BvCoSaMP is generated. In fact, by taking into account the sparsity order of the channel, CoSaMP is one of the most successful algorithms to estimate the support. By applying block behavior into the recovery algorithm, BCoSaMP is changed to the more accurate algorithm for estimating the support, as well. Support estimation determines the nonzero paths of channel estimation vector. Furthermore, to increase the accuracy we used LS estimating, as well. Further to estimation algorithm, the pilot allocation approaches are also proposed.
We have proposed three different pilot allocation methods based on probabilitybased mutual coherence optimization for DCSbased MIMO relay channel estimation, as well. The mutual coherence of the measurement matrix is considered as the fitness function in this paper. It is caused by the impact of mutual coherence on the mean square error of estimation in DCSbased approaches which is addressed in the literature. Moreover, orthogonal pilot allocations are considered in channel estimation, i.e., each antenna transmits neutral subcarrier in pilot subcarriers of the other antennas. At first, we have shown that shifting the measurement matrix rows and columns have not affected the mutual coherence. Furthermore, it is proved that mutual coherence of a block measurement matrix can be calculated using smaller and simpler nonblock measurement matrix. Utilizing these phenomena, we have defined the new fitness function for pilot allocation. Modeling the mutual coherence of measurement matrix implies that the fitness function is combinatorial and needs to be optimized using different pilot sequences. Herein, in order to minimize the fitness function, we have used probabilitybased approaches. Specifically, Sequential CrossEntropy (SCE)based pilot allocation, Extended Estimating of Distribution Approximation (EEDA)based pilot allocation, and Parallel CrossEntropy (PCE)based pilot allocation are three different pilot allocation approaches which have been proposed in this paper. SCEbased pilot allocation works based on the sampling from a Probability Density Function (PDF) which is updated in each iteration. Updating the PDF in each iteration is related to the current samples and previous iterations’ samples, as well. The relation can be controlled using a control constant τ. On the other hand, considering the marginal value for control constant (τ=1), the algorithm will update the PDF utilizing current samples. This approach is called EEDA which is an extension of [18]. EEDA approach will increase the convergence speed, but it will increase the risk of local minima trapping. Subsequently, to increase the rate of convergence together with the decrease of local minima trapping, we have proposed PCEbased pilot allocation which considers multiple local extrema in the iterations. It will increase the speed of convergence expectedly, moreover decrease the risk of local minima trapping.

The compressed channel estimation using proposed BvCoSaMP algorithm which uses block representation of the channel vectors to increase the accuracy of support estimation and excess LS estimation stage in the positions determined by estimated support to enhance the channel coefficients accuracy.

The pilot allocation is optimized using proposed SCEbased sequence determination algorithm and extended to MIMO pilot allocation.

In order to increase the speed of convergence of SCEbased pilot allocation, a special case is used called EEDAbased pilot allocation.

To increase the accuracy together with the speed of convergence for pilot allocation, PCEbased pilot allocation algorithm is proposed in which multiple PDFs are used to generate the pilot sequences.
The remainder of the paper is as follows. The system model is represented in Section 2. Section 3 covers channel estimation approach using BvCoSaMP, and pilot allocation scheme for SISO and MIMO systems and proposed pilot allocation methods are described in Sections 4 and 5, respectively. Numerical results are expressed in Section 6. Eventually, concluding remarks are demonstrated in Section 7.
Notations: Matrices and vectors are denoted by uppercase and lowercase boldfaced letters, respectively. . and (.)^{∗} denote the complex modulus and the conjugate of a complex number. For a given matrix A, A^{ T } and A^{ H } denote its transpose and conjugate transpose, respectively, and A_{i,j} denotes the (i,j)th element of A. For a given vector x with its element denoted by x_{ l }, \(\\mathbf {x}\_{2} = \sqrt {\mathbf {x}^{H}\mathbf {x}}\) represents the Euclidean norm, \(\\mathbf {x}\_{1} = \sum _{l}x_{l}\) is the l_{1}norm, and diag(x) denotes a diagonal matrix with x on its main diagonal. For two vectors x and y, <x,y> denotes their inner product. For a given set Λ, n(Λ) is the number of elements in Λ. \(\mathbb {C}^{m~\times ~ n}\) stands for the set of all complexvalued m × n matrices, and ∅ denotes the null set.
