1 Introduction

There has been an increased interest on the massive multiple-input and multiple-output uplink system that has emerged in recent years. However, various recent studies have showed that the Massive MIMO system has many advantages over the old spread spectrum system in terms of the high data rate and performance. The main advantage of massive MIMO system is the ability to have a large number of antennas since it can be manufactured in high numbers. In the contrary to the massive use of spectrum system is limited since the spectrum itself is limited and expensive [1].

Massive Multiple-input multiple-output (MIMO) is one of the most challenging and emerging areas in wireless communication systems. This is mainly due to its high spectral efficiency, strength alongside declining in addition to non-multiuser-interference competence. Such a move on massive MIMO uplink aimed at providing the Base Station (BS) acting on a large number of antennas to approach higher data rates in cellular multi-user systems, without increasing the bandwidth.

In Massive Multiple-input multiple-output up-linking systems, the receivers require to estimate the channel state information (CSI) to be able to use the linear decoding methods, e.g., MMSE decoding method and zero forcing (ZF) decoding method. The performance of these linear methods relies on how accurate CSI estimation is determined. However, the performance will be degraded if the CSI estimation is inaccurate [2, 3]. In the literature [4, 5], the CSI is typically estimated using the pilot signals in order to obtain the channel coefficients from the received signals. Furthermore, in the pilot signals reuse in cellular system causes associated inter-cell intervention named equally to pilot contamination that degrades the approximated presentation and efficiency of the spectrum [4, 6,7,8].

In practical, the CSI estimation using the pilot signals is typically incorrect because of directly above restraints on Channel State Information achievement in addition to its flexibility and deferment. As well, the inaccuracy of the CSI can significantly degrade the enactment advances obtained through Massive Multiple Input Multiple Output uplink systems.

To that end, pilot contamination is the limiting feature of the systems capacity. This problem occurs due to reuse the pilot signals in neighboring cells. To alleviate pilot contamination, a variety of decoding approaches have been proposed; whereas, most of them require the pilot signals, when it is not straightforwardly conceivable from the presence of the algorithmic computation interruptions or time constraints. Furthermore, for a longer length in the pilot signals diminishes the transmission efciency [9].

In [10], the mechanism presents the channel estimation method based on Bayesian approaches. They assumed to know the channel statistics. Also, the approaches estimate the channel matrix using the eigenvalue decomposition methods. However, the channel estimation algorithm is reported based on shifting the location of pilots in time frames. As well in [7], a channel estimation algorithm based on second-order covariance matrix is proposed. The performance of their estimation method still relies on how accurate the covariance information estimation. Nonetheless, the uplink power control had been projected so as to address issue of pilot contamination in multi-cell MIMO uplink systems through improving energy efciency. However, in for a perfect CSI and multi-cell coordination are required to reduce the uplink power consumption [8, 11].

Even though, in non-linear iterative algorithms to estimates channels and transmitted symbols based on statistics power control and mutual piloting techniques are projected that shows outcomes of the pilot contaminations are removable whenever the power established from inner-cell operators is superior to the receiving end from outer cell handlers. On the other hand, the operation assumption necessitates for both power regulation alongside regular cell geometry, which might not embrace the complete simplification [12, 13]. There are also decoding methods that depend on the asymptotic orthogonality of channels from different users. However, based on the orthogonality, the channels estimations are likely to be realized from the covariance matrices of the acknowledged signals. However, it has residual scale multiplicative ambiguity. To address this ambiguity problem, pilot sequences are still needed [14,15,16,17,18]. In [5] and from a blind pilot decontamination method is proposed. In their method, as the rst step, the pre-whitening techniques are used to project the received signals into a signal subspace. Then, as the second step, power-controlled transmission is used to identify the desired user. It is noticed that this method depends on the asymptotic orthogonality.

This paper suggests a blind decoding algorithmic system established on the cumulant matrix fourth order that assures the stoutness of the decoding algorithm with reference to performance. Also, the proposed method based on Higher Order Cumulants uses the statistical independence of sources earlier to the encrypting process without using pilot sequences. From side to side simulations can be used in the verification of the substantial Bit Error Rate (BER) enhancements realized through the projected method. Such a method is deliberated as blind due to the pilot systems is presumed to be unidentified. However, the simulation situations are anticipated so as to make observation on the variations encountered in the BER in the functional expression of signal to noise ratio (SNR), number of antennas and the length of symbols per user.

