# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## 2 Background
### Space-Time Block Coding and OFDM  
MIMO systems are mainly designed for narrowband or flat channels. Applying MIMO systems in the wideband frequency selective channel implies a constant penalty factor in the coding gain compared with that in flat-frequency channels. Furthermore, at high SNRs, an irreducible error rate floor is inevitable [51]. This irreducible error rate _floor_ is due to the existence of multipath delay spread, and it persists even if we increase the number of antennas. Since the ISI is the root cause of the error _floor_, in principle, it can be mitigated by resorting to adaptive equalization, but this can be too complex to implement in such an environment. Another option that is widely used is to resort to OFDM, which naturally converts a frequency-selective fading channel into a frequency-nonselective fading channel. The subcarriers (i.e., tones) in an OFDM symbol are essentially narrowband signals. Since these tones fit perfectly as vehicles for space-time codes, OFDM is an enabler for this efficient coding technique [11]. Figure 1 shows the coding scheme for a generic coding matrix \(\mathbf{X}[k]\in\mathbb{C}^{M_{T}\times P}\), where \(P\) is the number of time samples for the transmission of one block of coded symbols and \(k=1,~{}2,\cdots,~{}K\) is the carrier index of the \(k\)th MIMO-STBC encoded symbol matrix \(\mathbf{X}[k]\) along the OFDM symbol. \(\mathbf{R}[k]\in\mathbb{C}^{M_{\mathbf{S}}\times P}\) is the matrix of received symbols, \(\mathbf{\hat{s}}\in\mathbb{C}^{M_{\mathbf{S}}}\) is the decoded vector, and \(M_{\mathbf{S}}\) is the number of modulated symbols in a MIMO-STBC matrix. The transmitting space-time block coder (STBC) encodes the data symbol vector \(\mathbf{s}[k]\in\mathbb{C}^{M_{\mathbf{S}}}\) using the code matrix to construct the transmitting matrix \(\mathbf{X}[k]\) of length \(K\). The streams \(\mathbf{X}_{m_{T},p}[k]\) are fed to the IFFT modulator of each \(m_{T}\) transmitting antenna, at each \(p\) period of time relative to the OFDM symbol sequence. In this manner, the information is transmitted in \(\mathbf{X}[k]\) blocks of \(M_{T}\) antennas and \(P\) OFDM symbols in each \(k\)-th carrier. To illustrate this scheme, Figure 2 shows an example for a two-transmitting antenna system using Alamouti coding. Consequently, the channel is given by \(\mathbf{H}\in\mathbb{C}^{M_{T}\times M_{\mathbf{S}}\times P\times K}\). It should be emphasized that in the simulation in Section 4 of this work, the channel is not assumed to be static over the entire MIMO-STBC block since it spreads over time in \(P\)-consecutive OFDM symbols. This is particularly necessary for the proposed work, as the receiver will fit and adapt to the characteristics and variations of the channel over time. This is also necessary for a massive number of broadcast antennas due to the length of the long block coding \(P\) that transmits over time in consecutive OFDM symbols [11]. As in OFDM systems, MIMO-OFDM also requires the channel state information to decode the received symbols. One of the most popular and widely used approaches to MIMO channel estimation is to employ pilot signals (also referred to as training sequences) and then estimate the channel based on the received data and the knowledge of the training sequence, as detailed in Figure 3. Based on pilot signals, in [52; 53], the least-squares (LS) channel estimation technique is applied for orthogonal frequency-division multiplexing systems with multiple transmit antennas [11]. A generalized coding scheme referred to as space-time block codes (STBCs) [16; 44; 54], based on the theory of orthogonal matrix designs, can achieve the full-transmit diversity of \(M_{T}M_{R}\) employing the maximum likelihood decoding algorithm at the receiver [44]. The idea is to transmit \(M_{T}\) orthogonal streams, which implies that the receiver antennas receive \(M_{T}\) orthogonal streams. This special class of space-time block codes is the so-called orthogonal STBC (OSTBC) [11; 54; 55].