Nyquist Theorem adding two same frequency near to Nyquist Frequency with phase shift

Question

This is my first question on this platform. Sorry if I made mistakes.

What happens if we add two or more same frequency signals near to Nyquist Frequency with phase shift and sample them?

For example, assume that we have a signal of 18000Hz. And add another signal with same frequency, but having 180 degree phase with our first signal. Then, sample this total signal at 36000Hz. What happens if we reconstruct this signal? Does this situation result in aliasing?

Note: I'am trying to figure out how exactly sound records are sampled when musicians playing their instruments. For example, assume that two keyboard players playing a note at 20000Hz which will be sampled at 44100Hz. Since they are human-being they are not supposed to hold the rhythm like in electronic music. Therefore, there will be a phase shift in their playing and wouldn't this cause aliasing in reconstructed signal?

Thanks for the answers.

Answer 1

When you add 2 or more sinusoids at the same frequency $f_o$ but with different phase shifts you get a sinusoid at same frequency but an additional attenuation term. Mathematically, you will have following: $$cos(2\pi f_ot + \phi)+cos(2\pi f_ot + \theta) = 2cos(\frac{\phi - \theta}{2}).cos(\frac{2\pi f_ot + \phi + 2\pi f_ot + \theta}{2})$$$$= 2cos(\frac{\phi - \theta}{2}).cos(2\pi f_ot + \frac{\phi + \theta}{2}),$$You see basically, you have an attenuation term $2cos(\frac{\phi-\theta}{2})$ which will change the amplitude of the sum of the sinusoids, but the frequency of the sinusoid remains at $f_o$. Hence, if you were sampling it with $f_s>2f_o$, then even after adding them you will not have any aliasing.

Only thing of concern is that because of attenuation term driving the amplitude of the sinusoidal, your signal might become 0 now, as in you example of $2^{nd}$ sinusoidal at the same frequency $f_o$ but having phase difference($\phi - \theta$) of $\pi$ compared to $1^{st}$ signal. That means $2cos(\frac{\pi}{2})$ which is equal to 0, and hence you have actually nothing to sample, both signals are completely out of phase and cancelling each other.

Answer 2

Practically speaking, a sine wave at 20k Hz is mostly inaudible. Eventhough the musical audio band is defined to be from 20 Hz to 20k Hz, for very most people the upper frequency of hearing is about 16k Hz.

Apart form this practical side note, assuming your musicians are playing their instruments at 20k Hz and the audio is sampled at 44.1k Hz, there won't be aliasing provided that the musical signal is sufficiently lowpass filtered with a cutoff frequncy of $f_c = 44100 / 2 = 22050$ Hz.