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It is known that if we only have $n$ samples and take DFT, we only get at most $n$ distinct frequency data. But let's say that there is a continuous periodic signal with more than $n$ frequencies, with highest frequency $f_h$. Then DFT should be cutting off some frequencies. What is happening here?

Also, as the signal is periodic, with samples $[x_1,x_2,..x_n]$ also being periodic, we can just create samples $[x_1,x_2,..,x_n,..x_{2n}]$. By DFT, there are at most $2n$ distinct-frequency data, but because this set of samples is basically equivalent to the first set of samples, it seems that there are at most $n$ distinct-frequency data. What's happening here also?

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  • $\begingroup$ You want to search and read on aliasing. Also note that the "samples" must be within a single period, so you can't increase the number of samples like you described. $\endgroup$
    – AlexR
    Commented Mar 6, 2015 at 16:37

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If you have a continuous signal then you can also take $2n$ or better $4n$ samples inside one period.

Since the harmonic frequencies are integer multiples of $2\pi/T$, sampling at $n$ points $kT/n$ for the frequency $m=ln+j$ results in exponents $(ln+j)2\pi/T\cdot kT/n=l\,2\pi+j/n\,2\pi$ and the leading multiple of the period $2\pi$ of the trigonometric functions has no influence on the resulting trigonometric polynomial. Which is also known as aliasing, since the higher frequency appears under the alias of a lower frequency.

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