How to detect "noisiness" of sound wave?

Some phonemes like "ssss" are basically white noise. How would you determine which parts of a wave are white noise?

From frequency analysis the white noise will have no tones so just using this would suggest it would be indistinguishable from no noise at all! (multiplying it by a sine wave would cancel out at all frequencies).

How do our ears hear white noise? How does it know it's not just a high frequency?

Is there a way to detect, say the entropy of a sound wave to determine which bits are white noise?

I want to write some software that detects things like the phoneme "ssss" so it needs to work out which bits of a wave are white noise.

Edit

I kind of get that you need to do an FFT to find the noise. So say $f(x)$ is a digital wave for x=0...1000 bytes.

Is there some kind of functional such as $F[f]$ which will give a number between 0..1 to say how close to white noise the signal is?

From the top of my head maybe something like:

$$F[f] = 1 - \frac{1}{1000^2}\int_0^{1000} \int_0^{1000} |f(x)-f( (x+a)_{mod 1000})|^2 da dx$$

but this is a double integral an would take a long time. My idea is that white noise would have no repetition so the integrand would never be zero. Just a guess. Don't think it would work in practice. Is there a simpler formula perhaps with just one integral or a sum?

• Multiplying does not result in zero. The intensity of white noise fluctuates at all scales. There are Fourier components at all frequencies. Oct 25 '15 at 1:12
• Say the sound is f(x)=random(x). Then $\int sin(ax)*f(x) dx \approx 0$ for all $a$ more or less. So I don't think this is a good test. Is there a better formula to test for white noise? It may fluctuate at all frequencies but minutely. Plus you'd have to test ALL frequencies. Oct 25 '15 at 1:21
• Right. And approximately equal is not the same as equal. And what do you mean by test? It's not clear to me what your ultimate goal is, but @DanielGriscom might be spot-on when he suggests the FFT. Warning: the FFT (or more fundamentally the DFT --- Discrete Fourier Transform ---, the FFT being an implementation of the DFT) takes some study to use and interpret correctly. Oct 25 '15 at 2:45
• Suggestion: play a bit with Audacity, or any software letting you see a frequency "image" of your speech. BTW it's false to say that this noise has to tone, because those noise are not white, they are "colored". Since even with highly distorted electronic guitare sound, you can still hear a musical scale. ;-) Oct 25 '15 at 9:32

The definition of (discrete) white noise includes the property that the autocovariance is zero for all nonzero time lags. (For zero time lag, it is the variance.) Conceptually, white noise is completely uncorrelated with itself. This property leads to the uniform density in Fourier space, but the autocovariance is easy to estimate from a record of fixed length. Nearly-white noise should have a very narrow peak in the autocovariance at zero lag.

A more rigorous approach, which would give you a less subjective measure of the "whiteness" of your process, would be to fit the time series to an autoregressive moving-average (ARMA) model. R, Matlab, and Mathematica all have packages for this, and I'm sure you could find open-source versions. The closer the process constants are to zero, the closer your series is to white noise.

However: To my mind, this isn't really an acoustics question so much as a time series analysis question. You may get better answers if you migrate over to the Signal Processing or Mathematics Stack Exchange sites.

Use an FFT to characterize the sound. With a pure tone, there will be lots of strength in one or two adjacent bins, with all others having very little signal. With white noise, every bin will have strength, as there will be elements of every frequency present. Yes, the bin strengths will be noisy, and you'll have to deal with that, but a little bit of statistics should go a long way.

• You must be an engineer. :) For some reason, engineers call the points of reciprocal space bins. If a frequency is just a bit off from a frequency in the reciprocal space, but still lies in the "bin", we do not find that that "bin" has amplitude and others don't. Instead we find amplitude all over the spectrum. This phenomenon confuses novices. Oct 25 '15 at 2:44
• @garyp, notice that the answer specifically refers to the FFT, which is an implementation of the discrete fourier transform (DFT). Therefore its natural to talk about bins. Oct 25 '15 at 5:07
• Wavelet analysis is another tool to learn more cabout the signal. Actually it uses the FFT in the calculations but you get more information about both the temporal and frequency behaviour. Oct 25 '15 at 9:18
• @ThePhoton FFT, DFT, same thing. You find bins in histograms, and the DFT/FFT is not a histogram. There are no bins. The word is common, but unfortunate for novices. The DFT assumes the function is periodic. A pure sine signal having a frequency incommensurate with the assumed period produces discontinuities at the period boundary, and FT amplitude appears across the entire spectrum, even though the signal is a pure sine. One must choose the period carefully, and use windowing functions to reduce the effect. But windows cause "leakage" of their own. The FFT is not a simple thing to use. Oct 25 '15 at 14:21
• @garyp, The word "bin" actually helps remind us that one number in the DFT spectrum covers a range of frequencies, not just a single precise frequency. I'm not sure why you think that each word (like "bin") can only be used in one context (like histograms). Oct 25 '15 at 16:11