Is it possible to, like white noise, excite all audio frequencies equally, but with a more immediate & less random burst? Like a signal discontinuity?

Question

I am trying to generate an audio signal that, like white noise, has "equal intensity at different frequencies, giving it a constant power spectral density", but unlike white noise, can be expressed without any random character and in a representative way even when just maximally brief burst.

What is the energy distribution of a an immediate discontinuity (typically experienced in the real world as a pop/click) where the signal goes for example from 0 to 1 over a one sample, then either stays there or gradually returns to 0 at an inaudible rate (eg. with a high pass filter for example at low frequency of say 1-5 Hz to return back to 0)?

I believe this must be like a square wave, since it is 1 half of a square wave:

Where the amplitude of any given excited frequency relative to the base frequency $nf_0$ is $1/n$.

So does the discontinuity have an exponentially higher energy distribution towards 0 Hz, and near 0 energy at the 1/2 sample rate?

A white noise signal has equal energy at all frequencies but, in my experience, only as it is averaged over time. If the signal is too brief, it cannot express this due to the random evolving nature.

Is there any type of signal that can be expressed more immediately than white noise (like a discontinuity) but that has equal energy at all frequencies? If not, why is this not possible?

Is it a matter of needing more samples of random data to express the high frequency excitation?

For example, could one pass a discontinuity through some type of filter to raise the higher frequency energy and decrease the lower frequency energy to "even it out"?

Or is white noise only capable of doing this?

If white noise is the only way, at a given sample rate (say 44100 Hz) what is the minimum sample duration of white noise typically required to get an even excitation at all audio frequencies (to 20 kHz)?

Is it possible to write a sequence of the minimum effective duration of sample data that will do this reliably? Ie. Shortest pseudo-"white noise" burst effective at all audio frequencies of interest? Or effective up to 1/2 sample rate?

Thanks for any help.

Answer 1

A step has a $f^{-2}$ power spectrum, not white. A delta function (impulse) has a white noise spectrum. You may approximate it with a narrow pulse.

A pseudorandom binary sequence or pseudonoise sequence is white up to the cutoff determined by its pulse width, but composed of discrete lines at multiples of its repeat frequency (which may be below your resolution).

There are infinite possibilities here, since any signal created by arbitrary phase adjustment of the components of a white noise signal has a white spectrom.

Answer 2

An impulse function in the time domain, which is basically a very narrow pulse of energy, will excite all frequencies within the bandwidth given by the reciprocal of the pulse width of the impulse and any other bandwidth limiting components in the system.