Introduction

The presence of correlation among antenna branches in multi-antenna systems negatively affects signal reception, especially in modern compact wireless hand-held devices where antenna spacing is limited. Existing literature assumes independent antenna branches, but in reality, closely spaced antennas are often correlated [1,2,3,4,5,6,7,8]. Hence, it is crucial to tackle the consequences of unwanted correlations between antenna branches. A preferred approach would involve incorporating a decorrelator at the receiver [6,7,8,9],.

There are various methods available for decorrelation, and one such technique is eigenvalue decomposition (EVD) [2, 10, 11]. EVD can be employed to decompose a covariance matrix of correlated received signals or channels. When EVD is applied to the covariance matrix, it transforms into a set of mutually orthogonal eigenvectors with dimensions devoid of correlation. Another method for decorrelation is the principal component analysis (PCA). PCA reduces data dimensionality and is grounded in the EVD [12] and [13]. It uses the principal components/eigenvectors from the covariance matrix to obtain decorrelated representations while retaining the variance of the original data.

Due to various limitations such as hardware constraints, space restrictions, and cost factors, it may not always be practical to deploy multiple physical antennas. However, virtual diversity decorrelators, also referred to as virtual antenna arrays or blind decorrelators, provide an alternative solution by simulating the advantages of multiple physical antennas using just one or a few physical antennas [5, 12, 14,15,16]. These methods remain an active and evolving focus of research and development in the wireless communications field. For instance, previous studies have explored decorrelation receivers, with the Karhunen-Loeve Transformation (KLT) to enhance performance through generating independent virtual antennas [2, 17, 18]. However, the KLT approach requires perfect knowledge of the covariance matrix, which relies on accurate channel state information (CSI) that is often unavailable in practical scenarios.

Furthermore, discrete Fourier transform (DFT) and fast Fourier transform (FFT) techniques have been employed to implement blind decorrelators in wireless communications [19] and [20]. However, a challenge persists in mitigating signal errors caused by correlation. In addition, DFT and FFT may necessitate specific designs, such as uniform circular arrays (UCAs) with uniform circular receive antenna spacing. Despite utilizing the current invention, these approaches yield limited decorrelation gains.

A dual diversity decorrelator that achieves decorrelation by adding and subtracting the received signals from correlated branches, proving effective in dual Selection Combining (SC) and Switch and Stay Combining (SSC) branches has been used in [21]. However, it does not improve equal gain combining (EGC) and square law combining (SLC) techniques [18].

Newly developed techniques have emerged that bypass the need for calculating covariance matrices by updating estimates for individual input training vectors [5]. However, these methods exhibit certain drawbacks in terms of convergence and stability. They may also not be essential for achieving coding gains. This research seeks to illustrate that coding gains can be achieved using a transformation matrix that synthesizes excess virtual signals from received correlated signals under arbitrarily correlated Nakagami-m fading channels with EGC. The essential statistical equations that are simple and converge rapidly, such as the SER and the coding gains, are derived. Analytical and simulation approaches support the obtained results. Furthermore, the effectiveness of the investigated decorrelator is confirmed to approach maximum-ratio combining (MRC), showcasing its superior performance compared to existing approaches.

The motivation for conducting this study stems from [18] and [21], which collectively indicate that no discernible benefits have been observed regarding the de-correlation of signals through EGC combining. [18] and [21] also limit themselves to dual correlated antenna branches. In contrast, this study will provide empirical evidence to demonstrate that the practice of de-correlating \(N_r\) antenna branches through EGC combining does, in fact, yield tangible benefits. Subsequently, the following paragraphs outline the accomplishments in this article:

  1. 1.

    Firstly, we derive the coding gains ((21)) for the EGC combining scheme in arbitrarily correlated M-PSK cojoint signals under Nakagami-m fading channels. To do so, the signal-to-noise ratio (SNR) at the output of the combiner and the SNR at the output of the decorrelator are analysed and used to evaluate the gain of the decorrelator system.

  2. 2.

    Secondly, we show that the coding gains for the EGC combiner in arbitrarily correlated fading channels that follows a Nakagami-m distribution depends on the fading parameter, m.

  3. 3.

    Thirdly, we show that the proposed decorrelator achieves decorrelation. In addition, it is blind, thus simple because it does not estimate the correlation matrix, but depends on the coding gains inherent in the conjoint signals, which are synthesized from the received signals.

  4. 4.

    Finally, we demonstrate that the SER performance of EGC equals the performance of MRC after decorrelation.

The following text describes the structure of this document. The first part serves as an introduction. The second part discusses the modeling of the system. Section three elaborates on signal combining and the resulting SNR. The fourth section examines the performance of the system. A summary of the findings is presented in the fifth section, and the work concludes in the sixth section.

In this study, the symbols used can be explained as follows: Matrices are denoted by bold uppercase letters, scalar quantities are represented by regular letters, and bold lowercase letters represent vectors. The Q-function with a Gaussian distribution is denoted as Q(.). Furthermore, the operators E[.], \((.)^*\), and \((.)^H\) represent the expectation, complex conjugate, and Hermitian operators, respectively. The gamma function is denoted as \(\Gamma (.)\), and \(J_0\left( \cdot \right)\) represents the Bessel function of the first kind with zero order.

