# On Beamforming with the Single-Sideband Transform
## 3 Signal Model and Beamforming  
Let a device in a reverberant environment consist of \(M\) sensors and a loudspeaker. We assume the presence of a desired source and undesired uncorrelated noise at each sensor of the device. For simplicity, we assume that the environment and sources are spatially stationary, although this condition can be relaxed. Let \(y_{m}[n]\) be the observed signal at the \(m\)-th sensor, given by  
\[y_{m}[n]=x_{m}[n]+s_{m}[n]+r_{m}[n] \tag{30}\]  
where \(x_{m}[n]\) is a desired component, \(s_{m}[n]\) is the undesired interfering signal (from the loudspeaker) captured by the sensors, and \(r_{m}[n]\) is uncorrelated noise present in the sensors. The index \(m\) (\(1\leq m\leq M\)) refers to the different sensors. Regarding the environment as an LTI system, then  
\[ \begin{split} x_{m}[n]&=h_{m}[n]*x[n]\\ &=a_{m}[n]*x_{1}[n]\end{split}  \tag{31}\]  
with \(h_{m}[n]\) being the frequency response between the desired source \(x[n]\) and the \(m\)-th sensor, and \(a_{m}[n]\) is the relative impulse response between the reference (assumed to be \(m=1\)) and the \(m\)-th sensors. In the time-frequency domain, Equation (30) becomes  
\[Y_{m,k}[l]=X_{m,k}[l]+S_{m,k}[l]+R_{m,k}[l]. \tag{32}\]  
Hereafter, the notation is different than it was previously to ease reading and emphasize the frame index \(l\). From Equation (31), using Equation (26) and assuming the CTF model, we can write \(X_{m,k}[l]\) as  
\[X_{m,k}[l]\approx A^{\prime}_{m,k}[l]*X^{\prime}_{1,k}[l]+A^{\prime\prime}_{m,k}[l]*X^{\prime\prime}_{1,k}[l] \tag{33}\]  
with \(X^{\prime}_{1,k}[l]=X_{1,k}[l]\) and \(X^{\prime\prime}_{1,k}[l]=X_{1,K-k}[l]\), and \(A^{\prime}_{m,k}[l]\) being the RFRs between each sensor and the reference for same- and conjugate-frequencies. Note that \(A^{\prime}_{m,k}[l]\) is not strictly causal, depending on the direction of arrival and features of the reverberant environment, as well as relative delays between the sources at each sensor. It is trivial to see that, for Equation (33) to be respected, \(A^{\prime}_{1,k}[l]=A^{\prime\prime}_{1,k}[l]=\delta_{0,l}\), which is a Kronecker delta at \(l=0\). By approximating \(A^{n}_{m,k}[I]\) through a truncated RFR \(D^{n}_{m,k}[I]\) with \(L_{D}\) samples, we obtain  
\[ \begin{split} X^{n}_{m,k}[I]&=A^{n}_{m,k}[I]*X^{n}_{1, k}[I]\\ &\approx D^{n}_{m,k}[I]*X^{n}_{1,k}[I]\end{split}  \tag{34}\]  
in which \(D^{n}_{m,k}[I]\) has \(\Delta\) non-causal and \(\Lambda\) causal samples, with \(L_{D}=\Delta+\Lambda+1\). By defining \(L_{D}\times 1\) vectors \(\mathbf{x}^{n}_{1,k}[I]\) and \(\mathbf{d}^{n}_{m,k}\)  
\[\mathbf{x}^{n}_{1,k}[I] =\Big{[}\ X^{n}_{1,k}[I+\Delta]\,\ \cdots\,\ X^{n}_{1,k}[I]\,\ \cdots\,\ X^{n}_{1,k}[I-\Lambda]\ \Big{]}^{\mathsf{T}} \tag{35a}\] \[\mathbf{d}^{n}_{m,k} =\Big{[}\ A^{n}_{m,k}[-\Delta]\,\ \cdots\,\ A^{n}_{m,k}[0]\,\ \cdots\,\ A^{n}_{m,k}[\Lambda]\ \Big{]}^{\mathsf{T}}, \tag{35b}\]  
we can rewrite the convolution of Equation (34) as a vector multiplication, given by  
\[D^{n}_{m,k}[I]*X^{n}_{1,k}[I]=\mathbf{d}^{n}_{m,k}\ \mathbf{x}^{n}_{1,k}[I] \tag{36}\]  
with \((\cdot)^{\mathsf{T}}\) being the transpose operator. Then, our observed signal \(Y_{m,k}[I]\) becomes  
\[Y_{m,k}[I]\approx\mathbf{d}^{\prime}_{m,k}\ \mathbf{x}^{\mathsf{T}}_{1,k}[I]+ \mathbf{d}^{\prime\prime}_{m,k}\ \mathbf{x}^{\prime\prime}_{1,k}[I]+S_{m,k}[I]+R_{m,k}[I]. \tag{37}\]  
The signal vectors \(\mathbf{x}^{\prime}_{1,k}[I]\) and \(\mathbf{x}^{\prime\prime}_{1,k}[I]\) are still frame-dependent, while the RFR vectors \(\mathbf{d}^{\prime}_{m,k}\) and \(\mathbf{d}^{\prime\prime}_{m,k}\) are not due to the assumption of spatial stationarity. Taking the \(L_{Y}\) most recent samples of \(Y_{m,k}[I]\) implies  
\[ \begin{split}\mathbf{Y}_{m,k}[I]&=\Big{[}\ Y_{m,k}[ I]\,\ Y_{m,k}[I-1]\,\ \cdots\,\ Y_{m,k}[I-L_{Y}+1]\ \Big{]}^{\mathsf{T}}\\ &=\bar{\mathbf{D}}^{\prime}_{m,k}\bar{\mathbf{x}}^{\prime}_{1,k} [I]+\bar{\mathbf{D}}^{\prime}_{m,k}\bar{\mathbf{x}}^{\prime\prime}_{1,k}[I]+ \mathbf{S}_{m,k}[I]+\mathbf{R}_{m,k}[I]\end{split}