How to check if a signal is power signal or energy signal?

Question

What is the procedure to check the signal type?

example:

$ x(t) = A \sin (\omega t) $

$ y(t) = A e^ {-\lambda |t|} $

Answer 1

Let's assume the signal, $x(t)$ is not identically zero for all $t$.

An "energy signal" (what I would prefer to call a "finite energy signal") is such a signal, $x(t)$ with a finite energy: $$ 0 \ < \ \int\limits_{-\infty}^{\infty} \big|x(t)\big|^2 \ \mathrm{d}t \ < \ \infty $$

BTW, sometimes for mathematical ease, we require a stricter sense of finite "energy": $$ 0 \ < \ \int\limits_{-\infty}^{\infty} \big|x(t)\big| \ \mathrm{d}t \ < \ \infty $$

And a "power signal" (what I would prefer to call a "finite power signal") is such a signal, $x(t)$ with finite power: $$ 0 \ < \ \lim_{T \to \infty} \frac{1}{T}\int\limits_{-\frac{T}2}^{\frac{T}2} \big|x(t)\big|^2 \ \mathrm{d}t \ < \ \infty $$

I think that is the most fundamental definitions of the two classes of continuous-time signals.

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You can do a very similar definitions for discrete-time signals, $x[n]$. Let's assume the signal, $x[n]$ is not identically zero for all $n$.

An "energy signal" (what I would prefer to call a "finite energy signal") is such a signal, $x[n]$ with a finite energy:

$$ 0 \ < \ \sum\limits_{n=-\infty}^{\infty} \big|x[n]\big|^2 \ < \ \infty $$

And for mathematical ease, we sometimes require a stricter sense of finite "energy": $$ 0 \ < \ \sum\limits_{n=-\infty}^{\infty} \big|x[n]\big| \ < \ \infty $$

And a "power signal" (what I would prefer to call a "finite power signal") is such a signal, $x[n]$ with finite power:

$$ 0 \ < \ \lim_{N \to \infty} \frac{1}{2N+1}\sum\limits_{n=-N}^N \big|x[n]\big|^2 \ < \ \infty $$

I think that is the most fundamental definitions of the two classes of discrete-time signals.

Answer 2

All bounded periodic signals are power signals, because they do not converge to a finite value so their energy is infinite and their power is finite. So we say that a signal is a power signal if its power is finite and its energy is infinite. And the signal is an energy signal if its energy is finite and power is zero.

Answer 3

Of course, all real world signals are bounded in power and energy, as infinite time is unobservable, and infinite power is unrealizable, at least by mere mortals.

Answer 4

The assumption is that the signals under consideration are deterministic.

Power signals

A periodic signal is always a power signal if its Fourier transform is a set of discrete components comprising of a fundamental and its harmonics. In case of sinusoid it has only the fundamental component.

Energy Signal

An aperiodic signal is an energy signal if its Fourier transform is continuous.