Conjoint diversity system modeling

Synthesizer operation and signal combining

A Single-Input Multiple-Output (SIMO) diversity system with \(N_r\) receive branches is examined. To process the received signals, a diversity decorrelator is employed, which generates multiple copies of the received signals. These newly generated signals are then fed into a diversity combiner to obtain an approximate version of the transmitted symbol. In the case of \(N_r\) diversity branches, \(N_r\) signals from the receive antennas produce \(N_{r}(N_{r}-1)\) additional decorrelated signals. This results in a total of \(N^{2}_{r}\) input signals to the diversity combiner. For instance, if we have 4 receive antennas as shown in Fig. 1, where \(N_r=4\), there will be 12 additional decorrelated outputs. To generate these decorrelated signals, a combination of sum, difference, and conjugate operations is performed on selected pairs of signals. This process introduces coding gains rather than diversity gains because the variances of the sum signal \(({{\widehat{s}}}_{i}+{{\widehat{s}}}_{k})\), difference signal \((j{{\widehat{s}}}_{k} - j{{\widehat{s}}}_{i})\), and branch signal \(({{\widehat{s}}}_{i})\) differ from those of the received branch signals. However, due to some correlation between the sum and difference signals, the resulting decorrelated outputs, known as conjoint signals, are not entirely mutually independent. Finally, the \(N^{2}_{r}\) outputs are combined using EGC diversity combiner.

Fig. 1
figure 1

Decorrelation synthesizer system with \(N_{r}= 4\), \(\theta =45^{0}\)

Full size image

Diversity synthesizer analysis

The suggested diversity decorrelator is presented as,

$$\begin{aligned} {\textbf{Q}}^{{\textbf{H}}} =\begin{pmatrix} cos \theta &{} cos \theta \\ - jsin \theta &{} jsin \theta \\ \end{pmatrix} \end{aligned}$$
(1)

where \(\theta =45^0\)

Consider \(z_k\) as the symbol that has been transmitted. In the case of a receive system with inputs \(\widehat{{\textbf{s}}}\), the de-correlator produces an output \(\mathbf {y =}{\textbf{Q}}^{{\textbf{H}}}\widehat{{\textbf{s}}}\). When we examine a two-branch system with two inputs, \({{\widehat{s}}\ }_{1}\ \text {and}\ {{\widehat{s}}\ }_{2}\), the signal at the decorrelator’s output can be expressed as,

$$\begin{aligned} \begin{pmatrix} y_{3}\\ y_{4}\\ \end{pmatrix} = \begin{pmatrix} cos \theta &{} cos \theta \\ - jsin \theta &{} jsin \theta \\ \end{pmatrix} \begin{pmatrix} {\widehat{{\textbf{s}}}\ }_{1}\\ {\widehat{{\textbf{s}}}\ }_{2}\\ \end{pmatrix} \end{aligned}$$
(2)

Correlated signal model

The signal received at the \(i_{th}\) branch, which is independent, is expressed as,

$$\begin{aligned} { {s}_i}={{ {\alpha }_i}}A_ke^{j{\psi }_i}e^{j{\phi }_k}{\mathbf +}{ n_i}, \ \ i=1,2, \cdots , N_r \end{aligned}$$
(3)

where the transmitted signal amplitude and phase angle are denoted by \(A_k\) and \({\phi }_k\), respectively. The dependence on the index k can be eliminated for symbol-by-symbol detection. \(n_i\) signifies the AWGN. The phase of the channel, denoted by \({{\psi }_i}\), follows a uniform distribution over the interval \([0,2\pi ]\) for M-PSK during a single signal interval. Additionally, \({ {\alpha }_i}\) refers to the Nagakami-m fading envelope, which has a probability distribution function for a statistically independent and identically distributed fading channel, as expressed in [22, 23], which is given as,

$$\begin{aligned} P_{\alpha }\left( \alpha \right) = 2\left( \frac{m}{\mathrm {\Omega }} \right) ^{m}\frac{\alpha ^{2m - 1}}{\Gamma \left( m \right) }\exp \left( - \frac{m\alpha ^{2}}{\mathrm {\Omega }} \right) ,\ \alpha \ge 0 \end{aligned}$$
(4)

The severity of fading, determined by the parameter m, varies from 0.5 to \(\infty\). \(\Omega =E( \alpha ^2)\) represents the average power or the mean squared value of \(\alpha\). The Gamma function, represented by the symbol \(\Gamma (m)\), is defined as [9]; \(\Gamma \left( m \right) = \int _{0}^{\infty }y^{m - 1}exp( - y)dy.\)

Additionally, the signal received from a correlated Nakagami-m channel, characterized by the correlation coefficient \(\rho\), is expressed as follows.

$$\begin{aligned} {{\widehat{s}}}_{i} = \left( I_{i} + \text {jD}_{i} \right) .e^{j\varphi _{k}} + n_{i},\ i = 1,2,\ \ldots ,\ N_{r} \end{aligned}$$
(5)

where \(I_{i}\) and \(D_{i}\) are random variables that conform to a Gaussian distribution, exhibiting a zero mean and a variance of \(\sigma ^{2}\) with \(E\left[I_{i}D_{k} \right]= 0\ \ \ \ \ \ i,k = 1,2,\ \ldots ,\ N_{r}\), \(C_{I_{i}I_{k}} = E\left[I_{i}I_{k} \right]= \rho \sigma ^{2}\) and \(C_{D_{i}D_{k}} = E\left[D_{i}D_{k} \right]= \rho \sigma ^{2}\).

