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The time signal which i'm trying to find the aliases for is:

$$x:{\mathbb R}\rightarrow {\mathbb R}\\\ x(t)=\cos(50t) +2\cos(70t).$$

If the sample period is $T_s = \frac{\pi}{60}$ then according to Nyquist -Shannon sampling theorem (which btw my professor failed to prove) there is/are a signal(s) which after sampling will be equal to the sampled version of the above signal, if we sample those with the same sample frequency in the frequency band $[-55, 55]$.

I don't understand the meaning of the last sentence , what does a frequency band means here?

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Let's use a simple example: A sine wave of 400 Hz. The Fourier spectrum has two components: one at 400 Hz and one at -400 Hz (since it's a real valued time signal).

Now let's sample this at 1 kHz. Sampling in one domain is periodic repetition in the other domain. So the sampled signal consists of your two original frequencies, -400 Hz and +400 Hz repeated again and again in 1 kHz intervals. So you get components at +-400 Hz, +-600 Hz, +-1400 Hz, +-1600 Hz, etc.

Any of these mirror frequencies (600, 1400, 1600, 2400, 2600, ...) will give you the exact same samples as the original 400 Hz.

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  • $\begingroup$ Hi, I'm still struggling here, does the frequency band make sense in frequency domain or i can give it some meaning in time domain as well? So i need to look at its spectrum to find out what's happening? And i don't understand when you say So the sampled signal consists of your two original frequencies, -400 Hz and +400 Hz repeated again and again in 1 kHz intervals, to me the sampled signal is just a bunch of discrete points over time. And i'm sorry if my questions appears to be very idiotic, but i'm very confused. $\endgroup$ – Sam B Jan 1 at 20:43
  • $\begingroup$ I remember a part of the conversation between me and my professor, he told me if you sample the signal, you'll see impulses at -70, 70, 50, -50, which is completely vague to me. $\endgroup$ – Sam B Jan 1 at 20:47

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