How to determine sampling frequency for an x(t) signal avoiding aliasing?

The antitransform of the function is given:

$$\hat{x}(f) = \frac{123 + i246\pi f}{246 - 24600 \pi^2 f^2 + i4920\pi f}$$

I'm asked to determine which frequency can I sample using the function x(t) avoiding aliasing? (or which range of values)

Update the problem actually suggest five answers: 100, 10, 1, 0.4 or none of these. Turns out the answer is none of these.

• draw a bode plot for given transfer function and the bandwidth comes close to 0.275 Hz hence sampling frequency is grater than 0.55 Hz, if your calculations is based on cutoff frequency (0.2 Hz) then sampling frequency becomes 0.4 Hz. Commented Dec 12, 2019 at 16:47
• “losing aliasing” is a peculiar usage. do you mean avoiding aliasing?
– user28715
Commented Dec 12, 2019 at 18:08
• @StanleyPawlukiewicz yes, sorry I'm correcting it. Commented Dec 12, 2019 at 18:12

The function you show has a frequency range of $$-\infty$$ to $$\infty$$ so it isn’t strictly band limited. To completely avoid aliasing, it needs infinite sampling.

Of course, you can use tables or Fourier identities to obtain the exact inverse.

Usually some acceptable criteria is specified for tolerable aliasing when specifying a sample rate.

• How did you determine the frequency was -\infty to \infty Commented Dec 12, 2019 at 18:30
• Probably by inspection. But the formal way to do it is to note that for any finite $f$, $\left | \hat x (f) \right | > 0$. Commented Dec 12, 2019 at 18:31
• @Someone, by inspection,
– user28715
Commented Dec 12, 2019 at 18:38
• Thank you I accepted your answer. Now what if my function was actually finite, is there a way to obtain the frequencies using the fourier transform? Commented Dec 12, 2019 at 19:03
• @Someone you would use the Nyquist criteria.
– user28715
Commented Dec 12, 2019 at 19:30