Due to the independence of the noise components and lack of correlation with the signals, \(E\left[n_{i}n_{k} \right]= E\left[n_{i}I_{k} \right]= E\left[n_{i}D_{k} \right]= 0\).

The correlation coefficient \(\rho _{\text {kl}}\) can be used to express the spatial correlation between two received signals from any two branches [24] and [25] as,

$$\begin{aligned} \rho _{\text {kl}} = \frac{E\left[{{\widehat{h}}}_{k}{{{\widehat{h}}}^{*}}_{l} \right]}{\sqrt{\sigma _{{{\widehat{h}}}_{k}}^{2}\sigma _{{{\widehat{h}}}_{l}}^{2}}},\ l,k = 1,2,\ \ldots \ N_{r} \end{aligned}$$
(6)

The random variables \({{\widehat{h}}}_{k}\) and \({{\widehat{h}}}_{l}\), with variances of \(\sigma _{{{\widehat{h}}}_{k}}^{2}\) and \(\sigma _{{{\widehat{h}}}_{l}}^{2}\) respectively, are included. If \({\textbf{R}}\) is a correlation matrix that has been normalized, it can be expressed as [24] [26]:

$$\begin{aligned} {\textbf{R}}=\begin{pmatrix} 1 &{} \rho _{12}\ \ \ \ \cdots &{} \rho _{1N_{r}} \\ \rho _{21} \vdots &{} 1\ \ \ \ \ \ \ddots &{} \rho _{2N_{r}} \vdots \\ \rho _{N_{r}} &{} \ldots \ \ \ \ \ \ \cdots &{} 1 \\ \end{pmatrix} \end{aligned}$$
(7)

Based on the explanations provided in (6), it is clear that the correlation coefficients \({\rho }_{k,l}\) and \({\rho }_{l,k}\) are complex conjugates, meaning \({\rho }_{k,l}= {\rho }^*_{l,k}\). Consequently, the correlation matrix, when normalized, exhibits Hermitian properties, implying that it is equal to its conjugate transpose ( \({{\textbf{R}}}^H={{\textbf{R}}}\)).

The process of eigenvalue decomposition (EVD) can be employed to decompose \({\textbf{R}}\) into orthogonal eigenvectors and eigenvalues. This decomposition can be achieved through an eigen-filter operation, as explained in [2, 26,27,28,29,30] as,

$$\begin{aligned} \begin{aligned} {\varvec{\Lambda }}&={{\textbf{Q}}}{{\textbf{R}}}{{{\textbf{Q}}}}^H\\ {{\textbf{R}}}&={{{\textbf{Q}}}}^H{\varvec{\Lambda }}{{\textbf{Q}}} \end{aligned} \end{aligned}$$
(8)

The matrix \({{\textbf {Q}}}\) consists of orthonormal eigenvectors, and the matrix \(\varvec{\Lambda }\) contains positive eigenvalues. When the EVD is applied, \({{\textbf {Q}}}\) is obtained, and its rows represent orthonormal basis elements, making \({{\textbf {Q}}}\) a unitary transformation. Additionally, \({{\textbf {Q}}}^H={{\textbf {Q}}}^{-1}\). \({{\textbf {Q}}}\) creates a reciprocal network, and therefore, it is the designed eigen-filter/diversity de-correlator that modifies the input in a unitary manner to produce an output. The functioning of this filter is thoroughly analyzed in reference [24, 27]. Referring to the output of the \(i_{th}\) branch, the equivalent channel coefficient is expressed as:

$$\begin{aligned} {{\widehat{h}}}_{i} = \sqrt{\epsilon _{i}}{{\widehat{\textbf{Q}}}}_{i}{{\widehat{\textbf{Q}}}}_{i} = \sqrt{\epsilon _{i}}{{\widehat{\textbf{Q}}}}_{i}\widehat{{\widehat{\textbf{h}}}} \end{aligned}$$
(9)

where \(\epsilon _{i} \in \varvec{\Lambda }\) and \({{\widehat{\textbf{Q}}}}_{i}\) refers to the \(i_{th}\) column of \({\widehat{\textbf{Q}}}\). \(\widehat{{\widehat{\textbf{h}}}}_{i}\) represents an approximation of the correlated Nakagami-m fading channel, which is characterized by a complex channel coefficient. The phase, on the other hand, is uniformly distributed random variable between \((0,2\pi )\) for M-PSK.

