1 Introduction

Since from last 5 years, there is continuously growing requirements for higher data rates on constrained resources and available bandwidth. This demand of higher data rate is resulting into increased interest of researchers to initiate the working towards 5G (fifth generation) wireless communications [1]. For such future wireless communications, multiple input multiple output (MIMO) is technique which can utilize the resources efficiently as compared to other techniques, hence massive MIMO is key technology for future communication systems like 5G. In this paper, our concentration is designing channel estimation technique for MIMO-OFDM system. The MIMO was integrated with communication or transmission systems such as orthogonal frequency division multiplexing (OFDM) as well as code division multiple access (CDMA). MIMO-OFDM transmission methods are widely studied since from last one decade [2]. In MIMO, basically the transmitter antennas are employed in order to gain the higher data rates using spatial multiplexing and optimize the link reliability using either of three coding standards such as (1) space–time, (2) space–frequency and (3) space–time–frequency. The basic characteristic of all three coding standards is assumption of accurate channel information at the side of receiver. In case of practice, when the channel information is not available, design of receiver is basically depends on the suboptimum equalization differentiation solutions in order to track and acquire the data at receiver using training sequence. But the training sequence is leads to be overhead limitation which may be prohibitive [1].

With view of 5G communications, the MIMO-OFDM transmission systems are now considered as strong contender for designing the future wireless communication systems. This is because of distinct benefits of both OFDM and MIMO. The channel evaluation approach of MIMO-OFDM are divided in the three important types thus as semi blind, blind channel and training based evaluation methods. In first category, prepare known training samples in manner to execute perfect channel evaluations. The least square (LS) and MMSE are the examples of training based channel evaluations methods. The second category is based on combined properties of training based and blind based channel estimation methods and used with MIMO-OFDM communication systems. Third channel estimation approach is called as blind method in which second order stationary statistics (SOS) or higher order statistical (HOS) are used for delivering higher spectral efficiency. In wireless communication systems, the wireless channel frequently designed as sparse channel with the higher delay spread [8], however number of significant non zero paths basically very small. Depending on assumption of sparsity of equivalent discrete-time channel in which only some taps in line of longer tapped delay are significantly considered. In CDMA and OFDM systems, sparse structure of wireless channel is widely used in manner to improve the channel evaluations performance [8]. There are different sparse channel based estimation techniques which are utilizing the training sequence and command with two main steps such as (1) position detection of most significant taps (MSTs), also called as non-zero taps, (2) effective channels estimation fetching by using the MSTs position. There is increasing research interest in designing blind channel estimation techniques. There are number of recent blind channel estimation techniques claims that increasing researchers interest [9]. There are number reasons due which there is increasing researches on blind channel estimation approaches. In OFDM systems, basically symbols are transmitted in the form of blocks, therefore approach of iterative channel estimation and block based is enabled in 4th generation wireless systems and same will be applicable in 5G systems. Therefore, first obtain the initial symbol estimates by using the blind channel estimation method and then utilized the starting symbol evaluations to increases the larger loyalty channel evaluations.

This methods iteratively repeated with the soft information exchange in order improves the both data symbol estimations as well as channel estimates [8]. This leads in rapid increase in low mobility based applications and hence motivating to design blind channel estimation which basically needs the large number of samples for good performance based on quasi static channel conditions.

