# How to standardize the energy of a Dirac delta function relative to sample rate (width) and amplitude?

## Background

I was instructed that a Dirac delta function (impulse from $$0$$ to $$A$$ then back to $$0$$ at short duration) has a white noise audio frequency type excitation distribution here

ie. It should provide equal excitation of amplitude/energy at each frequency.

This is good, as this is what I want for my application. However, I am puzzled on two related points as I have experimented with these before and struggled around understanding their practical use.

## 1) Amplitude, Width, & Energy

How can one standardize the energy or resulting output of a Dirac delta function across different sample rates?

For example, in audio, one might have $$44.1 kHz$$ sampling on one system and $$88.2 kHz$$ on another. On the $$44.1 kHz$$ system, a Dirac (minimal single sample width spike from and to 0 with amplitude $$A$$) is now twice the sample/time width as one on a $$88.2 kHz$$ system.

If they are both still "white" in their energy, then I presume one can standardize by adjusting the amplitudes relative to the expected width. But how? What would be the formula to get an equal amplitude excitation?

Let's say specifically you want to excite all audio frequencies ($$20 Hz$$ to $$20 kHz$$) to an amplitude of $$1.0$$ from $$0$$. What height should the single sample Dirac be at any given sample rate?

## 2) Limit of "White"?

Relating further to sample widths, at what threshold does a Dirac stop being "white" and start being more like a square wave or step function?

As noted also in that thread, a square or step excites at $$1/f$$. If a Dirac is wide enough it will presumably stop being a Dirac and start being a step/square. But what is that threshold?

Whether at $$44100 Hz$$ sampling or $$88200 Hz$$ sampling, for audio, we want $$20 Hz$$ to $$20 kHz$$ frequencies. What would be the widest $$n$$ of samples the impulse (straight from $$0$$ to $$A$$ then back to $$0$$) could be before it would stop behaving "white" between that range?

Thanks for any thoughts or explanation.