frequency - Discrete Time Signal Property

Property:

The frequency of oscillation of discrete time sinusoids sequence increases as $\omega$ increases from $0$ to $\pi$. If $\omega$ is increased from $\pi$ to $2\pi$ then frequency of oscillation decreases.

My question:

What is meant by "frequency of oscillation" ?

How can a frequency of a signal vary (increase) from $0$ to $\pi$ and (decrease) from $\pi$ to $2\pi$. Isn't it should be same from $0$ to $2\pi$ ?

Answer

Take a complex exponential

$$x[n]=e^{jn\omega}\tag{1}$$

Let's assume that $\omega=\pi$. This gives

$$x[n]=e^{jn\pi}=(-1)^n\tag{2}$$

(because $e^{j\pi}=-1$). Eq. $(2)$ shows that a signal with frequency $\pi$ is an alternating signal, so $\omega=\pi$ clearly is the maximum possible frequency of a discrete-time signal.

Now assume that $\omega>\pi$. Let's write $\omega=\pi+\Delta\omega$ with $0<\Delta\omega<\pi$. The signal $x[n]$ from $(1)$ can be written as

$$x[n]=e^{jn\omega}=e^{jn(\pi+\Delta\omega)}\tag{3}$$

Since the complex exponential function is $2\pi$-periodic, we can subtract a multiple of $2\pi$ ($2\pi n$, $n\in\mathbb Z$) from its argument without changing anything:

$$x[n]=e^{jn(\Delta\omega-\pi)}=e^{-jn(\pi-\Delta\omega)}\tag{4}$$

Eq. $(4)$ shows that increasing $\Delta\omega$ from $0$ to $\pi$ corresponds to decreasing the frequency from its maximum value $\pi$ down to zero.