# In what cases can you get aliasing below the Nyquist frequency?

I took the one-sided FFT of a signal and plotted up until the Nyquist frequency. Then, I took the real part of this FFT multiplied by $$i\omega$$ following a calculation that I'm trying to do of a physics quantity related to the complex shear modulus (see link in the comments below; I don't think the physics is directly relevant, though).

This latter plot, $$Re(i\omega FFT)$$, follows my intuition for what this physical quantity should be, except for the large dip at high frequencies just below the Nyquist. About half the values are captured in this dip in log-log space. Since this is the FFT multiplied by $$i\omega$$ and then taking the real part, could doing this affect the value of the Nyquist frequency? Or could there be aliasing in certain cases below the Nyquist? I am trying to understand if this dip is physical or not.

Here is the FFT:

And here is $$Re(i\omega FFT)$$:

• Why did you multiply by jw?
– Ben
May 2, 2022 at 20:11
• Take $Real(i\omega FFT)$ is the calculation for a physical quantity related to the shear modulus of a material.
– user62718
May 2, 2022 at 20:14
• If I understand correctly, the top graph is the absolute value of the FFT, right? While the bottom graph is the real part of the FFT multiplied by jw?
– Ben
May 2, 2022 at 20:21
• Yes, that is correct.
– user62718
May 2, 2022 at 20:22
• See bottom of page 13 here (ucl.ac.uk/~ucahhwi/GM05/lecture3.pdf) but hopefully that's not necessary for my question, though I guess it's good to give some context.
– user62718
May 2, 2022 at 21:04

The dip in the real magnitude of the product with $$j\omega$$ indicates that the phase for the underlying process is close to ±90° (or $$\pm \pi/2$$ radians) at Nyquist: Assuming the first plot is a magnitude of the complex response and the horizontal axis is frequency in radians/sec (plot is not labeled so we can only guess), the magnitude at Nyquist = 5e3 radians/sec is approximately 0.06, and multiplied by 5e3 is 300. So that would be the magnitude for the real component if the phase at this frequency was close to 0° (response there would then be all real, as we see is the case for $$\omega = 100$$). Given the real magnitude is only 0.02, then the phase must be close to 90° such that the response is mostly imaginary at that frequency. This would be confirmed by plotting the phase of the original response, or the imaginary component: $$Im\{i\omega FFT\}$$