The recent studies presented the SOS and HOS based blind channel estimation methods, but none of such method is capable in interference signals cancellation for MIMO-OFDM [11]. Interference signals are caused by either other mobile users or fading channel in MIMO-OFDM wireless systems in blind manner. The HOS depends on ICA method is currently schemes for interference cancellation in MIMO systems, but not clearly designed and addressed for blind channel estimation. Additionally, inter carrier interference (ISI) also having major impact of performance of blind channel estimation and spectral efficiency, which is not yet addressed. Therefore, our first objective is to design novel spectral efficient channel estimation. As stated in abstract, in our previous research we designed hybrid ICA (HICA) approach by using pulse shaping for efficient blind channel estimation in manner to increase the efficiency and less error rates for MIMO-OFDM. Although, the large numbers of restriction by using OFDM system with MIMO is that higher PAPR and hence energy consumption performance, this is also called energy inefficient approach. Therefore, HICA method still requires to study with respect to energy efficiency in terms of PAPR and resource computations to achieve the tradeoffs between spectral and energy efficiency. We re-presented the architecture and algorithm of HICA and then present extensive simulation study for spectral and energy efficiency. The research gaps of HICA have been discussed in this paper. Section 2.1, presents the review of some PAPR analysis studies and channel estimation methods. Section 3, shown the system model for PAPR and MIMO-OFDM equations. In Sect. 3.1, represent the description on HICA method. Section 4, presents the simulation parameters and results, also presents the research problems for HICA, finally Sect. 5 presents the conclusion and future work.

2 Related Work

In this section, we are describing the related method presented previous for PAPR analysis of MIMO-OFDM and some channel estimation techniques.

2.1 PAPR Analysis

2.1.1 Leila Sahraoui et al. (2013)

In [2], they presented the study on PAPR analysis reduction of space–time block-coded (STBC) MIMO-OFDM system for 4G wireless networks. There were multiple numbers of approaches have been utilized to decreases the (STBC) MIMOOFDM system such as SLM, clipping and filtering and partial transmit sequence. Their simulation outcomes claim that clipping and filtering delivered efficient PAPR reduction than the others methods and only SLM technique conserves the PAPR reduction in reception part of signal.

2.1.2 P. Sunil Kumar et al. (2013)

In [3], author presented the efficiency differentiates for reduction of PAPR in space–time block coding. MIMO-OFDM approach defined using with SLM, CF, PTS and the Tone Reservation (TR) design. Their simulation results claims that CF technique is more efficient as compared to other techniques.

2.1.3 Ying-Che Hung et al. (2014)

In [4], author presented analysis of PAPR performance of MIMO OFDM systems that accepted one or two decent beam forming design equal gain transmission and maximum ratio transmission conceptually. Examines the changing numbers of channel taps after sampling. Finally they designed new PAPR reduction system for EGT OFDM and MRT OFDM systems.

2.1.4 S. Sujitha et al. (2014)

In [5], author introduced a solution to tackle high PAPR problem in MIMO-OFDM approach by the using MCMA. They simulated this approach using MATLAB libraries. The getting simulations output declared that the PAPR reduction efficiency of MCMA is well than of Scrambling Schemes.

2.1.5 Sadhana Singha et al. (2016)

In [6], adaptive clipping technique was proposed to minimize the PAPR of MIMO-OFDM systems. The successive peaks were clipped according to an adaptive algorithm. From the simulation outcomes their proposed clipping technique provides effective reduction in PAPR and good BER efficiency as compared to other methods. Additionally they also compared the BER performance and PAPR reduction capability with SISO-OFDM systems. Power spectral density (PSD) for conventionally clipped, adaptively clipped, and normal SISO-OFDM systems was analyzed and plotted.

2.1.6 Gourav Misra et al. (2016)

In [7], another PAPR analysis study presented recently. They presented examines of various approach to decrease the large PAPR, thus are SLM, PTS, SFBC, MIMOOFDM and SFBC MIMO-OFDM with integrated PTS.

2.2 Channel Estimation

In [8], author designed blind channel evaluations methods by using repeated index.

They proposed subspace blind channel evaluations approach depends on repeated index with same output as differentiates to lastly methods with several numbers of symbols.

Author does not evaluate the computational complexity for MIMO-OFDM.

In [9], proposed the blind channel estimation method in MIMO-OFDM systems with the orthogonal space–time block code. They designed the new weighted covariance matrix of the information accepted in manner to achieve the repetition in code. They proposed this approach with aim of resolving all the non-scalar ambiguities.

