Datasets:
| \begin{array}{rl}{s_{\mathrm{a}}(t)}&{\triangleq{\mathcal{F}}^{-1}[S_{\mathrm{a}}(f)]}\\&{={\mathcal{F}}^{-1}[S(f)+\operatorname{sgn}(f)\cdot S(f)]}\\&{=\underbrace{{\mathcal{F}}^{-1}\{S(f)\}}_{s(t)}+\overbrace{\underbrace{{\mathcal{F}}^{-1}\{\operatorname{sgn}(f)\}}_{j{\frac{1}{\pi t}}}*\underbrace{{\mathcal{F}}^{-1}\{S(f)\}}_{s(t)}}^{\mathrm{convolution}}}\\&{=s(t)+j\underbrace{\left[{\frac{1}{\pi t}}*s(t)\right]}_{\operatorname{\mathcal{H}}[s(t)]}}\\&{=s(t)+j{\hat{s}}(t),}\end{array} |