# On Beamforming with the Single-Sideband Transform
## 5 Conclusions  
We conducted a comprehensive investigation into the Single-Sideband Transform within the context of beamforming, examining its mathematical properties and interaction with key processes such as convolution, relative frequency response estimation, and the distortionless constraint. Our theoretical study reveals that despite its interesting real-valued representation, the SSBT exhibits higher susceptibility to errors in RFR estimation, requiring stricter constraints for its proper application. We found that in scenarios with longer time windows and under the multiplicative transfer function model, the estimation errors with the SSBT are reduced. Furthermore, we established that the SSBT and the Short-Time Fourier Transform are interchangeable in the time-frequency domain without  
Figure 5: Output metrics for the beamformers for varying iSERs, with \(N=512\) samples.  
the need for their respective inverse transforms, enabling a seamless conversion between the two.  
To validate our theoretical findings, we employed both transforms in the design of a convolutive Minimum-Power Distortionless Response beamformer within a reverberant environment across various scenarios. These practical results support our theoretical claims, showing that the SSBT-based filter slightly underperforms the STFT-based one in optimal conditions and significantly underperforms in non-ideal situations. While these findings highlight challenges in directly applying the SSBT for beamforming, the interchangeability of these transforms allows for filtering in the STFT domain--even when signals are initially in the SSBT domain. This allows for the combined use of the transformations, taking advantage of their respective strengths as applicable. Future research could explore integrating the SSBT into more robust beamformers, further comparing it to the STFT, and studying the cooperative integration of the two transforms in greater depth.  
Conceptualization, I.C. and V.P.C.; Methodology, V.P.C.; Formal analysis, V.P.C.; Software, V.P.C.; Writing--original draft V.P.C.; Writing--review and editing, I.C. and V.P.C.; Supervision, V.P.C. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the Pazy Research Foundation and the Israel Science Foundation (grant no. 1449/23).  
The source-code for the simulations developed here is available at [[https://github.com/VCuratrelli/py-ssb-ctf-bf](https://github.com/VCuratrelli/py-ssb-ctf-bf)]([https://github.com/VCuratrelli/py-ssb-ctf-bf](https://github.com/VCuratrelli/py-ssb-ctf-bf)) (accessed on 21 August 2024).  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} CTF & Convolutive Transfer Function \\ DI & Directivity Index \\ DSRF & Desired Signal Reduction Factor \\ FT & Fourier Transform \\ MPDR & Minimum-Power Distortionless-Response \\ MTF & Multiplicative Transfer Function \\ RFR & Relative Frequency Response \\ RFT & Real Fourier Transform \\ SER & Signal-to-Echo Ratio \\ SNR & Signal-to-Noise Ratio \\ SSBT & Single-Sideband Transform \\ STFT & Short-Time Fourier Transform \\ \end{tabular}