According to [19], the product \(({{\widehat{\textbf{Q}}}_{i}\widehat{{\widehat{h}}}})\) has entries that are independent of each other. As a result, the correlated paths can be treated as separate branches that have been scaled by eigenvalues. The matrix \({\textbf{Q}}\) is typically computed from matrix \({\textbf{R}}\) or given. Thus, \({\textbf{Q}}\) can be determined without measuring \({\textbf{R}}\), making \({\textbf{Q}}\) blind while \({\textbf{R}}\) does not need to be measured.

Combiner average output SNR

The expression for the probability density function (PDF) of the instantaneous branch SNR for independent signals in channels that undergo fading, as described by the Nakagami-m model, can be stated as

$$\begin{aligned} f_{{\overline{\gamma }}_{k}}\left( \gamma _{k} \right) = \left( \frac{m}{{\overline{\gamma }}_{k}} \right) ^{m}\frac{{\gamma _{k}}^{m - 1}}{\Gamma \left( m \right) }\exp \left( - \frac{m\gamma _{k}}{{\overline{\gamma }}_{k}} \right) ,\ \gamma _{k} \ge 0 \end{aligned}$$
(10)

where \({\gamma }_{k}{= \alpha }_{k}^{2}\frac{E_{s}}{N_{0k}}\)

The SNR, \({\overline{\gamma }}_{k,s}\), of the channel described in Eq. (9) can be depicted as;

$$\begin{aligned} {\overline{\gamma }}_{k,s} = \epsilon _{k}E\left\{ \gamma _{k} \right\} = \epsilon _{k}E\left[\alpha _{k}^{2}\frac{E_{s}}{N_{0k}} \right]= \epsilon _{k}{\overline{\gamma }}_{0,c} \end{aligned}$$
(11)

The average output SNR, denoted as \({\overline{\gamma }}_{0,c}\), is calculated to determine the SNR of the combiner’s output. By utilizing the diversity decorrelator, the power distribution of the branch signals is adjusted. Consequently, the output SNRs from the decorrelator, represented as \(({\widehat{{\widehat{\gamma }}}}_{1}\, {\widehat{{\widehat{\gamma }}}}_{2}\,..\ \ {\widehat{{\widehat{\gamma }}}}_{N_{r}})\), contain eigenvalues that are scaled in proportion to the average output SNR (\({\overline{\gamma }}_{0,c}\)) of the particular \(c_{th}\) combiner in use. Thus,

$$\begin{aligned} {\widehat{{\widehat{\gamma }}}}_{k,c} = \epsilon _{k}{\overline{\gamma }}_{0,c} \end{aligned}$$
(12)

We can express the average output SNR using the individual signals from each branch for MRC and EGC as \({\overline{\gamma }}_{0,c}= \sum _{k = 1}^{N_{r}}{\widehat{{\widehat{\gamma }}}}_{k}\). Moreover, as indicated by Eq. (12), we can express the PDF of the SNR for each branch after passing through the decorrelator as

$$\begin{aligned} f_{{\overline{\gamma }}_{0}}\left( \gamma _{k} \right) = \left( \frac{m}{{\widehat{{\widehat{\gamma }}}}_{k,c}} \right) ^{m}\frac{{\gamma _{k}}^{m - 1}}{\Gamma \left( m \right) }\exp \left( - \frac{m\gamma _{k}}{{\widehat{{\widehat{\gamma }}}}_{k,c}} \right) ,\ \gamma _{k} \ge 0{\ \ \ } \end{aligned}$$
(13)

This represents the SNR of the correlated system.

Synthesizer gains for EGC

Maximum-ratio combining

The average output SNR for MRC, \({\overline{\gamma }}_{0, \text {MRC}} = \sum _{k = 1}^{N_{r}}{\overline{\gamma }}_{k} = {N_{r}\overline{\gamma }}_{k}\) is given in [4, 31, 32], where \({\overline{\gamma }}_{k}\) is the average SNR of the output signal at the \(k_{th}\) branch and is optimal.

Average SNR approximation for equal-gain combining (EGC)

Given independent fading paths and a total of \(N_{r}\) receive antennas, and considering the symbol received at the \(i_{th}\) antenna, \(r_{i}\), the \(i_{th}\) received symbol obtained at the output of the EGC combiner can be expressed as follows:

$$\begin{aligned} \acute{r_{i}} = \sum _{i = 1}^{N_{r}}r_{i} \end{aligned}$$
(14)

\(r_{i} = \propto _{i}\sqrt{E_{s}} + n_{i}\) and \(\propto _{i}\) is the stochastic amplitude of the symbol on the \(i_{th}\) path of the \(i_{th}\) antenna. \(E_{s}\) represent the symbol energy, and \(n_{i}\) is the AWGN component at the receiver, with a mean of zero and variance \(\sigma ^{2} =N_0/2\) [33].

The conditional SNR per symbol, \(\gamma _{0, \text {EGC}}\), at the output of the EGC combiner can be calculated according to [5] when symbols are transmitted with equal power as:

$$\begin{aligned} \gamma _{0, \text {EGC}} = \frac{\left( \sum _{i = 1}^{N_{r}} \propto _{i} \right) ^{2}E_{s}}{\sum _{i = 1}^{N_{r}}N_{i}} \end{aligned}$$
(15)

Where \(N_{i}\) denotes AWGN that has been added to the signal and has a power spectral density.

