# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## 3 Proposed Approach  
### Coding Scheme  
Similarly to the work of [56], the present work is derived from the full-rate full-diversity complex-valued space-time block code scheme proposed by [PERSON] [61]. The transmission matrix proposed in [61] is given by [11]  
\[\mathbf{A}_{i,j}= \begin{bmatrix}s[i]&s[j]\\ -s[j]^{*}&s[i]^{*}\end{bmatrix} , \tag{6}\]  
in which \(s[i]\) is the \(i\)th input symbol to be encoded. Based on [61], [PERSON] [56] proposed a quasi-orthogonal coding scheme using four antennas and consequently four encoded symbols as [11]  
\[\mathbf{S}_{4}^{4}= \begin{bmatrix}\mathbf{A}_{1,2}&\mathbf{A}_{3,4}\\ -[\mathbf{A}_{3,4}]^{*}&[\mathbf{A}_{1,2}]^{*}\end{bmatrix} = \begin{bmatrix}s[ 1]&s[2]&s[3]&s[4]\\ -s[2]^{*}&s[1]^{*}&-s[4]^{*}&s[3]^{*}\\ -s[3]^{*}&-s[4]^{*}&s[1]^{*}&s[2]^{*}\\ s[4]&-s[3]&-s[2]&s[1]\end{bmatrix} , \tag{7}\]  
where \(\mathbf{S}\) is the quasi-orthogonal coding matrix. The main idea behind the work of [56] is to build a \(4\times 4\) matrix from two \(2\times 2\) matrices, keeping a fixed transmission rate [11]. In the present paper, we generalize the idea presented in [56] to a new recursive method of generating coding schemes, as given by [11]  
\[\mathbf{S}_{M_{s}}^{M_{T}}= \begin{bmatrix}\mathbf{S}_{M_{s}-M_{T}/2}^{M_{T}/2 }&\mathbf{S}_{M_{s}}^{M_{T}/2}\\ -[\mathbf{S}_{M_{s}}^{M_{T}/2}]^{*}&[\mathbf{S}_{M_{s}-M_{T}/2}^{M_{T}/2}]^{* }\end{bmatrix} , \tag{8}\]  
in which \(M_{T}=2^{n}\), \(\forall\;n\geq 1\) is the number of transmitting antennas and \(M_{s}\) is the number of encoded symbols. In the proposed scheme, \(M_{s}\triangleq M_{T}\) and the code rate is \(R=M_{T}/M_{s}=1\)[11]. The recurrence is performed until we find \(\mathbf{S}_{n}^{1}=s[n]\), \(\forall n\in[1,\,2,\cdots,M_{S}]\) in Equation (8). For example, with four transmitting antennas, Equation (8) results in [11]\[\mathbf{S}_{4}^{4}= \begin{bmatrix}\mathbf{S}_{4-4/2}^{4/2}&\mathbf{S}_{4}^{4/2}\\ -[\mathbf{S}_{4}^{4/2}]^{*}&[\mathbf{S}_{4-4/2}^{4/2}]^{*}\end{bmatrix} = \begin{bmatrix}\mathbf{S}_{2}^{2}&\mathbf{S}_{4}^{2}\\ -[\mathbf{S}_{2}^{2}]^{*}&[\mathbf{S}_{2}^{2}]^{*}\end{bmatrix} . \tag{9}\]  
From the recurrent structure of Equation (8), \(\mathbf{S}_{2}^{2}\) and \(\mathbf{S}_{2}^{4}\) are  
\[\mathbf{S}_{2}^{2}= \begin{bmatrix}\mathbf{S}_{1}^{1}&\mathbf{S}_{1}^{1}\\ -[\mathbf{S}_{2}^{1}]^{*}&[\mathbf{S}_{1}^{1}]^{*}\end{bmatrix} , \tag{10}\]  
and  
\[\mathbf{S}_{4}^{2}= \begin{bmatrix}\mathbf{S}_{1}^{1}&\mathbf{S}_{4}^{1}\\ -[\mathbf{S}_{4}^{1}]^{*}&[\mathbf{S}_{3}^{1}]^{*}\end{bmatrix} . \tag{11}\]  
Thus, substituting Equations (10) and (11) into Equation (9),  
\[\mathbf{S}_{4}^{4}= \begin{bmatrix}\begin{bmatrix}\mathbf{S}_{1}^{1}&\mathbf{S }_{2}^{1}\\ -[\mathbf{S}_{2}^{1}]^{*}&[\mathbf{S}_{1}^{1}]^{*}\end{bmatrix}^{*}&\begin{bmatrix} \mathbf{S}_{3}^{1}&\mathbf{S}_{4}^{1}\\ -[\mathbf{S}_{1}^{1}]^{*}&[\mathbf{S}_{3}^{1}]^{*}\end{bmatrix}^{*}\end{bmatrix}  = \begin{bmatrix}\mathbf{S}_{1}^{1}&\mathbf{S}_{2}^{1}&\mathbf{S}_{3}^{1}& \mathbf{S}_{4}^{1}\\ -\mathbf{S}_{1}^{1}&\mathbf{S}_{1}^{1}&-\mathbf{S}_{1}^{1}&\mathbf{S}_{1}^{1} \\ -\mathbf{S}_{1}^{1}&-\mathbf{S}_{4}^{1}&\mathbf{S}_{1}^{1}&\mathbf{S}_{2}^{1} \\ \mathbf{S}_{4}^{1}&-\mathbf{S}_{3}^{1}&-\mathbf{S}_{2}^{1}&\mathbf{S}_{1}^{1} \end{bmatrix} . \tag{12}\]  
Replacing \(\mathbf{S}_{n}^{1}=s[n]\), \(\forall n\in[1,\,2,\,3,\,4]\), into Equation (12),  
\[\mathbf{S}_{4}^{4}= \begin{bmatrix}s[1]&s[2]&s[3]&s[4]\\ -s[2]^{*}&s[1]^{*}&-s[4]^{*}&s[3]^{*}\\ -s[3]^{*}&-s[4]^{*}&s[1]^{*}&s[2]^{*}\\ s[4]&-s[3]&-s[2]&s[1]\end{bmatrix} . \tag{13}\]  
Note that Equation (13) is equal to the coding scheme proposed by [56] with four antennas, as in Equation (4). However, in contrast to the work of [56], our coding scheme, presented in Equation (8), can generate coding matrices for any \(M_{\mathrm{T}}=2^{n},\,\,\forall\,\,n\geq 1\), and \(M_{\mathrm{s}}\triangleq M_{\mathrm{T}}\)[11]. For the case of \(n=1\), Equation (8) is equal to Equation (6), the full-rate full-diversity complex-valued space-time block code scheme proposed in [61].