# On Beamforming with the Single-Sideband Transform
## 2 Frequency and Time-Frequency Transforms
### RFR Estimation for Time-Frequency Transforms  
Under the same assumptions as in Section 2.2, we can write the output of our systems with the CTF model in the SSBT domain as  
\[X_{m;\mathbb{S}}[I,k]=X_{1;\mathbb{S}}^{\prime}[I,k]*A_{m;\mathbb{S}}^{\prime }[I,k]+X_{1;\mathbb{S}}^{\prime\prime}[I,k]*A_{m;\mathbb{S}}^{\prime\prime}[I,k] \tag{26}\]  
where \(X_{1;\mathbb{S}}^{\prime}[I,k]\), \(X_{1;\mathbb{S}}^{\prime\prime}[I,k]\), \(A_{m;\mathbb{S}}^{\prime}[I,k]\) and \(A_{m;\mathbb{S}}^{\prime\prime}[I,k]\) are defined similarly to their counterparts from Section 2.2. In particular, given \(X_{m;\mathcal{I}}[I,k]\) and \(X_{1;\mathcal{I}}[I,k]\) under the STFT, we have  
\[\mathbf{E}\left\{X_{m;\mathcal{I}}[I+\lambda,k]X_{1;\mathcal{I}}^ {*}[I,k]\right\} =\mathbf{E}\left(\sum_{\tau}A_{m;\mathcal{I}}[\tau,k]X_{1; \mathcal{I}}[I+\lambda-\tau,k]X_{1;\mathcal{I}}^{*}[I,k]\right\} \tag{27}\] \[=\sum_{\tau}A_{m;\mathcal{I}}[\tau,k]\mathbf{E}\left\{X_{1; \mathcal{I}}[I+\lambda-\tau,k]X_{1;\mathcal{I}}^{*}[I,k]\right\}\] \[=\sum_{\tau}A_{m;\mathcal{I}}[\tau,k]\sigma_{X_{1;\lambda-\tau}}^{2 }[k]\] \[=A_{m;\mathcal{I}}[\lambda,k]\sigma_{X_{1;0}}^{2}[k]+\sum_{\tau \
eq\lambda}A_{m;\mathcal{I}}[\tau,k]\sigma_{X_{1;\lambda-\tau}}^{2}[k].\]  
This is equal to \(A_{m;\mathcal{I}}[\lambda,k]\sigma_{X_{1;0}}^{2}\) iff \(\sigma_{X_{1;\lambda-\tau}}^{2}[k]=0\;\forall\;\lambda\
eq\tau\), which is equivalent to different samples of \(X_{1;\mathcal{I}}[I,k]\) being uncorrelated. This contradicts the fact that \(X_{1;\mathcal{I}}[I,k]\) is the output of a convolution, thus having correlated samples. The overlap between windows in the transform also contributes to the correlation of different windows. With this, the summation over \(\tau\
eq\lambda\) is an error term in the RFR estimation.  
Applying this same process to the SSBT, we have that  
\[\begin{split}&\mathbf{E}\big{\{}X_{1;S}\left[l,k\right]X_{m;S}\left[l+ \lambda,k\right]\big{\}}\\ &=A_{m;S}\left[\lambda,k\right]\sigma_{X_{1;0}}^{2}[k]+\sum_{ \tau\
eq\lambda}A^{\prime}_{m;\mathcal{S}}\left[\tau,k\right]\sigma_{X^{ \prime},\lambda-\tau}^{2}[k]+A^{\prime\prime}_{m;S}\left[\tau,k\right]\sigma_{ X^{\prime}X^{\prime\prime},\lambda-\tau}^{2}[k].\end{split} \tag{28}\]  
In this scenario, there is an error term for the same-frequency component but also one for the cross-frequency interference. Expanding \(\sigma_{X^{\prime}X^{\prime\prime},\lambda-\tau}^{2}[k]\) leads to  
\[\begin{split}\sigma_{X^{\prime}X^{\prime\prime},\lambda-\tau}^{2 }[k]&=\mathbf{E}\left\{X_{1;S}^{\prime}[l+\tau-\lambda,k]X_{1;S}^{ \prime\prime}[l,k]\right\}\\ &=\sigma_{X}^{2}[k]\sum_{\mu\geq 0}(H^{\prime}_{1,\mu}-H^{\prime}_{ 1,-\mu})(H^{\prime\prime}_{1,\mu+(\lambda-\tau)}-H^{\prime\prime}_{1,\mu-( \lambda-\tau)}).\end{split} \tag{29}\]  
Three variables are of interest: the frame delay between different sensors \(\lambda\), the RFR index \(\tau\), and the delay between same- and cross-frequency \(\mu\). In all cases where \(\mu\
eq 0\) and \(\tau\
eq\lambda\), the summation term of Equation (29) is not trivially null, and, therefore, the summation over \(\tau\) in Equation (28) is also not identically zero. Comparing Equations (27) and (28) reveals that the SSBT estimation introduces error terms due to the cross-frequency cross-frame correlation, an issue absent in the STFT. This difference diminishes the robustness and performance of beamformers designed in the SSBT domain, potentially increasing output distortion due to inaccuracies in estimating relative frequency responses.