Moreover, the performance of the presented algorithm is computed and a comparison evaluation is prepared among the minimum-mean square-error (MMSE) decoding and zero-forcing (ZF) decoding with imperfect channel state information (CSI) based on their performances and computation involvedness. Though, the remaining parts of this paper are systematized into Sect. 2, in a short depiction of the massive MIMO uplink signaling exemplary together with multi-path declining is obtainable. While for Sect. 3, there is a proposal on the new-fangled decoding algorithms established on cumulant matrices in fourth order. As well, Sect. 4 gives a presentation on the algorithm implementation in which the comparative simulative outcomes and suppositions are assumed for both the fifth and sixth Sections correspondingly.

2 Signal Model

Let us consider an K-cell massive MIMO uplink system. Let us also assume a microcell Rician–Rayleigh fading environment. However, the environment of Rician–Rayleigh fading means that the received signal from the first cell is Rician faded and the interference signals from co-channels are Rayleigh faded. In each cell, let us assume that a base station (BS) composed of M multiple antennas, whereas each terminal user in the cell composed of one single transmit antenna. The received signals at the BS in the first cell with T sample vectors, denoted by the M x T matrix \(\mathbf{X}\), is given by [5, 11,12,, 19, 20]

$$\begin{aligned} {\mathbf{X}} = \sqrt{\alpha }{\mathbf{HS}} + {{\mathbf{B}}_0} \end{aligned}$$
(1)

where \(\alpha = \rho {\sigma ^2}\) is the average power of each user, \(\rho \) is the SNR, \({\sigma ^2}\) is the variance of Gaussian noise. \({\mathbf{S}} = {\left[ {{\mathbf{s}_1},{\mathbf{s}_2},\; \ldots ,{\mathbf{s}_{\mathbf{K}}}} \right] ^T}\) is an M x T transmitted signals, where T-dimensional vector \(\mathbf{s}_i\) represents the signal sent by the transmitter terminal in the ith cell. Through this paper, we assume \({{\mathrm{s}}_{\mathrm{i}}}\left( n \right) \in \left\{ { \pm 1} \right\} \) to be the nth sample of \(\mathbf{s}_i\). Furthermore, the M x K mixing matrix \(\mathbf{H}\) represents the channel coefficients \((h_{mk})\) from the K users to BS. However, one can express the channel matrix \(\mathbf{H}\) as follows:

$$\begin{aligned} {\mathbf{H}}= \sqrt{{\beta _k}} {\left[ {{h_{mk}}} \right] _{M \times K}} \end{aligned}$$
(2)

where \({\beta _k}\) represents the path loss and the large scale fading between the kth cells user and the BS. \((h_{mk})\) denotes the channel gain from the user in the kth cell to the mth antenna at the BS. Nonetheless, the Rician fading channel is expressed by \({H_1} = {\left[ {{h_{m1}}} \right] _{M \times 1}}\), where \((h_{m1})\) is independent identical distributed (i.i.d) taken from the Gaussian random variables with mean \(\mu \) and unite variance. \(\mu = \sqrt{2F}\) is the value of the direct line-of-sight signal and F is the Rician factor. The other co-channel coefficients are assumed to be Rayleigh faded. These Rayleigh fading channel is denoted by \({{\mathbf{H}}_k} = {\left[ {{h_{mk}}} \right] _{M \times k}}\;\forall \;k = 2,\;3,\; \ldots ,K\), where \((h_{mk})\) is independent identical distributed (i.i.d) taken from the Gaussian random variables with zero-mean and unite variance. Lastly, \({{\mathbf{B}}_0}\) represents the M T noise matrix. The vector of the noise is both spatial and temporal white alongside a variance of \({\sigma ^2}\).

3 The Proposed Decoding Scheme Based on Higher Order Cumulants

For the proposed decoding scheme, a blind channel estimation method based on the higher order cumulants framework is presented to decode a user signal. On the contrary to training based approaches. The proposed scheme estimates the channel matrix without using a training sequence or pilot signals to avoid pilot contamination problem in massive MIMO uplink systems. In such a case, it is important to start by recalling the obtained vector-form in the second equation (2),

$$\begin{aligned} {\mathbf{x}} = \sqrt{\alpha }{\mathbf{Hs}} + {{\mathbf{b}}_0} \end{aligned}$$

The aim is to estimate the channel matrix \(\mathbf{H}\) of the conventional received vector \(\mathbf{x}\) under the following assumptions.

