# Low-Complexity Convolutional Neural Network for Channel Estimation
## 2 5G NR System Model  
Considering the downlink in a single-cell environment, the waveforms are sent by the transmitter equipped with M antennas over the channel, and a corrupted version of them are collected by K antennas at the receiver. The propagation over the channel can bring noise, attenuation, distortion, fading, and interference [16]. Mathematically, this is Equation (1). \[Y=\mathbf{H}\,X+N \tag{1}\]  
where \(Y=(y_{1}\ y_{2}\ \dots\ y_{K})^{T}\in\ \mathbb{C}^{K\times 1}\), is the received signal in vector form. The channel environment is captured by \(\mathbf{H}\in\ \mathbb{C}^{K\times M}\), the transmitted signal is represented by \(X=(x_{1}\ x_{2}\ \dots\ x_{M})^{T}\in\ \mathbb{C}^{M\times 1}\), and \(N=(n_{1}\ n_{2}\ \dots\ n_{K})^{T}\in\ \mathbb{C}^{K\times 1}\) is the additive white Gaussian noise in the transmission. The operator \((\dots)^{T}\) defines the transposed vector form. Focusing on the 5G NR standard, before transmitting the signal, the information is built along the Downlink Shared Channel (DL-SCH) and Physical Downlink Shared Channel (PDSCH). The first stage involves error detection, segmentation, channel-coding operations based on Low-Density Parity-Check (LDPC) technique, rate-matching, and finally reconstructing the codeblocks into codewords [17]. The subsequent phase of the downlink chain is the PDSCH. The codeword first becomes subject to a scrambling operation and then undergoes symbol modulation. The symbols are mapped onto \(L\) layers. The number of layers determines how many independent streams can be transmitted in parallel, \(S=(s_{1}\ s_{2}\ \dots\ s_{L})^{T}\). Because every signal can be considered as the product of a linear combination, the data layers are not directly transmitted, but they are input into the precoding module. As a consequence, the input-output model can be rewritten as Equation (2). \[Y=\mathbf{H}\,\mathbf{P}S+N \tag{2}\]  
where \(\mathbf{P}\in\ \mathbb{C}^{M\times L}\) identifies the precoding matrix and it is a function of the wireless channel \(\mathbf{H}\), and \(S\in\ \mathbb{C}^{L\times 1}\) is composed of the L layers. Therefore, the transmission can be improved in the form of increased achievable capacity and interference cancellation defining an appropriate precoding matrix [18]. The improvement depends on the technique selected to obtain the precoding matrix (e.g., Maximum-Ratio Transmission (MRT) Precoder, Zero Forcing (ZF) Precoder, and Diagonalization, but all of them require knowledge of the channel [19]. The described model is well-known to explain the input-output relationship in the system, and highlights the improvement given by channel knowledge and interference cancellation. However, in practice, the signals involved in the transmission are multidimensional, and every signal on each antenna is defined in the frequency and time domains, that is, subcarriers and symbols. This structure composes the resource grid where one resource element is defined as the intersection of one subcarrier with a symbol. Therefore, the real dimensions of the channel are given by the number of subcarriers (\(N_{sc}\)), the number of symbols (\(N_{symb}\)), and the antennas at the receiver (\(K\)) and transmitter (\(M\)). To collect the channel knowledge, the reference signal is allocated in some dedicated resource elements across the transmitted grid. In the 5G NR standard, the reference signal is the DM-RS. The DM-RS provides high flexibility in the transmission configuration to cater to different deployment scenarios. This is possible thanks to multiple configurations in the resource grid. A first classification can be done considering the mapping type in the time domain. For mapping type A, the first DM-RS is located in symbol 3 or 4 of the slot regardless of where the data transmission starts, while mapping type B places the reference signal in the first symbol of the data allocation. In case of rapid change of the channel, additional reference symbols are included in the resource grid in order to ensure an accurate channel estimation.