# What does zero-mean noise mean?

I don't know where to start from. and this question might be very silly,

I have found such as comment from someone at Poisson noise and curve fitting - denoise first?,

"I think if your noise is zero-mean it doesn't matter, if not it does (you should de-noise first). I'm not entirely sure though so I won't put it in an answer yet."

Q1. Actually, I want to know what does zero-mean noise mean? and what does it not matter means?

Q2. Why we presume that the noise in Gaussian filter have sigma 1 and mean 0 ?

It is an easy concept but a sometimes it is hard to express it compactly. The difficulty arises because of additional concepts to be known for any formal description.

As you may know, mathematical analysis of noise variables and signals are handled via the tools of probability theory, in particular by the stochastic (random) processes.

The usual progress is such that first properties of random variables are described and then properties of random signals are derived based on it.

The mean for a random variable, X, is defined to be its Expected Value E(X) and computed based on its Probability Density Function $f_X(x)$ such as $\mu_X = \int {xf_X(x) dx}$ with the limits from minus to plus infinity for a continuous random variable.

So a zero mean random variable is that one for which the above integral is zero. (for a discrete random variable the integral is replaced by a sum and pdf is replaced by a pmf). This is also called as the average value.

Random signals are considered to be made up of an infinite set of random variables each making a single value at a single instance of observation of the instance being created. Hence there is the concept of ensemble and time averages.

Now coming to random signals X(t) (noise) expected value of a random signal is also expressed as E(X(t)) which for a Stationary (or at east Weakly Stationary up to second order) Process is a fixed value.

Your answer is that a zero mean noise is that one for which E(X(t)) is zero for all t.

However above is a theoretical description of Mean. In practice to compute mean and other theoretical parameters, you will refer to statistical computation which deal with practically observed data and computations based on that.

The key which connects statistically made (practical) computations to theoretical computations is the Ergodicity Theorem which states that for an ergodic random process theoretical computations are equal to statistical ones.

• Thank yo so much Sir, Briefly, Do you want to say what if we get all noise signals then the sum of noise signal is zero? If so, I am correct, this is only in Gaussian noise model? or Poisson noise model? – gmotree May 30 '15 at 8:09
• @gmotree Practically (due to the Ergodicity Theorem) the sum of the samples (or subsequently the average) will be very close to zero for a zero mean random signal whatsoever its model is. – Fat32 May 30 '15 at 8:38
• @gmtree I saw, thanks ;) I assume std = standart deviation. It is the square root of Variance, which is = $E( (X-E(X))^2)$ – Fat32 May 30 '15 at 8:50
• I have already know the meaning of std what std is standard deviation. But I want to know why typically the noise model have std=1? what does std=1 mean at noise model? if you want to make me another question then plz let me know. i'll make another question. – gmotree May 30 '15 at 8:52
• i think Bulent S. that the questin refered to poisson noise and relation to gaussian normalised noise and you have did explained wikipedia way mean and stantart deviation definition. – Black Yasmin May 30 '15 at 10:43