  • AS1) The number of BS antennas is much larger than the user number, i.e., \(M\; \gg \;K\)

  • AS2) the M x K channel matrix, \(\mathbf{H}\), is of full column rank, which equals K. Furthermore, the elements of channel matrix are i.i.d. taken from the Gaussian random variable.

  • AS3) The elements of the transmitted signals, \(\mathbf{s}\), are white, independent and identically distributed (i.i.d). and the binary phase-shift keying (BPSK) modulation is assumed.

  • AS4) The additive white Gaussian noise vector is independent of source signals with covariance \( {\sigma ^2}\).

  • AS5) The average transmit power on each antenna is normalized to unity.

The method composed of two steps:

3.1 Preprocessing (Whitening of Data)

The initial part is comprise the preprocessing step, where the channel are estimated to the extent of unitary matrices through the second order statistics (SOS). Moreover, pre-whitening of the received data aims at the mitigation of the noise vector as well as the elimination of data redundancy. Under aforementioned assumptions, the M x M covariance matrix (\({\mathbf{R}}\)) of the noiseless received signals can be expressed by

$$\begin{aligned} {\mathbf{R}} = {\mathrm{E}}\left[ {{\mathbf{x}}{{\mathbf{x}}^{\mathrm{T}}}} \right] - {{\mathrm{\sigma }}^2}{{\mathbf{I}}_{\mathrm{M}}} \end{aligned}$$
(3)

By substituting \({\mathbf{x}}\) in (3), one gets \({\mathbf{R}}\) as follows

$$\begin{aligned} {\mathbf{R}} = \alpha {\mathbf{H}}{\mathrm{E}}\left[ {S{S^{\mathrm{T}}}} \right] {{\mathbf{H}}^{\mathrm{T}}} = \alpha {\mathbf{H}}{{\mathbf{H}}^{\mathrm{T}}} \end{aligned}$$
(4)

Based on the singular value decomposition (SVD), \({\mathbf{R}}\) is then simplified as

$$\begin{aligned} {\mathbf{R}} = {\mathbf{V}{\varvec{\Lambda }}}{{\mathbf{V}}^{\mathrm{T}}} \end{aligned}$$
(5)

where \({\mathbf{V}}\) is a M x K matrix satisfying

$$\begin{aligned} {{\mathbf{V}}^{\mathrm{T}}}{\mathbf{V}} = {{\mathbf{I}}_{\mathrm{K}}} \end{aligned}$$
(6)

Also, \({{{\varvec{\Lambda }} }}\) is given as K x K diagonal matrix comprised of real entries. Therefore, in equation (4), the M x K matrix \({\mathbf{R}}\) can be stated in the unitary significance disintegration based on its singular value decomposition as follows:

$$\begin{aligned} {\mathbf{H}} = \frac{1}{{\sqrt{\alpha }}}{\mathbf{V}}{{{{\varvec{\Lambda }} }}^{ - \frac{1}{2}}}{{\mathbf{U}}^{\mathrm{T}}} \end{aligned}$$
(7)

While, the \({\mathbf{U}}\) matrix represents a K x K full rank unitary matrix as \({\mathbf{U}}{{\mathbf{U}}^{\mathrm{T}}} = {{\mathbf{I}}_{\mathrm{K}}}\).

As a result, the whitening process of obtaining matrix \({\mathbf{V}}\) can be integrated on the K x 1 whitened data vector \({\mathbf{z}}\) with the covariance of identity matrix, \({{\mathbf{R}}_{{\mathrm{zz}}}} = {{\mathbf{I}}_{\mathrm{K}}}\), that can be calculated as illustrated below:

$$\begin{aligned} {\mathbf{z}}= & {} \frac{1}{{\sqrt{\alpha }}}{{{{\varvec{\Lambda }} }}^{ - \frac{1}{2}}}{{\mathbf{V}}^{\mathrm{T}}}{\mathbf{x}} \end{aligned}$$
(8)
$$\begin{aligned} {\mathbf{z}}= & {} {{\mathbf{U}}^{\mathrm{T}}}s + {{{{\varvec{\Lambda }} }}^{ - \frac{1}{2}}}{{\mathbf{V}}^{\mathrm{T}}}{b_0} \end{aligned}$$
(9)

