Periodicity of complex exponential in continuous and discrete time (Eq 1.51, Signals and Systems by Oppenheim & Wilsky)

Question

Hi All: This is very basic but I've always wondered about it and now I see it in print in a textbook so I may as well ask. In Signals and Systems on page 26, it says

$$e^{j(\omega_0 + 2\pi)n} = e^{j2\pi n} e^{j\omega_0 n} = e^{j\omega_0 n} \tag{1.51} $$

Then it says:

From eq. (1.51), we see that that the exponential at frequency $\omega_{0} + 2\pi$ is the same as that at frequency $\omega_{0}$. Thus, we have a very different situation from the continuous time case in which the signals $e^{j\omega_{0} t}$ are all distinct for distinct values of $\omega_{0}$. In discrete time, these signals are not distinct, as the signal with frequency $\omega_{0}$ is identical to the frequencies with $\omega_{0} \pm 2\pi$, $\omega_{0} \pm 4\pi$ and so on.

The reason why the concept above has always confused me is because if we replace $n$ with $t$, then it is true for integer values of $t$ also. Why does this not matter ? I know that concept also has to do with why aliasing cannot exist in the continuous case so, if I can understand this, a lot of things would get clear for me. Thanks for your help and the great answers in general from many.

Answer 1

The simple answer is that

$$e^{j\omega_0t}\neq e^{j(\omega_0+2\pi)t}\tag{1},\qquad t\notin\mathbb{Z}$$

Since $t$ is a real variable, the inequality is true for uncountably many values of $t$. Equality is only achieved for countably many integer values of $t$. From $(1)$ it follows that for real $t$, the function $e^{j\omega_0t}$ is not $2\pi$-periodic in $\omega_0$, unlike in the discrete-time case.

Two functions of a real variable can be equal on a countably infinite set of values of the independent variable, even though they're not the same function.

Answer 2

But $t$ can also have fractional values. If $t=n=$ integer then $\omega_0$ and $\omega_0+2\pi$ are aliases for the same frequency.