# What is special about the frequency $\omega_0=\pi$ that suddenly causes rate of oscillation decrease?

In Alan Oppenheim's book Signals and Systems a comparison is made between the properties of discrete-time and continuous-time complex exponential signals in section 1.3 pg. 26. Specifically it says:

The continuous-time complex exponential $e^{j\omega_0t}$ has two properties: 1. the larger the magnitude of $\omega_0$, the higher is the rate of oscillation in the signal 2. $e^{j\omega_0t}$ is periodic for any value of $\omega_0$

Consider the discrete-time signal $e^{j(\omega_o+2\pi)n}= e^{j2\pi n} e^{j\omega_0 n}=e^{j\omega_0 n}$. Because of the periodicity implied by this equation, the signal $e^{j\omega_0 n}$ does not have a continually increasing rate of oscillation as $\omega_0$ is increased in magnitude. Rather, as illustrated in Figure 1.27, as we increase $\omega_0$ from 0, we obtain signals that oscillate more and more rapidly until we reach $\omega_0=\pi$. As we continue to increase $\omega_0$, we decrease the rate of oscillation until we reach $\omega_0=2\pi$, which produces the same constant sequence as $\omega_0=0$ My confusion is with the part that is in bold. For the continuous-time case, we know that

$\omega_0 = \frac{2\pi}{T_0}$ and it makes sense to me that decreasing $T_0$ (more frequent oscillations) would cause $\omega_0$ to increase. Similarly, for the discrete-time case we know that

$\omega_0 N = 2\pi m$ for $N,m \in \mathbb{Z}$ Solving this equation for $\omega_0$, we get

$\omega_0=\frac{2\pi m}{N}$. Just like in the continuous-time case, decreasing N should cause $\omega_0$ to increase. So I am not seeing what is so special about $\omega_0=\pi$, the rate of oscillations suddenly start decreasing. I don't see that in the math if you decrease N.

## 1 Answer

As a consequence of the sampling theorem, the frequency spectrum of a discrete random variable is periodic, but it turns out that signals that are inherently discrete in time and not derived from an A/D converted continuous time signal are also periodic in frequency. As a consequence, as the frequency is increased, it goes from the positive portion of one periodic interval, to the next, i.e. it transitions to the negative image of the next interval.

In the old days, DSP was mostly a sub discipline of Electrical Engineering, and DSP was focused on doing things digitally that were previously done analog. In order to talk about DSP, without having to reference, a sampling frequency, a convention called unit sampling was commonly used, and a period of the spectrum is from -1/2 to 1/2.

As time progressed it was noted that DSP wasn't directly tied to processing analog derived signals and unit sampling was somewhat artificial, so it was recognized that $2\pi$ was a natural periodic interval, an $2\pi$ fits between $-\pi$ and $\pi$, which is why $\omega=\pi$ is a transition point in the spectrum.