To that end, the transmitted vector s would be obtained with a Zero-Forcing (ZF) equalizer approach. Where the estimates of M x 1 transmitted vector \(\hat{\mathbf{s}}\) are expressed as:

$$\begin{aligned} \hat{\mathbf{s}}= {\mathbf{Uz}} \end{aligned}$$
(10)

The subsequent step on completion of the preprocessing stage, the estimation of the M x K channel matrix \({\mathbf{H}}\) is reduced to assist in the determination of the K x K unitary matrix denoted by \({\mathbf{U}}\).

3.2 Determination the Unitary Matrix \({\mathbf{U}}\).

In this subsection, the independence of equal symbols is utilized for the estimation of the unitary matrix \({\mathbf{U}}\). In particular, the rotational matrix \({\mathbf{U}}\) in the framework of fourth order cumulant tensor is established, which has been successfully applied under various setting in [6, 8, 21]. A cumulant matrix of the transmitted vector\(\hat{\mathbf{s}}\), in association with an arbitrary K x K matrix \({\mathbf{U}} \equiv {{\mathbf{u}}_{l,k}}\;\;\forall \;\left( {1 \le l,k \le K} \right) \) is dened as [21,22,23]

$$\begin{aligned} {{\mathbf{Q}}_{\hat{S}}}\left( {\mathbf{U}} \right) = \mathop {\sum }\limits _{l,k = 1}^{K{\mathrm{\;}}} cum\left( {{{\hat{\mathbf{s}}}_i},{{\hat{\mathbf{s}}}_j},\;{{\hat{\mathbf{s}}}_l},{{\hat{\mathbf{s}}}_k}} \right) {{\mathbf{u}}_{l,k}}\;\;\forall \;\left( {1 \le i,j \le K} \right) \end{aligned}$$
(11)

However, the fourth order cumulant (FOC) matrix from the independent zero-mean transmitted vector \({\hat{\mathbf{s}}}\) for \(\left( {1 \le i,j \le K} \right) \), is given by

$$\begin{aligned} {\mathrm{cum}}\left( {{{\hat{\mathbf{s}}}_i},{{\hat{\mathbf{s}}}_j},\;{{\hat{\mathbf{s}}}_l},{{\hat{\mathbf{s}}}_k}} \right) = \left\{ \begin{array}{l} {\mathrm{E}}\left[ {{{\hat{\mathbf{s}}}_i},{{\hat{\mathbf{s}}}_j},\;{{\hat{\mathbf{s}}}_l},{{\hat{\mathbf{s}}}_k}} \right] - {\mathrm{E}}\left[ {{{{{\hat{\mathbf{s}}}}}_{\mathrm{i}}}{{{{\hat{\mathbf{s}}}}}_{\mathrm{j}}}} \right] {\mathrm{E}}\left[ {{{{{\hat{\mathbf{s}}}}}_{\mathrm{k}}}{{{{\hat{\mathbf{s}}}}}_{\mathrm{l}}}} \right] \\ - {\mathrm{E}}\left[ {{{{{\hat{\mathbf{s}}}}}_{\mathrm{i}}}{{{{\hat{\mathbf{s}}}}}_{\mathrm{k}}}} \right] {\mathrm{E}}\left[ {{{{{\hat{\mathbf{s}}}}}_{\mathrm{j}}}{{{{\hat{\mathbf{s}}}}}_{\mathrm{l}}}} \right] - {\mathrm{E}}\left[ {{{{{\hat{\mathbf{s}}}}}_{\mathrm{i}}}{{{{\hat{\mathbf{s}}}}}_{\mathrm{l}}}\left] {\mathrm{E}} \right[{{{{\hat{\mathbf{s}}}}}_{\mathrm{j}}}{{{{\hat{\mathbf{s}}}}}_{\mathrm{k}}}} \right] \end{array} \right\} \end{aligned}$$
(12)

