# Review of Channel Estimation for Candidate Waveforms of Next Generation Networks
## 4 System Model Description of FBMC OFDM-OQAM and GFDM-OQAM
### OFDM-OQAM Transceiver
#### 4.1.3 Channel Estimation for OFDM-OQAM  
The performances of two linear and three adaptive based CE algorithms are analysed for the described filter bank OFDM-OQAM based transceiver. These linear algorithms are LS and LMMSE, and the adaptive estimation schemes are LMS, normalised LMS (NLMS) and recursive LS (RLS). Least Square  
The LS channel estimate is given as [100]:  
\[ \begin{split}\hat{\mathbf{H}}_{LS}[n,k]&=\left[ \hat{H}_{LS}[n,0],\hat{H}_{LS}[n,1],\ldots,\hat{H}_{LS}[n,(L_{ch}-1)]\right]^{T }\\ &=\left[\frac{L_{ch}}{M}\left(\sum_{k=0}^{\frac{M}{L_{ch}}-1} \frac{s[n,k]}{x[n,k]},\sum_{k=ML_{ch}}^{\frac{2M}{L_{ch}}-1}\frac{s[n,k]}{x[n,k]},\ldots,\sum_{k=M(L_{ch}-1)L_{ch}}^{M-1}\frac{s[n,k]}{x[n,k]}\right)\right]^{T }.\end{split}  \tag{32}\]  
LMMSE  
The simplified LMMSE channel estimate \(\hat{\mathbf{H}}_{LMMSE}[n,k]\) is generally given as [73; 86; 137]:  
\[\hat{\mathbf{H}}_{LMMSE}[n,k]=\mathbf{R}_{Hs}[n,k]\mathbf{R}_{ss}^{-1}[n,k]s[ n,k], \tag{33}\]  
where \(\mathbf{R}_{Hs}[n,k]\) is the cross-covariance matrix between \(H[n,k]\) and \(s[n,k]\) expressed as:  
\[\mathbf{R}_{Hs}[n,k]=E[Hs^{H}]=\mathbf{R}_{HH}[n,k]\mathbf{X}^{H}[n,k], \tag{34}\]  
and \(\mathbf{R}_{ss}[n,k]\) is the auto-covariance matrix of \(s[n,k]\) expressed as:  
\[\mathbf{R}_{ss}[n,k]=E[ss^{H}]=\mathbf{X}[n,k]\mathbf{R}_{hh}[n,k]\mathbf{X}^ {H}[n,k]+\sigma_{\eta}^{2}\mathbf{I}[m]. \tag{35}\]  
In Equation (34), \(\mathbf{X}[n,k]=diag[\mathbf{x}[n,k]]\). Therefore, substituting Equations (34) and (35) into Equation (33) gives:  
\[\hat{\mathbf{H}}_{LMMSE}[n,k]=\frac{\mathbf{R}_{HH}[n,k]s[n,k]}{\mathbf{X}[n, k]\Big{[}\mathbf{R}_{HH}[n,k]+(\mathbf{X}^{H}[n,k]\mathbf{X}[n,k])^{-1} \sigma_{\eta}^{2}\mathbf{I}[n,k]\Big{]}}, \tag{36}\]  
where \(\mathbf{R}_{HH}\) is the autocorrelation matrix of the channel impulse response given by Equation (11). The notation \(\sigma_{\eta}^{2}\) denotes the noise variance \(E|\eta[n,k]|^{2}\) defined in Equation (21), \(\mathbf{I}[n,k]\) is an identity matrix, and \(E\) is the expectation notation. Least Mean Square  
This cost function to be minimised for LMS based CE is expressed as [139; 136; 19; 19]:  
\[\mathbf{J}(\mathbf{n})=E\{|\mathbf{e}[n,k]|^{2}\}=E\{\mathbf{e}[n,k]\mathbf{ e}^{*}[n,k]\} \tag{37}\]  
where \(E\{\cdot\}\) is the expectation operator, \(*\) denotes complex conjugation, and \(e^{*}[n,k]\) is the CE error given as:  
\[\mathbf{e}^{*}[n,k]=s^{*}[n,k]-\hat{H}[n,k]x^{H}[n,k]. \tag{38}\]  
The LMS for the filter bank OFDM-OQAM system is given as:  
\[\hat{\mathbf{H}}_{LMS}[n+1,k]=\hat{H}[n,k]+\mu\mathbf{e}^{*}[n,k]x[n,k], \tag{39}\]  
where \(\mu\) is the fixed positive step size parameter that satisfies the condition \(0<\mu<2\). Normalised Least Mean Square  
In NLMS CE, the estimation error is normalised by the power of the input signal. The CIR for this algorithm is represented as [136; 137; 138; 139; 140]:  
\[\mathbf{\hat{H}}_{NLMS}[n+1,k]=\hat{H}[n,k]+\frac{\mu\mathbf{e}^{*}[n,k]x[n,k] }{||x[n,k]||^{2}}, \tag{40}\]  
where \(||x[n,k]||^{2}\) represents the Euclidean norm of the staggered input QQAM symbols, \(\mu\) is the step size parameter and the error \(e^{*}[n,k]\) is expressed as in Equation (38). Recursive Least Square  
The index of performance to be minimised for RLS is [136; 141]:  
\[\mathbf{J}(\mathbf{n})=\sum_{k=0}^{n-1}\lambda^{n-k}|\mathbf{e}[n,k]|^{2}, \tag{41}\]  
where \(\lambda\) is the forgetting/weighting factor, which assumes value in the range \(0<<\lambda<1\). The inverse autocorrelation matrix (\(\mathbf{R}^{-1}[n,k]\)) of input vector \(x[n,k]\) is given as:  
\[\mathbf{R}^{-1}[n,k]=\lambda^{-1}\mathbf{R}^{-1}[n-1,k]-\lambda^{-1}\mathbf{ \hat{g}}[n,k]x^{H}[n,k]\mathbf{R}^{-1}[n-1,k], \tag{42}\]  
where \(\mathbf{\hat{g}}[n,k]\) is the gain vector expressed as:  
\[\mathbf{R}_{Hs}[n,k]=E[Hs^{H}]=\mathbf{R}_{HH}[n,k]\mathbf{X}^{H}[n,k], \tag{34}\]0  
Therefore, the RLS channel estimate can be expressed as:  
\[\mathbf{R}_{Hs}[n,k]=E[Hs^{H}]=\mathbf{R}_{HH}[n,k]\mathbf{X}^{H}[n,k], \tag{34}\]1  
where \(\mathbf{e}^{*}[n,k]\) is defined by Equation (38).