# Machine Learning-Based 5G-and-Beyond Channel Estimation for MIMO-OFDM Communication Systems
## 2 System Model
### 5G-and-Beyond Channel Model  
In this paper, we consider the 5G-and-beyond channel model, which is defined by the 3 GPP standard in [33]. The 5G-and-beyond channel model includes the effect of multi-path and Doppler shifting, which cause frequency-selective fading and time-selective fading, respectively. In particular, we exploit the TDL-C model defined for the NLOS channel for the full frequency range from 0.5 GHz to 100 GHz [33] with Rayleigh fading distribution. Figure 1: The illustration of the considered MIMO-OFDM system model with the proposed DNN-aided module in blue. In the figure, CP denotes cyclic prefix; S/P denotes serial to parallel; P/S denotes parallel to serial; IFFT denotes inverse fast Fourier transform; and FFT denotes fast Fourier transform. The Doppler spectrum of each tap is characterized by a classical [PERSON]'s spectrum shape, which is expressed as  
\[S(f)=\frac{1}{\pi f_{d}\sqrt{1-\left(\frac{f}{f_{d}}\right)^{2}}},\quad|f|<f_{d}, \tag{5}\]  
where \(f_{d}\) (Hz) is the maximum Doppler shift; i.e., \(f_{d}=\frac{vf_{c}}{c}\), for a given speed \(v\)(m/s) and a carrier frequency \(f_{c}\)(Hz), with \(c\approx 3\times 10^{8}\) being the light speed. The auto-correlation of [PERSON]'s Doppler spectrum is [34]  
\[R(\tau)=\int_{-f_{d}}^{f_{d}}S(f)e^{2\pi\tau}df=J_{0}(2\pi f_{d}\tau), \tag{6}\]  
where \(J_{0}(. )\) is the first kind of Bessel function of order 0. From the continuous form in (6), the discrete form of the auto-correlation function is defined as follows:  
\[R[l]=J_{0}(2\pi f_{d}|l|T_{sym}), \tag{7}\]  
where \(l\) and \(T_{sym}\) are the symbol index and the symbol duration, respectively. We denote \(h_{a,b}(\tau_{i},t)\) as the time-variant channel impulse response from the \(a\)-th transmission antenna (\(a=1,\cdots,N_{T}\)) to the \(b\)-th receiver antenna (\(b=1,\cdots,N_{R}\)), where \(\tau_{i}\) is the transmission delay at the \(l\)-th tap of the propagation channels. A mathematical description of the frequency-selective and time-variant channel model is given in [35] as follows  
\[h_{a,b}(\tau_{l},t)=\sum_{l=0}^{L-1}h_{l}\delta(\tau_{l}-t)\times\exp j[2\pi f _{D,l}(t-\tau_{l})-2\pi f_{c}\tau_{l}], \tag{8}\]  
with \(l\) is the index of taps, \(h_{l}\) represents the \(l\)-th resolved amplitude, and \(\tau_{l}\) represents the express delay of the \(l\)-th tap. \(f_{D,l}=v(t)f_{c}\text{cos}[\theta_{l}]/c\) is the Doppler frequency induced by the relative movement of the Tx and Rx, \(v(t)\) represents the relative velocity, \(\theta_{l}\) denotes the aggregate phase angle of all components arriving in the \(l\)-th tap, and \(c\) is the speed of light. To model the propagation channels in this paper, we exploit the Matlab 5G toolbox [33] to simulate the instantaneous channels. The 5G-and-beyond channels have the TDL-C profile displayed in Figure 2, with the color-map displaying the channel gain. In more detail, the channel gain varies from \(-12\) dB to \(-47\) dB. This figure indicates that the considered channel profile is not sparse, which is a consequence of the mobile communication carrier frequency at sub-6 GHz; i.e., here, the carrier frequency is set to 4 GHz (the channel estimation quality can be enhanced if a proper domain, in which the channels are spares, is determined, and thus a sparse channel estimation technique is effectively utilized. This work is left for the future). In addition, Figure 3 plots the expectation \(\mathbb{E}\{\mathbf{H}\mathbf{H}^{H}\}\), where \(\mathbf{H}\) is the channel matrix of a subcarrier. It shows that all the coefficients are non-zero, therefore verifying the spatial correlation among the channels. Figure 2: The 2 time\(-\)varying channel profile with \(f_{d}=200\) Hz in the 20 OFDM symbols.