Since, the \(\hat{\mathbf{s}}\) vector are presumed to have unitary variance. Consequently, the fourth order cumulant matrix becomes:

$$\begin{aligned} {\mathrm{cum}}\left( {{{{{\hat{\mathbf{s}}}}}_{\mathrm{i}}},{{{{\hat{\mathbf{s}}}}}_{\mathrm{j}}},{{{{\hat{\mathbf{s}}}}}_{\mathrm{k}}},{{{{\hat{\mathbf{s}}}}}_{\mathrm{l}}}} \right) = {\mathrm{\;E}}\left[ {{{{{\hat{\mathbf{s}}}}}_{\mathrm{i}}}{{{{\hat{\mathbf{s}}}}}_{\mathrm{j}}}{{{{\hat{\mathbf{s}}}}}_{\mathrm{k}}}{{{{\hat{\mathbf{s}}}}}_{\mathrm{l}}}} \right] - {{\mathrm{\delta }}_{{\mathrm{ij}}}}{{\mathrm{\delta }}_{{\mathrm{kl}}}} - {{\mathrm{\delta }}_{{\mathrm{ik}}}}{{\mathrm{\delta }}_{{\mathrm{jl}}}} - {{\mathrm{\delta }}_{{\mathrm{il}}}}{{\mathrm{\delta }}_{{\mathrm{jk}}}} \end{aligned}$$
(13)

where \({{\mathrm{\delta }}_{{\mathrm{ij}}}}\) symbolizes the Kronecker delta, which is equal to \(1{\mathrm{\;\;\;\;}}\forall {\mathrm{\;\;i}} = {\mathrm{j}}\) ; or else zero. Due to the independent features on the symbols (\(\hat{\mathbf{s}}\)), most of the cumulant variables are zero. Thus, the fourth order cumulant set (\({\mathrm{cum}}( {{{{{\hat{\mathbf{s}}}}}_{\mathrm{i}}},{{{{\hat{\mathbf{s}}}}}_{\mathrm{j}}},{{{{\hat{\mathbf{s}}}}}_{\mathrm{k}}},{{{{\hat{\mathbf{s}}}}}_{\mathrm{l}}}})\)) for symbols \(\hat{\mathbf{s}}\) can be expressed as:

$$\begin{aligned} {\mathrm{cum}}\left( {{{{{\hat{\mathbf{s}}}}}_{\mathrm{i}}},{{{{\hat{\mathbf{s}}}}}_{\mathrm{j}}},{{{{\hat{\mathbf{s}}}}}_{\mathrm{k}}},{{{{\hat{\mathbf{s}}}}}_{\mathrm{l}}}} \right) = \left\{ {\begin{array}{c} { \ne 0\;\;\;\;\;\;\;\;\;\;\forall \;i = j = k = l}\\ { = 0\;\;\;\;\;\;\;\;\;\;\;\;Otherwsie\;\;\;\;\;\;\;} \end{array}} \right\} {\mathrm{\;\;}}\forall {\mathrm{\;\;}}1 \le {\mathrm{i}},{\mathrm{j}},{\mathrm{k}},{\mathrm{l}} \le {K} \end{aligned}$$
(14)

Therefore, one is capable of expressing the diagonal structure of the functional expression in equation (11) dependent on the Joint Approximate Diagonal Eigen matrices technique as stated below:

$$\begin{aligned} {{\mathbf{Q}}_{\hat{\mathbf{S}}}}\left( {\mathbf{U}} \right) = \mathop \sum \limits _i diag\;{\left( {{\mathbf{U}}{{\mathbf{M}}_i}{{\mathbf{U}}^T}} \right) ^2} \end{aligned}$$
(15)

where \({{{\mathbf{M}}_i}}\) denotes the ith of the cumulant tensor (Eigen-matrices) and \({\mathbf{U}}\) represents a unitary matrix. On the other hand, the Eigenvalue decomposition is diagonalization. Thus, the objective of the equation is to take a \(K^2\) of substantial eigen-matrices \({{{\mathbf{M}}_i}}\) forming the \({{\mathbf{Q}}_{\hat{\mathbf{S}}}}\left( {\mathbf{U}} \right) \) matrix to be as diagonal matrix as possible. Therefore, the minimization of equation (15) is equivalent to minimize the sum of the off-diagonal entries as follows:

$$\begin{aligned} {J_{Cost}}\left( {\mathbf{U}} \right) = \mathop \sum \limits _{l = 1}^{{K^2}} Off\left( {{\mathbf{U}}{{\mathbf{M}}_i}{{\mathbf{U}}^T}} \right) \end{aligned}$$
(16)