In [10], another SOS based blind channel estimation method proposed. They proposed the algorithm of blind repeated for the purse emerging time changing wireless channel in MIMO-OFDM systems which are pre-coded. With this approach subspace based tracking was designed for fast time varying wireless channels. Their approach called the data from the time as well as frequency domain as the frequency correlation of the wireless channels in order to faster the required SOSs updates.

In [11], author proposed the recent blind channel evaluations method depends on subspace and SOS models for MIMO-OFDM. Their approach exploited the null space introduced by the OSTBC. This approach worked with only single receiver antenna as well. Additionally they proposed the modified proposed approach with goal of requirement of less received blocks.

In [12], the recent novel blind channel estimation technique designed for long term evaluation (LTE) wireless networks based on advantages of wavelet transform denoising characteristics with ICA capability of blind estimation in LTE networks. They called this approach as WD-ICA. The denoising approach was designed in order to handle the blind interference cancellation. This was the first attempted approach for blind channel estimation using ICA. The ISI cancellation is not performed in this method.

3 System Model

The system model of proposed HICA is elaborated and redefined in Figs. 1 and 2 below. Figure 1 indicating the MIMO-OFDM transmitter’s block and Fig. 2 is representing the MIMO-OFDM receiver’s block. Finally Fig. 3 shows the complete MIMO-OFDM transmitter and receiver system using HICA blind channel estimation technique.

Fig. 1
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MIMO-OFDM transmitter

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Fig. 2
figure 2

MIMO-OFDM receiver

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Fig. 3
figure 3

Proposed MIMO-OFDM system model

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In proposed system model, at transmitter data X from users is randomly generated and then forwarded to modulation process on each symbol on which pulse shaping algorithm is applied to inter symbol interference cancellation. After pulse shaping, first symbol mapping, IFFT and CP operations are performed at each transmitter. Once symbols are ready transmit over the AWGN channel through transmitter’s antennas towards receiver’s antennas, the measurement of PAPR is done on each symbol. Here we are not using any PAPR reduction technique. At the receiver side, we then first applied proposed HICA method for efficient channel estimation and signal detection. Figure 2 showing the complete functionality of HICA block. After estimation, the reverse operations applied at receiver side to get the original data at receiver side. The modulation and demodulation is performed by either of techniques such as QAM, QPSK, and BPSK etc. Let’s consider that there are M transmitted signals with every signal consisting of samples S. the attenuation for nth channel path is represented by An. Attenuation factor is complex number. Therefore we introduced the pulse shaping here to remove the ISI effects before performing the modulation in order to minimize the error rates and optimize the spectral efficiency. At end of receiver and during the transmission process designed the proposed HICA block based blind interference cancellation approach. Based on above system models, two algorithms are discussed in upcoming section. After algorithms, we present the HICA block mathematical representation and PAPR measurement equations.

Algorithm 1: Transmitter block

Input: Data X

Output: Transmitted signals

Step 1: Random data generation X

Step 2: Representation of X in different symbols as per the number of transmitter and receiver antennas X1…Xn

Step 3: Apply the Pulse shaping on input symbols of each user in order inter symbol cancellations.

Step 4: Apply the modulation on X1…Xn.

Step 5: IFFT on X1…Xn

Step 6: Cyclic Prefix on X1…Xn

Step 7: Measure PAPR using Eq. (13) of X1…Xn

Step 8: Transmit data X1…Xn through AWGN channel

Once data is sent from transmitter antennas via AWGN channel, next we applied our second algorithm of HICA to estimate signals. This functionality is elaborated in Algorithm 2.

Algorithm 2: HICA

Input: Transmitter signals X1…Xn

Output: Estimated signals X′1…X′n

Step 1: Initialization HICA iterations in IT and set it = 0;

Step 2: Random W initialization

Step 3: Defining the objective function Jold ← J (W).

Step 4: Gradient computation of objective function using Eq. (4).

Step 5: W updating according to negative gradient direction, W ← W − µJw.

Step 6: W normalization according to unitary constraint, W ← W/||W||.

