# Values of angular frequency of discrete-time sinusoidal signal

Why are the values of angular frequency of discrete-time sinusoidal signals in between -pi and pi?

$$x[n] = A \cos( \omega n)$$
will produce exactly the same samples for all discrete-time frequencies such as $$\omega = \pi/3$$ or $$\omega = \pi/3 + 2\pi$$ or $$\omega = \pi/3 + 6\pi$$ or $$\omega = \pi/3 -8 \pi$$... In other words every frequency of the form $$w = w_0 + 2\pi k$$ for all integers $$k$$ will yield the same set of samples; they are indistinguishable, and we chose to use the range $$[0,2\pi]$$ or $$[-\pi,\pi]$$, when referring to the discrete-time sinusoidal frequencies.
Note that the periodicity of the discrete-time frequency $$\omega$$ relies on the fact that the index $$n$$ is an integer. Otrherwise, like in the continuous-time case, the frequency $$\omega$$ would range from $$-\infty$$ to $$\infty$$.