# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## 2 Background  
The main goal in multi-antenna systems is to increase the channel capacity with \(M_{T}\) transmit and \(M_{R}\) receive antennas by a factor of \(\min(M_{T},M_{R})\) without using additional transmit power or spectral bandwidth [43]. Considering the MIMO digital communication system \(\mathbf{r}(k)=\mathbf{H}^{T}\mathbf{x}(k)+\boldsymbol{\eta}(k)\), with transmitted signal \(\mathbf{x}(k)\in\mathbb{C}^{M_{T}}\), received signal \(\mathbf{r}(k)\in\mathbb{C}^{M_{R}}\), and additive white Gaussian noise (AWGN) vector \(\boldsymbol{\eta}(k)\in\mathbb{C}^{M_{R}}\), the channel capacity of \(\mathbf{H}\in\mathbb{C}^{M_{T}\times M_{R}}\) is expressed as [44]  
\[C=\log_{2}\biggl{[}\det\biggl{(}\mathbf{I}_{M_{R}}+\frac{E_{s}}{M_{T}E_{0}} \mathbf{H}^{H}\mathbf{R}_{xx}\mathbf{H}\biggr{)}\biggr{]}, \tag{1}\]  
in which \(\mathbf{I}_{M_{R}}\) is an \(M_{R}\times M_{R}\) identity matrix, \([\cdot]^{T}\) is the transpose operator, \([\cdot]^{H}\) is the conjugate transpose operator, \(E_{s}\) is the total transmitted signal power, \(E_{0}\) is the AWGN power, \(\mathbf{R}_{xx}=\mathrm{E}\{\mathbf{x}(k)\mathbf{x}^{H}(k)\}\) is the correlation matrix of \(\mathbf{x}(k)\), and \(\mathrm{E}\{\cdot\}\) is the expectation operator. However, if no channel state information (CSI) is available at the transmitter, we can assume that the channel components are equally probable. In this case, we consider that power is equally divided among the transmitting antennas, which implies \(\mathbf{R}_{xx}=\mathbf{I}_{M_{T}}\). The capacity in such a case is then given by [11; 45]  
\[C=\log_{2}\biggl{[}\det\biggl{(}\mathbf{I}_{M_{R}}+\frac{E_{S}}{M_{T}N_{0}} \mathbf{H}^{H}\mathbf{H}\biggr{)}\biggr{]}. \tag{2}\]  
Note that Equation (2) can be outperformed if the channel information is available at the transmitter (leading to a coding gain). However, Equation (2) is the maximum diversity capacity without channel knowledge at the transmitter. Furthermore, if \(M_{T}=M_{R}=1\), Equation (2) represents the Shannon capacity for SISO systems [11]. In order to increase capacity, the concepts of diversity [46], coding [47], and array [48] gains play key roles in MIMO systems. The array gain is the average increase in the signal-to-noise ratio (SNR) at the receiver that arises from the coherent combining effect of multiple antennas at the transmitter, receiver, or both. Multiple antenna systems require perfect channel knowledge at the transmitter, receiver, or both to achieve this array gain [48]. On the other hand, diversity gain is obtained by the provision of replicas of the transmitted signal at the receiver [46]. Diversity gain techniques are used to mitigate degradations in the error performance due to wireless fading channels (e.g., due to multipath). Since the probability that statistically independent fading channels simultaneously experience deep fading is insignificant, there are various ways of performing diversity gain and space diversity. To accomplish this, it is necessary to use sufficiently separated antennas in the array (by more than \(10\lambda\) on base stations and \(2\lambda\) to \(5\lambda\) on mobile devices [44]) to guarantee independent wireless channels. In contrast, coding gain is usually provided by temporal channel coding, e.g., convolutional and block codes [11]. Space-time code is a digital communication technique used to transmit multiple copies of a data stream via multiple antennas to compensate for fading and AWGN. At the receiver side, these multiple copies of the signal are received by one or more antennas, improving the communication reliability. Depending on the encoder algorithm at the transmitter, we can have different space-time codes. Space-time trellis codes (STTCs) combine modulation and trellis coding to transmit signals over a MIMO channel. Although STTCs provide both coding gain and diversity gain, the computational complexity is higher than other space-time codes, mainly in the receiver, where a Viterbi decoder is necessary [49].