Step 7: If Jold − J (W) < ε (where ε is a very small threshold Value), then go back to step 4

Step 8: Set of signals estimation ŝ [n] = WH x Yr (where, Yr is the set of received signal)

Step 9: Form every estimated signal ŝ [n] as vector which represented by Vit[n]

Step 10: If it < 1 goes to step 2 else, continue.

Step 11: Find the set of the common vectors for all runs of algorithm up to itth run.

Step 12: If there is no common vectors does not appear for all the HICA executions, then go to step 2, else iteration is terminated.

Step 13: Apply Eq. (5) to prioritize the common estimated signals with J (ŝit).

Step 14: Selection of desired signals (m) with largest J (ŝit) in order perform the blind interference cancellation.

Step 15: Ambiguity Elimination using Eq. 9.

Step 16: Return estimated signals X′1…X′n

Algorithm 3: Receiver

Input: Estimated signals X′1…X′n

Output: Data X′

Step 1: FFT on estimated signals X′1…X′n

Step 2: CP Removal X′1…X′n

Step 3: Demodulation X′1…X′n

Step 4: Pulse Reshaping X′1…X′n

Step 5: Return X′

The equations used in HICA algorithm are discussed below.

3.1 HICA Blind Estimation

The idea of the proposed method is to first apply the pulse shaping method in each user’s symbols in manner to reduce the internal symbol interference. Pulse shape is light weight ISI cancellation approach. We used the square root raised cosine filter in up sampling domain to realize the pulse shaping filter. In this section we are just presenting the core equations those are added for efficient blind channel estimation. In first step, let’s consider gt(t) transmit side pulse shape filter for each user symbols and gr(t) is receive antenna side matched filter. The composite channel is represented as T * R with matrix H(t). The (iR, iT) channel using pulse phase filtering is represented by:

$$h_{{i R^{\prime}}} i_{T} \left( {\text{t}} \right) = h_{{i R^{\prime}}} i_{{T^{\prime}c}} \left( t \right)*g_{t} \left( t \right)*g_{r} \left( t \right)$$
(1)

where \(h_{i R'} i_{{T^{\prime}c}} \left( t \right)\) is the (iR, iT) element of H(t).

Here the channel can be represented as the L tap FIR filters array for blind channel estimation.

In second step, for blind channel estimation is which is done by ICA block. In proposed HICA, first signals sources from receiver observation mixture are identified by using determining separation matrix represented as (W). In third step, at the every iteration of HICA, the estimated signals order is different due to random initialization of HICA method. In spite of that, if there is significant information in estimated signal, then it will appear at every repetition always. The estimated common signals further prioritize depends on their Higher Order Statistics in order to select the estimated desire signals (m) using proposed HICA method by leaving the interference related components.

In second step, W is performed using the maximizing the non-gaussianity of the observation signals principle. The non-gaussianity for random variable (v) containing the complex data is measured by the Kurtosis K[s] as:

$$K \left[ v \right] = E[|v|^{4} ] - 2(E[|v|^{2} ])^{2} - E\left[ {vv} \right]E\left[ {v^{*} v^{*} } \right]$$
(2)

where (.)* is represents the complex conjugate.

The estimation of W is performed by the minimization of J (W) objective function within the unitary constraint (WWH = IR) due to negative results of kurtosis value on different modulation schemes. This objective function is based on estimated signals ŝi [n] kurtosis values represented as:

$$W = \left\{ {\min_{W} } \right.J\left( W \right) = \mathop \sum \limits_{j = 1}^{M} K[\hat{s}[n]]$$
(3)

The objective function J (W) minimization is done by gradient computation of objective function as:

$$Jw = \frac{\partial J\left( W \right)}{\partial W} = K\left( {W^{H} \hat{s}\left[ n \right]} \right)\left[ {E\left\{ {\hat{s}\left[ n \right])^{3} } \right\}} \right]$$
(4)

where (w) is represents the one vector from the separation matrix W.

As the objective function (Jw) optimization is in constraint of WWH = IR, object function gradient must be complemented using projecting W over the interval after each step performed by dividing the W by its norm.

