# Discrete Real Sinusoid Formula

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?

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 $$N$$.