# Advanced Noncoherent Detection in Massive MIMO Systems via Digital Beamspace Preprocessing
## 2 System Model
### Noncoherent Detection
#### 2.4.1 Full-Array
The optimal solution in terms of spatial filtering and maximizing the SINR is obtained via [PERSON]'s method (or a minimum variance distortionless response filter (MVDR)) [36, 40, 41], where the signal of interest is passed undistorted while the power from all other spatial frequencies and directions is minimized. Based on the knowledge of the covariance matrix in (13), this data-dependent approach leads to a higher resolution and thus power focusing capability than the conventional beamforming. However, mismatches in the array response due to multi-path propagation and imprecise knowledge of the covariance matrix, which has to be estimated in practice, lead to a severe performance degradation [40]. The same applies to subspace-based methods such as multiple signal classification (MUSIC) [36, 42] or rotational invariance techniques (ESPRIT) [43], which offer high resolution capabilities. Figure 2: Spatial power distribution at \(N_{\text{rc}}=12\) receive branches (\(\bullet\)) in case of Bartlett beamforming (\(\blacklozenge\)) and in case of a combination with eigenspace-based post-processing (\(\blacklozenge\)) considering two incident waves (\(\blacksquare\)/ \(\blacksquare\)) and a field of view (FoV) of \(\Phi=\pi/2\). Alternatively, a combination of Bartlett and eigenspace-based processing [44], denoted as Bartlett-eigenspace beamforming (BEBF) in the following, can provide a more robust approach against aforementioned issues. The corresponding overall weighting matrix can be expressed as  
\[\mathbf{W}_{\text{BE,BE}}=\mathbf{W}_{\text{BE,B}}\mathbf{W}_{\text{BF,E}}\,, \tag{22}\]  
where  
\[\mathbf{W}_{\text{BF,E}}=[\mathbf{w}_{\text{E,1}},\mathbf{w}_{\text{E,2}},\dots,\mathbf{w}_{ \text{E,N}_{\text{rx}}}] \tag{23}\]  
denotes the eigenspace-based beamforming matrix. In an initial step, conventional beamforming is performed for spatial prefiltering according to  
\[\bar{\mathbf{R}}=\mathbf{W}_{\text{BE,B}}^{\text{H}}\mathbf{R}=[\mathbf{r}_{0},\mathbf{r}_{1}, \dots,\mathbf{r}_{N_{\text{bl}}-1}]\,. \tag{24}\]  
Due to the beamspace transformation, the array response vector changes to  
\[\bar{\mathbf{a}}_{N_{\text{rx}}}(\phi_{m})=\mathbf{W}_{\text{BF,B}}^{\text{H}}\mathbf{a}_ {N_{\text{rx}}}(\phi_{m})\,. \tag{25}\]  
The covariance matrix of (24) can be computed via its sample estimate and an eigenvalue decomposition is applied as follows:  
\[\Delta\theta=\frac{2d_{\text{a}}}{N_{\text{rx}}\lambda}\sin(\Phi/2). \tag{20}\]0  
where the diagonal matrices \(\mathbf{\Lambda}_{s}\) and \(\mathbf{\Lambda}_{n}\) contain the \(N_{\text{s}}\) largest (signal-subspace) and the \(N_{\text{rx}}-N_{\text{s}}\) smallest (noise-subspace) eigenvalues, respectively. \(\mathbf{V}_{s}\) and \(\mathbf{V}_{n}\) denote the corresponding eigenvectors. The respective weight vectors in (23) for eigenspace-based post-processing are composed according to  
\[\Delta\theta=\frac{2d_{\text{a}}}{N_{\text{rx}}\lambda}\sin(\Phi/2). \tag{20}\]1  
Comparing the resultant spatial power distribution to conventional beamforming as illustrated in Figure 2, the additional eigenspace-based post-processing restores the orthogonality of the selected beams and achieves a higher resolution as expected. This provides more flexibility since the beams do not need to be selected from a regular grid anymore. Most importantly, the energy of the incoming wave is focused almost entirely to one receive branch reducing the spatial interference. Noteworthy, this approach is more computational complex than conventional beamforming, which can be implemented via a fast Fourier transform (FFT) later on. Moreover, the knowledge of the number of sources \(N_{\text{s}}\) is required, i.e., the number of users \(N_{\text{u}}\), and this method can suffer from instabilities when performing the inverse of the estimated covariance matrix. Diagonal loading can counteract this issue [40].