In third step, the execution of HICA method number of times with various random initialization of W at each time in order to estimate the common signals for HICA executions. The two \(\hat{s}_{i} [n]\) and \(\hat{s}_{i}\) estimated signals for different executions are assumed different if the spectral angle Mapper (SAM) among their related vectors VI[n] and VJ[n] is more than the estimated threshold value ε. After that, common estimated signals are prioritized based on their 3rd and 4th HOSs as:

$$J\left( {\hat{s}_{q} } \right) = \left( {\frac{1}{12}} \right)\left[ {Q_{q}^{3} } \right]^{2} + \left( {\frac{1}{48}} \right)\left[ {Q_{q}^{4} - 3} \right]^{2}$$
(5)

where \(Q_{q}^{3} = E\left\{ {\hat{s}_{q}^{3} } \right\} = \left( {\frac{1}{T}} \right)\sum\nolimits_{n = 1}^{T} {(\hat{s}_{q} [n])^{3} }\) is 3rd order of statistics and \(Q_{q}^{4} = E\left\{ {\hat{s}_{q}^{4} } \right\} = \left( {\frac{1}{T}} \right)\sum\nolimits_{n = 1}^{T} {(\hat{s}_{q} [n])^{4} }\). Is 4th order of statistics of estimated signals?

And q is nothing but execution index.

Ambiguity elimination The estimation of estimated users signals still to the permutation as well as phase rotation ambiguities due to ICA algorithm ambiguity issues. The ŝi[n] is nothing similar to the original transmitted signal s[n], and there is presence of ambiguity matrix a comparing with the s[n]. This can be represented as:

$$\check{S}i[n] = AX\hat{s}_{i} [n]$$
(6)

Two indeterminacies forming the A as:

$$A = {\text{P}} \times {\text{D}}$$
(7)

where P is nothing but the permutation ambiguity matrix and (D) is phase rotation ambiguity matrix.

In proposed blind channel estimation method, ambiguities elimination is performed by multiplying the all estimated signals by the ambiguities elimination step of the proposed method is represent by post-multiplying the estimated signals \(L_{F}\) which is represented as:

$$L_{F} = \arg { \hbox{min} }_{{L{\mathcal{E}}G}} \parallel s\left[ n \right] - \hat{s}\left[ n \right] x {\text{A}}\parallel^{2}$$
(8)

The final estimated signal is:

$${\hat{\text{S}}\text{iF}}[{\text{n}}] = {\text{L}}_{\text{F}} \times {\hat{\text{s}}}[{\text{n}}]$$
(9)

3.2 PAPR Measurement

The PAPR of the signal, x(t), is then provided the peak prompt power to the average power, following are shown the formula,

$${\text{PAPR}} = \frac{{max\,0 \le t \le T|x(t)|^{2} }}{{E[|x(t)|^{2} ]}}$$
(10)

where E[.] is the anticipated operator. From the central limit formulas, for most values of N, the X (t) is a real and imaginary values.

The amplitude of the OFDM signal, therefore, has a Rayleigh distribution with zero mean and a variance of N times the variance of one complex sinusoid. In practice, it calculates the probability of PAPR surpass as determent index to shown the PAPR distribution. These can be discussing as complementary cumulative distribution function (CCDF). The CCDF formulas shown in below,

$$CCDF\left( {PAPR\left( {x\left( n \right)} \right)} \right) = p_{{r\left( {PAPR\left( {x\left( n \right)} \right)} \right) > PAPR_{0} }}$$
(11)

Due to the independence of the N samples, the CCDF of the PAPR of a data block with Nyquist rate sampling is given by

$$P = P_{{r\left( {PAPR\left( {x\left( n \right)} \right)} \right) > PAPR_{0} = 1 - \left( {e^{{ - PAPR_{0} }} } \right)}} {\text{N}}$$
(12)

This expression considers that, the domain signals samples are N times are commonly free and un-integrated and does not perfect for low numbers of subcarriers. Hence, there have been multiple efforts to acquire the correct distribution of PAPR.

