1. Discrete Time Sinusoidal Signal

A discrete time sinusoid has this expression:
\[A\cos(\omega_0n + \theta),\qquad n \in]-\infty, +\infty[\tag{1.1}\]
where:
$A$ is the Amplitude
$\omega_0$ is the Frequency in radians per sample
$\theta$ is the Phase in radians.
We can express frequency with the variable $f_0$ defined by $\omega_0 = 2\pi f_0$
and the (1) becomes:
\[A\cos(2\pi f_0 n + \theta),\qquad n \in]-\infty, +\infty[\tag{1.2}\]
Now the frequency $f_0$ has dimensions of cycles per samples.

A discrete time sinusoid is periodic only if its frequency $f_0$ is a rational number.

A discrete time signal x(n) is periodic with period N (N>0) if and only if:
\[x(n+N) = x(n), \qquad \forall n\tag{1.3}\]
The smallest value of N is called fundamental period.For a sinusoid with frequency $f_0$ we should have:
\[\cos[2\pi f_0(N+n) + \theta] = \cos(2\pi f_0n+\theta)\]
This relation is true if and only if
\[2\pi f_0N=2k\pi\Rightarrow f_0=\frac{k}{N}\]
in other words the sinusoid is periodic if we can express the frequency as a rational.
The fundamental period is N (the value we obtain cancelling common factors so that k and N are relatively prime).
The absolute value of k is the number of cycles the phasor makes before going to the initial position.

EXAMPLE 1.a
Let's consider the discrete sequence $x(n) = \cos(0.3\pi n)$, we can see that $\omega_0=0,3\pi$ and so since $f_0=\frac{\omega_0}{2\pi}$ we obtain that $f_0=\frac{0.3\pi}{2\pi}\Rightarrow f_0=\frac{3}{20}$
The sequence x(n) is periodic and N=20

EXAMPLE 1.b
Let's consider the discrete sequence $x(n) = \cos(0.3n)$, we can see that $\omega_0=0,3$ and so since $f_0=\frac{\omega_0}{2\pi}$ we obtain that $f_0=\frac{0.3}{2\pi}\Rightarrow f_0=\frac{3}{20\pi}$
The sequence x(n) is not periodic

A discrete time sinusoid is periodic in frequence with period of $2\pi$
To prove this assertion, let us consider a sinusoid $\cos[\omega_0n+ \theta]$ and add $2\pi$ to $\omega_0$, we obtain:
\[\cos[(\omega_0+2\pi)n+ \theta]=\cos[\omega_0n+2\pi n+ \theta]=\cos[\omega_0n+ \theta] \]
in fact n is an integer number.
This result is more important because it permits us to study the sinusoid only in the frequency range of $2\pi$ (i.e. $-\pi\leq\omega\leq\pi$ or $-\frac{1}{2}\leq f \leq\frac{1}{2}$)