# Sampling $x(t)=\cos(4\pi t)+\cos(2\pi t)$

Imagine that we sample the signal $$x(t)=\cos(4\pi t)+\cos(2\pi t)$$ with a certain sample frequency $$f_s$$ and we obtain $$x[n]$$. Now, by ideal interpolation, we get $$y(t)$$ from $$x[n]$$.

How can we know the used sample frequency $$f_s$$ by looking at $$y(t)?$$

As a particular case, which $$f_s$$ verifies that $$y(t)=2\cos(2\pi t)?$$

How can we know the used sample frequency fs by looking at y(t)?

You can't.

The sampling theorem states that any sampling higher than twice the highest signal frequency allows for perfect reconstruction. In you case any sampling frequency higher than 4 would result in the same $$y(t)$$.

I'm struggling with this a bit. Is this a posed problem somewhere? Maybe you didn't quote it quite right. For instance the relationship between $$n$$ and $$t$$ is not given but assumed to be $$n=t$$ at integer values.

The only sampling rate (assuming sampling starts at zero) at which $$\cos(4\pi t)+\cos(2\pi t)$$ appears the same as $$2 \cos(2\pi t)$$ would be once per second, or whole number multiples of seconds. But that would also appear to be a DC signal with a value of 2. So this fails "ideal interpolation" unless that term has some kind of trick meaning.

There is no other $$f_s$$ which would "verify" your assertion. A sampling rate of four per second would allow reconstruction of the original function, but only because you have pure cosine functions with zero phase. The Nyquist criteria is you must sample faster than four per second.

P.S. Personal pet peeve: Sampling is a rate with a typical unit of samples per second, not a frequency with units of Hz. Yes, most textbooks are sloppy on this.