Signal Time Domain Properties

Casual Signal

If a signal is only non-zero for $t \geq 0$ , it is called a casual signal.

If,

f(t) = f(t + T)

Where $T$ is the minimum positive number that makes this true.

Then $f(t)$ is periodic with $T$ .

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For common signals like,

f(t) = Re(e^{j\omega t})

Obviously, the period is $\frac{2\pi}{\omega}$ .

However, consider the discrete signal,

f[t] = Re(e^{j\omega t})

Let's solve,

\omega T = 2\pi k

The $T$ doesn't guarantee to be $\frac{2\pi}{\omega}$ . This is because in discrete signal, the minimum value of $T$ is one instead of zero.

To solve the equation, obvious, $\omega$ must be a rational scale of $2\pi$ , thus assum,

\omega = 2\pi \frac{p}{q}

Where $gcd(p, q) = 1$ .

Thus,

T = \frac{qk}{p}

So,

T = q

It is important to note that higher $\omega$ doesn't imply shorter $T$ for discrete signals.

The energy of a signal is defined as,

E = \int_{-\infty}^{+\infty} f(t)^2 \mathrm{d}t

If the energy of a signal exists, it is called a energetic signal.

The power of a signal is defined as,

P = \lim_{T \to +\infty} \frac{1}{2T} \int_{-T}^{+T} f(t)^2 \mathrm{d}t

If the power of a signal exists, it is called a power signal.