To that end, one can estimate the unitary matrix \({\mathbf{U}}\) as follows:

$$\begin{aligned} {\mathbf{U}}:\left\{ {\begin{array}{l} {\mathop {\min }\limits _{\mathbf{U}} {J_{Cost}}\left( {\mathbf{U}} \right) = \mathop \sum \limits _{l = 1}^{{K}^2} Off\left( {{\mathbf{U}}{{\mathbf{M}}_i}{{\mathbf{U}}^T}} \right) \;\;}\\ {subject\;to\;{\mathbf{U}}{U^T} = {I_L}} \end{array}} \right\} \end{aligned}$$
(17)

In this work, we minimize the cost function in (17) using the Jacobi Optimization as in [21, 22] to provide fast and accurate channel estimation. It should be noted that criterion in (17) has already appeared in literature for ICA estimation problems [21,22,23,24,25,26]. In our work, an extension to Massive MIMO uplink systems is obtained by applying criterion in (17) on the Zero-Forcing equalized transmitted signals s in (10).

4 Algorithm Implementation

For the implementation of the algorithm, the section reports the minimization of the cost-effective functionality provided by the unitary constraint in (20). To lower the computational complexity of the proposed method, we used the Jacobi Optimization method [14, 15, 27] to minimize the criterion in (20). Owning the Optimization of Jacobi, where the unitary matrices are expressed as a function of the Rotational matrix stated below:

$$\begin{aligned} {\mathbf{U}} = {{\mathbf{R}}_1}{{\mathbf{R}}_2} \ldots {{\mathbf{R}}_{\mathrm{q}}} \end{aligned}$$
(18)

where \({{\mathbf{R}}_i}\) is the matrix for rotation corresponds with the i-th sweep, also, \({{\mathbf{R}}_i}\) shows an n x n identity matrices excluding the entries below,

$$R_{{ij}} = \left[ {\begin{array}{*{20}c} {r_{{ii}} } & {r_{{ij}} } \\ {r_{{ji}} } & {r_{{jj}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } \\ {\sin \theta } & {\cos \theta } \\ \end{array} } \right]\quad \forall \;1 \le i \le n - 1;\;\left( {i + 1} \right) \le j \le n$$
(19)

while \(\theta \) denotes angle of rotational.

Though, following stages represent the steps to be followed in the proposed algorithm, dependent upon the fourth order cumulant matrix:

  • Pre-processing

    • Zero-mean of received signal \({\mathbf{x}}\).

    • Compute the covariance Matrix \({\mathbf{R}}\)

    • Compute the whiten vector \({\mathbf{z}}\)

  • Replication

    • Compute Cumulant Matrices \({{\mathbf{Q}}_{\hat{S}}}\left( {\mathbf{U}} \right) \) of equation (16)

    • Call Joint Diagonlization \({\mathbf{U}}\) in (17)

  • Progression till convergence.

  • End

Once the rotational matrix \({\mathbf{U}}\) has been estimated. The least square estimation of the mixing matrix \({\mathbf{H}}\) is given by

$$\begin{aligned} {{\mathbf{H}}_{{\mathrm{LS}}}} = {\mathbf{x}}{{{{\hat{\mathbf{s}}}}{\mathrm{\;}}}^{\mathrm{T}}}{\left( {{{{\hat{\mathbf{s}}}}{\mathrm{\;}}}{{{{\hat{\mathbf{s}}}}{\mathrm{\;}}}^{\mathrm{T}}}} \right) ^\dag } \end{aligned}$$
(20)

where \({{\hat{\mathbf{s}}}}= {\mathbf{Uz}}\). However, one can express the estimated mixing matrix \({{\mathbf{H}}_{{\mathrm{LS}}}}\) in terms of the ideal mixing matrix \({\mathbf{H}}\) as follows:

$$\begin{aligned} {{\mathbf{H}}_{LS}} = {\mathbf{H}}{{\mathbf{D}}^{ - 1}}{{\mathbf{\Gamma }}^{ - 1}} \end{aligned}$$
(21)

