# On Beamforming with the Single-Sideband Transform
## 2 Frequency and Time-Frequency Transforms
### Discrete Time-Frequency Transforms  
The Short-Time Fourier Transform (STFT) [11; 12] of a discrete-time signal \(x[n]\) is  
\[X_{\mathcal{F}}[l,k]=\sum_{m=0}^{K-1}w[n]x[n+l\cdot O]e^{-|2\pi k\frac{(nt+O)} {K}} \tag{20}\]  
where \(w[n]\) is an analysis window of length \(K\), and \(O\) is the hop size between successive windows of the transform, usually \(O=\lfloor\frac{K}{2}\rfloor\). \(l\) is the frame index, and \(k\) is the frequency bin index. The STFT is generally seen as a discretization of the FT, being applied over sequential time \"snippets\".  
The Single-Sideband Transform (SSBT) [16; 24] is similarly defined, being the RFT's windowed-discrete-time equivalent. The SSBT of \(x[n]\) is  
\[X_{S}[l,k]=\sqrt{2}\mathbb{R}\left\{\sum_{n=0}^{K-1}w[n]x[n+l\cdot O]e^{-|2\pi k \frac{(nt+O)}{K}+\frac{1}{4}}\right\}. \tag{21}\]  
One advantage of using the STFT is the need for only \(\lfloor(K+1)/2\rfloor+1\) frequency bins, given its complex-conjugate behavior for real-time-domain signals. Meanwhile, the SSBT requires all \(K\) bins to correctly capture all the information of \(x[n]\), but its coefficients are real.  
Assuming that all \(K\) bins of the STFT are available (even though they are not all necessary), similarly to Equations (4), (5a) and (5b) we now have  
\[X_{S}[l,k] =\sqrt{2}\mathbb{R}\Big{\{}X_{\mathcal{F}}[l,k]e^{\frac{3\pi}{4}} \Big{\}} \tag{22}\] \[=-\mathbb{R}\{X_{\mathcal{F}}[l,k]\}-\mathbb{I}\{X_{\mathcal{F}}[l,k]\}.\]  
For the abuse of notation, we let \(X_{S}[l,K]\equiv X_{S}[l,0]\), and equally for \(X_{\mathcal{F}}[l,K]\).  
As is the case for the RFT, the SSBT also does not hold the convolution theorem in the same way as the STFT. However, similarly to what is shown in Equation (7) (and in Property A4), the convolution on the SSBT domain through the MTF model [25] can be given by  
\[h[n]*x[n]\rightleftharpoons X_{\mathbb{S}}[I,k]H_{\mathbb{S}}^{\prime}[k]+X_{ \mathbb{S}}[I,K-k]H_{\mathbb{S}}^{\prime\prime}[k] \tag{23}\]  
or, with the CTF model,  
\[h[n]*x[n]\rightleftharpoons X_{\mathbb{S}}[I,k]*H_{\mathbb{S}}^{\prime}[I,k]+X _{\mathbb{S}}[I,K-k]*H_{\mathbb{S}}^{\prime\prime}[I,k] \tag{24}\]  
in which this convolution is performed over \(l\), with \(H_{\mathbb{S}}^{\prime}[I,k]=H_{\mathcal{I}}^{\mathrm{R}}[I,k]\) and \(H_{\mathbb{S}}^{\prime\prime}[I,k]=H_{\mathcal{I}}^{\mathrm{I}}[I,k]\). Note that this is an approximation, as cross-band interference [26; 27] is necessary to model the convolution perfectly; however, this effect is not considered here for either transform. The conjugate frequency's presence is a byproduct distinct to the cross-band interference, the latter happening due to aliasing and the windowing processes, and the former coming from the continuous-time convolution theorem for the SSBT (see Section 2.1 and Property A3).  
Similarly to the relationship presented in Equation (5), we have  
\[X_{\mathbb{S}}[I,k] =\frac{1}{\sqrt{2}}\left(e^{\mathrm{i}\frac{\pi}{4\pi}}X_{ \mathcal{I}}[I,k]+e^{-\mathrm{i}\frac{\pi}{4}}X_{\mathcal{I}}[I,K-k]\right), \tag{25a}\] \[X_{\mathcal{I}}[I,k] =\frac{1}{\sqrt{2}}\left(e^{-\mathrm{i}\frac{\pi}{4\pi}}X_{ \mathbb{S}}[I,k]+e^{\mathrm{i}\frac{\pi}{4\pi}}X_{\mathbb{S}}[I,K-k]\right). \tag{25b}\]  
Equations (25a) and (25b) give us a bijective relationship between time-frequency signals in the SSBT and STFT domains. That is, for the same transform parameters (window type and size, overlap, etc.), it is possible to convert from one to the other without using their inverses and going into the time-domain, which is resource-consuming. This interchangeability means they can be used together, transforming from one to the other for situations where each is more advantageous.