2 System model
where Ψ=ΦW and e=W^{ T }h.
3 Distributed compressed sensing channel estimation
CSbased approaches try to estimate channels individually while DCSbased channel estimation tries to estimate jointly sparse channels altogether. There are different channels in MIMO systems between the various transmitting and receiving antennas. Since all the transmit antennas gathered near each other and receive antennas as well, the significant scatterers which are encountered by the transmitted signals are the same. As a consequence, the sparsity pattern of different transmitreceive pairs would be the same while the channel coefficient would be completely different. Thus, utilizing DCS would be very useful. As a result, we have used the CoSaMP method and define it in a distributed manner to estimate MIMO channels.
3.1 BvCoSaMP algorithm
In this subsection, we represent the BvCoSaMP. Apparently, the CoSaMP algorithm utilizes the channel sparsity. Here, we used BCoSaMP algorithm to estimate the joint support of the channel responses. Finally, after calculating the support of the channels, the channel impulse responses are estimated using LSbased estimation. The algorithm is represented in Algorithm 1.
Obviously, exchanging the formation of the vectors between DCS and CS are repeated in Algorithm 1. Consequently, we try to explain it here. As explained before, e=[e_{1},e_{2},…,e_{Q(2L−1)}]^{ T } is the DCSformed channel model where \(Q = N_{\mathbb {S}}N_{\mathbb {D}}\). Actually, e_{ i } for i=1,2,…,Q are related to the first path of different channel pairs and i=Q + 1,Q + 2,…,2Q are related to the second path of different channel pairs. Consequently, we can represent it by e_{ i } for i=(l − 1)Q + [1:Q] and l=[1:2L − 1] to clarify the DCSformed representation. Hence, in Algorithm 1, to sum up all the paths in one vector, we should sum every Qelement of DCSformed vector and consider it as the corresponding paths. This reformation is represented in Step 2 of Algorithm 1. Actually, in Step 2, we try to accumulate the calculated measurements coherently in order to organize the proxy. Moreover, to represent the DCS form into CS form, we should extend the DCS form selections by using \((\hat {T}~~1)Q ~+~ [\!1:Q]\), where \(\hat {T}\) is selected path from DCS form representation.
We consider the BCoSaMP algorithm to represent the support of the vectors. Actually, support estimation determines the nonzero paths of channel estimation vector. To improve the accuracy of support estimation, block representation helps CoSaMP algorithm. Consequently, using BCoSaMP we determine the significant paths of all the channel pairs in the network. Besides, at the end of the algorithm, we decide to calculate channel coefficients by LS equation to improve the accuracy of the estimation. LS equation is solved based on the support of the algorithm which is extracted from BCoSaMP. In other words, BCoSaMP represented that LS estimation should be solved only on the significant paths which are determined. Hence, we combine LS channel estimation and BCoSaMP together and generate the BvCoSaMP. We should consider that in the LS part of the channel estimation, the LS equation is only solved for K indices of the measurement matrix which are estimated by BCoSaMP. In fact, we used two interactive tools called block representation and LS estimation to increase the accuracy of estimation. Block representation enhances the precision of support estimation, and LS estimation improves the accuracy of coefficient estimation. Accordingly, two main features of channel estimation are increased attractively by exploiting the block representation and LS estimation. Consequently, in simulation results, the performance is developed expectedly.