The SNR of the output signal in EGC, \(\overline{\gamma }_{0, \text {EGC}}\), which is calculated as an average and expressed as the expectation of (15), is determined for the uncorrelated channels with identical power spectral density of AWGN (\(N_{0}\)) as follows [1]:

$$\begin{aligned} {\overline{\gamma }}_{0, \text {EGC}} = \frac{E_{s}}{N_{r}N_{0}}\left[\sum _{i = 1}^{N_{r}}{E\left\{ \left| \propto _{i} \right. \ |^{2} \right\} +}\sum _{i = 1\ j \ne i}^{N_{r}}{\sum _{j = 1}^{N_{r}}{E\left( \propto _{i} \right) E\left( \propto _{j} \right) }} \right]\end{aligned}$$
(16)

The expression for \(E\left[{\acute{r}}^{2} \right]\), where \(r_{i} = \propto _{i}\sqrt{E_{s}} + n_{i}\) represents the \(i_{th}\) symbol received at the \(i_{th}\) antenna and \(r_{j} = \propto _{j}\sqrt{E_{s}} + n_{j}\) represents the \(j_{th}\) symbol received at the \(j_{th}\) antenna (with the superscript removed for simplicity), can be written as:

$$\begin{aligned} \begin{aligned} E\left[{\acute{r}}^{2} \right]= \sum _{i = 1\ }^{N_{r}}{\sum _{j = 1}^{N_{r}}{E\left( r_{i}r_{j} \right) }} = \sum _{i = 1\ }^{N_{r}}\sum _{j = 1}^{N_{r}}E\left( \propto _{i} \propto _{j}E_{s} + \propto _{i}\sqrt{E_{s}}n_{j} + \propto _{j}\sqrt{E_{s}}n_{i}+ n_{i}n_{j}\right) \end{aligned} \end{aligned}$$
(17)

The aim is to come up with a method that can blindly estimate the SNR without the requirement of transmitting pre-known training symbols. Consequently, we devise a SNR estimator that relies on the symbols received. Given that \(\propto\) follows a Nakagami-m distribution and considering the value of \(E\left[\propto ^{k} \right]= \frac{\Gamma \left( m + \frac{k}{2} \right) }{m^{k/2}\Gamma \left( m \right) }\), we can infer that \(\ \text {E}\left[\propto ^{2} \right]= \mathrm {\Omega } = 1\) and the expected value of \(E\left[\propto \right]= \frac{\Gamma \left( m + \frac{1}{2} \right) }{\sqrt{m}\Gamma \left( m \right) }\).

Using (17), it can be shown that Eq. (16) can be expressed as

$$\begin{aligned} \begin{aligned} {\overline{\gamma }}_{0, \text {EGC}}&= \left[1 + \left( N_{r} - 1 \right) \frac{\Gamma ^{2}\left( m + \frac{1}{2} \right) }{m\Gamma ^{2}\left( m \right) } \right]\overline{\gamma } \end{aligned} \end{aligned}$$
(18)

where \(\overline{\gamma }\) is the average SNR in the different diversity branches.

Decorrelation gain for EGC

In a two-branch system, the synthesizer generates four outputs. To calculate the variance of these outputs, [5] can be used, and the average SNR at the output of the EGC combiner can be expressed as

$$\begin{aligned} \begin{aligned} {\overline{\gamma }}_{0, \text {S}} = \frac{N_{r}\left( \left[1 + \rho \right]+ \left[1 - \rho \right]+ 1 + 1 \right) \overline{\gamma }}{4}=N_{r}\overline{\gamma } \end{aligned} \end{aligned}$$
(19)

The coefficient \(\rho\) represents the correlation between two random variables, with its range falling between 0 and 1. Consequently, the gains obtained from de-correlation is given as

$$\begin{aligned} \begin{aligned} G_{\overline{\gamma }}\left( \text {EGC} \right) = \frac{N_{r}\overline{\gamma }}{\left[1 + \left( N_{r} - 1 \right) \frac{\Gamma ^{2}\left( m + \frac{1}{2} \right) }{m\Gamma ^{2}\left( m \right) } \right]\overline{\gamma }}= \frac{N_{r}}{\left[1 + \left( N_{r} - 1 \right) \frac{\Gamma ^{2}\left( m + \frac{1}{2} \right) }{m\Gamma ^{2}\left( m \right) } \right]} \end{aligned} \end{aligned}$$
(20)

As the synthesizer has to perform its operation on two branches at any given instance, then \(N_r=2\).

$$\begin{aligned} G_{\overline{\gamma }}\left( \text {EGC} \right) = \frac{2}{\left[1 + \frac{\Gamma ^{2}\left( m + \frac{1}{2} \right) }{m\Gamma ^{2}\left( m \right) } \right]} \end{aligned}$$
(21)

