## 8 Multi-Segmental Channel Estimation
### Least-Squares-Based MSCE  
For MSCE, we split the received signal \(r_{n},n=0,1,\ldots,N_{s}-1\) into \(T\) segments, and each segment has \(M=N_{s}/T\) samples. The FFT can be performed partially as expressed as follows:  
\[\begin{split} y_{t}(d)&=\frac{1}{\sqrt{N_{s}}}\sum_{n= (t-1)N_{t}}^{tN_{t}-1}r_{n}e^{-j\frac{2\pi}{N_{s}}dn}\\ &=\frac{1}{N_{s}}\sum_{n=(t-1)N_{t}}^{tN_{t}-1}\sum_{l=0}^{N_{h}- 1}h_{tl}(n,l)e^{-j\frac{2\pi}{N_{s}}kl}\cdot\sum_{k=0}^{N_{s}-1}s_{k}e^{-j\frac {2\pi}{N_{s}}nk}e^{-j\frac{2\pi}{N_{s}}nd}\\ &=\frac{1}{N_{s}}\sum_{k=0}^{N_{s}-1}s_{k}\sum_{n=(t-1)N_{t}}^{tN _{t}-1}h_{f}(n,k)e^{-j\frac{2\pi}{N_{s}}n(d-k)}\end{split} \tag{55}\]  
where the quantity \(h_{f}(n,k)\) denotes the channel frequency response for the \(k\)-th subcarrier at time index \(n\). We assume that the channel remains constant during segment \(t\), so Equation (55) can be simplified as [7]:  
\[\begin{split} y_{t}(d)&=\sum_{k=0}^{N_{s}-1}s_{k}h_{ f}(t,k)\frac{1}{N_{s}}\sum_{n=(t-1)N_{t}}^{tN_{t}-1}e^{-j\frac{2\pi}{N_{s}}n(d-k)} \\ &=\sum_{k=0}^{N_{s}-1}s_{k}h_{f}(t,k)\delta_{t}(d-k),\end{split} \tag{56}\]  
where:  
\[\delta_{t}(d-k)=\frac{1}{T}e^{j2\pi(d-k)\frac{2t-1}{2T}}\text{sinc}(\frac{ \pi(d-k)}{T}). \tag{57}\]  
By defining \(\mathbf{y}_{t}=[y_{t}(0),\ldots,y_{t}(N_{s}-1)]^{T}\), \(\mathbf{h}_{df}(t)=[h_{df}(t,0),\ldots,h_{df}(t,N_{s}-1)]^{T}\), \(\mathbf{h}_{tl}(t)=[h_{tl}(t,0),\ldots,h_{tl}(t,l),\ldots,h_{tl}(t,L-1)]^{T}\), and the term \(\delta_{k}(t)\) in Equation (57) is re-organized in a matrix form as:  
\[\mathbf{\Delta}_{t}=\left[\begin{array}{ccc}\delta_{0}(t)&\ldots&\delta_{N _{s}-1}(t)\\ \vdots&\ddots&\vdots\\ \delta_{1-N_{s}}(t)&\ldots&\delta_{0}(t)\end{array}\right], \tag{58}\]  
Equation (55) for the \(t\)-th segment can be rewritten in the following form:  
\[\mathbf{y}_{t}=\underbrace{\sqrt{N_{s}}\mathbf{\Delta}_{t}\text{diag}( \mathbf{s})\mathbf{F}_{L}}_{\mathbf{A}}\mathbf{h}_{tl}(t), \tag{59}\]  
where \(\mathbf{h}_{df}(t)=\sqrt{N_{s}}\mathbf{F}_{L}\mathbf{h}_{tl}(t)\). Using LS estimation in the time domain [26], the channel estimates in the \(t\)-th segment can be obtained by:  
\[\mathbf{\hat{h}}_{tl}(t)=(\mathbf{A}^{H}\mathbf{A})^{-1}\mathbf{A}^{H} \mathbf{y}_{t}. \tag{60}\]  
Note that there are \(N_{p}\) pilot symbols already known to the receiver, and the zeroes can be set for the unreliable symbol estimates of \(\mathbf{\hat{s}}\) in each iteration, the reliability of which can be evaluated by LLRs fromthe channel decoder. In the following steps, we introduce the piece-wise linear model to approximate the channel impulse responses for the time-varying channels between the \(t\)-th segment and \((t+1)\)-th segment, as in [34]. Hence, the channel estimates \(\mathbf{\hat{h}}_{tl}(n)=[\hat{h}_{tl}(n,0),\ldots,\hat{h}_{tl}(n,l),\ldots,\hat {h}_{tl}(n,L-1)]^{T}\) for different time indices can be obtained, and the corresponding channel frequency response \(\hat{h}_{df}(d,k)\) can be obtained. The matrix \(\mathbf{\Delta}_{t}\) can be pre-computed, once the number of segments \(T\) is determined. In addition, the differences of channel estimates between the \((p-1)\)-th and \(p\)-th iteration can be measured, which can help us to terminate the MSCE at an earlier iteration.