3.2 Complexity analysis
Here, we discuss the complex multiplications which are loaded in each of the states in Algorithm 1 to represent the complexity of the algorithm. Firstly, in Step 1 of Algorithm 1, the number of complex multiplications to proceed the proxy is \(N_{\mathbb {D}}N_{p}\). Furthermore, in Step 6\(N_{\mathbb {D}}N_{p}\) complex multiplications are performed. Besides, \(N_{\mathbb {S}}N_{\mathbb {D}}(2L~~1)\) complex numbers are multiplied in Step 9 and finally in Step 11 number of complex multiplications is \(N_{\mathbb {D}}N_{p}\). Hence, the overall complexity of the proposed algorithm is of order \(3N_{\mathbb {D}}N_{p} ~+~ N_{\mathbb {S}}N_{\mathbb {D}}(2L~~1)\).
4 Pilot allocation for compressed channel estimation
CSbased approaches try to estimate channels individually while DCSbased channel estimation tries to determine jointly sparse channels altogether. As a result of adjacent antenna sitting, the sparsity pattern of different transmitreceive pairs would be the same while the channel coefficient would be completely different. Thus, utilizing DCSbased channel estimation would be very useful. The measurement matrix could be generated using random pilot subcarriers to estimate the channels. Moreover, increasing the accuracy and bandwidth efficiency of channel estimation could be guaranteed using optimal pilot allocation based on RIP optimization. As stated, since there is no polynomial time approach to evaluating RIP, instead, mutual coherence is used which is computationally tractable. Hence, we introduce the mutual coherence at first and then minimize it using proposed populationbased algorithms.
where λ_{ i } and λ_{ j } are pilot subcarriers among N available subcarriers. In other words, mutual coherence can be expressed as the maximum offdiagonal entry of Gram matrix G{Φ}=Φ^{ H }Φ if Φ is orthonormal [20]. Obviously, to minimize the mutual coherence, we have to choose pilot subcarriers which minimize Eq. (7). Moreover, in this equation, the pilot positions λ_{ i } and λ_{ j } are selectable and could be considered as the design criteria. Besides, this problem is combinatorial and should be optimized using iterative search methods. Here we use probabilitybased methods to minimize Eq. (7). In the following section, the proposed algorithms are explained. To estimate channel using DCSbased approach, we should extend the mutual coherence to MIMO case.
4.1 Extension to MIMO
In order to design optimal pilot sequences for different transmit antennas in MIMOOFDM relay networks, we should define the mutual coherence for measurement matrix Ψ in Eq. (5). Since Ψ is a columnwise permutation of Φ, at first, we should consider the impact of permutation of columns on the mutual coherence.
Theorem 1.
Consequently, \(\mu \{ \bar {\boldsymbol {\Phi }} \} = \max _{i\neq j, 1 \leq i,j \leq n} \phi _{i}^{H} \phi _{j} = \mu \{ \boldsymbol {\Phi }\}\) which indicates the equality of both mutual coherence. Therefore, the permutation of measurement matrix columns will not effect the mutual coherence value.
where μ{Γ_{ i },Γ_{ i }}=μ{Γ_{ i }} and μ{Γ_{ i },Γ_{ j }} is the largest offdiagonal absolute value of \( \boldsymbol {\Gamma }_{i}^{H} \boldsymbol {\Gamma }_{j}\). Hence, in order to calculate the mutual coherence of the MIMOOFDM mutual coherence, it is sufficient to calculate all the available SISOOFDM pairs and consider the largest value as the mutual coherence.
5 Proposed pilot allocation algorithms
In this section, we will propose three algorithms based on the population and probability, to optimize the combinatorial problem in Eq. (6). As mentioned, these algorithms work based on the probability function and try to sample the PDF using different populations and do this until convergence of the PDF to the steady state. Here, \(\max _{i,j; i\neq j} \left  \sum _{l=0}^{2L2}e^{j2\pi (\lambda _{m}  \lambda _{n})l / N} \right \) is the fitness function which should be optimized. In order to decrease the computational complexity in fitness function we used Eq. (14). Evidently, there are lots of zero elements in matrix Ψ which are unnecessary to be multiplied. Hence, we used Eq. (14) to optimize the computations in fitness function evaluation. Moreover, this optimization is done over the search space \(\mathcal {S}\) which is composed of \(\binom {N}{N_{p}}\) candidate. Since the exhaustive search over search space \(\mathcal {S}\) is intractable, we consider probabilitybased approaches to optimize it using initial generation \(\mathcal {G}^{0}\) where I different individuals are found in each generation.
