# On Beamforming with the Single-Sideband Transform
## 1 Introduction  
Filtering is an essential tool available in the modern world, being necessary in fields such as communications [1], biomedical applications [2; 3; 4], and system control [5; 6], among other areas [7; 8; 9]. At its core, filtering mitigates the effects of undesired sources and can use temporal and spatial information regarding the signals, sources, sensors, and environment to enhance the signals. Given time samples, these filtering processes can be implemented in the time-, frequency-, and time-frequency domains [10], each offering distinct advantages. In particular, time-frequency methods exploit frequency-related information while dynamically adapting to signal and environmental changes in time, providing a tradeoff between strictly time or frequency information. Generically, transforms are the primary tool for achieving desired time-frequency domain data, with the Short-Time Fourier Transform (STFT) [11; 12] being prominent in the literature due to its widespread use. However, alternative transforms can also be employed [13; 14; 15], each offering a unique perspective and insight regarding the signal, leading to different performances depending on specific application requirements. Among these alternatives, an approach based on single-sideband modulation has occasionally been proposed [16] for various applications in signal enhancement [17; 18], from acoustic echo cancellation [19] to speech de-reverberation [20] and machine learning signal enhancement [21], showing varying levels of success. The Single-Sideband Transform (SSBT) holds promise in the field due to its real-valued representation. This characteristic simplifies mathematical developments by avoiding complex-valued coefficients or matrices, and it also leads to a more straightforward hardware implementation, resulting in more cost-efficient devices. Previous research shows that this transform works best with short time-frequency frames [20; 21]. Despite this, the research on applying this transform for beamforming is limited. In particular, its potential for spatio-temporal multisensorbeamforming in reverberant scenarios [22] under a convolutive transfer function (CTF) model [23] is still open to be explored. However, it is necessary to understand that the transform's properties is crucial to ensure accurate and meaningful outputs. The SSBT lacks comprehensive examination in the literature, resulting in a limited understanding of its features and limitations compared to other methodologies. This knowledge gap complicates the appropriate application and utilization of this transform. This paper aims to explore the SSBT properties, how they interact with traditional beamforming assumptions, and how to properly implement a beamformer, specifically the Minimum-Power Distortionless Response, under this new transform. We begin by examining a continuous-frequency version of the SSBT, comparing its properties to those of the Fourier Transform (FT) and the STFT. We study how these properties impact basic beamforming concepts such as the convolution theorem and relative frequency response estimation for the SSBT. While the SSBT is more error-prone and restrictive than the STFT regarding beamforming design, we also demonstrate a bijective interchangeability between them. This allows for their joint usage, potentially enhancing beamformer design within the SSBT domain by converting into the STFT and back without depending on inverse transforms, which are computationally intensive. We also test its direct implementation for beamforming in a real-life-like reverberant scenario, comparing it to the STFT in this regard. Our theoretical findings matched the experimental results: the beamformers based on SSBT were slightly less effective than the ones based on STFT in ideal conditions, and significantly less effective in non-ideal scenarios. This emphasizes the limitations of the SSBT compared to the STFT in practical beamforming.