New combiner output SNR

To calculate the average output SNR of the new combiner, we replace \({\widehat{{\widehat{\gamma }}}}_{k,c}\) with an expression that incorporates the average SNR of the received branch, \(\overline{\gamma }\). By using Eqs. (12) and (18), we obtain

$$\begin{aligned} \begin{aligned} {\widehat{{\widehat{\gamma }}}}_{k,EGC} = G_{\overline{\gamma }}\left( \text {EGC} \right) \ \ \epsilon _{k}\frac{{\overline{\gamma }}_{0,EGC}}{N_{r}} = G_{\overline{\gamma }}\left( \text {EGC} \right) \ \ \epsilon _{k}\left( \frac{1 + \left( N_{r} - 1 \right) \frac{\Gamma ^{2}\left( m + \frac{1}{2} \right) }{m\Gamma ^{2}\left( m \right) }}{N_{r}}\ \right) \overline{\gamma } \end{aligned} \end{aligned}$$
(22)

Similarly, the MRC average output SNR for each correlated branch can be expressed as

$$\begin{aligned} \begin{aligned} {\widehat{{\widehat{\gamma }}}}_{k,MRC} = \epsilon _{k}\frac{{\overline{\gamma }}_{0,MRC}}{N_{r}}= \epsilon _{k}\overline{\gamma } \end{aligned} \end{aligned}$$
(23)

An overall formula can be obtained to describe the MRC and EGC combiners by taking into account the average signal-to-noise ratio of each branch. We can merge Eqs. (22) and (23) as

$$\begin{aligned} {\overline{\gamma }}_{0,\ \ C} = \epsilon _{k}\beta _{c}\overline{\gamma } \end{aligned}$$
(24)

Where

$$\begin{aligned}\beta _{c} = \left\{ \begin{matrix} \beta _{\text {MRC}} = 1\ for\ MRC \\ \beta _{\text {EGC}} = \frac{1 + \left( N_{r} - 1 \right) \frac{\Gamma ^{2}\left( m + \frac{1}{2} \right) }{m\Gamma ^{2}\left( m \right) }}{N_{r}}\ \text {for EGC} \\ \end{matrix} \right. \ \end{aligned}$$

Performance analysis of SER for MPSK under the synthesizer system with correlation

The SER analysis, caused by the fading channel, can be evaluated by averaging the SER of the AWGN channel using the PDF of the fading envelope. To estimate the SER for coherent MPSK signaling over the AWGN channel, one can obtain an approximate expression by employing the nearest neighbor approximation method [32, 34,35,36].

$$\begin{aligned} \begin{aligned} \text {SER}_{\text {MPSK}} \approx 2Q\left( \frac{d_{min}}{\sqrt{2N_{0}}} \right) =2Q\left( \frac{2A sin\dfrac{\pi }{M}}{\sqrt{2N_{0}}} \right) =2Q\left( \sqrt{2\gamma }sin \left( \dfrac{\pi }{M} \right) \right) \end{aligned} \end{aligned}$$
(25)

where A is commonly determined by the signal energy and \(d_{min}\) is the minimum distance of the constellation.

The Q-function, represented as \(Q\left( \sqrt{b\gamma } \right) = \frac{1}{\pi }\int _{0}^{\frac{\pi }{2}}\exp \left( - \frac{b\gamma }{2}\sin ^2\theta \right) d\theta\), where \(b =2 sin^2(\pi /M)\), is defined in [4]. By employing the trapezoidal rule to solve for \(Q\left( \sqrt{b\gamma }\right)\), a simplified closed form of (25) is stated as:

$$\begin{aligned} \begin{aligned} SER_{MPSK}&= \frac{1}{t}\left[ \frac{e^{- b\gamma /2}}{2}+\sum _{i=1}^{t-1}e^{-b\gamma /S_{i}}\right] \end{aligned} \end{aligned}$$
(26)

In this context, t corresponds to the count of summations, where \(S_i = 2\sin ^2\theta _i\) and \(\theta _i=\frac{i\pi }{2t}\).

To determine the SER for MPSK in the new system, the SER of an AWGN channel is averaged over the PDF of a Nakagami-m fading channel with \(N_{r}\) receive antennas at a given instantaneous SNR [36].

$$\begin{aligned} P_{ser}= E\left[SER_{MPSK}\right]=\int _{0}^{\infty }SER_{MPSK}f_{\gamma }\left( \gamma \right) d\gamma \end{aligned}$$
(27)

Since \(f_{\gamma }\left( \gamma \right) =\left( \frac{m}{\overline{\gamma }} \right) ^{mN_{r}}\frac{\varvec{\gamma }^{mN_{r}-1}}{\Gamma \left( mN_{r} \right) }\exp \left( -\frac{m\gamma }{\overline{\gamma }} \right) ,\gamma \ge 0\) for the MRC [37], by substituting this expression in (27), we obtain

$$\begin{aligned} \begin{aligned} P_{ser}= \int _{0}^{\infty }\frac{1}{t}\left[ \frac{e^{- b\gamma /2}}{2} +\sum _{i=1}^{t-1}e^{- b\gamma /S_i} \right] \left( \frac{m}{\overline{\gamma }} \right) ^{mN_r}\frac{\gamma ^{mN_r-1}}{\Gamma \left( mN_r \right) }\exp \left( -\frac{m\gamma }{\overline{\gamma }} \right) \ d\gamma \end{aligned} \end{aligned}$$
(28)

