# On Beamforming with the Single-Sideband Transform
## 3 Signal Model and Beamforming
### Filtering and the MPDR Beamformer  
Our objective is to recover the desired signal at the reference sensor \(X_{1,k}\{l\}\) without any distortion while minimizing the output signal's power. For this, a linear filter \(\mathbf{f}_{k}\{l\}\) is employed, yielding an estimate \(Z_{k}\{l\}\) of our desired signal:  
\[\begin{split} Z_{k}^{n}[l]&\approx X_{1,k}\{l\}\\ &=\mathbf{f}_{k}^{\mathrm{H}}[l]\mathbf{y}_{k}\{l\}\end{split} \tag{42}\]  
with \((\cdot)^{\mathrm{H}}\) being the transposed-complex-conjugate operator. Using Equation (40b) in Equation (42) yields  
\[\begin{split} Z_{k}\{l\}&=\mathbf{f}_{k}^{\mathrm{H}} [l]\mathbf{D}_{k}^{\prime}\mathbf{\tilde{x}}_{1,k}^{\prime}[l]+\mathbf{f}_{k}^ {\mathrm{H}}[l]\mathbf{D}_{k}^{\prime\prime}\mathbf{\tilde{x}}_{1,k}^{\prime \prime}[l]+\mathbf{f}_{k}^{\mathrm{H}}[l]\mathbf{s}_{k}\{l\}+\mathbf{f}_{k}^ {\mathrm{H}}[l]\mathbf{r}_{k}\{l\}\\ &=X_{f,k}\{l\}+S_{f,k}\{l\}+R_{f,k}\{l\} \tag{43}\]  
where \(X_{f,k}\{l\}\) is the filtered desired signal, \(S_{f,k}\{l\}\) is the filtered interference signal, and \(R_{f,k}\{l\}\) is the filtered noise signal. In particular, the filtered desired signal is  
\[X_{f,k}\{l\}=\mathbf{f}_{k}^{\mathrm{H}}\{l\}\mathbf{D}_{k}^{\prime}\mathbf{ \tilde{x}}_{1,k}^{\prime}[l]+\mathbf{f}_{k}^{\mathrm{H}}\{l\}\mathbf{D}_{k}^{ \prime\prime}\mathbf{\tilde{x}}_{1,k}^{\prime\prime}[l] \tag{44}\]  
where each component of the desired signal is exposed. The distortionless constraint on the desired signal is translated as  
\[X_{f,k}\{l\}=X_{1,k}\{l\} \tag{45}\]  
which is equivalent to requiring that each component of \(X_{1,k}\{l\}\) is perfectly recovered. This means maintaining the same-frequency component while nulling the cross-frequency parcel that appears in Equation (44). From Equation (39b), we have that the desired signal for the current index \(l\) is the \((\Delta+1)\)-th element of \(\mathbf{\tilde{x}}_{1,k}^{n}\{l\}\); thus, the constraints are  
\[\mathbf{f}_{k}^{\mathrm{H}}\{l\}\mathbf{D}_{k}^{\prime} =\mathbf{i}_{\Delta}^{\mathrm{T}} \tag{46a}\] \[\mathbf{f}_{k}^{\mathrm{H}}\{l\}\mathbf{D}_{k}^{\prime\prime} =\mathbf{0}^{\mathrm{T}} \tag{46b}\]  
where \(\mathbf{i}_{\Delta}\) is an \(L\times 1\) vector of zeros except for the \((\Delta+1)\)-th entry, which is \(1\), and \(\mathbf{0}\) is an \(L\times 1\) vector of zeros. For the STFT, only the first constraint of Equation (46) is considered since \(\mathbf{D}_{k}^{\prime\prime}\) is identically zero by definition, and, therefore, the second condition is trivially satisfied. With this, we write our constraint matrix as  
\[\mathbf{f}_{k}^{\mathrm{H}}\{l\}\mathbf{C}_{k}=\mathbf{i}^{\mathrm{T}} \tag{47}\]  
where for the STFT, \(\mathbf{C}_{k}=\mathbf{D}_{k}^{\prime}\) and \(\mathbf{i}=\mathbf{i}_{\Delta}\); and for the SSBT, \(\mathbf{C}_{k}=\left[\mathbf{D}_{k}^{\prime}\,\ \mathbf{D}_{k}^{\prime\prime}\right]\), and \(\mathbf{i}=\left[\mathbf{i}_{\Delta}\right]\).  
To minimize the output signal's power while satisfying the distortionless constraint, a Minimum-Power Distortionless Response (MPDR) beamformer is used, defined by  
\[\mathbf{f}_{\mathrm{mpd}r;k}\{l\}=\min_{\mathbf{f}_{k}\{l\}}\mathbf{f}_{k}^{ \mathrm{H}}\{l\}\mathbf{\Gamma}_{\alpha;k}\{l\}\mathbf{f}_{k}\{l\}\text{ s.t. }\mathbf{f}_{k}^{\mathrm{H}}\{l\}\mathbf{C}_{k}=\mathbf{i}^{\mathrm{T}} \tag{48}\]  
where \(\mathbf{\Gamma}_{\alpha;k}\{l\}=(1-\alpha)\mathbf{\Gamma}_{\mathbf{y}_{k}