# Machine Learning-Based 5G-and-Beyond Channel Estimation for MIMO-OFDM Communication Systems
## 3 Deep Learning-Based Channel Estimation
### Motivations  
As long as no inter-carrier interference occurs, each subcarrier can be expressed as an independent channel, therefore preserving the orthogonality among the subcarriers. The orthogonality allows each subcarrier component of the signal in (10) to be expressed as the Hadamard product of the transmitted signal and channel frequency response at the subcarrier [34] as  
\[\mathbf{y}_{b}(t)=\sum_{a=1}^{N_{T}}\mathbf{h}_{a,b}(t)\odot\mathbf{x}_{a}(t)+ \mathbf{n}_{b}(t), \tag{21}\]  
where \(\mathbf{n}_{b}(t)\), \(\mathbf{h}_{a,b}(t)\), and \(\mathbf{x}_{a}(t)\) are the Fourier transforms of the noise, channel, and signal, respectively (unless we are working in the frequency domain). Of all the traditional channel estimation methods, LS estimation is one of the most common approaches. We denote by \(\mathbf{\hat{h}}_{\text{LSb}}\) the channel estimate from the transmission antennas at the \(b\)-th receiver antenna obtained by this estimation method. LS estimation gives the closed-form expression of the channel estimate as [8]  
\[\mathbf{\hat{h}}_{\text{LSb}}(t)=\left([\mathbf{X}(t)^{H}\mathbf{X}(t)]\right) ^{-1}\mathbf{X}^{H}(t)\mathbf{y}_{b}(t), \tag{22}\]  
where \((\cdot)^{H}\) denotes the Hermitian transpose, and  
\[\mathbf{X}(t)=\left[\text{diag}(\mathbf{x}_{1}(t)),\cdots,\text{diag}(\mathbf{ x}_{N_{T}}(t))\right]^{T} \tag{23}\]  
Figure 4: The pilot structure used in the considered MIMO-OFDM system. is the \(N_{P}\times(N_{T}N_{P})\) matrix, denoting the transmitted signal from the transmission antennas; \(N_{P}\) is the number of the pilot signals in an OFDM symbol; and \((\cdot)^{T}\) is the regular transpose. The channel estimate from each transmission antenna can be formulated as  
\[\hat{\mathbf{h}}_{\text{LSbi}}(t)=\Big{[}\big{[}\hat{\mathbf{h}}_{\text{LSbi}}(t )\big{]}_{(i-1)N_{P}},\dots,\big{[}\hat{\mathbf{h}}_{\text{LSbi}}(t)\big{]}_{iN _{P}-1}\Big{]}^{T},i=1,\cdots,N_{T}. \tag{24}\]  
Then, the channel responses from all sub-carriers can be obtained by applying a linear interpolation method. It should be noted that LS estimation is a widely-used estimation approach because of its simplicity. Nevertheless, this technique does not exploit the side information from noise and statistical channel properties, such as the spatial correlation among antennas, in the estimation, and thus high channel estimation error can occur when applying LS estimation for propagation environments with a high mobility. To cope with the above drawbacks, one can utilize the LMMSE estimation approach, which minimizes the mean square error. For LMMSE estimation, the channel estimate is formulated in the closed form expression as [34]  
\[\hat{\mathbf{h}}_{\text{LMMSEbi}}(t)=\mathbf{R}_{\mathbf{h}\hat{\mathbf{h}}_ {\text{LSbi}}}\bigg{(}\mathbf{R}_{\mathbf{hh}}+\frac{\sigma_{n}^{2}}{\sigma_ {x}^{2}}\mathbf{I}_{N_{P}}\bigg{)}^{-1}\hat{\mathbf{h}}_{\text{LSbi}}(t),\quad i =1,\cdots,N_{T}, \tag{25}\]  
where \(\hat{\mathbf{h}}_{\text{LMMSEbi}}(t)\) is the LMMSE estimated channel from the \(i-\)th transmission antenna at the \(b-\)th receiver antenna, \(\mathbf{R}_{\mathbf{hh}}=\mathbb{E}\{\mathbf{h}\mathbf{h}^{H}\}\) is the auto-correlation matrix of the channel response in the frequency domain with the size of \(N_{P}\times N_{P}\); \(\mathbf{R}_{\mathbf{hh}\hat{\mathbf{h}}_{\text{LSbi}}}=\mathbb{E}\{\mathbf{h} \hat{\mathbf{h}}_{\text{LSbi}}^{H}\}\) is the cross-correlation between the actual channel and the channel estimate obtained by the LS estimation with the size of \(N_{\text{FFT}}\times N_{P}\); \(\sigma_{x}^{2}\) is the variance of the transmitted signals, respectively; \(\mathbf{I}_{N_{P}}\) is the identity matrix of size \(N_{P}\times N_{P}\). The impacts of both noise and spatial correlation among the antennas are taken into account by LMMSE estimation, which is able to improve the channel estimation accuracy. However, LMMSE estimation requires the prior knowledge of channel statistical properties; thus, the computational complexity is higher than LS estimation.