# Computationally Efficient Soft Detection Schemes for Coded Massive MIMO Systems +
## 3 Proposed Method
### MMSE-Filtering-Matrix Approximation  
In this subsection, we focus on reducing the computational complexity of the MMSE-PIC-based JIDD scheme caused by the calculation of the Gram matrix, filtering matrix, and PE-SINR. First, by using the property that the Gram matrix, \(\mathbf{G}=\mathbf{H}^{H}\mathbf{H}\), can be approximated to a diagonal-like matrix in a massive MIMO system, \(\mathbf{W}^{H}\) can also be approximated as follow [13; 14]:\[\mathbf{W}^{H} =(\mathbf{G}\mathbf{\Lambda}+\sigma^{2}\mathbf{I}_{M})^{-1}\mathbf{H }^{H} \tag{12}\] \[\approx\begin{bmatrix}\widetilde{w}_{1,1}&0&\cdots&0\\ 0&\widetilde{w}_{2,2}&\cdots&0\\ 0&0&\ddots&0\\ 0&0&0&\widetilde{w}_{M,M}\end{bmatrix}^{-1}\mathbf{H}^{H},\]  
where \(\widetilde{w}_{i,i}=g_{i,i}E_{i}+\sigma^{2}\), and \(g_{i,i}\) denotes the \(i\)th diagonal element of \(\mathbf{G}\). This method reduces the complexity of calculating matrix inversion and produces an approximating performance to the conventional MMSE-PIC-based JIDD system. Nevertheless, this method still requires not a small amount of computational complexity to calculate diagonal elements of \(\mathbf{G}\).  
A previous study proposed a method to reduce the computational complexity of the PE-SINR estimation, by approximating the diagonal elements of \(\mathbf{G}\) with a universal constant value [15]. However, this method cannot be directly applied to the JIDD system. To solve this problem, we propose an approximating method in the calculation of both the Gram matrix and PE-SINR as follows. We use the fact that the column vectors of the channel matrix are almost orthogonal as the number of antennas increases [13]. Then \(\mathbf{G}\) can be approximated by \(N/\text{MI}_{M}\), by considering transmit power normalization. Consequently, the filtering matrix \(\mathbf{W}^{H}\) can be estimated without any matrix inversion as follows:  
\[\mathbf{W}^{H} =(\mathbf{G}\mathbf{\Lambda}+\sigma^{2}\mathbf{I}_{M})^{-1} \mathbf{H}^{H} \tag{13}\] \[\approx\begin{bmatrix}\frac{N}{M}E_{1}+\sigma^{2}&0&\cdots&0\\ 0&\frac{N}{M}E_{2}+\sigma^{2}&\cdots&0\\ 0&0&\ddots&0\\ 0&0&0&\frac{N}{M}E_{M}+\sigma^{2}\end{bmatrix}^{-1}\mathbf{H}^{H}.\] \[=\begin{bmatrix}(\frac{N}{M}E_{1}+\sigma^{2})^{-1}&0&\cdots&0\\ 0&(\frac{N}{M}E_{2}+\sigma^{2})^{-1}&\cdots&0\\ 0&0&\ddots&0\\ 0&0&0&(\frac{N}{M}E_{M}+\sigma^{2})^{-1}\end{bmatrix}\mathbf{H}^{H}.\]  
With the above approximation, \(\mathbf{w}_{i}^{H}\) in (6) can be expressed as follows:  
\[\mathbf{w}_{i}^{H}\approx\frac{1}{\frac{N}{M}E_{i}+\sigma^{2}}\mathbf{h}_{i}^{ H}, \tag{14}\]  
where, \(\mathbf{h}_{i}^{H}\) is the \(i\)th row of \(\mathbf{H}^{H}\). Therefore, (6) can be simplified as follows:  
\[\hat{z}_{i}=\mathbf{w}_{i}^{H}\hat{\mathbf{y}}_{i}\approx\frac{1}{\frac{N}{M}E _{i}+\sigma^{2}}\mathbf{h}_{i}^{H}\hat{\mathbf{y}}_{i}, \tag{15}\]  
Finally, we show that the PE-SINR estimation can be simplified to a layer independent universal value. First, \(\mu_{i}\) in (10) can be expressed as follows, by inserting (14).  
\[\mu_{i}=\mathbf{w}_{i}^{H}\mathbf{h}_{i}\approx\frac{1}{\frac{N}{M}E_{i}+ \sigma^{2}}\mathbf{h}_{i}^{H}\mathbf{h}_{i}=\frac{\frac{N}{M}}{\frac{N}{M}E_{i }+\sigma^{2}}. \tag{16}\]  
Then, the layer dependent PE-SINR in (9) can be expressed as a constant value as follows:  
\[\rho_{i}\approx\frac{N}{M\sigma^{2}}. \tag{17}\]