# Spectral function value of a frequency above the nyqist limit

this was the question on my today's exam on signals and systems:

*We have a continuous function $$f(t)$$. Its value of the spectral function at an angular frequency of $$8000\pi\ rad/s$$ is $$\frac{1}{32000}e^{j\frac{\pi}{4}}$$.

Afterwards, the same signal is sampled at sampling frequency $$64000\pi\ rad/s$$. Aliasing did not occur. Determine whether it is possible to tell the value of the spectral function of the sampled signal at an angular frequency of $$72000\pi\ rad/s$$.*

I realized that the maximum frequency I can recover from the signal is $$32000\pi\ rad/s$$. Therefore, I said that we could not recover the spectral function value at angular frequency of $$72000\pi\ rad/s$$.

The correct answer, however, was $$e^{j\frac{\pi}{4}}$$. I don't see an error in what I said, so I will appreciate any push towards the right answer. Thank you.

When you sample the signal in time it becomes periodic in frequency. The spectrum gets from -32k$$\pi$$rad/s to +32k$$\pi$$rad/s gets repeated with repetition rate of 64k$$\pi$$rad/s. Since $$72 = 64+8$$ the value at 72k$$\pi$$rad/s is the same as at 8k$$\pi$$rad/s (and at 136k$$\pi$$rad/s and -56k$$\pi$$rad/s etc)
• I thought so too after the exam, but as you can see, the coefficients are different, the first one is multiplied by $\frac{1}{32000}$ and the second one isn't, but according to what you say, they should be identical. Jan 17, 2022 at 18:09