The convergence of (28) can be achieved by utilizing the property of the Gamma function as provided in [9, 37, 38]. For \(\propto \ > 0\),

$$\begin{aligned} \int _{0}^{\infty }{\ \gamma ^{\propto - 1}e^{- \gamma \lambda }d\gamma = \frac{\Gamma \left( \propto \right) }{\lambda ^{\propto }}}\,\ \ \ for\ \lambda > 0; \end{aligned}$$
(29)

Upon solving Eq. (28) with the help of (29), (28) transforms into

$$\begin{aligned} \begin{aligned} P_{ser\_ MRC} = \frac{1}{t}\left[ \frac{1}{2}\left( \frac{2m}{b\overline{\gamma } + 2m} \right) ^{mN_{r}} + \sum _{i = 1}^{t - 1}\left( \frac{mS_{i}}{b\overline{\gamma } + mS_{i}} \right) ^{mN_{r}} \right] \end{aligned} \end{aligned}$$
(30)

The SER for the new system can be demonstrated to be

$$\begin{aligned} \begin{aligned} SER = \frac{1}{t}\left[ \frac{1}{2}\left( \frac{2m}{b\overline{{\widehat{{\widehat{\gamma }}}}_{k,c}} + 2m} \right) ^{mN_{r}} + \sum _{i = 1}^{t - 1}\left( \frac{mS_{i}}{b{\widehat{{\widehat{\gamma }}}}_{k,c} + mS_{i}} \right) ^{mN_{r}} \right] \end{aligned} \end{aligned}$$
(31)

By substituting \({\widehat{{\widehat{\gamma }}}}_{k,c}\) with the expressions provided in (12) and (24), results in

$$\begin{aligned} \begin{aligned} SER = \frac{1}{t}\left[ \frac{1}{2}\left( \frac{2m}{b\epsilon _{k}\beta _{c}\overline{\gamma } + 2m} \right) ^{mN_{r}}+ \sum _{i = 1}^{t - 1}\left( \frac{mS_{i}}{b\epsilon _{k}\beta _{c}\overline{\gamma } + mS_{i}} \right) ^{mN_{r}} \right] \end{aligned} \end{aligned}$$
(32)

Theoretical and simulation results

The study utilizes Monte-Carlo simulations to verify the validity of the decorrelation gain equation derived in (21) and the SER equation derived in (32). Additionally, we demonstrate the performance of the virtual receive diversity system using various antenna configurations for the MPSK modulation scheme.

In the legends of the figures, the labels ’Corr’ and ’Decorr’ represent the correlated system and the proposed decorrelated(virtual receive diversity) system, respectively. The term ’Sim’ corresponds to the results obtained from simulations, while ’Theory’ represents outcomes from the analytical framework described in Eq. (32). This study considers fading channels that conform to the Nakagami-m distribution and assumes arbitrary correlation, which is modeled using the Bessel model. The correlation model assumes a linear antenna array with a uniform distribution of the angle of arrival (AoA). The correlation coefficient, denoted by \(\rho\), is expressed as [21].

$$\begin{aligned} \rho = J_{0}\left( 2\pi \dfrac{d}{\lambda }\right) \end{aligned}$$
(33)

\(J_{0}\) corresponds to the Bessel function of the first kind with a zero order, d signifies the separation distance between antennas, and \(\lambda\) symbolizes the wavelength of the carrier signal.

Fig. 2
figure 2

Comparing simulated and analytical SER for the decorrelator with \(m=1\), \(N_r=4\), and \(d=0.2\lambda\)

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Figure 2. compares the SER performance of BPSK, 4-PSK, 16-PSK, and 64-PSK modulation schemes with a diversity order of four and an antenna separation distance of \(0.2\lambda\) when \(m=1\). BPSK exhibits the best performance, followed by 4-PSK, 16-PSK, and 64-PSK in terms of SER. For a given SNR, such as 15 dB, the SER values are \(3.666X10^{-6}\), \(5.93X10^{-5}\), \(2.977X10^{-2}\), and 0.236 for BPSK, 4-PSK, 16-PSK, and 64-PSK, respectively. This observation aligns with the well-established fact that the SER performance generally deteriorates as the number of phases and symbols increases. This is due to the higher susceptibility of higher-order M-PSK schemes to noise and phase errors. Consequently, BPSK typically exhibits better SER performance when compared to higher-order M-PSK schemes like 4-PSK, 16-PSK, and 64-PSK.

Fig. 3
figure 3

Comparing SER between correlated and decorrelated systems with \(Nr=4\), \(m=1\), and \(d=0.2\lambda\)

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In Fig. 3., we demonstrate the performance of the virtual receiver diversity by comparing theoretical results of a correlated system and a virtual receiver diversity system for BPSK, 4-PSK, 16-PSK, and 64-PSK, where \(N_r=4\), \(m=1\), and \(d=0.2\lambda\). It can be deduced that the proposed virtual receiver diversity achieves decorrelation gains of approximately 1 dB for the EGC systems. A greater increase in decorrelation gain implies that a reduced SNR is required to achieve an equivalent SER, suggesting improved performance even in the presence of noise.