There have been many attempts to find the exact probability distribution of PAPR for analysis MIMO-OFDM system PAPR efficiency is the similar as the SISO single antenna. The overall system, the PAPR is states as the higher along entire PAPR transmit antennas:

$$PAPR_{MIMO - OFDM} = max_{{i \le i \le n_{t} }} PAPR_{i}$$
(13)

where PAPR i denotes the PAPR of transmit antenna. Particularly, hence in MIMO-OFDM, the Mt N numbers of time domain samples are examined differentiates to N in the SISO-OFDM approach, the CCDF of PAPR in MIMO-OFDM is shown in below:

$$P_{r} (PAPR_{MIMO - OFDM} ) > PAPR_{0} = (1 - 1 - e^{{ - PAPR_{0} }} )M_{{T^{N} }}$$
(14)

4 Simulation Results and Discussion

The simulation is performed for this study under different network conditions such as varying number of sender antennas and accepted antennas. The following table display the simulation arguments used to evaluate the proposed method (Table 1).

Table 1 MIMO-OFDM simulation parameters
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Figures 4 and 5 are showing the performance results of ICA and proposed HICA methods for BER and PAPR analysis for 2 × 2 MIMO-OFDM respectively. Similarly Figs. 6 and 7 for 4 × 4 MIMO-OFDM, Figs. 8 and 9 for 6 × 6 MIMO-OFDM respectively.

Fig. 4
figure 4

QPSK-BER analysis for 2 × 2 MIMO OFDM

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Fig. 5
figure 5

QPSK-PAPR analysis for 2 × 2 MIMO OFDM

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Fig. 6
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QPSK-BER analysis for 4 × 4 MIMO OFDM

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Fig. 7
figure 7

QPSK-PAPR analysis for 4 × 4 MIMO OFDM

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Fig. 8
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QPSK-BER analysis for 6 × 6 MIMO OFDM

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Fig. 9
figure 9

QPSK-PAPR analysis for 6 × 6 MIMO OFDM

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Tables are showing the average performances for MSE, BER and PAPR simulation results (Tables 2, 3, 4).

Table 2 Average BER using QPSK
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Table 3 Average MSE using QPSK
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Table 4 Average PAPR using QPSK
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The figures and tabular results showing for QPSK modulation, the increasing number of antennas is having major impact on the performance of BER and MSE, as the number of antennas increases, the BER and MSE performances are decreases. Whereas the PAPR performance does not have any significance based on transmitter and receiver antennas numbers. Rather the PAPR is decreasing as the number of antennas increases. The BER and MSE is efficient using HICA for all types of MIMO networks, whereas PAPR does showing any reduction for HICA as compared to ICA, in fact PAPR is nearly similar in both methods. Similarly we simulated the QAM 16 modulation technique using ICA and proposed HICA channel estimation methods. The Tables 5, 6 and 7 are showing the average performances for BER, MSE and PAPR respectively. The similar significance is observed from the QAM modulation based results. In QAM and QPSK, the BER performance of QPSK is better, MSE performance of QAM is better and PAPR performance of QAM is better. Therefore QAM is superior than using QPSK modulation.

Table 5 Average BER using QAM
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Table 6 Average MSE using QAM
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Table 7 Average PAPR using QAM
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5 Conclusion and Future Work

This paper showed the hybrid blind channel estimation technique and its evaluation in terms of BER, MSE and PAPR results. HICA method is showing the BER and MSE efficiency as compared to ICA technique with different antennas with both QPSK and QAM modulation methods. A modulation method does not have any impact on HICA and ICA methods performances. However the PAPR performance is not reduced in both ICA and HICA techniques. This means that there is excessive energy consumption while transmission of data in MIMO-OFDM. The simulation results showing that HICA optimized the channel estimation efficiency however failed to achieve the trade off between BER and PAPR rates and hence will be the appealing concept for future work.