Can you please explain: why is a specific type of noise called "Gaussian noise"? Why is it relevant to call it Gaussian? Please, explain in layman's terms.
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1$\begingroup$ Noise is a random process and a random process $x(t)$ or $x[n]$ is a collection of random variables $X_t$ or $X_n$ for each $t$ or $n$. As you know random variables are characterised by their Probability Density Functions (pdf), such as Uniform, Bernoulli, Binomial, Multinomial, Poisson, Exponential, Rayleigh, Gamma and Gaussian. Now if the collection of random variables associated with a random process all have thier pdfs as Gaussian type, then that process is called as a Gaussian random process. $\endgroup$– Fat32Commented Mar 16, 2016 at 10:52
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2 Answers
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Noise is random, but like most random phenomena, it follows a certain pattern. Different patterns are given different names.
Consider rolling a die. This is clearly random. Roll the die 1000 times, keeping track of each result. Then, calculate the histogram of the result; you'll find that you got each of 1, 2, 3, 4, 5 and 6 approximately the same number of times. This pattern is called "uniform", and throwing a die can be modeled by a "uniform random variable".
The same experiment can be repeated with thermal noise. Heat up a resistor, amplify the resulting voltage and measure its instantaneous power multiple times. Then calculate the histogram. This time, you won't find a uniform histogram; it will be shaped like a bell curve, with values near zero more common than values far away from zero. This kind of histogram is called Gaussian, after K. F. Gauss.
Gaussian random phenomena are very common in nature. It turns out that whenever the random thing you observe is the aggregate of many, independent random events, the overall random variable is Gaussian (this is technically called the central limit theorem). In the case of thermal noise, you're measuring the aggregate effect of millions or billions of randomly oscillating electrons, excited by the heat.
There's an easier way to create Gaussian randomness at home (or simulated in a computer): take many dice, say 100, throw them all many times, and keep track of the total sum of each throw. If you find the histogram again, you'll see it follows a bell curve. The reason is intuitively easy to grasp: with 100 dice, you're very unlikely to get a total of 100 (all dice would have to land in 1), but it's very easy to get a number around 350, because many different combinations add up to such a number.
To summarize, there are many different kinds of noise that can affect a signal or an image, each with different statistical properties. Gaussian noise is a particularly important kind of noise because it is very prevalent. It is characterized by a histogram (more precisely, a probability density function) that follows the bell curve (or Gaussian function). As you study it more, you'll find that it also has several other important statistical properties.
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$\begingroup$ Good summary. Gaussian random variables also have several unique and often useful properties. One, for example, is that the sum of two independent Gaussian random variables is also Gaussian distributed. $\endgroup$– Jason RCommented Mar 16, 2016 at 0:59
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$\begingroup$ @JasonR Thanks! And you're right, of course; I sort of hinted at that sort of thing at the end, when I mentioned studying Gaussian RVs in more depth. I thought my answer was long enough already. $\endgroup$– MBazCommented Mar 16, 2016 at 1:06
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It is named after extremely influential and famous german-speaking mathematician Carl Friedrich Gauss who lived during the 18th and 19th centuries and influenced among other things early statistics in which the Gaussian distribution (normal distribution) is related to the central limit theorem, a sum or mean value of enough of equidistributed random numbers will be (close to) gaussian if the underlying distribution is nice enough (in practice it needs to be rather nasty behaving distribution for it not to be true).
In pronouncing Gauss, if you start out with a g the rest keeps going like a mouse, but in the english speaking world usually the "ou" in mouse turns into an "o" in german "hallo".
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1$\begingroup$ There seems to be an automatic downvote on me after just a second of posting, can someone check it? Is this a common problem? $\endgroup$ Commented Mar 16, 2016 at 8:30
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1$\begingroup$ There is no way even the mods can see downvoting.. Rest assured that, if it's systematic, then your rep will be restored. +1 from me, anyway. :-) $\endgroup$– Peter K. ♦Commented Mar 16, 2016 at 11:51