5.1 Sequential CrossEntropybased pilot allocation approach
where τ∈(0,1] is a parameter of algorithm. Since the probability behavior of populations is evident, possibility of getting stuck in a local minimum is lower. The stagebystage algorithm is represented in Algorithm 2. The stop criteria is based on the PDF, and when the probability function contains zero and \(1/N_{\mathbb {S}}N_{p}\), it means that the best possible value for pilot allocation is obtained.
5.2 Special case of SCE (τ=1)—extension of [18]
As a result, the memory of PDF will be omitted. Wang et. al. in [18] proposed this approach called as the EDA for the SISOOFDM system. Here, we extend the EDAbased algorithm for generating \(N_{\mathbb {S}}\) different orthogonal sequences for the MIMOOFDM system. In [18], it is noted that matching the sequences with the best sequence in every stage is mandatory, but in the following theorem, it is proved that matching to the sequence is not mandatory at all, since matching the sequences is obtained using circular shifting of the unmatched sequences. This phenomenon is considered in the following Theorem.
Theorem 2.
Here, we consider Φ=[ϕ_{1};ϕ_{2};ϕ_{3};…;ϕ_{ n }] where ϕ_{ i } is 1 × L complex vector. On the other hand, we assume \(\bar {\boldsymbol {\Phi }} = \left [ \phi _{1}; \phi _{3}; \phi _{2}; \ldots ; \phi _{n} \right ]\), then mutual coherence for both of the measurement matrices can be calculated as the maximum offdiagonal entry of \(\sum _{i = 1}^{n}\phi _{i}^{H}\phi _{i}\). Consequently, the mutual coherence for both of the matrices are the same and rowwise permutation does not change the mutual coherence.
Specifically, in the EDAbased algorithm which is represented in [18], after circular shifting J best sequences, the PDF of the current stage is calculated by counting the number of used subcarriers in J selected placements. Obviously, the counting process is independent of the circular shifting. Another drawback of the algorithm can be considered as trapping in the local minima, since the current PDF is considered without taking into account previous ones. Apparently, the PDF is stuck in local minima more rapidly, and thus, the algorithm would be terminated. Although the speed of convergence is more than SCE algorithm, the accuracy is less since local minima trapping has occurred more rapidly. This algorithm is considered in Algorithm 3.
5.3 Parallel CrossEntropybased pilot allocation algorithm
As mentioned, in SCEbased pilot allocation algorithm, the risk of trapping in local minima is lower than the EEDAbased approach. But, the convergence speed is absolutely inferior to the other EEDAbased approach. In order to improve the convergence speed of the SCEbased algorithm, we have proposed PCEbased pilot allocation plan. PCE considers α number of points in search space \(\mathcal {S}\) and tries to converge the PDF of these points to steady state, simultaneously. Hence, since α points are considered simultaneously, trapping in one local minimum is lowered than before. Moreover, working on a different number of PDFs at the same time accelerates the convergence of the proposed method. This algorithm is represented in Algorithm 4.
As demonstrated in Algorithm 4, the difference of the PCE and SCE is considered in pipelining the update PDF and generations. In other words, in each iteration of the algorithm, generation update is performed on α pipelined branches. Moreover, these α pipelined branches are utilized to update α different PDFs. As a consequence, the random sampling is done over α different PDFs, where α different generations are constructed. This approach uses more computational calculations in each iteration which can be handled by parallelprocessing supporting hardware. In the other hand, we obtain two significant advantages including conservative behavior in impacting by local minima trapping and lower number of iterations in convergence. Thus, utilizing PCEbased pilot allocation will confirm two critical aspects of pilot allocation algorithms.
