# Advanced Noncoherent Detection in Massive MIMO Systems via Digital Beamspace Preprocessing
## 2 System Model
### Noncoherent Detection
#### 2.4.1 Full-Array  
Considering a full-array architecture, a first choice of the beamforming matrix is to select \(N_{\text{rx}}\) steering vectors defined in (2) as the respective columns, i.e.,  
\[\mathbf{W}_{\text{BF,B}}=[\mathbf{w}_{\text{B,1}},\mathbf{w}_{\text{B,2}},\ldots,\mathbf{w}_{ \text{B,}N_{\text{rx}}}] \tag{17}\]  
and  
\[\mathbf{w}_{\text{B,m}}=\mathbf{a}_{N_{\text{rx}}}(\phi_{m})=\frac{1}{\sqrt{N_{\text{ rx}}}} \begin{bmatrix}1\\ \text{e}^{-\text{i}kd_{\text{a}}\sin(\phi_{m})}\\ \vdots\\ \text{e}^{-\text{i}k(N_{\text{rx}}-1)d_{\text{a}}\sin(\phi_{m})}\end{bmatrix} \, \tag{18}\]  
where \(\phi_{m}\in[-\Phi/2,\Phi/2]\) is the corresponding beam direction of the \(m\)-th receive branch chosen from a given FoV \(\Phi\). This method is also known as conventional or Bartlett beamforming (BBF) [36]. It can also be interpreted as an \(N_{\text{rx}}\)-point discrete Fourier transform (DFT), where the weights \(\mathbf{w}_{\text{B,m}}\) represent a bank of narrowband filters each tuned to the spatial frequency  
\[\theta_{m}=\frac{d_{\text{a}}}{\lambda}\sin(\phi_{m})\, \tag{19}\]  
which is related to the respective angles \(\phi_{m}\) or rather the spatial location \(m\). A uniform sampling in the frequency domain given a FoV \(\Phi\) leads to a spatial frequency spacing  
\[\Delta\theta=\frac{2d_{\text{a}}}{N_{\text{rx}}\lambda}\sin(\Phi/2). \tag{20}\]  
Considering the entire visible range \(\Phi=\pi\) and the general antenna distance \(d_{\text{a}}=\lambda/2\), the \(N_{\text{rx}}\) spatial beams become orthogonal with spacing \(\Delta\theta=1/N_{\text{rx}}\). This corresponds to an angular spacing of \(\Delta\phi=\sin^{-1}(\lambda/D)\), which is a measure of the angular resolution of a uniform linear array with aperture size \(D=N_{\text{rx}}\,d_{\text{a}}\) also known as Rayleigh resolution limit [39]. Strictly speaking, this does not hold for reception at endfire directions, where the resolution drops due to a reduced array aperture size. Furthermore, the FoV is limited by the antenna pattern in practice. Thus, the orthogonality of the beams vanishes when uniformly chosen from a smaller angular range \(\Phi<\pi\). Alternatively, the antenna distance \(d_{\text{a}}\) could be optimized (increased) for a specific FoV such that \(\Delta\theta=1/N_{\text{rx}}\) holds in (20) in order to restore the orthogonality. The resultant spatial power distribution of a plane-wave phase front associated with direction \(\phi\) can be expressed as [39]  
\[\left|\mathbf{w}_{\text{B},m}^{\text{H}}\mathbf{a}_{N_{\text{rc}}}(\phi) \right|^{2} =\frac{1}{N_{\text{rc}}^{2}}\left|\sum_{n=1}^{N_{\text{rc}}}\text{ e}^{-\text{j}nkd_{\text{a}}(\sin(\phi)-\sin(\phi_{m}))}\right|^{2}\] \[=\frac{1}{N_{\text{rc}}^{2}}\left|\frac{\sin\left(\frac{N_{\text{ rc}}}{2}kd_{\text{a}}\left(\sin(\phi)-\sin(\phi_{m})\right)\right)}{\sin\left( \frac{1}{2}kd_{\text{a}}\left(\sin(\phi)-\sin(\phi_{m})\right)\right)}\right|^{2}\] \[\approx\text{sinc}^{2}\left(\frac{\pi D}{\lambda}\left(\sin(\phi )-\sin(\phi_{m})\right)\right)\,, \tag{21}\]  
which is approximately a squared sinc-function sampled at the chosen beam directions \(\phi_{m}\) of the \(m\)-th receive branch. Its peak is shifted towards the spatial location, where the direction of the incoming wave matches one of the generated beams, i.e., where \(\phi_{m}=\phi\). This is exemplary depicted in Figure 2, where two plane waves separated by two adjacent beams impinge at an array of \(N_{\text{rc}}=12\) elements considering an FoV of \(\Phi=\pi/2\). Apparently, the signals are not entirely resolvable, since the selected beams are not orthogonal in case of \(d_{\text{a}}=\lambda/2\). This in turn results into (strong) spatial interference. Hence, when the angular separation of the impinging signals is small, the spatial interference rejection capability using Bartlett beamforming is limited.