# The plot of instantaneous power of the Dirac function

I am very confused. I have tried researching this question for the last two weeks and I cannot get a conclusive answer.

I was wondering how would I go about plotting the instantaneous power in the time domain of the Dirac input signal. The problem becomes as I try to calculate the power in time domain, $$P(t) = \left\lvert\delta(t)\right\rvert^2$$. I know how to calculate power by using Rayleigh's theorem to convert the the signal to frequency domain where delta(f) = 1. to get infinity over an infinite period which is undefined value for power, but I just want to plot power over time for the Dirac function.

delta isn't square-able since as x goes to 0 the delta function goes to infinity and you cannot square infinity, also something about the delta function being a distribution and not able to be square-able.

I was wondering if someone could point me in the right direction to be able to plot this function.

Thank you

• Is there a reason you want to do this? A Dirac impulse is not a function. What could be the possible interpretation or utility of calculating its "power"?
– MBaz
Apr 12 at 18:38
• How are you trying to use the delta functional that you need to calculate its power? For engineering purposes, the delta functional is there to make your life easier -- if it's making your life impossible, you're misusing it. Apr 12 at 20:07
• the other thing that you should be aware of is that, while the notion of White Noise exists, the reality of White Noise does not. White Noise has equal power per Hz of bandwidth and an unlimited bandwidth. so White Noise (which has a power spectrum that is flat all the way to infinity and a dirac delta as an autocorrelation) has more power output than the sun. Apr 12 at 21:17
• @MBaz It's part of an assignment I was asked to plot the instantaneous power of the signal. Apr 14 at 7:01
• @lowFrequencyLearning Then I suggest asking your instructor to clarify the meaning of the assignment, given what you have learned here.
– MBaz
Apr 14 at 12:45

The problem becomes as I try to calculate the power in time domain, $$P(t) = \left\lvert\delta(t)\right\rvert^2$$

As you figured out yourself, that's impossible: the power of the Dirac impulse isn't defined, it diverges (goes to infinity).

Hence, you especially can't plot the power "correctly"; it's zero anywhere but at $$t=0$$, where it diverges.

But quite honestly, this should come as no surprise: you can't correctly plot $$\delta(t)$$, either (the value at $$t=0$$ isn't defined just as much as its square isn't).

So, whatever question can be answered by that plot: I'm afraid you'll have to find a "better" question.

• Would you say the function |δ(t)|^2 could be similar to the plot of the δ(t) function? Apr 14 at 7:04
• again, none of these have a plot. They are not functions. So, "is it similar" can be answered with "what are you even talking about?" Apr 14 at 9:54
• thanks for replying Apr 15 at 4:31