6 Numerical results
In this section, the numerical results are collected to represent the performance of the proposed SCE, EEDA, and PCEbased pilot allocation algorithms. In all the simulations, the number of OFDM subcarriers is N=512, and 16ary quadrature amplitude modulation (16QAM) with Gray coding is utilized. Among N available subcarriers, there are N_{ p } subcarriers as pilot subcarriers. Various values are assigned to N_{ p } in different simulations. Without loss of generality, in MIMO scenarios the number of transmitting and receiving antennas for all the nodes are 2, i.e., \(N_{\mathbb {S}} = N_{\mathbb {R}} = N_{\mathbb {D}} = 2\). Sparse Rayleigh fading channels in the networks are modeled using Finite Impulse Response (FIR) filters with L taps where K number of taps are nonzero and are i.i.d. using zero mean and unit variance complex Gaussian distribution. Furthermore, all the results are averaged over 1000 independent channel ensembles.
6.1 Comparison of the proposed algorithms
Sample pilot sequences and their mutual coherence which have been achieved by different proposed approaches and comparison to the method in [11]
Alg.  SCE  EEDA  PCE  GA  

μ  0.2360  0.2439  0.2227  0.2557  
N _{ s }  1  2  1  2  1  2  1  2 
6  9  31  36  29  31  18  55  
14  22  42  44  47  49  20  61  
33  37  52  71  57  63  36  99  
Pilot  40  44  80  87  64  71  44  132 
seq.  51  54  92  142  72  110  66  148 
56  59  149  150  115  116  86  160  
62  66  155  186  121  124  116  179  
95  99  187  194  198  200  124  196  
103  105  195  203  207  208  135  203  
112  135  205  210  217  218  145  222  
151  152  215  217  227  228  166  230  
158  212  223  225  236  237  177  238  
219  242  230  233  247  248  185  256  
243  249  237  241  255  257  197  263  
252  257  244  251  278  280  237  277  
276  318  263  291  287  289  252  295  
324  325  292  299  298  308  306  307  
344  349  311  329  317  318  323  331  
354  355  330  335  326  346  348  352  
372  373  349  409  348  355  357  381  
413  420  414  418  357  362  385  419  
430  431  423  426  363  412  421  428  
436  483  429  437  414  434  446  434  
487  493  445  469  439  466  484  447  
502  512  498  500  470  483  490  483 
6.2 Evaluating MSE and BER performance
where h is the complete channel vector and \(\hat {\mathbf {h}}\) is its estimation. Moreover, N_{ MC } is the Monte Carlo iteration which is 1000. BER is evaluated using Monte Carlo simulation using N_{ MC } individual simulation.
6.3 Performance comparison with the other estimation method
6.4 The effect of significant paths of channel
7 Conclusions
In this paper, we have considered the channel estimation problem in AF MIMOOFDM relay networks using DCSbased approaches. Firstly, the channel estimation method proposed is called BvCoSaMP where the channel estimation is improved utilizing block sparsity of MIMO channels and LS estimation. In order to improve the performance of the estimation, we design three different algorithms to minimize the mutual coherence of the resultant channel estimation measurement matrix. The proposed methods are based on the CrossEntropy optimization and include SCE, EEDA, and PCEbased approaches. Utilizing just one local minimum point in SCE and EEDA, the performance is worse than PCEbased approach. Multipoint tracking ability of PCE makes it possible to increase the accuracy and speed of the pilot sequence determination algorithm.
Declarations
Acknowledgements
The authors want to acknowledge the help of all the people who influenced the paper. Specifically, they want to acknowledge the anonymous reviewers for their reasonable comments.
Funding
There is no source of funding for this paper.
Availability of data and materials
Not applicable.
Authors’ contributions
AA and MA carried out the mathematical parts of the paper and simulations were done by AA. Both of authors read and approved the final manuscript.
Ethics approval and consent to participate
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Consent for publication
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Competing interests
The authors declare that they have no competing interests.
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