# On the Performance of Partial LIS for 6G Systems
## 2 System and Signal Characterization
#### System and Signal Model for the Receivers  
It is known that the ZF and MMSE receivers are complex, from a computation point of view, as they require a channel inversion for each frequency component. This translates into heavy battery use. The ZF receiver presents the advantage of intersymbol interference remotion that is undertaken in a very efficient manner. Nevertheless, it also has the disad  
Figure 1: Block diagram of an LIS System.  
Figure 2: Block diagram of a partial LIS.  
vantage of presenting enhanced noise for average-to-high noise levels, which translates into a performance degradation. On the other hand, MRC and EGC receivers avoid such complexity, leading to a very simple receiver that is especially useful in an LIS scenario in which the number of antenna elements is extremely high. Nevertheless, MRC and EGC receivers generate some level of interference in the decoding process, which should be suppressed with iterative receivers.  
Assuming a non-iterative receiver, the estimated frequency domain data symbols \(\mathbf{\tilde{S}}_{k}=\left[\mathbf{\tilde{S}}_{k}^{(1)},\ldots,\mathbf{ \tilde{S}}_{k}^{(R)}\right]^{T}\) become  
\[\mathbf{\tilde{S}}_{k}=\mathbf{B}_{k}\mathbf{Y}_{k} \tag{2}\]  
Depending on the algorithm, \(\mathbf{B}_{k}\) can be computed [16] as  
1. Using the ZF receiver, \(\mathbf{B}_{k}\) becomes \[\mathbf{B}_{k}=\mathbf{H}_{k}^{H}\left(\mathbf{H}_{k}\mathbf{H}_{k}^{H}\right) ^{-1}\] (3)
2. Using the MMSE receiver, \(\mathbf{B}_{k}\) becomes \[\mathbf{B}_{k}(\mathbf{H}_{k}\mathbf{H}_{k}^{H}+\beta\mathbf{I})^{-1}\mathbf{ H}_{k}^{H}\] (4) where \(\beta=\frac{\sigma_{N}^{2}}{\sigma_{S}^{2}}=\frac{\frac{\epsilon\left[\left|N_ {k}\right|^{2}\right]}{2}}{\frac{\epsilon\left[\left|S_{k}\right|\right]}{2}}\) and where \(\mathbf{I}\) is an \(R\times R\) identity matrix.
3. Using the MRC receiver, \(\mathbf{B}_{k}\) becomes \[\mathbf{B}_{k}=\mathbf{H}_{k}^{H}\] (5)
4. Using the EGC receiver, \(\mathbf{B}_{k}\) becomes \[\mathbf{B}_{k}=\exp\left\{j\arg\left(\mathbf{H}_{k}^{H}\right)\right\}\] (6)  
A very efficient receiver commonly associated with SC-FDE schemes is the iterative block-decision feedback equalization (IB-DFE) [15]. Such an iterative receiver uses feedforward and feedback coefficients to process the signals in the frequency domain, reaching a performance that is typically much better than that of a non-iterative receiver. IB-DFE can be viewed as turbo equalization [15]. As described above, the MRC and EGC receivers are very simple from a computation point of view, but generate some residual interference in the decoding process, especially for moderate values of \(T/R\). This can be mitigated by employing an iterative receiver, similar to the IB-DFE, that implements the following function [16]:  
\[\mathbf{\tilde{S}}_{k}=\mathbf{B}_{k}^{H}\mathbf{Y}_{k}-\mathbf{C}_{k} \mathbf{\tilde{S}}_{k} \tag{7}\]  
The interference cancellation matrix \(\mathbf{C}_{k}\) can be computed as [16]  
\[\mathbf{C}_{k}=\mathbf{A}_{k}^{H}\mathbf{H}_{k}-\mathbf{I} \tag{8}\]  
where \(\mathbf{I}\) is an \(R\times R\) identity matrix.