So my signal processing text book says that for the power of a signal to be calculated it must either be periodic or have statistical regularity. However, for the step function, which is neither periodic nor has statistical regularity, the power of that signal can be calculated. Why is that so ?

 

I believe the unit step function does meet the criterion of statistical regularity. Chosing random t values on the interval (-∞,∞) gives you a 50% chance of either x(t) = 0 or x(t) = 1. As sample size approaches ∞, the average approaches 1/2.

When in doubt, just go with the definition of average signal power:
\(P = \lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^{2}dt\)


As long as this expression is finite and non-zero the signal is a power signal. In the case of the unit step this is cetrainly the case.

Mike

 

Probably begs the question - "Is DC periodic?"

Probably also depends what other properties the signal is known to have. A step signal differs from DC to the extent that it has a "history" (for -∞<t<+∞) of a different value prior to the step time. Also a perfect step can probably only exist or propagate in a system with infinite bandwidth.

To an observer with no knowledge of the step history - beyond the step time the step signal is indistinguishable from a DC signal.

 

To calculate the power of a signal, it must be a power signal. Further,
average power of a signal can only be calculated over a period and hence
it must be a periodic signal, therefore a periodic signal is a power signal
if its energy content per period is finite. Periodic signal is statistically
regular since Statistical Regularity is defined in probability as random
events exhibit regularity when repeated enough times. The unit step signal
is a Power signal, since when we calculate its power it comes to 1/2
(i.e finite value), and when we calculate its energy, it comes out to infinity.
If a signal has energy as infinity and power as a finite non-zero value,
then it is a power signal, not an energy signal and hence its power
can be calculated.

 

Guys I don't get it how unit step signal is power signal though it is not periodic or statistically regular

 

So my signal processing text book says that for the power of a signal to be calculated it must either be periodic or have statistical regularity. However, for the step function, which is neither periodic nor has statistical regularity, the power of that signal can be calculated. Why is that so ?

That's my doubt too......