where \({\mathbf{D}}\) is an unknown diagonal matrix and \({\mathbf{\Gamma }}\) is an unknown permutation matrix. Therefore, we have to estimate the \({\mathbf{D}}\) and \({\mathbf{\Gamma }}\) matrices to solve the scaling and phase permutation ambiguities. However, since the binary phase-shift keying (BPSK) modulation is employed in this paper, the symbol of the signals can be easily recovered by taking the signum function of estimated symbol i.e. \({\mathbf{D}} = {\mathbf{sgn}}\left( {{{\hat{\mathbf{s}}}}} \right) \). The permutation ambiguity \({\mathbf{\Gamma }}\) can be resolved by using a reference bit. In other words, any sign ambiguity in the recovered symbols is fixed using the reference bit in each users dedicated data channel.

5 Simulation Results

In this section, simulations results are reported to show the performance of the proposed blind decoding approach compared to ZF decoding, MMSE decoding with imperfect channel state information (CSI) and the blind decoding method given in [7]. The MMSE-based channel estimation [8] is used in the MMSE decoding and ZF decoding. BPSK modulation is used, and the channel is assumed of block fading.

Monte Carlo Simulations have been run to verify the validity of the algorithms. Figure 1 presents the simulation results of BER vs. SNR of the presented detectors. The parameters are set as: Number of samples for each block N=400, the Rician fading factor F=0.45, the large-scale date parameter is considered as

$$\begin{aligned} {{\mathrm{\beta }}_{\mathrm{l}}} = \left\{ {\begin{array}{c} {1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall \;l = 1\;\;\;\;\;\;\;\;}\\ {0.6\;\;\;\;\;\;\;\;\;\;\;\;\forall \;l = 2, \ldots ,\;5\;\;} \end{array}} \right. \end{aligned}$$

Number of cells K=5, with various values of SNR from -20 dB to 5dB. Figure 1 demonstrates that the proposed decoding algorithm, based on the fourth order cumulant matrices, outperforms the MMSE decoding and ZF decoding methods with imperfect channel estimation at pilot sequence length equals 40. it also shows that the BER performance of the proposed method outperforms the decoding method given in [7]. Figure 2 shows the simulation results of BER vs. SNR with various numbers of block samples (N). It is obvious that the performance of the proposed blind decoding algorithm improves further as N increases.

Fig. 1
figure 1

BER performance as a function of SNR with various decoding methods

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Fig. 2
figure 2

BER performance as a function of SNR for different sample lengths N

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Fig. 3
figure 3

BER performance as a function of SNR for various number of antennas M

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Moreover, Fig. 3 shows the simulation results of BER vs. SNR with numbers of block samples, \(\hbox {M}= 800\), various number of BS antennas M. It is obvious that the proposed approach performs more consistently and exhibits improvement in the performance as M increases.

Figure 4 presents the BER performance of the various decoding methods. It again shows that the proposed blind decoding performs well and outperforms the MMSE decoding with imperfect channel estimation. It is also observed that the performance of the ZF-decoding and MMSE-Decoding is noticeably degrading with imperfect channel estimation. On the contrary, the blind proposed decoding obtains good BER performance without a pilot sequence. It is also worthwhile to study the performance of the presented algorithms with different reference bits. Thus, Fig. 5 shows the performance of the proposed blind decoding with different numbers reference bits. Obviously, the performance advances when more bits can be used.

Fig. 4
figure 4

BER performance as a function of SNR with various pilot sequence length

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Fig. 5
figure 5

BER performance as a function of SNR with various reference bits

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6 Conclusion

In this paper, we have presented a blind adaptive decoding algorithm for a multicell massive MIMO uplink system. The presented blind decoding method alleviates the pilot contamination problem caused by pilot sequence reuses in neighbouring cells. The results show the suggested algorithmic exhibitions as improved performance comparative to the other methods of decoding systems with imperfect channel estimation. Our simulation also shows that the proposed blind decoding algorithm method doesnt require pilot sequences to avoid the pilot contamination in massive MIMO uplink system, thus improving the performance of conventional decoding. The simulation results show that the presented blind decoding algorithm outperforms MMSE decoding and ZF decoding with imperfect CSI. Furthermore, the performance of the proposed blind decoding shows advance as N (the number of block samples) increases.