# Normalizing white noise to match pink noise's amplitude at a given frequency?

I have a pinking filter which I am running white noise through to get pink noise. I would like to then mix the white noise with this pink noise, where the amplitude of the white noise will match the pink noise at a given frequency.

My pinking filter is +/-0.5 dB above 10 Hz.

So let's say I want the white noise to match the amplitude of the pink noise at 100 Hz. Or at least be standardized to close to this frequency's amplitude.

I believe in terms of power which is amplitude squared for pink noise we have: power = 1/freq.

So would it be: amplitude = sqrt(1/freq)?

And thus amplitude of pink noise at 100 hz would be 0.316? I know that's not correct but I don't know what is.

Basically I'm looking for a multiplier for the white noise so that at that given frequency it will match the amplitude of the pink noise.

Thanks.

• white noise, in the continuous-time domain, has infinite energy – robert bristow-johnson Jan 3 '20 at 3:57
• Thanks RBJ. I'm not sure if I'm voicing things wrong but what I'm looking for is a multiplier for the unfiltered white noise so it will match or be standardized to the amplitude of the pink noise at a given frequency, eg. 100 Hz. – mike Jan 3 '20 at 4:20
• lemme see if i can "answer" you with an answer. but i dunno if it serves your question. – robert bristow-johnson Jan 3 '20 at 4:24
• Are you able to calculate the magnitude response of your pinking filter at the chosen frequency? That will give you the relative amplitude change at that frequency, then you can compensate for it. – kippertoffee Jan 3 '20 at 14:57

Okay, when discussing white noise or pink noise (or red noise or brown noise or flicker) or some other random process, there is this property called the power spectrum, in which white noise has a constant value for all frequencies. But we integrate the power spectrum over all frequencies (from $$-\infty$$ to $$+\infty$$) to get power. Integrating a constant, non-zero function over an infinite frequency range gives you infinite power.

So in reality, white noise does not exist. (It puts out more power than the Sun.)

But if you bandlimit white noise to a finite bandwidth, you have a finite power.

Sampled signals (in discrete time) must have a finite bandwidth in order to be sampled properly.

If you were to synthesize bandlimited white noise by use of a random number generator that produces good random numbers with a uniform distribution between the values of $$0$$ and $$\Delta$$, you would have a mean of $$\frac{\Delta}{2}$$ and a variance of $$\frac{\Delta^2}{12}$$. If you subtract from each uniform random number the mean of $$\frac{\Delta}{2}$$, then you would have the same variance, but a mean of zero.

Now if you were to add $$N$$ of those independent zero-mean uniform random numbers together, the variance would increase to $$\frac{N \Delta^2}{12} \triangleq \sigma^2$$ and the distribution of the random numbers would closely approach that of a normal distribution (sometimes called Gaussian).

Now variance of a zero mean (or DC free) random process is the power, which is the area (or integral) of the power spectrum. So from $$-\frac{f_\mathrm{s}}{2}$$ to $$+\frac{f_\mathrm{s}}{2}$$, you would have a flat power spectrum with an amplitude of $$\frac{\sigma^2}{f_\mathrm{s}}$$ so that the area or integral comes out to be $$\sigma^2$$. (The amplitude in the power spectrum is zero for all $$|f|>\frac{f_\mathrm{s}}{2}$$. This is what makes it bandlimited.)

So now you have "white" noise that is finite energy. When you run that through a pinking filter (is that what you are doing to make pink noise?), then you have to multiply that constant amplitude of $$\frac{\sigma^2}{f_\mathrm{s}}$$ times the magnitude-squared of the pinking filter to get the frequency-dependent amplitude of the pink noise. Then you have to integrate that from from $$-\frac{f_\mathrm{s}}{2}$$ to $$+\frac{f_\mathrm{s}}{2}$$ to get the power of the pink noise. Whatever is the ratio of that power to the power of the bandlimited white noise going in, you compute the square-root of the reciprocal, and that is the scaling constant to multiply times your pink noise to make it have the same power as the bandlimited white noise going in.