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I have a basic question about the formulas represented for discrete sinusoids. In some textbooks we see this form

$$x[n]= A\cos (2\pi f nT + \phi ) $$

and in others we have

$$x[n]= A\cos (2\pi k/N n + \phi ) $$

What $N$ and $T$ stands for? And what is the differences between these two formulas?

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In the first formula, $x[n]$ is written as if it were a sampled version of the continuous-time function

$$x_c(t)=A\cos(2\pi ft+\phi)\tag{1}$$

If you sample $(1)$ at $t=nT$, $n\in\mathbb{Z}$, you get the first formula. $T$ is the sampling period, or the inverse of the sampling frequency $f_s=1/T$.

If you compare the two formulas in your question you see that $fT=k/N$. Assuming that $k$ and $N$ are integers, then the second formula is a special case of the first one. The discrete-time signal obtained by sampling $(1)$ is not necessarily periodic. However if $fT$ is rational, as in your second formula, then $x[n]$ is periodic with period $N$.

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