Teoría de la señal

Energy and power of a signal

\[ E_x = \sum_{n=-\infty}^{+\infty} \left| x\left[ n \right] \right|^2 \]
\[ P_x = \lim_{N\to\infty} \frac{1}{2N + 1} \sum_{n = -N}^{N} \left| x[n] \right|^2\]

Teoría de la señal

Not all sinusoids are periodic in discrete time

For a signal being periodic, it must fulfill the following condition:

\[ x[n] = x[n+M] \]
where \(M \in \mathbb{Z}\).

If \(x[n] = e^{j\left(\omega n + \phi\right)}\), then:

\[ e^{j\left(\omega n + \phi\right)} = e^{j\left(\omega \left(n+M\right) + \phi\right)}\]

\[ e^{j\omega n } e^{j\phi} = e^{j\omega n }e^{j\omega M} e^{j\phi}\]

\[ 1 =e^{j\omega M}\]

For \(e^{j \omega M}\) to be 1, the angle must be multiple of \(2\pi\). Then:

\[ \omega M = 2\pi N \]

where \( N \in \mathbb{Z}\) and \(2 \pi N\) represents any integer multiple of \(2\pi\).
Then, the frequency must meet:
\[ \omega = \frac{2\pi N}{M} \]

This is, \(\omega\) must be a rational multiple of \(2\pi\).

What does this mean?

Basically, that a periodic signal can’t take any arbitrary frequency in discrete time.