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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).

Thanks in advance.

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  • \$\begingroup\$ \$\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? \$\endgroup\$
    – a concerned citizen
    Commented Aug 4, 2021 at 19:51
  • \$\begingroup\$ @aconcernedcitizen well, this exponent treated as a distribution so its the same as treating cos as a distribution. \$\endgroup\$
    – FreeZe
    Commented Aug 4, 2021 at 21:17

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According to Wikipedia #309 says

This rule is useful in studying the Hilbert transform.

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.

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