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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.

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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)

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  • $\begingroup$ 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. $\endgroup$
    – ampersander
    Jan 17, 2022 at 18:09
  • $\begingroup$ That depends largely on the scaling conventions used for the continuous and discrete transforms and for the exact definitions of the "frequency function" that your Professor uses $\endgroup$
    – Hilmar
    Jan 17, 2022 at 18:32

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