1
\$\begingroup\$

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.

\$\endgroup\$
2
  • \$\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\$ 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
    Aug 4, 2021 at 21:17

1 Answer 1

0
\$\begingroup\$

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.

\$\endgroup\$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.