Fig. 4
figure 4

Analyzing fading parameter’s impact on the SER for the decorrelated system with \(N_r=3\) and \(d=0.1\lambda\)

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Figure 4. illustrates the influence of the fading parameter, m, which characterizes the Nakagami-m distribution, on the error performance of the M-PSK modulation scheme using the proposed virtual receiver diversity decorrelator. The study considered a diversity order of 3 and an antenna spacing equivalent to 0.1 of the wavelength. As anticipated, increasing m resulted in a decrease in SER, while decreasing m had the opposite effect, at a constant SNR. For instance, at a SNR of 20 dB, the SER values were 0.06164, 0.01662, \(1.196X10^{-3}\), and \(1.351X10^{-4}\) for 16-PSK when m was set to 0.5, 1, 3, and 7, respectively. This behavior can be attributed to the Nakagami-m distribution becoming more concentrated around its mean value. The increased concentration limits the dispersion of fading effects, resulting in reduced impact on the transmitted signal. Thus, higher values of m indicate a channel less prone to errors, leading to improved SER. Furthermore, it was observed that the simulation and analytical results closely matched for all tested values of fading parameter m, namely 0.5, 1, 3, and 7.

Fig. 5
figure 5

Comparing SER between correlated and decorrelated systems with varying m, \(N_r=4\), and \(d=0.1\lambda\)

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Figure 5. illustrates the effectiveness of the suggested diversity decorrelator by comparing its results with those of a conventional correlated system for 16-PSK. A diversity order of 4 and an antenna spacing of \(0.1\lambda\) are taken into account. Tests were conducted to examine varying fading parameters: m = 0.5, 1, 3, and 7. It was observed that the proposed diversity decorrelator achieves decorrelation gains of approximately 1 dB at a SER of \(10^{-3}\) for \(m=1\). However, as the value of m increases, the decorrelation gains decrease. For higher values of m, the decorrelation gains become negligible. This is attributed to less severe fading and fewer rapid fluctuations in the received signal power. As m approaches infinity, the channel experiences no fading.

Fig. 6
figure 6

Comparing SER for correlated and decorrelated EGC, with \(N_r=3\) and \(m=0.5\)

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Figure 6. showcases the effectiveness of the diversity decorrelator in terms of performance. The simulation results are compared for \(N_r = 3\), with different antenna spacings. It is anticipated that antennas positioned closer together would exhibit greater signal correlation, leading to a higher SER. This expectation is confirmed by the simulations, as the SER decreases when the antenna spacing changes from \(d=0.1\lambda\) to \(d=0.5\lambda\). Additionally, it is worth noting that the suggested diversity decorrelator achieves approximately 1 dB of decorrelation gain for the EGC combiner systems.

Fig. 7
figure 7

Comparing SER between EGC and MRC for \(m=3\), \(d=0.2\lambda\), and \(N_r=2, 3, 4\)

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Figure 7 compares the impact of diversity order on the performance of EGC and MRC SER with 16-PSK, with \(m=3\) and \(d=0.2\lambda\). According to the graph, the proposed virtual diversity receiver system achieves equivalent error performance to MRC in correlated systems. This can be explained as follows: In channels with significant fading, EGC combines branches with high noise levels, resulting in a significant penalty compared to MRC. However, in balanced channels with low fading, EGC takes full advantage of combining all the “good” branches, approaching the performance of MRC. This can also be attributed to the fact that as fading severity decreases, the relative weights of MRC approach unity, which are the EGC weights. One key finding is that instead of using the computationally complex MRC combining, EGC can be employed with the diversity decorrelator to achieve similar gains.

Conclusion

A virtual receiver diversity decorrelator that does not require the measurement of signal information from a correlated system has been analyzed for M-PSK signals using coherent EGC reception. The virtual receiver diversity decorrelator achieves decorrelation gains for EGC in channels that follow the Nakagami-m statistical distribution. The decorrelation gain obtained depends on the fading parameter m. Additionally, a decorrelation gain expression is derived for EGC reception in M-PSK correlated signals in channels experiencing fading according to the Nakagami-m distribution. Finally, the decorrelation gain expression derived is integrated into the derived SER expression for EGC to obtain the overall SER expression for the virtual receiver diversity decorrelator. The SER for the virtual receiver diversity decorrelator is validated through Monte Carlo simulation results. The expression results are found to be tightly bound to the simulation results. It is also demonstrated that the performance of EGC approaches the performance of MRC. An important outcome is that the utilization of the computationally complex MRC can be replaced by the implementation of EGC to achieve equivalent benefits, particularly in cases where the diversity decorrelator is engaged. Nevertheless, the developed decorrelator exhibits certain limitations. Owing to correlations between the sum and difference signals, the resultant decorrelated outputs, referred to as conjoint signals, do not demonstrate complete mutual independence. Thus, the decorrelator solely mitigates the extent of correlation.