# On Beamforming with the Single-Sideband Transform
## Appendix A Properties of the Real Fourier Transform
Taking the correlation between the two conjugate-frequencies yields  
\[\mathbb{E}\big{\{}X_{\mathcal{R}}(f)X_{\mathcal{R}}(-f)\big{\}}=\mathbb{E} \big{\{}X_{\mathcal{F}}^{\mathbf{R}}(f)^{2}-X_{\mathcal{F}}^{\mathbf{I}}(f)^{ 2}\big{\}}\,.\] (A34)  
Under the same assumptions that the real and imaginary parts of \(X_{\mathcal{F}}(f)\) are identically distributed, we obtain the same result as before, where  
\[\mathbb{E}\big{\{}X_{\mathcal{R}}(f)X_{\mathcal{R}}(-f)\big{\}}=0\,.\] (A35)  
Note that, with the RFT, we did not use the complex correlation since it is real-valued. Lastly, we take the correlation between conjugate frequencies of the output of a system according to Equations (A25) and (A26):  
\[\mathbb{E}\big{\{}Y_{\mathcal{R}}(f)Y_{\mathcal{R}}(-f)\big{\}} =H_{\mathcal{F}}^{\mathbf{R}}(f)^{2}\mathbb{E}\big{\{}X_{\mathcal{ R}}(f)X_{\mathcal{R}}(-f)\big{\}}-H_{\mathcal{F}}^{\mathbf{R}}(f)H_{ \mathcal{F}}^{\mathbf{I}}(f)\mathbb{E}\big{\{}X_{\mathcal{R}}(f)^{2}\big{\}}\] (A36) \[\tilde{Y}_{\mathcal{R}}=-H_{\mathcal{F}}^{\text{R}}X_{\mathcal{F}}^{\text{R}}- H_{\mathcal{F}}^{\text{R}}X_{\mathcal{F}}^{\text{I}}-H_{\mathcal{F}}^{\text{I}}X _{\mathcal{F}}^{\text{R}}+H_{\mathcal{F}}^{\text{I}}X_{\mathcal{F}}^{\text{I }}\]0  
Since \(X_{\mathcal{R}}(f)\) and \(X_{\mathcal{R}}(-f)\) are independent and zero-mean, the first and last terms are zero, and the other two also cancel each other out since \(\mathbb{E}\big{\{}X_{\mathcal{R}}(f)^{2}\big{\}}=\mathbb{E}\big{\{}X_{\mathcal{ R}}(-f)^{2}\big{\}}=2\sigma_{f}^{2}\). Therefore,  
\[\tilde{Y}_{\mathcal{R}}=-H_{\mathcal{F}}^{\text{R}}X_{\mathcal{F}}^{\text{R}}- H_{\mathcal{F}}^{\text{R}}X_{\mathcal{F}}^{\text{I}}-H_{\mathcal{F}}^{\text{I}}X _{\mathcal{F}}^{\text{R}}+H_{\mathcal{F}}^{\text{I}}X_{\mathcal{F}}^{\text{I }}\]1 (A37)  
Relative frequency responses with the RFT are even for frequency-to-frequency and odd for conjugate frequency. Proof. : First, given two real systems with a shared input \(x(t)\), each with an impulse response \(h_{m}(t)\) and an output \(y_{m}(t)\), such that  
\[\tilde{Y}_{\mathcal{R}}=-H_{\mathcal{F}}^{\text{R}}X_{\mathcal{F}}^{\text{R}}- H_{\mathcal{F}}^{\text{R}}X_{\mathcal{F}}^{\text{I}}-H_{\mathcal{F}}^{\text{I}}X _{\mathcal{F}}^{\text{R}}+H_{\mathcal{F}}^{\text{I}}X_{\mathcal{F}}^{\text{I }}\]2 (A38) \[\tilde{Y}_{\mathcal{R}}=-H_{\mathcal{F}}^{\text{R}}X_{\mathcal{F}}^{\text{R}}- H_{\mathcal{F}}^{\text{R}}X_{\mathcal{F}}^{\text{I}}-H_{\mathcal{F}}^{\text{I}}X _{\mathcal{F}}^{\text{R}}+H_{\mathcal{F}}^{\text{I}}X_{\mathcal{F}}^{\text{I }}\]3and, for their conjugate frequencies,  
\[ \begin{split} Y_{1}(-f)&=X(-f)H_{1,\mathcal{F}}^{\text{R} }(-f)+X(f)H_{1,\mathcal{F}}^{\text{I}}(-f)\\ Y_{2}(-f)&=X(-f)H_{2,\mathcal{F}}^{\text{R}}(-f)+X(f )H_{2,\mathcal{F}}^{\text{I}}(-f)\end{split}  \tag{109}\]  
where we omit the transform index when it is an RFT's domain value for compactness. If we define \(X_{1}^{\prime}(f)\) and \(X_{1}^{\prime\prime}(f)\) as  
\[ \begin{split} X_{1}^{\prime}(f)&=Y_{1}(f)\\ X_{1}^{\prime\prime}(f)&=Y_{1}(-f)\end{split}  \tag{110}\]  
then  
\[ \begin{split} Y_{m}(f)&=A_{m}^{\prime}(f)X_{1}^{ \prime}(f)+A_{m}^{\prime\prime}(f)X_{1}^{\prime\prime}(f)\\ Y_{m}(-f)&=A_{m}^{\prime}(-f)X_{1}^{\prime}(-f)+A_ {m}^{\prime\prime}(-f)X_{1}^{\prime\prime}(-f)\end{split}  \tag{111}\]  
where \(A_{m}^{\prime}(f)\) and \(A_{m}^{\prime\prime}(f)\) are given by  
\[ \begin{split} A_{m}^{\prime}(f)&=\frac{H_{1}^{ \prime}(f)H_{m}^{\prime}(f)+H_{1}^{\prime\prime}(f)H_{m}^{\prime\prime}(f)}{H_ {1}^{\prime 2}(f)+H_{1}^{\prime 2}(f)}\\ A_{m}^{\prime\prime}(f)&=\frac{H_{1}^{\prime}(f)H_{ m}^{\prime\prime}(f)-H_{1}^{\prime\prime}(f)H_{m}^{\prime\prime}(f)}{H_{1}^{ \prime 2}(f)+H_{1}^{\prime\prime 2}(f)}\end{split}  \tag{112}\]  
and, for their conjugate frequencies,  
\[