# Difference between "scaled white noise " vs. "unscaled white noise"

I have been searching Matlab user-guide for Wavelets. In their graphical user interface for denoising, there is an option for "unscaled white noise" and "scaled white noise" for the noise structure. What is the main difference between the two? Does the scaled white noise imply that the noise is amplitude changing with time? I have searched Google books and Google Scholar, nobody defines it. Thanks.

• Hi: in statistics, unscaled white noise would mean a random variable distributed normally with mean 0 and standard deviation equal to 1.0. Scaled would mean that it has a mean and a variance not necessarily equal to 0 and 1.0 respectively. Note that white noise is an idealized concept in that there really is no such thing because it's the derivative of brownian motion ( which doesn't exist ). But, for practical purposes, people use the normal distribution to simulate it. Note that this is statistical viewpoint. Could be some differences in DSP. Dec 25 '19 at 22:05
• Thanks. I am trying to simulate a noisy peak, as would be seen in analytical chemistry. I was trying to generate to generate random noise in Matlab using Random_noise=0.06*randn(1002,1), the 0.06 is just a number to reduce the magnitude of noise, 1002 is the length of the time vector, 1 indicate a single row. mathworks.com/help/matlab/math/… Dec 25 '19 at 22:15
• In this case, the standard deviation of the noise is 0.06 units, so it is a scaled white noise by your statistical definition and its mean is zero. Could you provide a reference which defines "scaled" noise this way? Thanks. Dec 25 '19 at 22:17
• Hi: I cant find any references either but I'm pretty certain that that's what it means. Note that it doesn't mean that it's changing with time. It's fixed but the mean can be non-zero or the variance can be made smaller or larger. Dec 26 '19 at 16:47
• @Mark, this scaled white noise and unscaled white noise was something very specific to Matlab. It has nothing to do with statistics. No wonder, I could not find it anywhere. The link is given below the comments. Dec 27 '19 at 21:57

## 1 Answer

Wavelets, as bases or frames, are linear like-decompositions. As such, signals and processes can be decomposed into a linear or weighed sum of coefficients multipling (wavelet) vectors:

$$s[n] = \sum a_n e[n]$$

The indexing above is not wavelet-specific. Traditionally, there was some natural order: low $$n$$ denoting low-scales, or high frequencies, and vice-versa. So traditional linear filtering often consisted in keeping indices above or below some cut-off indices (with weighting), assuming that data above and below could be noise (typically low-pass filtering).

This linear (order-related) scheme remains practically useful. However, and some cases, a non-linear selection can prove more efficient. Especially when indices, across time and scale, don't exhibit clear ordering anymore.

$$s[n] = \sum a_{j,k} \psi\left[\frac{n -k2^j}{2^j}\right]$$

The main idea is to select "cleaned signal" coefficient not within indices, but with respect to magnitude: the highest the magnitude, the most likely it belongs to data instead of noise.

Selecting the highest coefficients amounts to setting a threshold below which you discard coefficients. This threshold should be selected as cleverly of possible. nd here is the catch. In the early works on wavelet thresholding, for proofs and practice, it was often considering that the noise was Gaussian with (known) unit variance. This would yield nice theories and practical algorithms.

Yet, this requires a knowledge on the noise, which is often unknown (in shape and amplitude). If we forget the shape (Gaussian or not), if the noise level is $$\sigma$$ (instead on one), this should be taken into account, esp. by changing the threshold accordingly. Caveat, estimating this $$\sigma$$ remains tricky.

• Thanks for your time for a discussion on the concept thresholding which is also related to denoising. My main question is related to the fundamental difference between "scaled white noise" and "unscaled white noise" as used in MATLAB. What is the meaning of $scaled$ or $unscaled$ here? For example, if we are generating Gaussian noise by randn command as Random_noise=0.06*randn(length(t),1), where 0.06 is the standard deviation and the mean is zero. What should be call it, scaled noise or unscaled noise? Dec 26 '19 at 1:13
• The terminology of scaled white noise vs. unscaled white noise is very poor. There is no definition anywhere. Dec 26 '19 at 1:14
• I got the response from Matlab: "The meaning of "Unscaled white noise", "Scaled white noise", and "Non-white noise" correspond to the "one", "sln", and "mln" options in the legacy wden() function. The following documentation link will point you to the definitions. mathworks.com/help/releases/R2019a/wavelet/ref/… Dec 27 '19 at 21:55
• @M.Farooq: I couldn't see the link because you need to have an account but that's okay. I know what you mean by it not being official. Believe me, just being on this site, I've experienced terminology issues (I'm not DSP person at all ) so I getcha. But, amazing site nonetheless. Dec 28 '19 at 22:12