# What duration of a white noise burst is required for it to be "white" at a given frequency or frequency range?

I am running white noise bursts (with very short ramps on/off to prevent discontinuities) through underdamped resonant bandpasses which are tuned to any given $$f_0$$ and an underdamped $$Q$$.

Continuous white noise at an amplitude of $$1$$ if run through a completely underdamped resonant bandpass should bring that resonant bandpass to $$1$$ at its $$f_0$$.

But what about short bursts? And what about where a little damping is involved?

## Purpose

I would like to excite the bandpasses to the amplitude of the noise signal at their $$f_0$$ (say amplitude of $$1$$) with the shortest duration of noise possible in any given case.

For reference, I am not interested in Kronecker's delta as it does not perform what I need to - I need the smooth signal of white noise for this purpose.

## 1) Minimum duration of white noise for a given freq

I suspect that in all circumstances, a burst of white noise must have a duration of at least the period of the $$f_0$$ of the desired frequency to excite. If it is shorter than the period, and it does not have discontinuities (no steps) due to soft ramp on/off, then how can it adequately have the low frequency information needed?

When I test too short bursts (<1 period of $$f_0$$) they don't seem to activate the lower frequencies as much as longer bursts.

Due to random variation in the noise, I suspect even 2-3 periods may be needed to be reasonably sure you have reached amplitude of $$1$$ at $$f_0$$ but I am not sure.

Is this correct? Ie. What duration of white noise might typically be required for it to be reliably or at least reasonably "white" (full amplitude achieved) at a given frequency?

I originally thought I could reliably feed say 1-3 periods of white noise through each filter (duration tuned to $$n/f_0$$ for the bandpass) and get a reliable response across the spectrum. But this isn't happening. This seems reasonable on the low frequencies, but the high frequencies are not adequately excited. Unless I made an error.

So I am not sure how to standardize the duration to the minimum needed or what is happening.

What duration of white noise is needed to be "white" at a given frequency or across a frequency range?

## 2) The effects of damping

In any underdamped case, would the $$Q$$ of a bandpass be expected to affect the duration of a white noise burst required to excite it fully?

I suspect not, since a narrow bandwidth resonant bandpass still allows the signal through at full amplitude (in theory) at $$f_0$$. So I suspect the only problem I have is not knowing what the shortest duration of noise I can get away with is for each frequency target.

Thanks for any help. Hopefully that at least makes sense.

• "Continuous white noise at an amplitude of $1$ if run through a completely underdamped resonant bandpass should bring that resonant bandpass to $1$ at its $f_0$". That's not at all how filters act in response to noise. Your question contains many other such inaccuracies, and as such can't be answered, so much as rebutted point by point. It appears that you are trying to excite one or more bandpass filters with bursts of white noise and either measure the filters' or the noise's characteristics. If so reword your question to say what you really want to do. Feb 8 at 7:18