Energy and Power Signals

This is a sandbox for ruminations on Energy and Power Signals

Energy Signals

• Energy signals are defined as having finite energy:

$$E=\lim_{T\rightarrow\infty}\int_{-T/2}^{T/2}|x(\tau)|^2\,d\tau<\infty$$

• All finite-duration bounded signals are energy signals
• All absolutely integrable signals such that

$$\lim_{T\rightarrow\infty}\int_{-T/2}^{T/2}|x(\tau)|\,d\tau<\infty$$

are energy signals

• Not all energy signals are absolutely integrable (for example, $$x(t)=t^{-0.8}u(t-1)$$ is an energy signal, but not an absolutely integrable signal)
• Conjecture: the integral of an energy signal is an energy signal only if the average value of the original signal is 0.
• No periodic signals are energy signals
• Energy signals have zero average power

Power Signals

• Power signals are defined as having finite average power:

$$P_{avg}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(\tau)|^2\,d\tau<\infty$$

• All bounded periodic signals are power signals
• The integral of a periodic power signal is a power signal only if the average value of the original signal is 0.
• Power signals have infinite energy

Singularities

• The unit step is a power signal:

$$P_{avg}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2}|u(\tau)|^2\,d\tau$$

$$P_{avg}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T/2}\,d\tau$$

$$P_{avg}=\lim_{T\rightarrow\infty}\frac{1}{T}\frac{T}{2}=\frac{1}{2}$$

• The impulse function is neither an energy nor a power signal.