# Is there a way to calculate the Fourier transform of a Hilbert transform system?

Assume we have an LTI system with impulse response $$\frac{1}{\pi t}$$. Is there a way to calculate its Fourier transform to get the frequency response? I know that its classical Fourier transform is not defined since this function is not absolutely integrable, but, for example, $$\cos t$$ is also not absolutely integrable and we treat $$\pi\left(\delta\left(\omega-1\right)+\delta\left(\omega+1\right)\right)$$ as its Fourier transform. (We treat cos as a distribution).

• $\cos()$ is not treated as a distribution, it's used as the exponential form:$$\cos(x)=\dfrac{\mathrm{e}^{ix}+\mathrm{e}^{-ix}}{2}$$This is what gives you the $\delta(\omega\pm 1)$. Maybe math.ee, or dsp.ee are better fit for this? Aug 4, 2021 at 19:51
The transform given in the above page is $$\-i\pi \operatorname {sgn}(\xi )\$$. However, it is not mentioned if the trasnform is that of $$\1/x\$$ or $$\\frac{u(x)}{x}\$$. where $$\u(\cdot)\$$ is the unit step function.